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