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Repeated_Game - Repeated Games Econ 210C UCSB Example 1...

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Repeated Games Econ 210C UCSB March 30, 2011
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Example 1: Consider a firm that produces a good of quality θ [ 0, 1 ] . If consumers anticipate quality θ , their aggregate demand is Q = 4 + 6 θ - p . The firm’s marginal cost is MC ( θ ) = 2 + 6 θ 2 . In each period, the firm chooses a quality level θ 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 ¯ θ is chosen in each period. Does such a reputational equilibrium exist? Econ 210C UCSB Paper
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Example 2: Consider one-shot prisoner’s 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 2’s is u t 2 = 0. Thus, player 1 is better off. Suppose player 2 conditions his choice at time t on each other’s 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
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Basic Model Let Γ = { A i , u i } i N be a given one-shot game in strategic form. A repeated game based on Γ consists of a series of plays of Γ at times 0, 1, 2, · · · , T , which is denoted by Γ T . Γ T is finitely repeated if T < ; Γ 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 Γ be the prisoner’s 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
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Payoffs in Repeated Games How Do Players Evaluate Payoff Streams ( u 0 i , u 1 i , · · · , u t i , · · · ) ? Limit of Means: lim t -→ T ( t - 1 τ = 0 u τ 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. Discounted Payoffs: ( 1 - δ ) T t = 0 δ t u t i , where δ ( 0, 1 ) is the discount factor (e.g. δ = 1 / ( 1 + r ) ).
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