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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