Transcript problem set 9 from Osborne’s Introd. To G.T. Ex. 459.1, 459.2, 459.3 Repeated Games (a general treatment) What is the minimum that a player can.

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problem set 9
from Osborne’s
Introd. To G.T.
Ex. 459.1, 459.2, 459.3
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)
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 G1 (s,t)
C
D
C
2,2 0,3
D
3,0 1,1
s
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Repeated Games (a general treatment)
Player 2 can minimize the best that 1 can do by choosing t:
min max G1 (s,t)
t
s
In the P.D. :
In the P.D. it is a Nash equilibrium for
each to play the stratgy that minimaxes
the other.
max G1 (s,t)
s
In general playing
the strategy
that
min max
G1 (s,t)
holds the other tot his minimax
payoff is
s
NOT a Nash Equilibrium
C
D
C
2,2 0,3
D
3,0 1,1
3
1
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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.
A folk theorem:
approximately
Every feasible value of G, which gives each player at least
his minimax value, can be obtained as a Nash Equilibrium
payoff (for δ~1)
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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 subgame 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
A
B
C
A 4,4 3,0 1,0
B 0,3 2,2 1,0
C 0,1 0,1 0,0
max G1 (s,t)
s
4
3
1
min max G1 (s,t)
t
s
To ‘minimax’ the other one should play C
when both play C, each gets 0
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Repeated Games (a general treatment)
A
B
An Example
C
A 4,4 3,0 1,0
a sub-game perfect equilibrium strategy:
B 0,3 2,2 1,0
C 0,1 0,1 0,0
all
(B,B)
not (C,C)
B
not (B,B)
C
1
(C,C)
C
not (C,C) 2
(C,C)
C
.........
k
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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) ???
Let both firms play the following strategy (Sk):
All
½πm
pm
0
zero
profit
c
1
All
c
2
All
c
All
3
c
k
For which values of k is the pair (Sk, Sk)
a sub-game perfect equilibrium ???
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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 m
0
1
V = (1 - ρ)( π + δV )+ ρδV
2
1
k 0
V = V
m
0.5  1 - ρ  π
0
V =
k
1 - δ  1 - ρ  - ρδ
0
All
½πm
pm
0
zero
profit
c
1
All
c
2
All
c
3
All
c
k
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).
The best one can do at state 0, is to slightly undercut
the other, this will yield a payoff of:  1 - ρ πm + δV 1 
m
0.5
1
ρ
π


m
1
0
1
ρ
π
+
δV

V
=
 
k
1 - δ  1 - ρ  - ρδ
½πm
pm
zero
0 profit
All
c
c
c
All All
All
1
2
3
c
k
V 1 =  kV 0
Incomplete Monitoring
2δ  1 - ρ +  2ρ - 1 δ
when 2ρ - 1  0
1 > 2δ  1 - ρ +  2ρ - 1 δ
0+1
k+1
1
 2δ  1 - ρ +  2ρ - 1 δ
there exists no equilibrium of this form.
a more subtle equilibrium ???
when 2ρ - 1 < 0 and δ > 1/  2  1 - ρ   :
 1 - 2δ  1 - ρ  
1
k
log 

log δ
 2ρ - 1 
k+1
k
Social Contract
Overlapping Generations
A person lives for 2 periods
….
young
old
young
old
young
old
….
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Social Contract
A 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 she
he produces
produces
Is there a ‘better’ equilibrium ??
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Social Contract
Let 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 Contract
Let 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 Contract
more subtle strategies:
Punish your mother iff she is ‘bad’
A person is ‘bad’ if, either
1. 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’.
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Incomplete Information
meet
probability ½
B
X
avoid
probability ½
B
X
B 2 ,1 0 ,0
B 2 ,0 0 ,2
X 0 ,0 1 ,2
X 0 ,1 1 ,0
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Incomplete Information
0
½
B
X
½
B
X
B 2,1 0,0
B 2,0 0,2
X 0,0 1,2
X 0,1 1,0
meet
avoid
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0
½
meet
½
avoid
1
1
B
X
B
X
2
2
2
2
B X
B X
B X B X
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