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Naima Hammoud
March 9, 2017
Game Theory: Lecture 2
Prisoner’s Dilemma
Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Colin
Rose
Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Colin
Prisoner’s Dilemma
Rose
Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Colin
Prisoner’s Dilemma
Rose
Prisoner’s Dilemma
Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Colin
Rose
Prisoner’s Dilemma
Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Colin
Rose
Prisoner’s Dilemma
Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Colin
Rose
Prisoner’s Dilemma
Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Colin
Rose
Prisoner’s Dilemma
Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Colin
Rose
Prisoner’s Dilemma
Confess Don’t confess
Confess -1 -1 -4 0
Don’t confess 0 -4 -3 -3
Colin
Rose
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
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
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
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.
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
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.
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
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
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
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
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
Blonde or Brunette?
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
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.
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.
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
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
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
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
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
Game of Chicken
Game of Chicken
Two cars driving towards each other, the one who swerves first loses the game!
Swerve Straight
Swerve
Straight
Game of Chicken
Two cars driving towards each other, the one who swerves first loses the game!
Swerve Straight
Swerve
Straight
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
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
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
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
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
Battle of sexes
Tennis Basketball
Tennis
Basketball
p 1-p
q
1-q
player 2
player 1
Battle of sexes
Tennis Basketball
Tennis
Basketball
p 1-p
q
1-q
player 2
player 1
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
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
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
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
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
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
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
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
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
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