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problem set 9 from Osborne’s Introd. To G.T. Ex. 459.1, 459.2, 459.3

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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) CD C 2, 2 0, 3 D 3, 0 1, 1 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

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4 Repeated Games (a general treatment) Player 2 can minimize the best that 1 can do by choosing t : CD C 2, 2 0, 3 D 3, 0 1, 1 In the P.D. :31 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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5 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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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 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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7 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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8 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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9 Repeated Games (a general treatment) An Example ABC A 4, 4 3, 0 1, 0 B 0, 3 2, 2 1, 0 C 0, 1 0, 0 431 To ‘ minimax ’ the other one should play C when both play C, each gets 0

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10 Repeated Games (a general treatment) An Example ABC A 4, 4 3, 0 1, 0 B 0, 3 2, 2 1, 0 C 0, 1 0, 0 not (C,C) BC (B,B) not (B,B) C (C,C) C not (C,C) all (C,C) 12k......... a sub-game perfect equilibrium strategy:

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11 Incomplete Monitoring T wo firms repeatedly compete in prices à la Bertrand, δ the discount rate E ach observes its own profit but not the price set by the other. D emand 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 p m When demand is D(p), and both firms charge p m, each earns ½π m =½ (p m -c)D(p m ).

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12 Incomplete Monitoring Can the firms achieve cooperation (p m ) ??? For which values of k is the pair (S k, S k ) a sub-game perfect equilibrium ??? 123 k pmpm c ½πm½πm zero profit cc c All Let both firms play the following strategy (S k ): 0

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Incomplete Monitoring Let V 0, V 1 be the expected discouned payoffs at states 0,1 (respectively), when both players play S k. 123 k pmpm c ½πm½πm zero profit cc c All 0

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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). 123 k pmpm c ½πm½πm zero profit cc c All 0 The best one can do at state 0, is to slightly undercut the other, this will yield a payoff of:

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Incomplete Monitoring

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16 Social Contract young old young old …. young old Overlapping Generations A person lives for 2 periods

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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 (c 1, c 2 ), is given by: (1,1) (2,0) It is an equilibrium for each young person to consume the 2 units he produces she produces Is there a ‘better’ equilibrium ??

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18 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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19 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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20 Social Contract more subtle strategies: 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’. Punish your mother iff she is ‘bad’

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21 Incomplete Information BX B 2, 1 0, 0 X 1, 2 BXB 2, 0 0, 2 X 0, 1 1, 0 BXB20 X01 probability ½ BXB20 X01 meet avoid

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22 Incomplete Information BX B 2, 1 0, 0 X 1, 2 BXB 2, 0 0, 2 X 0, 1 1, 0 ½ ½ meet avoid 0

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23 ½ ½ meetavoid 0 1 1 2 2 XB 2 2 XB XB XBXBXB

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