1

22

problem set 9

from Osborne’sIntrod. To G.T.

Ex. 459.1, 459.2, 459.3

33

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)

44

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

55

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

66

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

?

77

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.

88

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.

99

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

1010

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:

11

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).

12

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

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

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

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 ???

16

Social Contract

young

old young

old

….

….young

old

Overlapping Generations

A person lives for 2 periods

17

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 ??

18

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.

19

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.

20

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’

2121

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

2222

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

2323

½ ½meet avoid

0

11

22

XB

22

XB

XBXBXBXB

2424