game theory: lecture 2 - university of...
TRANSCRIPT
![Page 1: Game Theory: Lecture 2 - University of Oxfordpeople.maths.ox.ac.uk/griffit4/Math_Alive/3/game_theory2.pdf · Game Theory: Lecture 2. Prisoner’s Dilemma Confess Don’t confess Confess-1](https://reader033.vdocuments.us/reader033/viewer/2022041909/5e66bd413ca11709904575cc/html5/thumbnails/1.jpg)
Naima Hammoud
March 9, 2017
Game Theory: Lecture 2
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Prisoner’s Dilemma
Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Colin
Rose
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Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Colin
Prisoner’s Dilemma
Rose
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Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Colin
Prisoner’s Dilemma
Rose
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Prisoner’s Dilemma
Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Colin
Rose
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Prisoner’s Dilemma
Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Colin
Rose
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Prisoner’s Dilemma
Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Colin
Rose
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Prisoner’s Dilemma
Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Colin
Rose
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Prisoner’s Dilemma
Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Colin
Rose
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Prisoner’s Dilemma
Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Colin
Rose
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Prisoner’s Dilemma
Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
In summary, no matter what Colin chooses, Rose is always better off not confessing. Similarly, no matter what Rose does, Colin is better off not confessing as well.
Colin
Rose
oneNashequilibrium
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Prisoner’s Dilemma
Colin
The prisoner problem has a Nash equilibrium which is a strictly dominant strategy. However, this strategy is NOT optimal.
Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Rose
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Matching Pennies
Heads Tails
Heads 1 -1 -1 1
Tails -1 1 1 -1
Two players, Colin and Rose, toss a penny each simultaneously: Rose wins if the pennies match; Colin wins if there is a mismatch.
Colin
Rose
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Mixed Strategy Nash Equilibrium
Heads Tails
Heads 1 -1 -1 1
Tails -1 1 1 -1
No single pair of deterministic strategies works for both players. So, there is no
pure strategy for both to follow. There is, however, a mixed strategy.
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Mixed Strategy Nash Equilibrium
Heads Tails
Heads 1 -1 -1 1
Tails -1 1 1 -1
No single pair of deterministic strategies works for both players. So, there is no
pure strategy for both to follow. There is, however, a mixed strategy.
What works then?• It would be a bad idea to play any single deterministic strategy in matching pennies• Idea: confuse the opponent by playing randomly• Define a strategy as a probability distribution over the actions• Pure strategy: only one action is played with positive probability• Mixed strategy: more than one action is played with positive probability
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Mixed Strategy Nash Equilibrium
Heads Tails
Heads 1 -1 -1 1
Tails -1 1 1 -1
No single pair of deterministic strategies works for both players. So, there is no
pure strategy for both to follow. There is, however, a mixed strategy.
Utility under mixed strategies:• What is a player’s payoff if all players follow a mixed strategy?• Can’t read the payoff from the game matrix anymore.• Extend the definition of utility and use the idea of expected utility.• The utility for a strategy profile will be the expected utility.
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Mixed Strategy Nash Equilibrium
Heads Tails
Heads 1 -1 -1 1
Tails -1 1 1 -1
Colin
Rose
• Suppose that Rose thinks Colin will play p(Heads)+(1-p)(Tails)• Recall the definition of expected value: given probabilities p1,...,pn of playing
events (or actions) with payoffs a1,...,an, the expected value is p1 a1 +...+pn an
p 1-p
q
1-q
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Mixed Strategy Nash Equilibrium
Heads Tails
Heads 1 -1 -1 1
Tails -1 1 1 -1
Colin
Rose
p 1-p
q
1-q
Rose’s Expectations for playing pure strategies
ERose
(Heads) = p(1) + (1� p)(�1) = 2p� 1
ERose
(Tails) = p(�1) + (1� p)(1) = �2p+ 1
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Mixed Strategy Nash Equilibrium
p 1-p
q
1-q
Colin
Rose
Suppose that Rose thinks Colin will play p(Heads)+(1-p)(Tails), in this case she should be indifferent about playing heads or tails.
ERose
(Heads) = ERose
(Tails)
p(1) + (1� p)(�1) = p(�1) + (1� p)(1) p = 1/2
Heads Tails
Heads 1 -1 -1 1
Tails -1 1 1 -1
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Mixed Strategy Nash Equilibrium
Heads Tails
Heads 1 -1 -1 1
Tails -1 1 1 -1
p 1-p
q
1-q
Colin
Rose
Suppose that Colin thinks Rose will play q(Heads)+(1-q)(Tails), in this case he should be indifferent about playing heads or tails
EColin
(Heads) = EColin
(Tails)
q(�1) + (1� q)(1) = q(1) + (1� q)(�1)q = 1/2
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Mixed Strategy Nash Equilibrium
Heads Tails
Heads 1 -1 -1 1
Tails -1 1 1 -1
p 1-p
q
1-q
Colin
Rose
So the mixed strategies (½ , ½), (½ , ½) are a Nash equilibrium
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Blonde or Brunette?
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Blonde or Brunette?Two friends at a bar. A blonde and two brunettes walk in. Both prefer the blonde, but if both pursue her, they will end up with nothing. If one pursues a brunette then the other one has a chance with the blonde.
Brunette Blonde
Brunette
Blonde
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Blonde or Brunette?
Brunette Blonde
Brunette
Blonde
Two friends at a bar. A blonde and two brunettes walk in. Both prefer the blonde, but if both pursue her, they will end up with nothing. If one pursues a brunette then the other one has a chance with the blonde.
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Blonde or Brunette?
Brunette Blonde
Brunette
Blonde
Certainlynotanoptimalsolution
Two friends at a bar. A blonde and two brunettes walk in. Both prefer the blonde, but if both pursue her, they will end up with nothing. If one pursues a brunette then the other one has a chance with the blonde.
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Blonde or Brunette?Two friends at a bar. A blonde and a brunette walk in. Both prefer the blonde, but if both pursue her, they will end up with none. If one pursues the brunette then the other one has a chance with the blonde.
Brunette Blonde
Brunette 1 1 1 3
Blonde 3 1 0 0
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Blonde or Brunette?Two friends at a bar. A blonde and a brunette walk in. Both prefer the blonde, but if both pursue her, they will end up with none. If one pursues the brunette then the other one has a chance with the blonde.
Brunette Blonde
Brunette 1 1 1 3
Blonde 3 1 0 0
twoNashequilibria
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Mixed Strategy Nash Equilibrium
p 1-p
q
1-q
Brunette Blonde
Brunette 1 1 1 3
Blonde 3 1 0 0
player 2
player 1
E1(Brunette) = E1(Blonde)
p(1) + (1� p)(1) = p(3) + (1� p)(0)
p = 1/3 1� p = 2/3
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Mixed Strategy Nash Equilibrium
p 1-p
q
1-q
player 2
player 1
E2(Brunette) = E2(Blonde)
q(1) + (1� q)(1) = q(3) + (1� q)(0)
q = 1/3 1� q = 2/3
Brunette Blonde
Brunette 1 1 1 3
Blonde 3 1 0 0
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Mixed Strategy Nash Equilibrium
p 1-p
q
1-q
player 2
player 1
�13 ,
23
�,�13 ,
23
�is a mixed strategy Nash equilibrium
Brunette Blonde
Brunette 1 1 1 3
Blonde 3 1 0 0
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Game of Chicken
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Game of Chicken
Two cars driving towards each other, the one who swerves first loses the game!
Swerve Straight
Swerve
Straight
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Game of Chicken
Two cars driving towards each other, the one who swerves first loses the game!
Swerve Straight
Swerve
Straight
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Game of Chicken
Two cars driving towards each other, the one who swerves first loses the game!
Swerve Straight
Swerve 0 0 -1 3
Straight 3 -1 -10 -10
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Game of Chicken
Two cars driving towards each other, the one who swerves first loses the game!
Swerve Straight
Swerve 0 0 -1 3
Straight 3 -1 -10 -10
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Mixed Strategy Nash Equilibrium
p 1-p
q
1-q
player 2
player 1
Swerve Straight
Swerve 0 0 -1 3
Straight 3 -1 -10 -10
E1(Swerve) = E1(Straight)
p(0) + (1� p)(�1) = p(3) + (1� p)(�10)
p = 3/4 1� p = 1/4
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Mixed Strategy Nash Equilibrium
p 1-p
q
1-q
player 2
player 1
Swerve Straight
Swerve 0 0 -1 3
Straight 3 -1 -10 -10
E2(Swerve) = E2(Straight)
q(0) + (1� q)(�1) = q(3) + (1� q)(�10)
1� q = 1/4q = 3/4
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Mixed Strategy Nash Equilibrium
p 1-p
q
1-q
player 2
player 1
Swerve Straight
Swerve 0 0 -1 3
Straight 3 -1 -10 -10
�34 ,
14
�,�34 ,
14
�is a mixed strategy Nash equilibrium
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Battle of sexes
Tennis Basketball
Tennis
Basketball
p 1-p
q
1-q
player 2
player 1
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Battle of sexes
Tennis Basketball
Tennis
Basketball
p 1-p
q
1-q
player 2
player 1
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Battle of sexes
Tennis Basketball
Tennis 2 1 0 0
Basketball 0 0 1 2
p 1-p
q
1-q
player 2
player 1
p = 1/3 1� p = 2/3 q = 2/3 1� q = 1/3and
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Battle of sexes
Tennis Basketball
Tennis 2 1 0 0
Basketball 0 0 1 2
p 1-p
q
1-q
player 2
player 1
p = 1/3 1� p = 2/3
�23 ,
13
�,�13 ,
23
�is a mixed strategy Nash equilibrium
q = 2/3 1� q = 1/3and
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Soccer Penalty KicksGoalie
Striker
Left Right
Left 0 1 1 0
Right 1 0 0 1
In this case both striker and goal keeper should play a mixed strategy with equal probabilities, i.e. 50-50
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Soccer Penalty KicksGoalie
Striker
Left Right
Left 0 1 1 0
Right 0.75 0.25 0 1
In this case the striker sometimes misses when they kick to the right
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Soccer Penalty Kicks
p 1-p
q
1-q
Goalie
Striker
Left Right
Left 0 1 1 0
Right 0.75 0.25 0 1
Estriker(Left) = Estriker(Right)
p(0) + (1� p)(1) = p(0.75) + (1� p)(0)
p = 4/7
Egoalie
(Left) = Egoalie
(Right)
q(1) + (1� q)(0.25) = q(0) + (1� q)(1)
q = 3/7
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Soccer Penalty Kicks: Data from 1417 games
p 1-p
q
1-q
Goalie
Striker
IgnacioPalacios-Huerta(2003) “Professionalsplayminimax”ReviewofEconomicStudies
Left Right
Left 0.58 0.42 0.95 0.05
Right 0.93 0.07 0.7 0.3
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Soccer Penalty Kicks: Data from 1417 games
p 1-p
q
1-q
Goalie
Striker
Left Right
Left 0.58 0.42 0.95 0.05
Right 0.93 0.07 0.7 0.3
Estriker(Left) = Estriker(Right)
p(0.58) + (1� p)(0.95) = p(0.93) + (1� p)(0.7)
p = 5/12 = 0.42IgnacioPalacios-Huerta(2003) “Professionalsplayminimax”ReviewofEconomicStudies
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Soccer Penalty Kicks: Data from 1417 games
p 1-p
q
1-q
Goalie
Striker
Left Right
Left 0.58 0.42 0.95 0.05
Right 0.93 0.07 0.7 0.3
Egoalie
(Left) = Egoalie
(Right)
q(0.42) + (1� q)(0.07) = q(0.05) + (1� q)(0.3)
q = 23/60 = 0.38IgnacioPalacios-Huerta(2003) “Professionalsplayminimax”ReviewofEconomicStudies
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Soccer Penalty Kicks: Data from 1417 games
p 1-p
q
1-q
Goalie
Striker
Left Right
Left 0.58 0.42 0.95 0.05
Right 0.93 0.07 0.7 0.3
IgnacioPalacios-Huerta(2003) “Professionalsplayminimax”ReviewofEconomicStudies
(0.38, 0.62) , (0.42, 0.58) is a mixed strategy Nash equilibrium
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Soccer Penalty Kicks: Data from 1417 games
Goalie Left Goalie Right Striker Left Striker Right
Nash frequency 0.42 0.58 0.38 0.62
Actual frequency 0.42 0.58 0.4 0.6
IgnacioPalacios-Huerta(2003) “Professionalsplayminimax”ReviewofEconomicStudies