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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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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Correlated EquilibriaB X
B 2 , 1 0 , 0
X 0 , 0 1 , 2
(2/3 , 2/3)(2 , 1) (1 , 2)
π1
π2
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π1
π2
Correlated EquilibriaB X
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
Correlated EquilibriaB X
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
Correlated EquilibriaB X
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
Correlated Equilibria
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
Correlated Equilibria
Nash Equilibria:
[H,D]
[D,H]
A mixed strategy equilibrium[ (1/2 , 1/2) , (1/2 , 1/2) ]π1
π2
Hawk Dove game‘Chicken’
D H
D 2 , 2 0 , 3
H 3 , 0 -1 , -1Not a Nash Equilibrium
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Correlated Equilibria
π1
π2
Hawk Dove game‘Chicken’
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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Correlated Equilibria
π1
π2
Hawk Dove game‘Chicken’
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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Correlated EquilibriaD H
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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Correlated EquilibriaD H
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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Correlated EquilibriaD H
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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Correlated Equilibria
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
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.
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 (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.
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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An infinitely repeated game
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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An infinitely repeated game
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
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