1

Chapter 4 Sequential Games

2

Extensive Form Games

H

H H

T

TT

(12) (40)(21) (21)

Any finite game of perfect information has a pure strategy Nash equilibrium It can be found by backward induction

Chess is a finite game of perfect information Therefore it is a ldquotrivialrdquo game from a game theoretic point of view

3

Extensive Form Games - Intro

bull A game can have complex temporal structurebull Information

ndash set of playersndash who moves when and under what circumstancesndash what actions are available when called upon to movendash what is known when called upon to movendash what payoffs each player receives

bull Foundation is a game tree

4

bull Big Monkey and Little Monkey eat warifruit which dangle from the extreme tip of a lofty branch of the waritree

bull A waritree produces only one fruit To get the warifruit at least one of the monkeys must climb the tree and shake the branch bearing the fruit until the fruit comes loose and falls to the ground

bull A warifruit is worth 10 calories of energy Climbing the tree uses 2 calories for Big Monkey but uses no energy for Little Monkey who is smaller If Little Monkey climbs the tree and shakes it down Big Monkey will eat 90 of the fruit (or 9 calories) before Little Monkey climbs back down and Little Monkey will get only 10 of the fruit (or 1 calorie)

bull If Big Monkey climbs the tree and Little Monkey waits Little Monkey will get 40 of the fruit and Big Monkey will get 60 If both monkeys climb the tree Big Monkey will get 70 of the fruit and Little Monkey will get 30 Assume each monkey is simply interested in maximizing his caloric intake

bull Each monkey can decide to climb the tree or wait at the bottom bull a What is likely to happen if Big Monkey makes his decision first bull b What is likely to happen if Little Monkey must decide firstbull c What if they both decide simultaneously

5

Fundamental Tools bull Big Monkey (BM) ndash Little Monkey (LM)bull Warifruit from waritree (only one per tree) = 10 Caloriesbull Climb the tree to get the fruitbull Cost to get the fruit

ndash 2 Calories for Big Monkeyndash zero for Little Monkey

bull Payoff ndash Both climb BM 7 Calories ndash LM 3 Caloriesndash BM climbs BM 6 Calories ndash LM 4 Caloriesndash LM climbs BM 9 Calories ndash LM 1 Calories

bull What will they do to maximize payoff taking into account cost

6

Fundamental Tools Extensive form games--Definition

bull An extensive form game G consists of ndash Playersndash Game treendash Payoffs

bull Terminal node t i(t) (payoffs)bull G has tree property only 1 path from root to any

terminal nodebull Occurrence of stochastic event fictitious player Nature probability assigned to each branch of which Nature is head node

7

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull 3 possibilities ndash BM decides first what to dondash LM decides first what to dondash Both decide simultaneously

bull BM decides first

Big Monkey

Little Monkey Little Monkey

w c

w c w c

00 91 44 53

8

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull Strategies ndash BM

bull Wait (w)bull Climb (c)

ndash LM Actions are ordered depending on (wc) of BMbull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

bull A series of actions that fully define the behavior of a player = strategy

bull A strategy for a player is a complete plan of how to plan the game and prescribes his choices at every information set (in this case node)

9

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull LM decides first

ndash The strategies are conversed Utility=(LM BM)

Little Monkey

Big Monkey Big Monkey

w c

w c w c

00 44 19 35

10

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull They choose simultaneously

bull Information Set a set of nodes at which

bull The same player choosesbull The player choosing does not know which node represents the

actual choice node ndash represented by dotted line

Big Monkey

Little Monkeyw c

w c w c

00 91 44 53

c w

c 53 44

w 91 00

LM

BM

11

bull The key to representing information in a game tree is realizing the connection between nodes and history

bull If you know which node you have reached you know precisely the history of the play

bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)

12

Composition of information sets

bull Each decision node is in exactly one information set

bull all nodes of an information set must belong to same player

bull every node of an information set must have exactly the same set of available actions

bull If every information set of every player is a singleton we have a game of perfect information

13

Fundamental Tools Normal form games--Definition

The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn

We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i

ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen

14

Fundamental Tools Normal form games--Illustration

bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM

bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

15

Fundamental Tools Normal form games--Illustration

bull Donrsquot get rid of weakly dominated as lose equilibrium

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

16

Sequential games

bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)

bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)

17

bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa

bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered

bull ndash We need to define the concept of an equilibrium in extensive form games

18

Problems with Nash equilibrium

bull Sequential nature of the game is lost when representing extensive form games in strategic form

bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do

bull Nash equilibrium does not distinguish between credible and non-credible threats

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

2

Extensive Form Games

H

H H

T

TT

(12) (40)(21) (21)

Any finite game of perfect information has a pure strategy Nash equilibrium It can be found by backward induction

Chess is a finite game of perfect information Therefore it is a ldquotrivialrdquo game from a game theoretic point of view

3

Extensive Form Games - Intro

bull A game can have complex temporal structurebull Information

ndash set of playersndash who moves when and under what circumstancesndash what actions are available when called upon to movendash what is known when called upon to movendash what payoffs each player receives

bull Foundation is a game tree

4

bull Big Monkey and Little Monkey eat warifruit which dangle from the extreme tip of a lofty branch of the waritree

bull A waritree produces only one fruit To get the warifruit at least one of the monkeys must climb the tree and shake the branch bearing the fruit until the fruit comes loose and falls to the ground

bull A warifruit is worth 10 calories of energy Climbing the tree uses 2 calories for Big Monkey but uses no energy for Little Monkey who is smaller If Little Monkey climbs the tree and shakes it down Big Monkey will eat 90 of the fruit (or 9 calories) before Little Monkey climbs back down and Little Monkey will get only 10 of the fruit (or 1 calorie)

bull If Big Monkey climbs the tree and Little Monkey waits Little Monkey will get 40 of the fruit and Big Monkey will get 60 If both monkeys climb the tree Big Monkey will get 70 of the fruit and Little Monkey will get 30 Assume each monkey is simply interested in maximizing his caloric intake

bull Each monkey can decide to climb the tree or wait at the bottom bull a What is likely to happen if Big Monkey makes his decision first bull b What is likely to happen if Little Monkey must decide firstbull c What if they both decide simultaneously

5

Fundamental Tools bull Big Monkey (BM) ndash Little Monkey (LM)bull Warifruit from waritree (only one per tree) = 10 Caloriesbull Climb the tree to get the fruitbull Cost to get the fruit

ndash 2 Calories for Big Monkeyndash zero for Little Monkey

bull Payoff ndash Both climb BM 7 Calories ndash LM 3 Caloriesndash BM climbs BM 6 Calories ndash LM 4 Caloriesndash LM climbs BM 9 Calories ndash LM 1 Calories

bull What will they do to maximize payoff taking into account cost

6

Fundamental Tools Extensive form games--Definition

bull An extensive form game G consists of ndash Playersndash Game treendash Payoffs

bull Terminal node t i(t) (payoffs)bull G has tree property only 1 path from root to any

terminal nodebull Occurrence of stochastic event fictitious player Nature probability assigned to each branch of which Nature is head node

7

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull 3 possibilities ndash BM decides first what to dondash LM decides first what to dondash Both decide simultaneously

bull BM decides first

Big Monkey

Little Monkey Little Monkey

w c

w c w c

00 91 44 53

8

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull Strategies ndash BM

bull Wait (w)bull Climb (c)

ndash LM Actions are ordered depending on (wc) of BMbull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

bull A series of actions that fully define the behavior of a player = strategy

bull A strategy for a player is a complete plan of how to plan the game and prescribes his choices at every information set (in this case node)

9

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull LM decides first

ndash The strategies are conversed Utility=(LM BM)

Little Monkey

Big Monkey Big Monkey

w c

w c w c

00 44 19 35

10

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull They choose simultaneously

bull Information Set a set of nodes at which

bull The same player choosesbull The player choosing does not know which node represents the

actual choice node ndash represented by dotted line

Big Monkey

Little Monkeyw c

w c w c

00 91 44 53

c w

c 53 44

w 91 00

LM

BM

11

bull The key to representing information in a game tree is realizing the connection between nodes and history

bull If you know which node you have reached you know precisely the history of the play

bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)

12

Composition of information sets

bull Each decision node is in exactly one information set

bull all nodes of an information set must belong to same player

bull every node of an information set must have exactly the same set of available actions

bull If every information set of every player is a singleton we have a game of perfect information

13

Fundamental Tools Normal form games--Definition

The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn

We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i

ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen

14

Fundamental Tools Normal form games--Illustration

bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM

bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

15

Fundamental Tools Normal form games--Illustration

bull Donrsquot get rid of weakly dominated as lose equilibrium

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

16

Sequential games

bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)

bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)

17

bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa

bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered

bull ndash We need to define the concept of an equilibrium in extensive form games

18

Problems with Nash equilibrium

bull Sequential nature of the game is lost when representing extensive form games in strategic form

bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do

bull Nash equilibrium does not distinguish between credible and non-credible threats

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

3

Extensive Form Games - Intro

bull A game can have complex temporal structurebull Information

ndash set of playersndash who moves when and under what circumstancesndash what actions are available when called upon to movendash what is known when called upon to movendash what payoffs each player receives

bull Foundation is a game tree

4

bull Big Monkey and Little Monkey eat warifruit which dangle from the extreme tip of a lofty branch of the waritree

bull A waritree produces only one fruit To get the warifruit at least one of the monkeys must climb the tree and shake the branch bearing the fruit until the fruit comes loose and falls to the ground

bull A warifruit is worth 10 calories of energy Climbing the tree uses 2 calories for Big Monkey but uses no energy for Little Monkey who is smaller If Little Monkey climbs the tree and shakes it down Big Monkey will eat 90 of the fruit (or 9 calories) before Little Monkey climbs back down and Little Monkey will get only 10 of the fruit (or 1 calorie)

bull If Big Monkey climbs the tree and Little Monkey waits Little Monkey will get 40 of the fruit and Big Monkey will get 60 If both monkeys climb the tree Big Monkey will get 70 of the fruit and Little Monkey will get 30 Assume each monkey is simply interested in maximizing his caloric intake

bull Each monkey can decide to climb the tree or wait at the bottom bull a What is likely to happen if Big Monkey makes his decision first bull b What is likely to happen if Little Monkey must decide firstbull c What if they both decide simultaneously

5

Fundamental Tools bull Big Monkey (BM) ndash Little Monkey (LM)bull Warifruit from waritree (only one per tree) = 10 Caloriesbull Climb the tree to get the fruitbull Cost to get the fruit

ndash 2 Calories for Big Monkeyndash zero for Little Monkey

bull Payoff ndash Both climb BM 7 Calories ndash LM 3 Caloriesndash BM climbs BM 6 Calories ndash LM 4 Caloriesndash LM climbs BM 9 Calories ndash LM 1 Calories

bull What will they do to maximize payoff taking into account cost

6

Fundamental Tools Extensive form games--Definition

bull An extensive form game G consists of ndash Playersndash Game treendash Payoffs

bull Terminal node t i(t) (payoffs)bull G has tree property only 1 path from root to any

terminal nodebull Occurrence of stochastic event fictitious player Nature probability assigned to each branch of which Nature is head node

7

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull 3 possibilities ndash BM decides first what to dondash LM decides first what to dondash Both decide simultaneously

bull BM decides first

Big Monkey

Little Monkey Little Monkey

w c

w c w c

00 91 44 53

8

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull Strategies ndash BM

bull Wait (w)bull Climb (c)

ndash LM Actions are ordered depending on (wc) of BMbull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

bull A series of actions that fully define the behavior of a player = strategy

bull A strategy for a player is a complete plan of how to plan the game and prescribes his choices at every information set (in this case node)

9

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull LM decides first

ndash The strategies are conversed Utility=(LM BM)

Little Monkey

Big Monkey Big Monkey

w c

w c w c

00 44 19 35

10

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull They choose simultaneously

bull Information Set a set of nodes at which

bull The same player choosesbull The player choosing does not know which node represents the

actual choice node ndash represented by dotted line

Big Monkey

Little Monkeyw c

w c w c

00 91 44 53

c w

c 53 44

w 91 00

LM

BM

11

bull The key to representing information in a game tree is realizing the connection between nodes and history

bull If you know which node you have reached you know precisely the history of the play

bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)

12

Composition of information sets

bull Each decision node is in exactly one information set

bull all nodes of an information set must belong to same player

bull every node of an information set must have exactly the same set of available actions

bull If every information set of every player is a singleton we have a game of perfect information

13

Fundamental Tools Normal form games--Definition

The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn

We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i

ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen

14

Fundamental Tools Normal form games--Illustration

bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM

bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

15

Fundamental Tools Normal form games--Illustration

bull Donrsquot get rid of weakly dominated as lose equilibrium

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

16

Sequential games

bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)

bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)

17

bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa

bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered

bull ndash We need to define the concept of an equilibrium in extensive form games

18

Problems with Nash equilibrium

bull Sequential nature of the game is lost when representing extensive form games in strategic form

bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do

bull Nash equilibrium does not distinguish between credible and non-credible threats

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

4

bull Big Monkey and Little Monkey eat warifruit which dangle from the extreme tip of a lofty branch of the waritree

bull A waritree produces only one fruit To get the warifruit at least one of the monkeys must climb the tree and shake the branch bearing the fruit until the fruit comes loose and falls to the ground

bull A warifruit is worth 10 calories of energy Climbing the tree uses 2 calories for Big Monkey but uses no energy for Little Monkey who is smaller If Little Monkey climbs the tree and shakes it down Big Monkey will eat 90 of the fruit (or 9 calories) before Little Monkey climbs back down and Little Monkey will get only 10 of the fruit (or 1 calorie)

bull If Big Monkey climbs the tree and Little Monkey waits Little Monkey will get 40 of the fruit and Big Monkey will get 60 If both monkeys climb the tree Big Monkey will get 70 of the fruit and Little Monkey will get 30 Assume each monkey is simply interested in maximizing his caloric intake

bull Each monkey can decide to climb the tree or wait at the bottom bull a What is likely to happen if Big Monkey makes his decision first bull b What is likely to happen if Little Monkey must decide firstbull c What if they both decide simultaneously

5

Fundamental Tools bull Big Monkey (BM) ndash Little Monkey (LM)bull Warifruit from waritree (only one per tree) = 10 Caloriesbull Climb the tree to get the fruitbull Cost to get the fruit

ndash 2 Calories for Big Monkeyndash zero for Little Monkey

bull Payoff ndash Both climb BM 7 Calories ndash LM 3 Caloriesndash BM climbs BM 6 Calories ndash LM 4 Caloriesndash LM climbs BM 9 Calories ndash LM 1 Calories

bull What will they do to maximize payoff taking into account cost

6

Fundamental Tools Extensive form games--Definition

bull An extensive form game G consists of ndash Playersndash Game treendash Payoffs

bull Terminal node t i(t) (payoffs)bull G has tree property only 1 path from root to any

terminal nodebull Occurrence of stochastic event fictitious player Nature probability assigned to each branch of which Nature is head node

7

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull 3 possibilities ndash BM decides first what to dondash LM decides first what to dondash Both decide simultaneously

bull BM decides first

Big Monkey

Little Monkey Little Monkey

w c

w c w c

00 91 44 53

8

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull Strategies ndash BM

bull Wait (w)bull Climb (c)

ndash LM Actions are ordered depending on (wc) of BMbull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

bull A series of actions that fully define the behavior of a player = strategy

bull A strategy for a player is a complete plan of how to plan the game and prescribes his choices at every information set (in this case node)

9

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull LM decides first

ndash The strategies are conversed Utility=(LM BM)

Little Monkey

Big Monkey Big Monkey

w c

w c w c

00 44 19 35

10

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull They choose simultaneously

bull Information Set a set of nodes at which

bull The same player choosesbull The player choosing does not know which node represents the

actual choice node ndash represented by dotted line

Big Monkey

Little Monkeyw c

w c w c

00 91 44 53

c w

c 53 44

w 91 00

LM

BM

11

bull The key to representing information in a game tree is realizing the connection between nodes and history

bull If you know which node you have reached you know precisely the history of the play

bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)

12

Composition of information sets

bull Each decision node is in exactly one information set

bull all nodes of an information set must belong to same player

bull every node of an information set must have exactly the same set of available actions

bull If every information set of every player is a singleton we have a game of perfect information

13

Fundamental Tools Normal form games--Definition

The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn

We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i

ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen

14

Fundamental Tools Normal form games--Illustration

bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM

bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

15

Fundamental Tools Normal form games--Illustration

bull Donrsquot get rid of weakly dominated as lose equilibrium

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

16

Sequential games

bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)

bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)

17

bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa

bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered

bull ndash We need to define the concept of an equilibrium in extensive form games

18

Problems with Nash equilibrium

bull Sequential nature of the game is lost when representing extensive form games in strategic form

bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do

bull Nash equilibrium does not distinguish between credible and non-credible threats

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

5

Fundamental Tools bull Big Monkey (BM) ndash Little Monkey (LM)bull Warifruit from waritree (only one per tree) = 10 Caloriesbull Climb the tree to get the fruitbull Cost to get the fruit

ndash 2 Calories for Big Monkeyndash zero for Little Monkey

bull Payoff ndash Both climb BM 7 Calories ndash LM 3 Caloriesndash BM climbs BM 6 Calories ndash LM 4 Caloriesndash LM climbs BM 9 Calories ndash LM 1 Calories

bull What will they do to maximize payoff taking into account cost

6

Fundamental Tools Extensive form games--Definition

bull An extensive form game G consists of ndash Playersndash Game treendash Payoffs

bull Terminal node t i(t) (payoffs)bull G has tree property only 1 path from root to any

terminal nodebull Occurrence of stochastic event fictitious player Nature probability assigned to each branch of which Nature is head node

7

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull 3 possibilities ndash BM decides first what to dondash LM decides first what to dondash Both decide simultaneously

bull BM decides first

Big Monkey

Little Monkey Little Monkey

w c

w c w c

00 91 44 53

8

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull Strategies ndash BM

bull Wait (w)bull Climb (c)

ndash LM Actions are ordered depending on (wc) of BMbull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

bull A series of actions that fully define the behavior of a player = strategy

bull A strategy for a player is a complete plan of how to plan the game and prescribes his choices at every information set (in this case node)

9

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull LM decides first

ndash The strategies are conversed Utility=(LM BM)

Little Monkey

Big Monkey Big Monkey

w c

w c w c

00 44 19 35

10

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull They choose simultaneously

bull Information Set a set of nodes at which

bull The same player choosesbull The player choosing does not know which node represents the

actual choice node ndash represented by dotted line

Big Monkey

Little Monkeyw c

w c w c

00 91 44 53

c w

c 53 44

w 91 00

LM

BM

11

bull The key to representing information in a game tree is realizing the connection between nodes and history

bull If you know which node you have reached you know precisely the history of the play

bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)

12

Composition of information sets

bull Each decision node is in exactly one information set

bull all nodes of an information set must belong to same player

bull every node of an information set must have exactly the same set of available actions

bull If every information set of every player is a singleton we have a game of perfect information

13

Fundamental Tools Normal form games--Definition

The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn

We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i

ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen

14

Fundamental Tools Normal form games--Illustration

bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM

bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

15

Fundamental Tools Normal form games--Illustration

bull Donrsquot get rid of weakly dominated as lose equilibrium

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

16

Sequential games

bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)

bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)

17

bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa

bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered

bull ndash We need to define the concept of an equilibrium in extensive form games

18

Problems with Nash equilibrium

bull Sequential nature of the game is lost when representing extensive form games in strategic form

bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do

bull Nash equilibrium does not distinguish between credible and non-credible threats

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

6

Fundamental Tools Extensive form games--Definition

bull An extensive form game G consists of ndash Playersndash Game treendash Payoffs

bull Terminal node t i(t) (payoffs)bull G has tree property only 1 path from root to any

terminal nodebull Occurrence of stochastic event fictitious player Nature probability assigned to each branch of which Nature is head node

7

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull 3 possibilities ndash BM decides first what to dondash LM decides first what to dondash Both decide simultaneously

bull BM decides first

Big Monkey

Little Monkey Little Monkey

w c

w c w c

00 91 44 53

8

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull Strategies ndash BM

bull Wait (w)bull Climb (c)

ndash LM Actions are ordered depending on (wc) of BMbull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

bull A series of actions that fully define the behavior of a player = strategy

bull A strategy for a player is a complete plan of how to plan the game and prescribes his choices at every information set (in this case node)

9

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull LM decides first

ndash The strategies are conversed Utility=(LM BM)

Little Monkey

Big Monkey Big Monkey

w c

w c w c

00 44 19 35

10

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull They choose simultaneously

bull Information Set a set of nodes at which

bull The same player choosesbull The player choosing does not know which node represents the

actual choice node ndash represented by dotted line

Big Monkey

Little Monkeyw c

w c w c

00 91 44 53

c w

c 53 44

w 91 00

LM

BM

11

bull The key to representing information in a game tree is realizing the connection between nodes and history

bull If you know which node you have reached you know precisely the history of the play

bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)

12

Composition of information sets

bull Each decision node is in exactly one information set

bull all nodes of an information set must belong to same player

bull every node of an information set must have exactly the same set of available actions

bull If every information set of every player is a singleton we have a game of perfect information

13

Fundamental Tools Normal form games--Definition

The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn

We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i

ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen

14

Fundamental Tools Normal form games--Illustration

bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM

bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

15

Fundamental Tools Normal form games--Illustration

bull Donrsquot get rid of weakly dominated as lose equilibrium

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

16

Sequential games

bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)

bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)

17

bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa

bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered

bull ndash We need to define the concept of an equilibrium in extensive form games

18

Problems with Nash equilibrium

bull Sequential nature of the game is lost when representing extensive form games in strategic form

bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do

bull Nash equilibrium does not distinguish between credible and non-credible threats

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

7

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull 3 possibilities ndash BM decides first what to dondash LM decides first what to dondash Both decide simultaneously

bull BM decides first

Big Monkey

Little Monkey Little Monkey

w c

w c w c

00 91 44 53

8

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull Strategies ndash BM

bull Wait (w)bull Climb (c)

ndash LM Actions are ordered depending on (wc) of BMbull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

bull A series of actions that fully define the behavior of a player = strategy

bull A strategy for a player is a complete plan of how to plan the game and prescribes his choices at every information set (in this case node)

9

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull LM decides first

ndash The strategies are conversed Utility=(LM BM)

Little Monkey

Big Monkey Big Monkey

w c

w c w c

00 44 19 35

10

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull They choose simultaneously

bull Information Set a set of nodes at which

bull The same player choosesbull The player choosing does not know which node represents the

actual choice node ndash represented by dotted line

Big Monkey

Little Monkeyw c

w c w c

00 91 44 53

c w

c 53 44

w 91 00

LM

BM

11

bull The key to representing information in a game tree is realizing the connection between nodes and history

bull If you know which node you have reached you know precisely the history of the play

bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)

12

Composition of information sets

bull Each decision node is in exactly one information set

bull all nodes of an information set must belong to same player

bull every node of an information set must have exactly the same set of available actions

bull If every information set of every player is a singleton we have a game of perfect information

13

Fundamental Tools Normal form games--Definition

The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn

We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i

ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen

14

Fundamental Tools Normal form games--Illustration

bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM

bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

15

Fundamental Tools Normal form games--Illustration

bull Donrsquot get rid of weakly dominated as lose equilibrium

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

16

Sequential games

bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)

bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)

17

bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa

bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered

bull ndash We need to define the concept of an equilibrium in extensive form games

18

Problems with Nash equilibrium

bull Sequential nature of the game is lost when representing extensive form games in strategic form

bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do

bull Nash equilibrium does not distinguish between credible and non-credible threats

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

8

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull Strategies ndash BM

bull Wait (w)bull Climb (c)

ndash LM Actions are ordered depending on (wc) of BMbull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

bull A series of actions that fully define the behavior of a player = strategy

bull A strategy for a player is a complete plan of how to plan the game and prescribes his choices at every information set (in this case node)

9

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull LM decides first

ndash The strategies are conversed Utility=(LM BM)

Little Monkey

Big Monkey Big Monkey

w c

w c w c

00 44 19 35

10

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull They choose simultaneously

bull Information Set a set of nodes at which

bull The same player choosesbull The player choosing does not know which node represents the

actual choice node ndash represented by dotted line

Big Monkey

Little Monkeyw c

w c w c

00 91 44 53

c w

c 53 44

w 91 00

LM

BM

11

bull The key to representing information in a game tree is realizing the connection between nodes and history

bull If you know which node you have reached you know precisely the history of the play

bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)

12

Composition of information sets

bull Each decision node is in exactly one information set

bull all nodes of an information set must belong to same player

bull every node of an information set must have exactly the same set of available actions

bull If every information set of every player is a singleton we have a game of perfect information

13

Fundamental Tools Normal form games--Definition

The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn

We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i

ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen

14

Fundamental Tools Normal form games--Illustration

bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM

bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

15

Fundamental Tools Normal form games--Illustration

bull Donrsquot get rid of weakly dominated as lose equilibrium

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

16

Sequential games

bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)

bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)

17

bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa

bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered

bull ndash We need to define the concept of an equilibrium in extensive form games

18

Problems with Nash equilibrium

bull Sequential nature of the game is lost when representing extensive form games in strategic form

bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do

bull Nash equilibrium does not distinguish between credible and non-credible threats

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

9

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull LM decides first

ndash The strategies are conversed Utility=(LM BM)

Little Monkey

Big Monkey Big Monkey

w c

w c w c

00 44 19 35

10

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull They choose simultaneously

bull Information Set a set of nodes at which

bull The same player choosesbull The player choosing does not know which node represents the

actual choice node ndash represented by dotted line

Big Monkey

Little Monkeyw c

w c w c

00 91 44 53

c w

c 53 44

w 91 00

LM

BM

11

bull The key to representing information in a game tree is realizing the connection between nodes and history

bull If you know which node you have reached you know precisely the history of the play

bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)

12

Composition of information sets

bull Each decision node is in exactly one information set

bull all nodes of an information set must belong to same player

bull every node of an information set must have exactly the same set of available actions

bull If every information set of every player is a singleton we have a game of perfect information

13

Fundamental Tools Normal form games--Definition

The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn

We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i

ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen

14

Fundamental Tools Normal form games--Illustration

bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM

bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

15

Fundamental Tools Normal form games--Illustration

bull Donrsquot get rid of weakly dominated as lose equilibrium

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

16

Sequential games

bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)

bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)

17

bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa

bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered

bull ndash We need to define the concept of an equilibrium in extensive form games

18

Problems with Nash equilibrium

bull Sequential nature of the game is lost when representing extensive form games in strategic form

bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do

bull Nash equilibrium does not distinguish between credible and non-credible threats

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

10

Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)

bull They choose simultaneously

bull Information Set a set of nodes at which

bull The same player choosesbull The player choosing does not know which node represents the

actual choice node ndash represented by dotted line

Big Monkey

Little Monkeyw c

w c w c

00 91 44 53

c w

c 53 44

w 91 00

LM

BM

11

bull The key to representing information in a game tree is realizing the connection between nodes and history

bull If you know which node you have reached you know precisely the history of the play

bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)

12

Composition of information sets

bull Each decision node is in exactly one information set

bull all nodes of an information set must belong to same player

bull every node of an information set must have exactly the same set of available actions

bull If every information set of every player is a singleton we have a game of perfect information

13

Fundamental Tools Normal form games--Definition

The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn

We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i

ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen

14

Fundamental Tools Normal form games--Illustration

bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM

bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

15

Fundamental Tools Normal form games--Illustration

bull Donrsquot get rid of weakly dominated as lose equilibrium

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

16

Sequential games

bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)

bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)

17

bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa

bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered

bull ndash We need to define the concept of an equilibrium in extensive form games

18

Problems with Nash equilibrium

bull Sequential nature of the game is lost when representing extensive form games in strategic form

bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do

bull Nash equilibrium does not distinguish between credible and non-credible threats

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

11

bull The key to representing information in a game tree is realizing the connection between nodes and history

bull If you know which node you have reached you know precisely the history of the play

bull To express uncertainty we use concept of information set (set of nodes you could be in at a given time)

12

Composition of information sets

bull Each decision node is in exactly one information set

bull all nodes of an information set must belong to same player

bull every node of an information set must have exactly the same set of available actions

bull If every information set of every player is a singleton we have a game of perfect information

13

Fundamental Tools Normal form games--Definition

The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn

We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i

ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen

14

Fundamental Tools Normal form games--Illustration

bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM

bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

15

Fundamental Tools Normal form games--Illustration

bull Donrsquot get rid of weakly dominated as lose equilibrium

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

16

Sequential games

bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)

bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)

17

bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa

bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered

bull ndash We need to define the concept of an equilibrium in extensive form games

18

Problems with Nash equilibrium

bull Sequential nature of the game is lost when representing extensive form games in strategic form

bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do

bull Nash equilibrium does not distinguish between credible and non-credible threats

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

12

Composition of information sets

bull Each decision node is in exactly one information set

bull all nodes of an information set must belong to same player

bull every node of an information set must have exactly the same set of available actions

bull If every information set of every player is a singleton we have a game of perfect information

13

Fundamental Tools Normal form games--Definition

The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn

We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i

ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen

14

Fundamental Tools Normal form games--Illustration

bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM

bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

15

Fundamental Tools Normal form games--Illustration

bull Donrsquot get rid of weakly dominated as lose equilibrium

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

16

Sequential games

bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)

bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)

17

bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa

bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered

bull ndash We need to define the concept of an equilibrium in extensive form games

18

Problems with Nash equilibrium

bull Sequential nature of the game is lost when representing extensive form games in strategic form

bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do

bull Nash equilibrium does not distinguish between credible and non-credible threats

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

13

Fundamental Tools Normal form games--Definition

The n-player normal form game consists of ndash Players i = 1hellipnndash A set Si of strategies for player i = 1hellipn

We call s = (s1 hellip sn) where si Si for i = 1hellipn a strategy profile for the game Each si is a strategy for player i

ndash A function i S for player i = 1hellipn where S is the set of strategy profiles so i(s) is player irsquos payoff when strategy profile s is chosen

14

Fundamental Tools Normal form games--Illustration

bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM

bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

15

Fundamental Tools Normal form games--Illustration

bull Donrsquot get rid of weakly dominated as lose equilibrium

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

16

Sequential games

bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)

bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)

17

bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa

bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered

bull ndash We need to define the concept of an equilibrium in extensive form games

18

Problems with Nash equilibrium

bull Sequential nature of the game is lost when representing extensive form games in strategic form

bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do

bull Nash equilibrium does not distinguish between credible and non-credible threats

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

14

Fundamental Tools Normal form games--Illustration

bull Another way to depict the BM-LM game (where BM chooses first) LM Actions are ordered depending on (wc) of BM

bull Climb no matter what BM does (cc)bull Wait no matter what BM does (ww)bull Do the same thing BM does (wc)bull Do the opposite of what BM does (cw)

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

15

Fundamental Tools Normal form games--Illustration

bull Donrsquot get rid of weakly dominated as lose equilibrium

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

16

Sequential games

bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)

bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)

17

bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa

bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered

bull ndash We need to define the concept of an equilibrium in extensive form games

18

Problems with Nash equilibrium

bull Sequential nature of the game is lost when representing extensive form games in strategic form

bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do

bull Nash equilibrium does not distinguish between credible and non-credible threats

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

15

Fundamental Tools Normal form games--Illustration

bull Donrsquot get rid of weakly dominated as lose equilibrium

LM

BMcc cw wc ww

w 91 91 00 00

c 53 44 53 44

16

Sequential games

bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)

bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)

17

bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa

bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered

bull ndash We need to define the concept of an equilibrium in extensive form games

18

Problems with Nash equilibrium

bull Sequential nature of the game is lost when representing extensive form games in strategic form

bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do

bull Nash equilibrium does not distinguish between credible and non-credible threats

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

16

Sequential games

bull If players take turns to move then we have a sequential game (sometimes called a dynamic game)

bull We model a sequential game by using a lsquogame treersquo (or an lsquoextensive form representationrsquo)

17

bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa

bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered

bull ndash We need to define the concept of an equilibrium in extensive form games

18

Problems with Nash equilibrium

bull Sequential nature of the game is lost when representing extensive form games in strategic form

bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do

bull Nash equilibrium does not distinguish between credible and non-credible threats

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

17

bull It can be shown that every strategic form game can be represented by an extensive game form game and vice versa

bull But strategies that are in equilibrium in strategic form games are not necessary equilibrium strategies in extensive form games games Ex Monopolist Made sense to fight as an equilibrium not not if other has already entered

bull ndash We need to define the concept of an equilibrium in extensive form games

18

Problems with Nash equilibrium

bull Sequential nature of the game is lost when representing extensive form games in strategic form

bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do

bull Nash equilibrium does not distinguish between credible and non-credible threats

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

18

Problems with Nash equilibrium

bull Sequential nature of the game is lost when representing extensive form games in strategic form

bull Some Nash equilibria rely on playing actions that are not rational once that action node has been reached In other words a choice only makes sense if you know what the opponent will do

bull Nash equilibrium does not distinguish between credible and non-credible threats

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

19

Solving sequential games

bull To solve a sequential game we look for the lsquosubgame perfect Nash equilibriumrsquo

bull For our purposes this means we solve the game using lsquorollbackrsquondash To use rollback start at the end of each branch and

work backwards eliminating all but the optimal choice for the relevant player

(Technical point ndash you can only use this trick if there are no information sets If you donrsquot know where you are it may be too difficult to decide)

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

20

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

21

Subgame

bull Its game tree is a branch of the original game tree

bull The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch

bull The payoff vectors are the same as in the original game

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

22

Subgame perfect equilibrium amp credible threats

bull Proper subgame = subtree (of the game tree) whose root is alone in its information set

bull Subgame perfect equilibrium ndash Strategy profile that is in Nash equilibrium in

every proper subgame (including the root) whether or not that subgame is reached along the equilibrium path of play

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

23

bull On October 22 1962 after reviewing newly acquired intelligence President John F Kennedy informed the world that the Soviet Union was building secret missile bases in Cuba a mere 90 miles off the shores of Florida

bull After weighing such options as an armed invasion of Cuba and air strikes against the missiles Kennedy decided on a less dangerous response

bull In addition to demanding that Russian Premier Nikita S Khrushchev remove all the missile bases and their deadly contents Kennedy ordered a naval quarantine (blockade) of Cuba in order to prevent Russian ships from bringing additional missiles and construction materials to the island

bull In response to the American naval blockade Premier Khrushchev authorized his Soviet field commanders in Cuba to launch their tactical nuclear weapons if invaded by US forces

bull Deadlocked in this manner the two leaders of the worlds greatest nuclear superpowers stared each other down for seven days - until Khrushchev blinked On October 28 thinking better of prolonging his challenge to the United States the Russian Premier conceded to President Kennedys demands by ordering all Soviet supply ships away from Cuban waters and agreeing to remove the missiles from Cubas mainland After several days of teetering on the brink of nuclear holocaust the world breathed a sigh of relief

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

24

Example Cuban Missile Crisis

Khrushchev

Kennedy

Arm

Retract

Fold

Nuke

-1 1

- 100 - 100

10 -10

Pure strategy Nash equilibria (Arm Fold) and (Retract Nuke)

Pure strategy subgame perfect equilibria (Arm Fold) Conclusion Kennedyrsquos Nuke threat was not credible

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

25

Backwards induction

bull Start from the smallest subgames containing the terminal nodes of the game tree

bull Determine the action that a rational player would choose at that action node ndash At action nodes immediately adjacent to terminal nodes the

player should maximize the utility This is because she no longer cares about strategic interactions Regardless of how she moves nobody else can affect the payoff of the game

Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played

bull Repeat until there are no action nodes left

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

26

(MDBK) payoff

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

27

The predation game

bull Nasty Guys is an incumbent firm producing bricks

bull SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market

bull Nasty Guys says that if SIC enters then it will ldquosquish them like a bugrdquo

bull What should SIC do

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

28

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

29

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

If SIC actually enters then lsquofightingrsquo is an incredible threat ndash it hurts SIC but also hurts NG So SIC knows the threat is just bluff

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

30

The predation game

SIC = -10 NG = -10

SIC = 30 NG = 30

Fight

Donrsquot fight

NGEnter

SIC

Donrsquot enter

SIC=0 NG=100

So the equilibrium is

SIC will enter

NG will not fight

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

31

Credible commitments

bull When Cortes arrived in Mexico he ordered that his ships should be burnt

bull This seems sillyndash His troops were vastly outnumberedndash Surely it is better to keep an lsquoescape routersquo

home

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

32

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

Think of Cortes trying to motivate his own soldiers

Fight Hard

Be careful

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

33

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

If no retreat possible will fight hard or die But if retreat is possible may fight less hard and lsquorun awayrsquo

Fight Hard

Be careful

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

34

C

Burn ships

Keep Ships

S

S Be careful

Fight Hard

C = 100 S = 0

C = -100 S = -100

C = 100 S = 0

C = 0 S = 10

So Cortes wants to burn his ships It is a credible commitment not to retreat ndash and this alters how his own troops behave

Fight Hard

Be careful

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

35

Hold up

bull Hold up occurs if one party has to incur sunk costs before they bargain with another party

bull For example hardware manufacturers and software developersndash Hardware manufacturers want software

manufacturers to make applications for their hardware

ndash But most of the cost of software is sunkndash So if bargain after the software is designed the

hardware manufacturer can seize most of the benefits

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

36

Holdup in equilibrium no-one designs software

payoffs = (software nintendo)

SoftwareDesigner

Nintendo(-$50000 $250000)

($100000 $100000)

(0 0)

design

Donrsquot design

Bargain hard (= Pay low price)

Bargainldquosoftrdquo

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

37

Strategies in extensive form

A strategy in an extensive form game is a complete description of the actions that player performs at any action node at which it is her turn to move turn to move

Key pointsndashIt is not sufficient to specify responses only at

those action nodes that are arrived at via some particular sequence of plausible play

ndashA strategy must prescribe an action at any action node where that player moves node where that player

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

38

Definition The strategy set of agent i is the cartesian product of the sets of children nodes of each information set belonging to iDefinition An information set I is a subset

of the nodes in a game tree belonging to player P such that

- All iI belong to P- For ijI there is no path from

i to j- All iI have the same number

of outgoing edges

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

39

Sequential Prisonerrsquos Dilemma

dotted line means P2 doesnrsquot know which state he is in

P1

P2 P2

Confess

Confess Confess Deny

Deny

Deny

(-5-5) (0-10) (-100) (-1-1)

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

40

bull With perfect information ndash each information set is a singleton (as you always know which state you are in)

bull A strategy profile (s1s2hellipsn) determines a unique path from the root to some terminal node (where s1 states what player 1 will do in every situation)

bull We say this unique path is supported by the strategy profile A path supported by a Nash equilibrium will be called an equilibrium path

bull A Nash equilbrium in sequential game (perfect or imperfect)

bull U(si s-i) gtU(si s-i) for all i

bull Note there can be two different strategy profiles which have the same path

bull Every path from the root to a terminal node is supported by at least one strategy profile

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

41

Example 49RLrdquo is best pathStategies (R RrsquoLrdquo) and (R LrsquoLrdquo) both support it(R RrsquoLrdquo) means P1 always takes R P2 takes Rrsquo if at node B and Lrdquo if at node C[Note Notation is confusing you always have to read to get meaning]

P1

P2 P2

A

(03) (41) (10)

B C

D E F

(21)G

L

LrsquoRrdquo

R

Rrsquo Lrdquo

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

42

Theorem (Kuhn) Every sequential game with perfect information has a Nash equilibrium (use backwards induction)

P1

P2 P2

A

(10) (01) (22)

B C

D E F

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

43

Stackelberg duopoly (like a monopoly but with exactly two players) corresponds to a sequential game first leader (firm 1) chooses how much to produce then follower (firm 2) chooses

can be solved by backward induction for each quantity q1 the follower chooses its best response q2

i (q1 q2) = qi[p(q) -ci]

where q = q1+q2

p(q) = A-q is the market clearing price when the total output in the market is q

ci is the marginal cost of the production of the product by firm i

That is the profit for each firm is

i (q1 q2) = qi[A-q1-q2 -ci]

Example 412 Stackelberg Duopoly Model

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

44

Solving by backwards induction

bull This is a two person game sequential game with two stages and perfect information

bull Find best response for each choice of q1

2 (q1 q2) = max q2[A-q1-q2 ndashc2]

2 (q1 q2) = -(q2)2 + q2[A-q1 ndashc2]

= -2q2 +A-q1-c2

Second derivative = -2

So the maximizer is (A-q1-c2)2

2

2

q

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

45

Continuing

bull Thus firm 1 should anticipate this result and choose q1 to maximize

1 (q1 q2) = q1[A-q1-(A-q1-c2)2 ndashc1] bull = frac12(-q1

2 + (A+c2-2c1)q1)

bull = -q1 +12(A+c2-2c1)

bull q1= frac12((A+c2-2c1) and q2 = frac14(A+2c1-3c2)

1

1

q

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

46

Type of games

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

47

subgame perfect equilibrium

bull A strategy profile of a sequential game is a subgame perfect equilibrium if it is a Nash equilbrium for every subgame of the original game In other words the strategy is perfect even if the play never goes to that part of the tree

bull An imperfect equilibrium is like a strategy that wouldnrsquot be optimal if the other player did something different

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

48

Imperfect information

bull 10487081048708 SPE is not an appropriate equilibrium concept because most games with imperfect information have too few proper subgames to rule out extraneous Nash equilibria of the game

bull Alternative equilibrium conceptsbull ndash Bayesian EquilibriumPerfect Bayesian Equilibriumbull ndash Sequentially rational equilibriumbull ndash Forwards inductionbull ndash Trembling hand equilibrium Topic of active research

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

49

bull A Nash equilibrium that fails to be subgame perfect is also known as Nash equilibrium supported by noncredible behavior

bull To find subgame perfect equilibrium use backward induction on the subgames of the original problem

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

50

Bob and Betty

bull Bob and Betty must cook wash dishes and vacuum Bob cant cook very well and just doesnt like to wash dishes so they have concocted the following game for allocating the tasks Betty moves first and she chooses between cooking and doing the dishes

bull If she chooses dishes then Bob chooses to Go Out or Cook

bull On the other hand if Betty chooses to cook then they simultaneously choose between the remaining two tasks vacuuming and doing the dishes The payoffs are at the end of the tree

bull You may conclude whatever you want about the relationships between the payoffs and the preferences of Bob and Betty for doing chores and being together

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

51

Note use normal form game to pick what Betty should do at CD

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

52

bull There are three subgames There are two proper subgames one beginning at node B and one beginning at node A The game itself is a subgame

bull There are two paths to Nash equilibriabull The first one (Path One) from the root through node A is BettyDishes

BobOut bull The other one (Path Two) from the root through node B to the information

set containing nodes C and D is BettyCook BobVacuum BettyVacuum You must use normal form on the subgame beginning at node B in order to find this second path

bull Path Two is supported by the strategy profile (Cook Vacuum Out Vacuum) That is Betty plays Cook at the root and Vacuum at C or D Bob plays Out at node A and Vacuum at node B This path is a subgame perfect equilibrium

bull For the subgame starting at A the proposed strategy profile reduces to ( Out) which is a Nash equilibrium For the subgame starting at node B the strategy profile reduces to (Vacuum Vacuum) which is a Nash equilibrium

bull Are there any other strategy profiles that will support Path Two

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

53

The strategy profile (Dishes Vacuum Out Vacuum) supports Path One the road to a Nash equilibrium

At the root node Betty plays Dishes and at node A Bob plays Out If they should happen to find themselves at the subgame starting at node B then they both play vacuum which is a Nash Equilibrium

But this strategy profile is not the only one that supports Path One Path One is also supported by the strategy profile (Dishes Dishes Out Dishes)

Betty plays Dishes at the root and Dishes at nodes C or D Bob plays Out at node A and Dishes at node B This is a Nash equilibrium profile since Out is Bobs best response to a play of Dishes by Betty at the root node

But this is not a subgame perfect equilibrium since a play of Dishes by Bob at node B and a play of Dishes by Betty at C or D could be improved upon In this case we have a strategy profile that involves a Nash equilibrium in one subgame but noncredible plays in another subgame

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

54

bull Consider the strategy profile (Dishes Vacuum Out Dishes) which also supports Path One

bull Is this profile a subgame perfect equilibrium No This profile does result in a Nash equilibrium in the subgame beginning at node A but there is a hitch In the subgame that includes the root Betty would never play Dishes at the root as called for by the profile

bull Are there any more strategic profiles that support Path One Consider (Dishes Dishes Out Vacuum) and explain why this is not a subgame perfect equilibrium

bull As in the normal form games we have seen there may be multiple Nash equilibria in an extensive form game The principle of subgame perfect equilibrium is to eliminate those Nash equilibria which may be based on non-credible or unreasonable promises or threats

bull In the analysis of the second Bob and Betty example we eliminated two strategy profiles that involved a Nash equilibrium in a subgame but which also included unreasonable behavior in other subgames

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

55

In Class ExerciseAsk the students to choose partners from the other side of the room or have

them imagine that each is playing with one person who is sitting on the other side of the room Each student will eventually be asked to write either pink or purple If both students in the real or imaginary pair write pink the person on the right-hand side of the room gets 50 points and the person on the left-hand side of the room gets 40 points (Right-hand and left-hand are defined from the studentsrsquo point of view) If both write purple the person on the lefthand side of the room gets 50 points and the person on the right-hand side of the room gets 40 points If the answers dont match neither player gets anything To play without the delay tactic simply ask the students to choose a color and write the choice Then play again immediately but explain that you will flip a coin first If it comes up heads those on the right-hand side of the room get to write their answers first otherwise those on the left-hand side of the room write first

What happens if we delay

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

56

bull Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions the tennis-point example has a second-mover advantage in its sequential-move version Other games show no change in equilibrium as a result of the change in rules games like the prisonersrsquo dilemma in which both players have dominant strategies fall into this category

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

57

Prove or disprove

bull When both players have a dominant strategy the dominant-strategy equilibrium will hold in both the simultaneous and the sequential versions of the game

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

58

Sequential Monopolist View

What are Nash equilibriaAre they subgame perfect

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

59

Thought Question

bull How do we change a game to our advantage

bull Use commitment threats and promises to change the nature of a game

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

60

Commitment

bull Reduce freedom of action by commitmentbull Thereby forcing a ldquofirst mover advantagerdquobull This move has to be ldquoobservablerdquobull Move has to be credible (believable)

ndash Reputation leads to credibility

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

61

Commitment ndash An Example

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

62

Commitment ndash An ExampleFor those that intend to teachhellip

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

NE

Tough 3 2 1 1

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

63

Commitment ndash An ExampleBut if we announce we are tough

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2 1 1

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

64

Commitment ndash An ExampleGet different NE

STUDENT

TE

AC

HE

R

Punctual Late

Weak 4 3 2 4

Tough 3 2NE

1 1

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

65

Strategic Moves and Threats

bull Deterrence and Compellence achieved through either a threat with an implied promise or a promise with an implied threat

bull Either deterrence or compellence requires credibilityndash Credibility involves some cost to the threatener as wellndash If threatener preferred the threat then she would carry it out anyway and

the promise part of the threat would never be carried outndash This cost is problematic because it might tempt the threatener to avoid

actually carrying out the threat thus making it less crediblendash Thus imposing a cost on herself is a necessary condition for a threat to

be successful but is not a sufficient conditionndash CAREFUL

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

66

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

67

Trade Negotiation

Japan

US Open Closed

Open 4 3 3 4

Closed 2 1 1 2

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

68

Changing the game ndash A threat

bull A threat -- ldquoWe will close our market if you close yoursrdquo

bull And a promise ndash ldquo We will open market if you open yoursrdquo

bull Effectively reduces Japans options tondash If Japan stays open then the US stays open

giving Japan 3 and the US 4 (favorable to the US over the no threat scenario)

ndash If Japan stays closed then US closes as well thus giving Japan 2 and the US 1

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

69

Trade Relations ndash Threats in Actionbull Credible partly because the threat is costly to the US If it were not costly

to the US to play close then Japan would not believe that US would play open otherwise and therefore would never open her own borders

bull If Japan calls US bluff then US has the temptation to not carry out the threat ndash cutting off freedom of action in the future (as threat will not be believed) That fact may keep threat credible

bull If Japanese market is already open then threat is part of deterrence strategy

bull If Japanese market is closed then threat is part of compellence strategy

bull A good threat is one that does not need to be carried outNOTE If the size of the threat is too big to be credible then a probabilistic element may be introduced (Gamble on whether will be carried out and on what amount of penalty will be)

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

70

Threats in action (cont)bull Japan may respond by agreeing in principle and then stalling ndash

banking on the USrsquos lack of desire to impose a cost on itself a technique called salami tactics

bull Salami tactics ndash fail to comply with the others wishes (particularly in compellence situations) in such a small way that it is not worth the while of the threatener to carry out her threat If that works you transgress a little morethe way to oppose salami tactics is by using a graduated threat

bull Used in politics a gradual process of threats and alliances as a means of overcoming opposition Consequently the player was able to exert hisher influence and eventually dominate the political landscape -- slice by slice

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

71

Prisoners Dilemma ndash Promises to Keep

bull Tit for Tat strategies are examples of promises that act as a deterrent to cheating

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

72

Promises or Threats

bull Deterrence ldquoPrevent you from doing somethingrdquo Threat ldquoI will punish you if you do itrdquo ndash requires me to only wait till you mess up ndash no monitoringPromise ldquoI will reward you if you donrsquot do itrdquo ndash requires me to monitor you constantlyThreats are cheaper than promises

bull Compellence ldquoI want you to do somethingrdquoThreat ldquoI will punish you if you donrsquot do itrdquo ndash requires me to monitor you constantlyPromise ldquoI will reward you if you do itrdquo ndash requires me to only wait till you do what I want you to doPromises are cheaper than threats

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

73

Countering Threatsbull Irrationality So nuts that any threats

will not have an effect on your behavior

bull Cut of communication so threats donrsquot reach you

bull Open escape routes for enemy thus tempting them to renege on threats

bull Undermine opponents motive to maintain reputation by promising secrecy if she does not punish you

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

74

Credibility Devices

bull Since credibility implies the temptation to be ldquonotrdquo credible certain devices are required to ensure credibility

Reducing the freedom of action through ndash automatic fulfillment (doomsday device)

According to a new book exclusively obtained by the Huffington Post Saudi Arabia has crafted a plan to protect itself from a possible invasion or internal attack It includes the use of a series of explosives including radioactive ldquodirty bombsrdquo that would cripple Saudi Arabian oil production and distribution systems for decades

ndash burning bridges ndash cutting off communication so nobody can argue with you

regarding your threat

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

75

Credibility Devicesbull Changing payoffs by using reputation bull make game part of larger game so that payoffs in the current

game are linked to repurcussions in other game bull dividing game into little subgames to allow reputation to work bull reduce monitoring costs and thus change payoffs by allowing

players to monitor each other bull irrationality ndash worry I just might Nuke as Irsquom irrationalbull contracts ndash have way of punishing if deal not keptbull Brinkmanship the policy or practice especially in politics and

foreign policy of pushing a dangerous situation to the brink of disaster (to the limits of safety) in order to achieve the most advantageous outcome by forcing the opposition to make concessions This might be achieved through diplomatic maneuvers by creating the impression that one is willing to use extreme methods rather than concede During the Cold War the threat of nuclear force was often used as such a deterrent

76

Solving Extensive Form Games

bull the usual procedure is to convert the extensive-form game to strategic form and find its equilibria

bull However some of these equilibria would have important drawbacks because they ignore the dynamic nature of the extensive-form

bull Reinchard Selten was the first to argue that some Nash equilibria are ldquomore reasonablerdquo than others in his 1965 article

77

Seltenrsquos Game

78

bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there

bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out

bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves

79

bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play

bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set

80

Little Horsey

bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D

81

Where are the NE

bull Convert to normal formbull Use standard techniques

82

Giving Gifts

83

bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it

84

bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted

bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1

bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing

bull Letrsquos construct the strategic form of this game

85

Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift

The action GN refers to ldquogive if get game theory not give if star trekrdquo

(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered

86

bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins

87

bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set

bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1

bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)

88

bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)

bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift

bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game

bull The belief will be a probability distribution over the nodes

89

bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached

bull A continuation game refers to the information set and all nodes that follow from that information set

90

bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ

bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x

91

bull Player irsquos expected utility in the continuation game starting at node x then is

bull Ui(σ|x) = zP(z|σx)ui(z)

bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin

92

bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q

bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1

bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold

bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set

bull In other words the unique sequentially rational strategy is to choose Y with certainty

• Chapter 4 Sequential Games
• Extensive Form Games
• Extensive Form Games - Intro
• Slide 4
• Fundamental Tools
• Fundamental Tools Extensive form games--Definition
• Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Composition of information sets
• Fundamental Tools Normal form games--Definition
• Fundamental Tools Normal form games--Illustration
• Slide 15
• Sequential games
• Slide 17
• Problems with Nash equilibrium
• Solving sequential games
• Slide 20
• Subgame
• Subgame perfect equilibrium amp credible threats
• Slide 23
• Example Cuban Missile Crisis
• Backwards induction
• Slide 26
• The predation game
• Slide 28
• Slide 29
• Slide 30
• Credible commitments
• Slide 32
• Slide 33
• Slide 34
• Hold up
• Holdup in equilibrium no-one designs software payoffs = (software nintendo)
• Strategies in extensive form
• Slide 38
• Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
• Slide 40
• Slide 41
• Slide 42
• Solving by backwards induction
• Continuing
• Type of games
• subgame perfect equilibrium
• Imperfect information
• Slide 49
• Bob and Betty
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• In Class Exercise
• Slide 56
• Prove or disprove
• Sequential Monopolist View
• Thought Question
• Commitment
• Commitment ndash An Example
• Commitment ndash An Example For those that intend to teachhellip
• Commitment ndash An Example But if we announce we are tough
• Commitment ndash An Example Get different NE
• Strategic Moves and Threats
• Trade Negotiation
• Slide 67
• Changing the game ndash A threat
• Trade Relations ndash Threats in Action
• Threats in action (cont)
• Prisoners Dilemma ndash Promises to Keep
• Promises or Threats
• Countering Threats
• Credibility Devices
• Slide 75
• Solving Extensive Form Games
• Seltenrsquos Game
• Slide 78
• Slide 79
• Little Horsey
• Where are the NE
• Slide 82
• Slide 83
• Slide 84
• Slide 85
• Slide 86
• Slide 87
• Slide 88
• Slide 89
• Slide 90
• Slide 91
• Slide 92

77

Seltenrsquos Game

78

bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there

bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out

bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves

79

bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play

bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set

80

Little Horsey

bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D

81

Where are the NE

bull Convert to normal formbull Use standard techniques

82

Giving Gifts

83

bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it

84

bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted

bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1

bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing

bull Letrsquos construct the strategic form of this game

85

Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift

The action GN refers to ldquogive if get game theory not give if star trekrdquo

(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered

86

bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins

87

bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set

bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1

bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)

88

bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)

bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift

bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game

bull The belief will be a probability distribution over the nodes

89

bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached

bull A continuation game refers to the information set and all nodes that follow from that information set

90

bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ

bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x

91

bull Player irsquos expected utility in the continuation game starting at node x then is

bull Ui(σ|x) = zP(z|σx)ui(z)

bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin

92

bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q

bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1

bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold

bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set

bull In other words the unique sequentially rational strategy is to choose Y with certainty

• Chapter 4 Sequential Games
• Extensive Form Games
• Extensive Form Games - Intro
• Slide 4
• Fundamental Tools
• Fundamental Tools Extensive form games--Definition
• Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Composition of information sets
• Fundamental Tools Normal form games--Definition
• Fundamental Tools Normal form games--Illustration
• Slide 15
• Sequential games
• Slide 17
• Problems with Nash equilibrium
• Solving sequential games
• Slide 20
• Subgame
• Subgame perfect equilibrium amp credible threats
• Slide 23
• Example Cuban Missile Crisis
• Backwards induction
• Slide 26
• The predation game
• Slide 28
• Slide 29
• Slide 30
• Credible commitments
• Slide 32
• Slide 33
• Slide 34
• Hold up
• Holdup in equilibrium no-one designs software payoffs = (software nintendo)
• Strategies in extensive form
• Slide 38
• Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
• Slide 40
• Slide 41
• Slide 42
• Solving by backwards induction
• Continuing
• Type of games
• subgame perfect equilibrium
• Imperfect information
• Slide 49
• Bob and Betty
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• In Class Exercise
• Slide 56
• Prove or disprove
• Sequential Monopolist View
• Thought Question
• Commitment
• Commitment ndash An Example
• Commitment ndash An Example For those that intend to teachhellip
• Commitment ndash An Example But if we announce we are tough
• Commitment ndash An Example Get different NE
• Strategic Moves and Threats
• Trade Negotiation
• Slide 67
• Changing the game ndash A threat
• Trade Relations ndash Threats in Action
• Threats in action (cont)
• Prisoners Dilemma ndash Promises to Keep
• Promises or Threats
• Countering Threats
• Credibility Devices
• Slide 75
• Solving Extensive Form Games
• Seltenrsquos Game
• Slide 78
• Slide 79
• Little Horsey
• Where are the NE
• Slide 82
• Slide 83
• Slide 84
• Slide 85
• Slide 86
• Slide 87
• Slide 88
• Slide 89
• Slide 90
• Slide 91
• Slide 92

78

bull The strategic form representation has two pure-strategy Nash equilibria (D L) and (UR) Look closely at the Nash equilibrium (UR) and what it implies for the extensive form In the profile (UR) player 2rsquos information set is never reached and she loses nothing by playing R there

bull But there is something ldquowrongrdquo with this equilibrium if player 2rsquos information set is ever reached then she would be strictly better off by choosing L instead of R In effect player 2 is threatening player 1 with an action that would not be in her own interest to carry out

bull we are interested in sequencing of moves presumably because players get to reassess their plans of actions in light of past moves

79

bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play

bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set

80

Little Horsey

bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D

81

Where are the NE

bull Convert to normal formbull Use standard techniques

82

Giving Gifts

83

bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it

84

bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted

bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1

bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing

bull Letrsquos construct the strategic form of this game

85

Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift

The action GN refers to ldquogive if get game theory not give if star trekrdquo

(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered

86

bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins

87

bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set

bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1

bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)

88

bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)

bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift

bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game

bull The belief will be a probability distribution over the nodes

89

bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached

bull A continuation game refers to the information set and all nodes that follow from that information set

90

bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ

bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x

91

bull Player irsquos expected utility in the continuation game starting at node x then is

bull Ui(σ|x) = zP(z|σx)ui(z)

bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin

92

bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q

bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1

bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold

bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set

bull In other words the unique sequentially rational strategy is to choose Y with certainty

• Chapter 4 Sequential Games
• Extensive Form Games
• Extensive Form Games - Intro
• Slide 4
• Fundamental Tools
• Fundamental Tools Extensive form games--Definition
• Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Composition of information sets
• Fundamental Tools Normal form games--Definition
• Fundamental Tools Normal form games--Illustration
• Slide 15
• Sequential games
• Slide 17
• Problems with Nash equilibrium
• Solving sequential games
• Slide 20
• Subgame
• Subgame perfect equilibrium amp credible threats
• Slide 23
• Example Cuban Missile Crisis
• Backwards induction
• Slide 26
• The predation game
• Slide 28
• Slide 29
• Slide 30
• Credible commitments
• Slide 32
• Slide 33
• Slide 34
• Hold up
• Holdup in equilibrium no-one designs software payoffs = (software nintendo)
• Strategies in extensive form
• Slide 38
• Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
• Slide 40
• Slide 41
• Slide 42
• Solving by backwards induction
• Continuing
• Type of games
• subgame perfect equilibrium
• Imperfect information
• Slide 49
• Bob and Betty
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• In Class Exercise
• Slide 56
• Prove or disprove
• Sequential Monopolist View
• Thought Question
• Commitment
• Commitment ndash An Example
• Commitment ndash An Example For those that intend to teachhellip
• Commitment ndash An Example But if we announce we are tough
• Commitment ndash An Example Get different NE
• Strategic Moves and Threats
• Trade Negotiation
• Slide 67
• Changing the game ndash A threat
• Trade Relations ndash Threats in Action
• Threats in action (cont)
• Prisoners Dilemma ndash Promises to Keep
• Promises or Threats
• Countering Threats
• Credibility Devices
• Slide 75
• Solving Extensive Form Games
• Seltenrsquos Game
• Slide 78
• Slide 79
• Little Horsey
• Where are the NE
• Slide 82
• Slide 83
• Slide 84
• Slide 85
• Slide 86
• Slide 87
• Slide 88
• Slide 89
• Slide 90
• Slide 91
• Slide 92

79

bull To anticipate a bit of what follows the problem with the (UR) solution is that it specifies the incredible action at an information set that is off the equilibrium path of play

bull Player 2rsquos information set is never reached if player 1 chooses U Consequently Nash equilibrium cannot pin down the optimality of the action at that information set

80

Little Horsey

bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D

81

Where are the NE

bull Convert to normal formbull Use standard techniques

82

Giving Gifts

83

bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it

84

bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted

bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1

bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing

bull Letrsquos construct the strategic form of this game

85

Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift

The action GN refers to ldquogive if get game theory not give if star trekrdquo

(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered

86

bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins

87

bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set

bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1

bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)

88

bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)

bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift

bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game

bull The belief will be a probability distribution over the nodes

89

bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached

bull A continuation game refers to the information set and all nodes that follow from that information set

90

bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ

bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x

91

bull Player irsquos expected utility in the continuation game starting at node x then is

bull Ui(σ|x) = zP(z|σx)ui(z)

bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin

92

bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q

bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1

bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold

bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set

bull In other words the unique sequentially rational strategy is to choose Y with certainty

• Chapter 4 Sequential Games
• Extensive Form Games
• Extensive Form Games - Intro
• Slide 4
• Fundamental Tools
• Fundamental Tools Extensive form games--Definition
• Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Composition of information sets
• Fundamental Tools Normal form games--Definition
• Fundamental Tools Normal form games--Illustration
• Slide 15
• Sequential games
• Slide 17
• Problems with Nash equilibrium
• Solving sequential games
• Slide 20
• Subgame
• Subgame perfect equilibrium amp credible threats
• Slide 23
• Example Cuban Missile Crisis
• Backwards induction
• Slide 26
• The predation game
• Slide 28
• Slide 29
• Slide 30
• Credible commitments
• Slide 32
• Slide 33
• Slide 34
• Hold up
• Holdup in equilibrium no-one designs software payoffs = (software nintendo)
• Strategies in extensive form
• Slide 38
• Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
• Slide 40
• Slide 41
• Slide 42
• Solving by backwards induction
• Continuing
• Type of games
• subgame perfect equilibrium
• Imperfect information
• Slide 49
• Bob and Betty
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• In Class Exercise
• Slide 56
• Prove or disprove
• Sequential Monopolist View
• Thought Question
• Commitment
• Commitment ndash An Example
• Commitment ndash An Example For those that intend to teachhellip
• Commitment ndash An Example But if we announce we are tough
• Commitment ndash An Example Get different NE
• Strategic Moves and Threats
• Trade Negotiation
• Slide 67
• Changing the game ndash A threat
• Trade Relations ndash Threats in Action
• Threats in action (cont)
• Prisoners Dilemma ndash Promises to Keep
• Promises or Threats
• Countering Threats
• Credibility Devices
• Slide 75
• Solving Extensive Form Games
• Seltenrsquos Game
• Slide 78
• Slide 79
• Little Horsey
• Where are the NE
• Slide 82
• Slide 83
• Slide 84
• Slide 85
• Slide 86
• Slide 87
• Slide 88
• Slide 89
• Slide 90
• Slide 91
• Slide 92

80

Little Horsey

bull Consider the following simple gamePlayer 1 gets to choose between U M or D If he chooses D the game ends If he chooses either U or M player 2 gets to choose between L and R without knowing what action player 1 has taken except that it was not D

81

Where are the NE

bull Convert to normal formbull Use standard techniques

82

Giving Gifts

83

bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it

84

bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted

bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1

bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing

bull Letrsquos construct the strategic form of this game

85

Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift

The action GN refers to ldquogive if get game theory not give if star trekrdquo

(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered

86

bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins

87

bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set

bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1

bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)

88

bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)

bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift

bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game

bull The belief will be a probability distribution over the nodes

89

bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached

bull A continuation game refers to the information set and all nodes that follow from that information set

90

bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ

bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x

91

bull Player irsquos expected utility in the continuation game starting at node x then is

bull Ui(σ|x) = zP(z|σx)ui(z)

bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin

92

bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q

bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1

bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold

bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set

bull In other words the unique sequentially rational strategy is to choose Y with certainty

• Chapter 4 Sequential Games
• Extensive Form Games
• Extensive Form Games - Intro
• Slide 4
• Fundamental Tools
• Fundamental Tools Extensive form games--Definition
• Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Composition of information sets
• Fundamental Tools Normal form games--Definition
• Fundamental Tools Normal form games--Illustration
• Slide 15
• Sequential games
• Slide 17
• Problems with Nash equilibrium
• Solving sequential games
• Slide 20
• Subgame
• Subgame perfect equilibrium amp credible threats
• Slide 23
• Example Cuban Missile Crisis
• Backwards induction
• Slide 26
• The predation game
• Slide 28
• Slide 29
• Slide 30
• Credible commitments
• Slide 32
• Slide 33
• Slide 34
• Hold up
• Holdup in equilibrium no-one designs software payoffs = (software nintendo)
• Strategies in extensive form
• Slide 38
• Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
• Slide 40
• Slide 41
• Slide 42
• Solving by backwards induction
• Continuing
• Type of games
• subgame perfect equilibrium
• Imperfect information
• Slide 49
• Bob and Betty
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• In Class Exercise
• Slide 56
• Prove or disprove
• Sequential Monopolist View
• Thought Question
• Commitment
• Commitment ndash An Example
• Commitment ndash An Example For those that intend to teachhellip
• Commitment ndash An Example But if we announce we are tough
• Commitment ndash An Example Get different NE
• Strategic Moves and Threats
• Trade Negotiation
• Slide 67
• Changing the game ndash A threat
• Trade Relations ndash Threats in Action
• Threats in action (cont)
• Prisoners Dilemma ndash Promises to Keep
• Promises or Threats
• Countering Threats
• Credibility Devices
• Slide 75
• Solving Extensive Form Games
• Seltenrsquos Game
• Slide 78
• Slide 79
• Little Horsey
• Where are the NE
• Slide 82
• Slide 83
• Slide 84
• Slide 85
• Slide 86
• Slide 87
• Slide 88
• Slide 89
• Slide 90
• Slide 91
• Slide 92

81

Where are the NE

bull Convert to normal formbull Use standard techniques

82

Giving Gifts

83

bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it

84

bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted

bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1

bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing

bull Letrsquos construct the strategic form of this game

85

Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift

The action GN refers to ldquogive if get game theory not give if star trekrdquo

(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered

86

bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins

87

bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set

bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1

bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)

88

bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)

bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift

bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game

bull The belief will be a probability distribution over the nodes

89

bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached

bull A continuation game refers to the information set and all nodes that follow from that information set

90

bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ

bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x

91

bull Player irsquos expected utility in the continuation game starting at node x then is

bull Ui(σ|x) = zP(z|σx)ui(z)

bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin

92

bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q

bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1

bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold

bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set

bull In other words the unique sequentially rational strategy is to choose Y with certainty

• Chapter 4 Sequential Games
• Extensive Form Games
• Extensive Form Games - Intro
• Slide 4
• Fundamental Tools
• Fundamental Tools Extensive form games--Definition
• Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Composition of information sets
• Fundamental Tools Normal form games--Definition
• Fundamental Tools Normal form games--Illustration
• Slide 15
• Sequential games
• Slide 17
• Problems with Nash equilibrium
• Solving sequential games
• Slide 20
• Subgame
• Subgame perfect equilibrium amp credible threats
• Slide 23
• Example Cuban Missile Crisis
• Backwards induction
• Slide 26
• The predation game
• Slide 28
• Slide 29
• Slide 30
• Credible commitments
• Slide 32
• Slide 33
• Slide 34
• Hold up
• Holdup in equilibrium no-one designs software payoffs = (software nintendo)
• Strategies in extensive form
• Slide 38
• Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
• Slide 40
• Slide 41
• Slide 42
• Solving by backwards induction
• Continuing
• Type of games
• subgame perfect equilibrium
• Imperfect information
• Slide 49
• Bob and Betty
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• In Class Exercise
• Slide 56
• Prove or disprove
• Sequential Monopolist View
• Thought Question
• Commitment
• Commitment ndash An Example
• Commitment ndash An Example For those that intend to teachhellip
• Commitment ndash An Example But if we announce we are tough
• Commitment ndash An Example Get different NE
• Strategic Moves and Threats
• Trade Negotiation
• Slide 67
• Changing the game ndash A threat
• Trade Relations ndash Threats in Action
• Threats in action (cont)
• Prisoners Dilemma ndash Promises to Keep
• Promises or Threats
• Countering Threats
• Credibility Devices
• Slide 75
• Solving Extensive Form Games
• Seltenrsquos Game
• Slide 78
• Slide 79
• Little Horsey
• Where are the NE
• Slide 82
• Slide 83
• Slide 84
• Slide 85
• Slide 86
• Slide 87
• Slide 88
• Slide 89
• Slide 90
• Slide 91
• Slide 92

82

Giving Gifts

83

bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it

84

bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted

bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1

bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing

bull Letrsquos construct the strategic form of this game

85

Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift

The action GN refers to ldquogive if get game theory not give if star trekrdquo

(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered

86

bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins

87

bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set

bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1

bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)

88

bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)

bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift

bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game

bull The belief will be a probability distribution over the nodes

89

bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached

bull A continuation game refers to the information set and all nodes that follow from that information set

90

bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ

bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x

91

bull Player irsquos expected utility in the continuation game starting at node x then is

bull Ui(σ|x) = zP(z|σx)ui(z)

bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin

92

bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q

bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1

bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold

bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set

bull In other words the unique sequentially rational strategy is to choose Y with certainty

• Chapter 4 Sequential Games
• Extensive Form Games
• Extensive Form Games - Intro
• Slide 4
• Fundamental Tools
• Fundamental Tools Extensive form games--Definition
• Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Composition of information sets
• Fundamental Tools Normal form games--Definition
• Fundamental Tools Normal form games--Illustration
• Slide 15
• Sequential games
• Slide 17
• Problems with Nash equilibrium
• Solving sequential games
• Slide 20
• Subgame
• Subgame perfect equilibrium amp credible threats
• Slide 23
• Example Cuban Missile Crisis
• Backwards induction
• Slide 26
• The predation game
• Slide 28
• Slide 29
• Slide 30
• Credible commitments
• Slide 32
• Slide 33
• Slide 34
• Hold up
• Holdup in equilibrium no-one designs software payoffs = (software nintendo)
• Strategies in extensive form
• Slide 38
• Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
• Slide 40
• Slide 41
• Slide 42
• Solving by backwards induction
• Continuing
• Type of games
• subgame perfect equilibrium
• Imperfect information
• Slide 49
• Bob and Betty
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• In Class Exercise
• Slide 56
• Prove or disprove
• Sequential Monopolist View
• Thought Question
• Commitment
• Commitment ndash An Example
• Commitment ndash An Example For those that intend to teachhellip
• Commitment ndash An Example But if we announce we are tough
• Commitment ndash An Example Get different NE
• Strategic Moves and Threats
• Trade Negotiation
• Slide 67
• Changing the game ndash A threat
• Trade Relations ndash Threats in Action
• Threats in action (cont)
• Prisoners Dilemma ndash Promises to Keep
• Promises or Threats
• Countering Threats
• Credibility Devices
• Slide 75
• Solving Extensive Form Games
• Seltenrsquos Game
• Slide 78
• Slide 79
• Little Horsey
• Where are the NE
• Slide 82
• Slide 83
• Slide 84
• Slide 85
• Slide 86
• Slide 87
• Slide 88
• Slide 89
• Slide 90
• Slide 91
• Slide 92

83

bull There are two players and player 1 receives a book which with probability p is a small game theory pocket reference and with probability 1 minus p is a Star Trek data manual The player sees the book wraps it up and decides whether to offer it to player 2 as a gift Player 2 hates Star Trek and is currently suffering in a graduate game theory course so she would prefer to get one but not the other Unfortunately she cannot know what is being offered until she accepts it

84

bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted

bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1

bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing

bull Letrsquos construct the strategic form of this game

85

Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift

The action GN refers to ldquogive if get game theory not give if star trekrdquo

(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered

86

bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins

87

bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set

bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1

bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)

88

bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)

bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift

bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game

bull The belief will be a probability distribution over the nodes

89

bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached

bull A continuation game refers to the information set and all nodes that follow from that information set

90

bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ

bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x

91

bull Player irsquos expected utility in the continuation game starting at node x then is

bull Ui(σ|x) = zP(z|σx)ui(z)

bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin

92

bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q

bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1

bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold

bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set

bull In other words the unique sequentially rational strategy is to choose Y with certainty

• Chapter 4 Sequential Games
• Extensive Form Games
• Extensive Form Games - Intro
• Slide 4
• Fundamental Tools
• Fundamental Tools Extensive form games--Definition
• Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Composition of information sets
• Fundamental Tools Normal form games--Definition
• Fundamental Tools Normal form games--Illustration
• Slide 15
• Sequential games
• Slide 17
• Problems with Nash equilibrium
• Solving sequential games
• Slide 20
• Subgame
• Subgame perfect equilibrium amp credible threats
• Slide 23
• Example Cuban Missile Crisis
• Backwards induction
• Slide 26
• The predation game
• Slide 28
• Slide 29
• Slide 30
• Credible commitments
• Slide 32
• Slide 33
• Slide 34
• Hold up
• Holdup in equilibrium no-one designs software payoffs = (software nintendo)
• Strategies in extensive form
• Slide 38
• Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
• Slide 40
• Slide 41
• Slide 42
• Solving by backwards induction
• Continuing
• Type of games
• subgame perfect equilibrium
• Imperfect information
• Slide 49
• Bob and Betty
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• In Class Exercise
• Slide 56
• Prove or disprove
• Sequential Monopolist View
• Thought Question
• Commitment
• Commitment ndash An Example
• Commitment ndash An Example For those that intend to teachhellip
• Commitment ndash An Example But if we announce we are tough
• Commitment ndash An Example Get different NE
• Strategic Moves and Threats
• Trade Negotiation
• Slide 67
• Changing the game ndash A threat
• Trade Relations ndash Threats in Action
• Threats in action (cont)
• Prisoners Dilemma ndash Promises to Keep
• Promises or Threats
• Countering Threats
• Credibility Devices
• Slide 75
• Solving Extensive Form Games
• Seltenrsquos Game
• Slide 78
• Slide 79
• Little Horsey
• Where are the NE
• Slide 82
• Slide 83
• Slide 84
• Slide 85
• Slide 86
• Slide 87
• Slide 88
• Slide 89
• Slide 90
• Slide 91
• Slide 92

84

bull Player 1 observes Naturersquos move and offers the wrapped gift to player 2 If the gift is accepted then player 1 derives a positive payoff because everyone likes when their gifts are accepted

bull Player 1 hates the humiliation of having a gift rejected so the payoff is minus1

bull Player 2 strictly prefers accepting the game theory book to not accepting it she is indifferent between not accepting this book and accepting the Star Trek manual but hates rejecting the Star Trek manual more than the game theory book because while dissing game theory is cool dissing Star Trek is embarrassing

bull Letrsquos construct the strategic form of this game

85

Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift

The action GN refers to ldquogive if get game theory not give if star trekrdquo

(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered

86

bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins

87

bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set

bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1

bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)

88

bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)

bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift

bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game

bull The belief will be a probability distribution over the nodes

89

bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached

bull A continuation game refers to the information set and all nodes that follow from that information set

90

bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ

bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x

91

bull Player irsquos expected utility in the continuation game starting at node x then is

bull Ui(σ|x) = zP(z|σx)ui(z)

bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin

92

bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q

bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1

bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold

bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set

bull In other words the unique sequentially rational strategy is to choose Y with certainty

• Chapter 4 Sequential Games
• Extensive Form Games
• Extensive Form Games - Intro
• Slide 4
• Fundamental Tools
• Fundamental Tools Extensive form games--Definition
• Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Composition of information sets
• Fundamental Tools Normal form games--Definition
• Fundamental Tools Normal form games--Illustration
• Slide 15
• Sequential games
• Slide 17
• Problems with Nash equilibrium
• Solving sequential games
• Slide 20
• Subgame
• Subgame perfect equilibrium amp credible threats
• Slide 23
• Example Cuban Missile Crisis
• Backwards induction
• Slide 26
• The predation game
• Slide 28
• Slide 29
• Slide 30
• Credible commitments
• Slide 32
• Slide 33
• Slide 34
• Hold up
• Holdup in equilibrium no-one designs software payoffs = (software nintendo)
• Strategies in extensive form
• Slide 38
• Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
• Slide 40
• Slide 41
• Slide 42
• Solving by backwards induction
• Continuing
• Type of games
• subgame perfect equilibrium
• Imperfect information
• Slide 49
• Bob and Betty
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• In Class Exercise
• Slide 56
• Prove or disprove
• Sequential Monopolist View
• Thought Question
• Commitment
• Commitment ndash An Example
• Commitment ndash An Example For those that intend to teachhellip
• Commitment ndash An Example But if we announce we are tough
• Commitment ndash An Example Get different NE
• Strategic Moves and Threats
• Trade Negotiation
• Slide 67
• Changing the game ndash A threat
• Trade Relations ndash Threats in Action
• Threats in action (cont)
• Prisoners Dilemma ndash Promises to Keep
• Promises or Threats
• Countering Threats
• Credibility Devices
• Slide 75
• Solving Extensive Form Games
• Seltenrsquos Game
• Slide 78
• Slide 79
• Little Horsey
• Where are the NE
• Slide 82
• Slide 83
• Slide 84
• Slide 85
• Slide 86
• Slide 87
• Slide 88
• Slide 89
• Slide 90
• Slide 91
• Slide 92

85

Two nash equilibruim GG-Y and NN-N But (NNN) is clearly irrational If the game ever reaches player 2rsquos information set accepting a gift strictly dominates non accepting a gift regardless of the gift

The action GN refers to ldquogive if get game theory not give if star trekrdquo

(GGY) is a Nash equilibrium for any value of p because p minus1 lt 0 lt p lt 1 Player 1 offersthe gift regardless of its type and player 2 accepts always In addition (NNN) is a Nashequilibrium Player 1 never offers any gifts and player 2 refuses any gifts if offered

86

bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins

87

bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set

bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1

bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)

88

bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)

bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift

bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game

bull The belief will be a probability distribution over the nodes

89

bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached

bull A continuation game refers to the information set and all nodes that follow from that information set

90

bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ

bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x

91

bull Player irsquos expected utility in the continuation game starting at node x then is

bull Ui(σ|x) = zP(z|σx)ui(z)

bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin

92

bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q

bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1

bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold

bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set

bull In other words the unique sequentially rational strategy is to choose Y with certainty

• Chapter 4 Sequential Games
• Extensive Form Games
• Extensive Form Games - Intro
• Slide 4
• Fundamental Tools
• Fundamental Tools Extensive form games--Definition
• Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Composition of information sets
• Fundamental Tools Normal form games--Definition
• Fundamental Tools Normal form games--Illustration
• Slide 15
• Sequential games
• Slide 17
• Problems with Nash equilibrium
• Solving sequential games
• Slide 20
• Subgame
• Subgame perfect equilibrium amp credible threats
• Slide 23
• Example Cuban Missile Crisis
• Backwards induction
• Slide 26
• The predation game
• Slide 28
• Slide 29
• Slide 30
• Credible commitments
• Slide 32
• Slide 33
• Slide 34
• Hold up
• Holdup in equilibrium no-one designs software payoffs = (software nintendo)
• Strategies in extensive form
• Slide 38
• Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
• Slide 40
• Slide 41
• Slide 42
• Solving by backwards induction
• Continuing
• Type of games
• subgame perfect equilibrium
• Imperfect information
• Slide 49
• Bob and Betty
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• In Class Exercise
• Slide 56
• Prove or disprove
• Sequential Monopolist View
• Thought Question
• Commitment
• Commitment ndash An Example
• Commitment ndash An Example For those that intend to teachhellip
• Commitment ndash An Example But if we announce we are tough
• Commitment ndash An Example Get different NE
• Strategic Moves and Threats
• Trade Negotiation
• Slide 67
• Changing the game ndash A threat
• Trade Relations ndash Threats in Action
• Threats in action (cont)
• Prisoners Dilemma ndash Promises to Keep
• Promises or Threats
• Countering Threats
• Credibility Devices
• Slide 75
• Solving Extensive Form Games
• Seltenrsquos Game
• Slide 78
• Slide 79
• Little Horsey
• Where are the NE
• Slide 82
• Slide 83
• Slide 84
• Slide 85
• Slide 86
• Slide 87
• Slide 88
• Slide 89
• Slide 90
• Slide 91
• Slide 92

86

bull Because the strategic form ignores timing Nash equilibrium only ensures optimality at the start of the game That is equilibrium strategies are optimal if the other players follow their equilibrium strategies But we cannot see whether the strategies continue to be optimal once the game begins

87

bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set

bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1

bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)

88

bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)

bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift

bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game

bull The belief will be a probability distribution over the nodes

89

bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached

bull A continuation game refers to the information set and all nodes that follow from that information set

90

bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ

bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x

91

bull Player irsquos expected utility in the continuation game starting at node x then is

bull Ui(σ|x) = zP(z|σx)ui(z)

bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin

92

bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q

bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1

bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold

bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set

bull In other words the unique sequentially rational strategy is to choose Y with certainty

• Chapter 4 Sequential Games
• Extensive Form Games
• Extensive Form Games - Intro
• Slide 4
• Fundamental Tools
• Fundamental Tools Extensive form games--Definition
• Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Composition of information sets
• Fundamental Tools Normal form games--Definition
• Fundamental Tools Normal form games--Illustration
• Slide 15
• Sequential games
• Slide 17
• Problems with Nash equilibrium
• Solving sequential games
• Slide 20
• Subgame
• Subgame perfect equilibrium amp credible threats
• Slide 23
• Example Cuban Missile Crisis
• Backwards induction
• Slide 26
• The predation game
• Slide 28
• Slide 29
• Slide 30
• Credible commitments
• Slide 32
• Slide 33
• Slide 34
• Hold up
• Holdup in equilibrium no-one designs software payoffs = (software nintendo)
• Strategies in extensive form
• Slide 38
• Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
• Slide 40
• Slide 41
• Slide 42
• Solving by backwards induction
• Continuing
• Type of games
• subgame perfect equilibrium
• Imperfect information
• Slide 49
• Bob and Betty
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• In Class Exercise
• Slide 56
• Prove or disprove
• Sequential Monopolist View
• Thought Question
• Commitment
• Commitment ndash An Example
• Commitment ndash An Example For those that intend to teachhellip
• Commitment ndash An Example But if we announce we are tough
• Commitment ndash An Example Get different NE
• Strategic Moves and Threats
• Trade Negotiation
• Slide 67
• Changing the game ndash A threat
• Trade Relations ndash Threats in Action
• Threats in action (cont)
• Prisoners Dilemma ndash Promises to Keep
• Promises or Threats
• Countering Threats
• Credibility Devices
• Slide 75
• Solving Extensive Form Games
• Seltenrsquos Game
• Slide 78
• Slide 79
• Little Horsey
• Where are the NE
• Slide 82
• Slide 83
• Slide 84
• Slide 85
• Slide 86
• Slide 87
• Slide 88
• Slide 89
• Slide 90
• Slide 91
• Slide 92

87

bull We shall require that player 2 form beliefs about the probability of being at any particular node in her information set

bull Obviously if the information set consists of a single node then if that information set is reached the probability of being at that node is 1

bull Letrsquos look at the game in the figure Let q denote player 2rsquos belief that she is at the left node in her information set (given that the set is reached this is the probability of player 1 having offered the game theory book) and 1 minus q be the probability that she is at the right node (given that the set is reached this is the probability of player 1 having offered the Star Trek book)

88

bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)

bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift

bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game

bull The belief will be a probability distribution over the nodes

89

bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached

bull A continuation game refers to the information set and all nodes that follow from that information set

90

bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ

bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x

91

bull Player irsquos expected utility in the continuation game starting at node x then is

bull Ui(σ|x) = zP(z|σx)ui(z)

bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin

92

bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q

bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1

bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold

bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set

bull In other words the unique sequentially rational strategy is to choose Y with certainty

• Chapter 4 Sequential Games
• Extensive Form Games
• Extensive Form Games - Intro
• Slide 4
• Fundamental Tools
• Fundamental Tools Extensive form games--Definition
• Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Composition of information sets
• Fundamental Tools Normal form games--Definition
• Fundamental Tools Normal form games--Illustration
• Slide 15
• Sequential games
• Slide 17
• Problems with Nash equilibrium
• Solving sequential games
• Slide 20
• Subgame
• Subgame perfect equilibrium amp credible threats
• Slide 23
• Example Cuban Missile Crisis
• Backwards induction
• Slide 26
• The predation game
• Slide 28
• Slide 29
• Slide 30
• Credible commitments
• Slide 32
• Slide 33
• Slide 34
• Hold up
• Holdup in equilibrium no-one designs software payoffs = (software nintendo)
• Strategies in extensive form
• Slide 38
• Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
• Slide 40
• Slide 41
• Slide 42
• Solving by backwards induction
• Continuing
• Type of games
• subgame perfect equilibrium
• Imperfect information
• Slide 49
• Bob and Betty
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• In Class Exercise
• Slide 56
• Prove or disprove
• Sequential Monopolist View
• Thought Question
• Commitment
• Commitment ndash An Example
• Commitment ndash An Example For those that intend to teachhellip
• Commitment ndash An Example But if we announce we are tough
• Commitment ndash An Example Get different NE
• Strategic Moves and Threats
• Trade Negotiation
• Slide 67
• Changing the game ndash A threat
• Trade Relations ndash Threats in Action
• Threats in action (cont)
• Prisoners Dilemma ndash Promises to Keep
• Promises or Threats
• Countering Threats
• Credibility Devices
• Slide 75
• Solving Extensive Form Games
• Seltenrsquos Game
• Slide 78
• Slide 79
• Little Horsey
• Where are the NE
• Slide 82
• Slide 83
• Slide 84
• Slide 85
• Slide 86
• Slide 87
• Slide 88
• Slide 89
• Slide 90
• Slide 91
• Slide 92

88

bull That is p is player 2rsquos initial belief (or the prior) of the book being a game theory reference and q is player 2rsquos conditional belief (updated belief or posterior)

bull For example I believe that if you receive a game theory book there is a 90 chance you will offer the book as a gift but if you receive star trek there is a 50 chance you will offer it as a gift

bull Requirement 1 (Beliefs) At each information set the player who gets to move must have a belief about which node in the information set has been reached by the play of the game

bull The belief will be a probability distribution over the nodes

89

bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached

bull A continuation game refers to the information set and all nodes that follow from that information set

90

bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ

bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x

91

bull Player irsquos expected utility in the continuation game starting at node x then is

bull Ui(σ|x) = zP(z|σx)ui(z)

bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin

92

bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q

bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1

bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold

bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set

bull In other words the unique sequentially rational strategy is to choose Y with certainty

• Chapter 4 Sequential Games
• Extensive Form Games
• Extensive Form Games - Intro
• Slide 4
• Fundamental Tools
• Fundamental Tools Extensive form games--Definition
• Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Composition of information sets
• Fundamental Tools Normal form games--Definition
• Fundamental Tools Normal form games--Illustration
• Slide 15
• Sequential games
• Slide 17
• Problems with Nash equilibrium
• Solving sequential games
• Slide 20
• Subgame
• Subgame perfect equilibrium amp credible threats
• Slide 23
• Example Cuban Missile Crisis
• Backwards induction
• Slide 26
• The predation game
• Slide 28
• Slide 29
• Slide 30
• Credible commitments
• Slide 32
• Slide 33
• Slide 34
• Hold up
• Holdup in equilibrium no-one designs software payoffs = (software nintendo)
• Strategies in extensive form
• Slide 38
• Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
• Slide 40
• Slide 41
• Slide 42
• Solving by backwards induction
• Continuing
• Type of games
• subgame perfect equilibrium
• Imperfect information
• Slide 49
• Bob and Betty
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• In Class Exercise
• Slide 56
• Prove or disprove
• Sequential Monopolist View
• Thought Question
• Commitment
• Commitment ndash An Example
• Commitment ndash An Example For those that intend to teachhellip
• Commitment ndash An Example But if we announce we are tough
• Commitment ndash An Example Get different NE
• Strategic Moves and Threats
• Trade Negotiation
• Slide 67
• Changing the game ndash A threat
• Trade Relations ndash Threats in Action
• Threats in action (cont)
• Prisoners Dilemma ndash Promises to Keep
• Promises or Threats
• Countering Threats
• Credibility Devices
• Slide 75
• Solving Extensive Form Games
• Seltenrsquos Game
• Slide 78
• Slide 79
• Little Horsey
• Where are the NE
• Slide 82
• Slide 83
• Slide 84
• Slide 85
• Slide 86
• Slide 87
• Slide 88
• Slide 89
• Slide 90
• Slide 91
• Slide 92

89

bull A strategy is sequentially rational for player i at the information set h if player i would actually want to chose the action prescribed by the strategy if h is reached

bull A continuation game refers to the information set and all nodes that follow from that information set

90

bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ

bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x

91

bull Player irsquos expected utility in the continuation game starting at node x then is

bull Ui(σ|x) = zP(z|σx)ui(z)

bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin

92

bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q

bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1

bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold

bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set

bull In other words the unique sequentially rational strategy is to choose Y with certainty

• Chapter 4 Sequential Games
• Extensive Form Games
• Extensive Form Games - Intro
• Slide 4
• Fundamental Tools
• Fundamental Tools Extensive form games--Definition
• Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Composition of information sets
• Fundamental Tools Normal form games--Definition
• Fundamental Tools Normal form games--Illustration
• Slide 15
• Sequential games
• Slide 17
• Problems with Nash equilibrium
• Solving sequential games
• Slide 20
• Subgame
• Subgame perfect equilibrium amp credible threats
• Slide 23
• Example Cuban Missile Crisis
• Backwards induction
• Slide 26
• The predation game
• Slide 28
• Slide 29
• Slide 30
• Credible commitments
• Slide 32
• Slide 33
• Slide 34
• Hold up
• Holdup in equilibrium no-one designs software payoffs = (software nintendo)
• Strategies in extensive form
• Slide 38
• Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
• Slide 40
• Slide 41
• Slide 42
• Solving by backwards induction
• Continuing
• Type of games
• subgame perfect equilibrium
• Imperfect information
• Slide 49
• Bob and Betty
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• In Class Exercise
• Slide 56
• Prove or disprove
• Sequential Monopolist View
• Thought Question
• Commitment
• Commitment ndash An Example
• Commitment ndash An Example For those that intend to teachhellip
• Commitment ndash An Example But if we announce we are tough
• Commitment ndash An Example Get different NE
• Strategic Moves and Threats
• Trade Negotiation
• Slide 67
• Changing the game ndash A threat
• Trade Relations ndash Threats in Action
• Threats in action (cont)
• Prisoners Dilemma ndash Promises to Keep
• Promises or Threats
• Countering Threats
• Credibility Devices
• Slide 75
• Solving Extensive Form Games
• Seltenrsquos Game
• Slide 78
• Slide 79
• Little Horsey
• Where are the NE
• Slide 82
• Slide 83
• Slide 84
• Slide 85
• Slide 86
• Slide 87
• Slide 88
• Slide 89
• Slide 90
• Slide 91
• Slide 92

90

bull Take any two nodes x y (that is y follows x in the game tree) and consider the mixed strategy profile σ Let P(y|σx) denote the probability of reaching y starting from x and following the moves prescribed by σ

bull That is P(y|σx) is the conditional probability that the path of play would go through y after x if all players chose according to σ and the game started at x

91

bull Player irsquos expected utility in the continuation game starting at node x then is

bull Ui(σ|x) = zP(z|σx)ui(z)

bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin

92

bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q

bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1

bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold

bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set

bull In other words the unique sequentially rational strategy is to choose Y with certainty

• Chapter 4 Sequential Games
• Extensive Form Games
• Extensive Form Games - Intro
• Slide 4
• Fundamental Tools
• Fundamental Tools Extensive form games--Definition
• Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Composition of information sets
• Fundamental Tools Normal form games--Definition
• Fundamental Tools Normal form games--Illustration
• Slide 15
• Sequential games
• Slide 17
• Problems with Nash equilibrium
• Solving sequential games
• Slide 20
• Subgame
• Subgame perfect equilibrium amp credible threats
• Slide 23
• Example Cuban Missile Crisis
• Backwards induction
• Slide 26
• The predation game
• Slide 28
• Slide 29
• Slide 30
• Credible commitments
• Slide 32
• Slide 33
• Slide 34
• Hold up
• Holdup in equilibrium no-one designs software payoffs = (software nintendo)
• Strategies in extensive form
• Slide 38
• Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
• Slide 40
• Slide 41
• Slide 42
• Solving by backwards induction
• Continuing
• Type of games
• subgame perfect equilibrium
• Imperfect information
• Slide 49
• Bob and Betty
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• In Class Exercise
• Slide 56
• Prove or disprove
• Sequential Monopolist View
• Thought Question
• Commitment
• Commitment ndash An Example
• Commitment ndash An Example For those that intend to teachhellip
• Commitment ndash An Example But if we announce we are tough
• Commitment ndash An Example Get different NE
• Strategic Moves and Threats
• Trade Negotiation
• Slide 67
• Changing the game ndash A threat
• Trade Relations ndash Threats in Action
• Threats in action (cont)
• Prisoners Dilemma ndash Promises to Keep
• Promises or Threats
• Countering Threats
• Credibility Devices
• Slide 75
• Solving Extensive Form Games
• Seltenrsquos Game
• Slide 78
• Slide 79
• Little Horsey
• Where are the NE
• Slide 82
• Slide 83
• Slide 84
• Slide 85
• Slide 86
• Slide 87
• Slide 88
• Slide 89
• Slide 90
• Slide 91
• Slide 92

91

bull Player irsquos expected utility in the continuation game starting at node x then is

bull Ui(σ|x) = zP(z|σx)ui(z)

bull where Z is the set of terminal nodes in the game and ui(middot) is the Bernoulli payoff function specifying player irsquos utility from the outcome z Zisin

92

bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q

bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1

bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold

bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set

bull In other words the unique sequentially rational strategy is to choose Y with certainty

• Chapter 4 Sequential Games
• Extensive Form Games
• Extensive Form Games - Intro
• Slide 4
• Fundamental Tools
• Fundamental Tools Extensive form games--Definition
• Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Composition of information sets
• Fundamental Tools Normal form games--Definition
• Fundamental Tools Normal form games--Illustration
• Slide 15
• Sequential games
• Slide 17
• Problems with Nash equilibrium
• Solving sequential games
• Slide 20
• Subgame
• Subgame perfect equilibrium amp credible threats
• Slide 23
• Example Cuban Missile Crisis
• Backwards induction
• Slide 26
• The predation game
• Slide 28
• Slide 29
• Slide 30
• Credible commitments
• Slide 32
• Slide 33
• Slide 34
• Hold up
• Holdup in equilibrium no-one designs software payoffs = (software nintendo)
• Strategies in extensive form
• Slide 38
• Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
• Slide 40
• Slide 41
• Slide 42
• Solving by backwards induction
• Continuing
• Type of games
• subgame perfect equilibrium
• Imperfect information
• Slide 49
• Bob and Betty
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• In Class Exercise
• Slide 56
• Prove or disprove
• Sequential Monopolist View
• Thought Question
• Commitment
• Commitment ndash An Example
• Commitment ndash An Example For those that intend to teachhellip
• Commitment ndash An Example But if we announce we are tough
• Commitment ndash An Example Get different NE
• Strategic Moves and Threats
• Trade Negotiation
• Slide 67
• Changing the game ndash A threat
• Trade Relations ndash Threats in Action
• Threats in action (cont)
• Prisoners Dilemma ndash Promises to Keep
• Promises or Threats
• Countering Threats
• Credibility Devices
• Slide 75
• Solving Extensive Form Games
• Seltenrsquos Game
• Slide 78
• Slide 79
• Little Horsey
• Where are the NE
• Slide 82
• Slide 83
• Slide 84
• Slide 85
• Slide 86
• Slide 87
• Slide 88
• Slide 89
• Slide 90
• Slide 91
• Slide 92

92

bull In the gift game Player 2 can calculate the expected payoff from choosing Y which is q(1)+(1minusq)(0) = q

bull the expected payoff from choosing N which is q(0)+(1minusq)(minus1) = qminus1

bull Since q gt qminus1 for all values of q (as q-1 is negative) it is never optimal for player 2 to choose N regardless of the beliefs player 2 might hold

bull Therefore the strategy N is not sequentially rational because there is no belief that player 2 can have that will make it optimal at her information set

bull In other words the unique sequentially rational strategy is to choose Y with certainty

• Chapter 4 Sequential Games
• Extensive Form Games
• Extensive Form Games - Intro
• Slide 4
• Fundamental Tools
• Fundamental Tools Extensive form games--Definition
• Fundamental Tools Extensive form gamesmdashIllustration (BM-LM)
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Composition of information sets
• Fundamental Tools Normal form games--Definition
• Fundamental Tools Normal form games--Illustration
• Slide 15
• Sequential games
• Slide 17
• Problems with Nash equilibrium
• Solving sequential games
• Slide 20
• Subgame
• Subgame perfect equilibrium amp credible threats
• Slide 23
• Example Cuban Missile Crisis
• Backwards induction
• Slide 26
• The predation game
• Slide 28
• Slide 29
• Slide 30
• Credible commitments
• Slide 32
• Slide 33
• Slide 34
• Hold up
• Holdup in equilibrium no-one designs software payoffs = (software nintendo)
• Strategies in extensive form
• Slide 38
• Sequential Prisonerrsquos Dilemma dotted line means P2 doesnrsquot know which state he is in
• Slide 40
• Slide 41
• Slide 42
• Solving by backwards induction
• Continuing
• Type of games
• subgame perfect equilibrium
• Imperfect information
• Slide 49
• Bob and Betty
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• In Class Exercise
• Slide 56
• Prove or disprove
• Sequential Monopolist View
• Thought Question
• Commitment
• Commitment ndash An Example
• Commitment ndash An Example For those that intend to teachhellip
• Commitment ndash An Example But if we announce we are tough
• Commitment ndash An Example Get different NE
• Strategic Moves and Threats
• Trade Negotiation
• Slide 67
• Changing the game ndash A threat
• Trade Relations ndash Threats in Action
• Threats in action (cont)
• Prisoners Dilemma ndash Promises to Keep
• Promises or Threats
• Countering Threats
• Credibility Devices
• Slide 75
• Solving Extensive Form Games
• Seltenrsquos Game
• Slide 78
• Slide 79
• Little Horsey
• Where are the NE
• Slide 82
• Slide 83
• Slide 84
• Slide 85
• Slide 86
• Slide 87
• Slide 88
• Slide 89
• Slide 90
• Slide 91
• Slide 92