simultaneous - move games - sabancı...

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SIMULTANEOUS - MOVE GAMESA game is simultaneous-move if the players choose their actions without any information on the other players’ actions.

This does NOT necessarily mean that the players are acting at exactly the same time (even though that may very well be the case).

Example: the interaction between Beckham and Rüstü

is a simultaneous-move game

(even though they do not act at exactly the same time)

because Rüstü does NOT observe where Beckham kicked the ball

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Example: the Rock-Paper-Scissors game

N = {1 , 2}

S1 = S2 = {r , p , s }

u1(r , s) = u1(p , r) = u1(s , p) = 1

u1(r , r) = u1(p , p) = u1(s , s) = 0

u1(r , p) = u1(p , s) = u1(s , r) = -1

u2 is defined similarly

A game (in strategic form or normal form) is

1. A set of players N

2. For each player i in N, a set of his strategies: Si

3. For each player i in N, his payoff function: ui

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Example 2

Guessing half of the average (10 players)

1. Explain the rules of the game

2. Write it in normal form

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To illustrate simple simultaneous-move games

we use a

Game Table

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PLAYER1

R P

PLAYER 2

R

P

S

S

Example: the Rock-Paper-Scissors game

draw a table:

Player 1 takes rows

Player 2 takes columns

Observe: each cell corresponds to a possible outcome

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PLAYER1

R P

PLAYER 2

0, 0 -1, 1R

P

S

1, -1 0, 0

-1, 1 1, -1

S

1, -1

-1, 1

0, 0

Then place in each cell

the payoffs agents get from that outcome

Row player : -1

Column player: 1

What about the half-of the average game?

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Conductor

Confess(Defect)

Deny(Cooperate)

10 yr, 10 yr

25 yr, 1 yr

1 yr, 25 yr

3 yr, 3 yr

Confess(Defect)

Deny(Cooperate)

Tchaikovsky

A tale of two prisoners:

ATTENTION: THIS IS NOT A GAME TABLE

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Conductor

Confess(Defect)

Deny(Cooperate)

-10, -10

-25, -1

-1, -25

-3, -3

Confess(Defect)

Deny(Cooperate)

Tchaikovsky

Prisoner’s Dilemma

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Student 1

Goof off(Defect)

Work hard(Cooperate)

1, 1

0, 3

3, 0

2, 2

Goof off(Defect)

Work hard(Cooperate)

Student 2

Students doing a project together

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Firm 1

Low(Defect)

High(Cooperate)

600, 600

-200, 1200

1200, -200

1000, 1000

Low(Defect)

High(Cooperate)

Firm 2

Two firms (duopolists), choosing prices

Other examples: the arms race, common property, etc.

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Husband

Football

Soap opera

2, 1

0, 0

0, 0

1, 2

Football Soap opera

Wife

Battle of the Sexes

Other examples: two politicians determining position on an issue

two merging firms choosing between PC and MAC

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Ali

Head

Tail

1, -1

-1, 1

-1, 1

1, -1

Head Tail

Veli

Matching Pennies

Example: leader and follower firms choosing product appearances

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Hunter 1

Stag

Hare

2, 2

1, 0

0, 1

1, 1

Stag Hare

Hunter 2

The Stag hunt ( Rousseau, in Discourse on the Origin and Foundations of Inequality Among Man )

Other example: arms race

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The Chicken game

JAMES

Swerve(Chicken)

DEAN

0, 0 –1, 1

1, –1 –2, –2

Straight(Tough)

Swerve(Chicken)

Straight(Tough)

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"The Gift of the Magi" (O. Henry)

is about a young couple who are short of money but desperately

want to buy each other Christmas gifts. Unbeknownst to Jim, Della

sells her most valuable possession, her beautiful hair, in order to

buy a platinum fob chain for Jim's watch; while unbeknownst to

Della, Jim sells his own most valuable possession, his watch, to

buy jeweled combs for Della's hair.

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The Gift of the Magi (O Henry)

Della

Don’t buypresent

Jim

0, 0 –1, 1

1, –1 –2, –2

Buypresent

Don’t buypresent

Buypresent

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Zero-sum game

A 2-player game is zero-sum if in each entry of the game table, the agents' payoffs add up to zero.

Constant-sum game

A 2-player game is constant-sum if in each entry of the game table, the agents' payoffs add up to the same constant.

Competitive game

A 2-player game is competitive if Player 1’s ranking of the strategy profiles is the opposite of Player 2’s ranking.

Every zero-sum game is a constant-sum game and every constant-sum game is competitive. However, every competitive game can be made zero-sum by changing the payoff numbers.

In these games, the players have totally opposing interests.

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PLAYER1

R P

PLAYER 2

0 -1R

P

S

1 0

-1 1

S

1

-1

0

(b) Payoffs in zero-sum notation

Zero-sum games:

it is sufficient to write one number in every box

the payoff of row player

Row player receives -1

Column player receives - ( -1 ) = 1

PLAYER1

R P

PLAYER 2

0, 0 -1, 1R

P

S

1, -1 0, 0

-1, 1 1, -1

S

1, -1

-1, 1

0, 0

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EMILY

Contribute Don't

NINA

5, 5, 5 3, 6, 3

6, 3, 3 4, 4, 1

Contribute

Don'tEMILY

Contribute Don't

NINA

3, 3, 6 1, 4, 4

4, 1, 4 2, 2, 2

Contribute

Don't

Contribute Don't Contribute

TALIA chooses:

Game tables with three players:

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ANALYZING GAMES

Which strategies will the agents choose?

What will be the outcome?

USE: the assumption of rationality

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In-class game

Guessing half of the average (10 players)

Play the game

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Discussion• dominance,

• iterated elimination of dominated strategies,

• culmination in a Nash equilibrium

• players getting close to the Nash equilibrium with more

experience

• if you expect the others not to play equilibrium strategies,

then your best response might be different

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Conductor

Defect

Cooperate

-10, -10

-25, -1

-1, -25

-3, -3

Defect Cooperate

Tchaikovsky

What will the players do?

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Strict Domination

• Take an agent (call him Bond, James Bond) and two of his strategies: U and D.

• The strategy U strictly dominates the strategy D (for Bond) if

1. for every possible strategy profile of the other agents, playing U yields a higher (>) payoff than playing D.

That is, independent of what the others do, playing U always gives a higher payoff

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Given a strategic form game

G = ( N , (S1,…,Sn) , (u1,…,un) )

for an Agent i, U strictly dominates D if

1. for every strategy profile s-i = ( s1 ,…, si-1 , si+1 ,…, sn )

of the agents other than i,

ui ( U , s-i ) > ui ( D , s-i )

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Conductor

Defect

Cooperate

-10, -10

-25, -1

-1, -25

-1, -3

Defect Cooperate

Tchaikovsky

What will the players do in this game?

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Weak Domination

• Take an agent (call him Bond, James Bond) and two of his strategies: U and D.

• The strategy U weakly dominates the strategy D (for Bond) if

1. for every possible strategy profiles of the other agents, playing U yields an at least as high (≥) payoff as playing D

and

2. for at least one strategy profile of the other agents, U yields a higher (>) payoff than playing D.

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Given a strategic form game G

For agent i, U weakly dominates D if

1. for every strategy profile s-i of the other agents,

ui ( U , s-i ) ≥ ui ( D , s-i )

and

2. for at least one s-i* of the other agents,

ui ( U , s-i* ) > ui ( D , s-i

* )

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U 5 -1 3M 2 -5 2D 3 2 1

Row Player’s payoffs

What are the dominance relations?

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NOTE: If U strictly dominates D

then U weakly dominates D

Definition:

If there is some strategy U that strictly (weakly) dominates D

then D is called a strictly (weakly) dominated strategy

Important: Rational players do NOT PLAY their

strictly dominated strategies

They can sometimes play their

weakly dominated strategies

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Conductor

Defect

Cooperate

-10, -10

-25, -1

-1, -25

-1, -1

Defect Cooperate

Tchaikovsky

Cooperate is weakly dominated in this game?

Can they sustain (Cooperate,Cooperate) ?

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QUESTION

• If M is strictly dominated by U and M is weakly dominated by D

what do I call M ?

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If a strategy of Bond strictly dominates all of his other strategies

it is called a strictly dominant strategy.

If Bond has a strictly dominant strategy, he always plays it

If a strategy of Bond weakly dominates all of his other strategies

it is called a weakly dominant strategy.

However, Bond does not always play a weakly dominant strategy.

Why? The answer lies in what Bond expects others to play.

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QUESTIONS

1. If U strictly dominates M and U weakly dominates D

what do I call U?

2. How many strictly dominant strategies can a player have?

How many weakly dominant strategies can a player have?

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Strictly Dominant Strategy for each agent

(it strictly dominates all of her other strategies)

Weakly Dominant Strategy for each agent

(it weakly dominates all of her other strategies)

BEST SITUATION

THEN WE KNOW WHAT TO DO !!!

We have a Dominant Strategy Equilibrium

or

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Definition: a Dominant Strategy Equilibrium of the game

G = ( N , (S1,…,Sn) , (u1,…,un) )

is a strategy profile (s1,…,sn) such that for every player i in N, siis a dominant strategy of player i.

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Conductor

Confess(Defect)

Deny(Cooperate)

-10, -10

-25, -1

-1, -25

-3, -3

Confess(Defect)

Deny(Cooperate)

Tchaikovsky

Both players have dominant strategies in the Prisoner’s Dilemma.

Thus (Confess, Confess) is a Dominant Strategy Equilibrium

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ROW

Left Center

COLUMN

1, 2

Right

2, 5 1, 3

0, 5 1, 3 1, 6

1, 1 3, 1 5, 2

Up

Level

Down

1. What are the domination relationships between players’ strategies?

2. Which are the dominant strategies? Strict or weak?

3. Which are the dominated strategies? Strict or weak?

4. Any dominant strategy equilibrium?

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U.S.AIR FORCES

North South

JAPANESE NAVY

2 2

1 3

North

South

What about the Battle of the Bismarck Sea?

Note that this is a zero-sum game

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What about the Chicken game?

JAMES

Swerve(Chicken)

DEAN

0, 0 –1, 1

1, –1 –2, –2

Straight(Tough)

Swerve(Chicken)

Straight(Tough)

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What do you do when there is no dominant strategy equilibrium ?????

Successively Eliminating Strictly Dominated Strategies

(because players never play them)

The table gets smaller and smaller and smaller and smaller and smaller

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DONNA’SDEEP DISH

High Medium

PIERCE’S PIZZA PIES

60, 60High

Medium

Low

Low

70, 36

35, 36

36, 70

50, 50

35, 30

36, 35

30, 35

25, 25

Successive elimination of strictly dominated strategies

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CAUTION: When dominance is weak, the result depends on the order of elimination.

HOW ABOUT GAMES LIKE THE CHICKEN GAME?

ROW

L C

COLUMN

1, 1T

B

R

0, 0

1, 1

1, 2

0, 0

1, 2

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Nash Equilibrium (J. Nash, 1951)

A strategy profile where an agent, given her opponents’ strategies, can NOT INCREASE her payoff by switching to another strategy

That is, a strategy profile whereeach agent’s strategy is a best response to the other agents’ strategies

Definition: a Nash Equilibrium of the game

G = ( N , (S1,…,Sn) , (u1,…,un) )

is a strategy profile (s1*,…,sn

*) such that for every player i in N,

ui( si* , s-i

* ) ≥ ui( si , s-i* )

for every si in Si.

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Interpretation:

Imagine a population of many decision-makers, who repeatedly interact (assume the role of a game player)

In time, they will form beliefs about how opponents behave.

With experience, they will find the best actions for themselves.

A Nash equilibrium is designed to model a steady state of socialinteraction (in a sense, a social norm) that emerges out of the players gaining experience in playing repeatedly.

Example:

Driving (more specifically, the crossroads game where you eitherstop or pass)

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Student 1

Goof off(Defect)

Work hard(Cooperate)

1, 1

0, 3

3, 0

2, 2

Goof off(Defect)

Work hard(Cooperate)

Student 2

The Prisoners’ Dilemma:

The incentive to free-ride eliminates the possibility of the mutually desirable outcome.

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Husband

Football

Soap opera

2, 1

0, 0

0, 0

1, 2

Football Soap opera

Wife

Battle of the Sexes

Both outcomes are stable social norms.

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Ali

Head

Tail

1, -1

-1, 1

-1, 1

1, -1

Head Tail

Veli

Matching Pennies

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SELES

DL CC

HINGIS

50 80

90 20

DL

CC

The Tennis Game

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Hunter 1

Stag

Hare

2, 2

1, 0

0, 1

1, 1

Stag Hare

Hunter 2

The Stag hunt

One outcome better than the other, but coordinated deviations are not allowed.

Will talk about focal equilibria.

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IN-CLASS game:

• Two players

• Each proposes a division of 1000 YTL

• If proposals match, they get what is proposed

• Otherwise, they get nothing

Assume: each player’s objective is to maximize her monetary gain.

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Consider the following coordination game:

ROWBACH

STRAVINSKY

2, 2

0, 0

0, 0

1, 1

BACH STRAVINSKY

COLUMN

Is one of the equilibria more likely?

It is called a focal equilibrium.

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The Chicken Game

JAMES

Swerve(Chicken)

DEAN

0, 0 –1, 1

1, –1 –2, –2

Straight(Tough)

Swerve(Chicken)

Straight(Tough)

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A

0 1

B

0, 0

2

0, 5 0, 0

5, 0 0, 0 –5, 0

0, 0 0, –5 –5, –5

0

1

2

FIGURE 4.13 Lottery Copyright © 2000 by W.W. Norton & Company

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ROW

Left Middle

COLUMN

3, 1

Right

2, 3 10, 2

4, 5 3, 0 6, 4

2, 2 5, 4 12, 3

Top

High

Low

5, 6 4, 5 9, 7Bottom

EXERCISE 4.7 Copyright © 2000 by W.W. Norton & Company

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A

1 2

B

10, 10

3

0, 0 0, 0

0, 0 15, 15 0, 0

0, 0 0, 0 15, 15

1

2

3

EXERCISE 4.9 Copyright © 2000 by W.W. Norton & Company