game theory lecture 7

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problem set 7

from Osborne’sIntrod. To G.T.

p.442 Ex. 442.1p.443 Ex. 443.1

p.342 Ex. 38,40(p.383 Ex. 8,9p. 388 Ex. 24,29)

from Binmore’sFun and Games

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Correlated EquilibriaB X

B 2 , 1 0 , 0

X 0 , 0 1 , 2

The equilibria of this game are:

[B,B]

[X,X]

and the mixed strategy equilibrium:

[(2/3 , 1/3),(1/3 , 2/3) ] (2/3 , 2/3)

(2 , 1)

(1 , 2)payoffs

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B 2 , 1 0 , 0

X 0 , 0 1 , 2

The equilibria of this game are:

[B,B]

[X,X]

and the mixed strategy equilibrium:

[(2/3 , 1/3),(1/3 , 2/3) ] (2/3 , 2/3)

(2 , 1)

(1 , 2)payoffs

5


B 2 , 1 0 , 0

X 0 , 0 1 , 2

(2/3 , 2/3)(2 , 1) (1 , 2)

π1

π2

6

π1

π2


B 2 , 1 0 , 0

X 0 , 0 1 , 2

A roulette wheelseen by both players

By varying the roulette one can obtain each

point in the convex hull

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π1

π2


B 2 , 1 0 , 0

X 0 , 0 1 , 2

If roulette stops on redred play BBif on greengreen play XXif on greygrey play (2/3,1/3)(2/3,1/3) or (1/3,2/3)

is a Nash equilibrium

The strategy pair

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π1

π2


B 2 , 1 0 , 0

X 0 , 0 1 , 2

Can one do more with this coordination device?

D H

D 2 , 2 0 , 3

H 3 , 0 -1 , -1

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π1

π2

Correlated Equilibria

Can one do more with this coordination device?

D H

D 2 , 2 0 , 3

H 3 , 0 -1 , -1

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D H

D 2 , 2 0 , 3

H 3 , 0 -1 , -1

Hawk Dove game‘Chicken’

D H

D 2 , 2 0 , 3

H 3 , 0 -1 , -1

D H

D 2 , 2 0 , 3

H 3 , 0 -1 , -1


Nash Equilibria:

[H,D]

[D,H]

A mixed strategy equilibrium[ (1/2 , 1/2) , (1/2 , 1/2) ]

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D H

D 2 , 2 0 , 3

H 3 , 0 -1 , -1


Nash Equilibria:

[H,D]

[D,H]

A mixed strategy equilibrium[ (1/2 , 1/2) , (1/2 , 1/2) ]π1

π2


D H

D 2 , 2 0 , 3

H 3 , 0 -1 , -1Not a Nash Equilibrium

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π1

π2


D H

D 2 , 2 0 , 3

H 3 , 0 -1 , -1

Nash Equilibria:

[H,D]

[D,H]

A mixed strategy equilibrium[ (1/2 , 1/2) , (1/2 , 1/2) ]

αβ

γ

A referee chooses one of the

three cells with the probabilities

α + β + γ = 1

The players agree on probabilities

α, β,γ

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π1

π2


D H

D 2 , 2 0 , 3

H 3 , 0 -1 , -1

αβ

γ

A referee chooses one of the

three cells with the probabilities

α + β + γ = 1

The players agree on probabilities

α, β,γ

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Correlated EquilibriaD H

D 2 , 2 0 , 3

H 3 , 0 -1 , -1

αβ

γWhen he chose a cell, the referee

tells player the row of the cell

and player the column of the cell.

1

2

This is interpreted as a recommendation

to play the row (column) strategy.

When is it an Equilibrium to follow the referees advice?

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D 2 , 2 0 , 3

H 3 , 0 -1 , -1

αβ

γ

When a player has heard ,

he prefers to play it H

!!!!

.

When player heard ,

he knows that player heard

with probability with probability

1 D

2

β αD , H

α + β α + β

For him, player 2 mixes

β α,

α + β α + β

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D 2 , 2 0 , 3

H 3 , 0 -1 , -1

αβ

γ

if player 2 mixes

β α,

α + β α + β

2β 0α

α + β α + β

3β α

α + β α + β

he prefers to play ifD α β

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D 2 , 2 0 , 3

H 3 , 0 -1 , -1

αβ

γ

β

β + γ

he prefers to play ifD

A similar argument for player 2 :

γ β

γ

β + γ

2β

β + γ

3β - γ

β + γ

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D H

D 2 , 2 0 , 3

H 3 , 0 -1 , -1

α

γ β

Correlated EquilibriaThe referee's recommendations are

self enforcing (Nash Equil.) if

α β

β

γ

α + β + γ = 1

What payoffs can be obtained ???

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D H

D 2 , 2 0 , 3

H 3 , 0 -1 , -1

α

Correlated Equilibriaβ

γ

π1

π2

γ βα β

α + β + γ = 1

All payoffs in this area can be achieved by choosing α,β,γ.

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D H

D 2 , 2 0 , 3

H 3 , 0 -1 , -1

α

Correlated Equilibriaβ

γ

π1

π2

γ βα β

α + β + γ = 1

5 5,

3 3

1α = β = γ =

3

With a simple Roulette

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Robert Aumann

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C D

C 2 , 2 0 , 3

D 3 , 0 1 , 1

The Prisoners’ Dilemma C D

C 2 , 2 0 , 3

D 3 , 0 1 , 1The unique Nash Equilibrium:

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The Prisoners’ Dilemma C D

C 2 , 2 0 , 3

D 3 , 0 1 , 1The unique Nash Equilibrium:

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C D

C 2 , 2 0 , 3

D 3 , 0 1 , 1

In a repeated game, there may be a possibility of coperation by

Playing D in every stage is a Nash equilibrium of the repeated game.

punishing

deviations from cooperation.

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C D

C 2 , 2 0 , 3

D 3 , 0 1 , 1

The grim (trigger) strategy

1. Begin by playing C and do not initiate a deviation from C2. If the other played D, play D for ever after.

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i.e. is the pair (grim , grim) a N.E. ??

C D

C 2 , 2 0 , 3

D 3 , 0 1 , 1



Is the grim strategy a Nash equilibrium?

If both play grim, they never defect

Not in a finitely repeated Prisoners’

Dilemma.

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C D

C 2 , 2 0 , 3

D 3 , 0 1 , 1



The grim strategy is Not a Nash equilibrium

in a finitely repeated Prisoners’ Dilemma.

Given that the other plays grim it pays to deviate in the last period and play D

Indeed, to obtain cooperation in the repeated P.D. it is necessary* to have infinite repetitions.

* It will be shown later

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An infinitely repeated game

1

2

D

C

2

1

22

CC D

1

22

1

22

1

22

D

1C D

1C D

history is:[D,D]history is:{ [C,D], [C,C] }

sub-games

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where ai is a vector of actions taken at time iai is [C,C] or [DC] etc.

A history at time t is: { a1, a2, ….. at }

A strategy is a function that assigns an action for each history.

for all histories

1 2 t

1 2 t

S a ,a , ......a C,D

a ,a , ......a

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The payoff of player 1 following a history { a1, a2, ….. at,...… }is a stream { G1(a1), G1(a2), ….. G1(at)...… }

one way of evaluating an infinite

stream of incomes

is as , where

0 1

t tt

t=0 t=0

w ,w , .....

c

δ c = δ w

t t

t=0 t=0

cδ c = c δ

1 - δ

tt

t=0

c = 1 - δ δ w

t0 1 t t

t=0

u w ,w , ...w , ...... = 1 - δ δ w

a discount factor 0 δ 1

game theory lecture 7

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