Repeated_Game

Repeated_Game - Repeated Games Econ 210C UCSB March 30,...

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Unformatted text preview: Repeated Games Econ 210C UCSB March 30, 2011 Example 1: Consider a firm that produces a good of quality [ 0, 1 ] . If consumers anticipate quality , their aggregate demand is Q = 4 + 6 - p . The firms marginal cost is MC ( ) = 2 + 6 2 . In each period, the firm chooses a quality level and a price. Consumers observe the price but not the quality until they have bought and consumed the good. Define a reputational equilibrium as one in which a particular quality level is chosen in each period. Does such a reputational equilibrium exist? Econ 210C UCSB Paper Example 2: Consider one-shot prisoners dilemma: C D C 3, 3 0, 5 D 5, 0 1, 1 Suppose that the game is repeated at times t = 0, 1, If each player chooses C each time regardless of what the other chooses, then the stage- t payoff for each is u t i = 3 for t = 0, 1, . Suppose player 2 makes choices as before but player 1 chooses D each time. Then, his stage- t payoff is u t 1 = 5 while player 2s is u t 2 = 0. Thus, player 1 is better off. Suppose player 2 conditions his choice at time t on each others choices at time 0, 1, , t- 1: choose C at time 0, continue to choose C at time t provided both chose C at times before t , but choose D if some chose D before time t . Will player 1 has any incentive to choose D? Econ 210C UCSB Paper Basic Model Let = { A i , u i } i N be a given one-shot game in strategic form. A repeated game based on consists of a series of plays of at times 0, 1, 2, , T , which is denoted by T . T is finitely repeated if T < ; T is infinitely repeated if T = . Stage- t Actions: a t i A i . Stage- t Action Profiles: a t A = A 1 A 2 A n . Player i s Stage- t Payoff at a t A : u t i = u i ( a t ) . Let be the prisoners dilemma in Example 2. Then stage- t actions: a t i A i = { C , D } , i = 1, 2, stage- t action profiles: a t A = A 1 A 2 = { ( C , C ) , ( C , D ) , ( D , C ) , ( D , D ) } . stage- t payoffs at a t A : u t i = 3 if a t = ( C , C ) ; 0 if a t = ( C , D ) ; 5 if a t = ( D , C ) ; 1 if a t = ( D , D ) . Econ 210C UCSB Paper Payoffs in Repeated Games How Do Players Evaluate Payoff Streams ( u i , u 1 i , , u t i , ) ? Limit of Means: lim t- T ( t- 1 = u i /t). Players are unconcerned with the timing of payoffs and their payoffs in any finite number of periods. The payoff sequences ( 1, 0, ) , ( 0, 1, 0, ) , ( 0, 0, 0, ) are equivalent....
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This note was uploaded on 12/26/2011 for the course ECON 210C taught by Professor Qin during the Fall '09 term at UCSB.

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Repeated_Game - Repeated Games Econ 210C UCSB March 30,...

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