1. problem set 9 from Osborne’s Introd. To G.T. Ex. 459.1, 459.2, 459.3.
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Transcript of 1. problem set 9 from Osborne’s Introd. To G.T. Ex. 459.1, 459.2, 459.3.
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problem set 9
from Osborne’sIntrod. To G.T.
Ex. 459.1, 459.2, 459.3
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Repeated Games (a general treatment)
What is the minimum that a player can guarantee?
In the Prisoners’ Dilemma it was the payoff of (D,D)
CC DD
CC 2 , 2 , 22
0 , 0 , 33
DD 3 , 3 , 00
1 , 1 , 11
By playing D, player 2 can ensure that player 1 does not get more than 1
For a general game G:Player 1 can always play the best response to the other’s action
max 1s
G (s,t)
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Repeated Games (a general treatment)
Player 2 can minimize the best that 1 can do by choosing t:
CC DD
CC 2 , 2 , 22
0 , 0 , 33
DD 3 , 3 , 00
1 , 1 , 11
In the P.D. :
minmax 1stG (s,t)
max 1sG (s,t) 3 3 1 1
minmax 1stG (s,t)
In the P.D. it is a Nash equilibrium for each to play the stratgy that minimaxes
the other.
In general playing the strategy that holds the other to his minimax payoff is
NOT a Nash Equilibrium
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Repeated Games (a general treatment)
In the infinitely repeated game of G, every Nash equilibrium payoff is at least the
minimax payoff
If a player always plays the best response to his opponent’s action, his payoff is at least his minmax
value.
Every feasible value of G, which gives each player at least his minimax value, can be obtained as a Nash Equilibrium
payoff
A folk theorem:
(for δ~1)
approximately
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Repeated Games (a general treatment)
If a point A is feasible, it can be (approximately) obtained by playing a cycle of actions.
Consider a the following strategy:• follow the cycle sequence if the sequence has been played in the past.
• If there was a deviation from it, play forever the action that holds the other to his minimax
It is an equilibrium for both to play this strategy
?
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Repeated Games (a general treatment)
• follow the cycle sequence if the sequence has been played in the past.
• If there was a deviation from it, play forever the action that holds the other to his minimax
If both follow the strategy, each receives more than his minimax
If one of them deviates, the other punishes him, hence the deviator gets at most his minimax.
hence, he will not deviate.
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Repeated Games (a general treatment)
Would a player want to punish after a deviation???
by punishing the other his own payoff is reduced
Playing these strategies is Nash but not sub-game perfect equilibrium.
To make punishment ‘attractive’:
• it should last finitely many periods
• if not all participated in the punishment the counting starts again.
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Repeated Games (a general treatment)An Example
AA BB CC
AA 4 , 44 , 4 3 , 03 , 0 1 , 01 , 0
BB 0 , 30 , 3 2 , 22 , 2 1 , 01 , 0
CC 0 , 10 , 1 0 , 10 , 1 0 , 00 , 0
max 1sG (s,t) 4 4 33 11
minmax 1stG (s,t)
To ‘minimax’ the other one should play C
when both play C, each gets 0
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Repeated Games (a general treatment)An Example AA BB CC
AA 4 , 44 , 4 3 , 03 , 0 1 , 01 , 0
BB 0 , 30 , 3 2 , 22 , 2 1 , 01 , 0
CC 0 , 10 , 1 0 , 10 , 1 0 , 00 , 0
not (C,C)
B C
(B,B)
not (B,B) C(C,C) C
not (C,C)
all
(C,C)
1 2 k.........
a sub-game perfect equilibrium strategy:
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Incomplete Monitoring
Two firms repeatedly compete in prices à la Bertrand, δ the discount rate
Each observes its own profit but not the price set by the other.
Demand is 0 with probability ρ, and D(p) with probability 1- ρ
Assume that production unit cost is c, that D(p)0, and that (p-c)D(p) has a unique maximum at pm
When demand is D(p), and both firms charge pm,each earns ½πm =½ (pm-c)D(pm).
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Incomplete Monitoring
Can the firms achieve cooperation (pm) ???
For which values of k is the pair (Sk, Sk) a sub-game perfect equilibrium ???
1 2 3 k
pm c
½πm
zero profit
c c cAll All All
All
Let both firms play the following strategy (Sk):
0
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Incomplete Monitoring
Let V0 , V1 be the expected discouned payoffs at states 0,1 (respectively), when both players play Sk.
1 2 3 k
pm c
½πm
zero profit
c c cAll All All
All
0
0 m 0 11V = (1 - ρ)( π + δV )+ ρδV
21 k 0V = V
m
0k
0.5 1 - ρ πV =
1 - δ 1 - ρ - ρδ
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Incomplete Monitoring
By the One Deviation Property, it suffices to check whether a deviation at state 0 can improve payoff.(At states 1,2,..k a deviation will not increase payoff).
1 2 3 kpm c
½πm
zero profit
c c cAll All All
All
0
0.5 m0
k
1 - ρ πV =
1 - δ 1 - ρ - ρδ
The best one can do at state 0, is to slightly undercut the other, this will yield a payoff of: m 11 - ρ π + δV m 11 - ρ π + δV
m
m 1 0k
0.5 1 - ρ π1 - ρ π + δV V =
1 - δ 1 - ρ - ρδ
1 k 0V = V
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Incomplete Monitoring
k+12δ 1 - ρ + 2ρ - 1 δ 1
k+1k2δ 1 - ρ + 2ρ - 1 δ 0+11 > 2δ 1 - ρ + 2ρ - 1 δ
when 2ρ - 1 0
there exists no equilibrium of this form.
:when and <2ρ - 1 0 δ > 1/ 2 1 - ρ
log
log
1 - 2δ 1 - ρ1 k
δ 2ρ - 1
a more subtle equilibrium ???
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Social Contract
young
old young
old
….
….young
old
Overlapping Generations
A person lives for 2 periods
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Social ContractA young person produces 2 units of perishable good. An old person produces 0 units.
A person’s preference for consumption over time(c1, c2), is given by: (1,1) (2,0)
It is an equilibrium for each young person to consume the 2 units he producesshe produces
Is there a ‘better’ equilibrium ??
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Social ContractLet each young person give 1 unit to her old mother, provided the latter
has, in her youth, given 1 unit to her own mother
If my mother was ‘bad’ I am required to punish her, but then I will be punished in my
old age.
It is better not to follow this strategy.
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Social ContractLet each young person give 1 unit to her old mother, provided ALL young persons in the past have contributed
to their mothers.
This is a sub-game perfect equilibrium:I am willing to punish my ‘bad’ mother, since I
will be punished anyway.
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Social Contractmore subtle strategies:
A person is ‘bad’ if, either1. She did not provide her mother, although the mother was not ‘bad’.or:2. She did not punish her mother, although the mother was ‘bad’.
Punish your mother iff she is ‘bad’
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Incomplete InformationIncomplete Information
BB XX
BB 2 , 12 , 1 0 , 00 , 0
XX 0 , 00 , 0 1 , 21 , 2
BB XX
BB 2 , 02 , 0 0 , 20 , 2
XX 0 , 10 , 1 1 , 01 , 0
BB XX
BB 22 00
XX 00 11
probability ½ probability ½
BB XX
BB 22 00
XX 00 11
meet avoid
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Incomplete InformationIncomplete Information
BB XX
BB 2 , 12 , 1 0 , 00 , 0
XX 0 , 00 , 0 1 , 21 , 2
BB XX
BB 2 , 02 , 0 0 , 20 , 2
XX 0 , 10 , 1 1 , 01 , 0
½ ½
meet avoid
0
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½ ½meet avoid
0
11
22
XB
22
XB
XBXBXBXB
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