2 problem set 9from Osborne’sIntrod. To G.T.Ex , 459.2, 459.3
3 For a general game G: Repeated Games (a general treatment) C D 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 1does not get more than 1CD2 , 20 , 33 , 01 , 1For a general game G:Player 1 can always play the best responseto the other’s action
4 Repeated Games (a general treatment) C D Player 2 can minimize the best that 1 can do by choosing t:CD2 , 20 , 33 , 01 , 1In the P.D. :In the P.D. it is a Nash equilibrium for each to play the stratgy that minimaxes the other.31In general playing the strategy that holds the other to his minimax payoff is NOT a Nash Equilibrium
5 Repeated Games (a general treatment) In the infinitely repeated game of G, every Nash equilibrium payoff is at least the minimax payoffIf a player always plays the best response to his opponent’s action, his payoff is at least his minmax value.A folk theorem:approximatelyEvery feasible value of G, which gives each player at least his minimax value, can be obtained as a Nash Equilibrium payoff(for δ~1)
6 ? 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 thesequence has been played in the past.If there was a deviation from it, playforever the action that holds the otherto his minimaxIt is an equilibrium for both to play this strategy?
7 Repeated Games (a general treatment) follow the cycle sequence if thesequence has been played in the past.If there was a deviation from it, playforever the action that holds the otherto his minimaxIf both follow the strategy, each receives more than his minimaxIf one of them deviates, the other punishes him, hence the deviator gets at most his minimax.hence, he will not deviate.
8 Playing these strategies is Nash but not sub-game perfect equilibrium. Repeated Games (a general treatment)Would a player want to punish after a deviation???by punishing the other his own payoff is reducedPlaying these strategies is Nash but not sub-game perfect equilibrium.To make punishment ‘attractive’:it should last finitely many periodsif not all participated in the punishment the countingstarts again.
9 Repeated Games (a general treatment) An Example A B C 4 , 43 , 01 , 00 , 32 , 20 , 10 , 0431To ‘minimax’ the other one should play Cwhen both play C, each gets 0
10 B C Repeated Games (a general treatment) A B C An Example 4 , 4 3 , 0 1 , 00 , 32 , 20 , 10 , 0An Examplea sub-game perfect equilibrium strategy:not (C,C)BC(B,B)not (B,B)(C,C)all12k
11 Incomplete Monitoring Two firms repeatedly compete in prices à laBertrand, δ the discount rateEach observes its own profit but not theprice set by the other.Demand is 0 with probability ρ, and D(p) withprobability 1- ρAssume that production unit cost is c, that D(p)0, and that (p-c)D(p) has a unique maximum at pmWhen 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) ???Let both firms play the following strategy (Sk):Allpmc½πmzeroprofitAll123kFor which values of k is the pair (Sk, Sk)a sub-game perfect equilibrium ???
13 Incomplete Monitoring Let V0 , V1 be the expected discouned payoffs at states 0,1 (respectively), when both players play Sk.123kpmc½πmzeroprofitAll
14 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:123kpmc½πmzeroprofitAll
16 …. …. Social Contract Overlapping Generations A person lives for 2 periods….youngoldyoungoldyoungold….
17 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 he producesshe producesIs 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 motherIf 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 This is a sub-game perfect equilibrium: 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 more subtle strategies: Social Contractmore subtle strategies:Punish your mother iff she is ‘bad’A person is ‘bad’ if, eitherShe did not provide her mother, althoughthe mother was not ‘bad’.or:She did not punish her mother, althoughthe mother was ‘bad’.