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# L5 - Infinitely Repeated Games We recall that an infinitely...

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Page 1 of 38 Infinitely Repeated Games We recall that an infinitely repeated game of ,( ),( ) i i G N A is an extensive game with perfect information and simultaneous moves * , , ,( ) i N H P in which     1 { } ( ) t t H A A ( ) P h N for each nonterminal h H * i is a preference relation on A that extends i and satisfies weak separability.

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Page 2 of 38 Infinitely Repeated Games We consider three forms of preference relations: Discounting 1 2 3 ( ) ( , , ) t i i i i v v v v is evaluated by 1 1 t t i t δ v Limit of Means 1 2 3 ( ) ( , , ) t i i i i v v v v is evaluated by 1 lim T t i t T v T  Overtaking 1 2 3 ( ) ( , , ) t i i i i v v v v is evaluated by 1 lim T t i t T v 
Page 3 of 38 Nash Equilibrium What are the Nash equilibria of Infinitely Repeated Games?

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Page 4 of 38 Nash Equilibrium Let ,( ),( ) i i G N A . The set of outcomes is i N i A A   . Consider the players play an indefinitely repetition of a cycle of outcomes of length γ , in which each outcome a A is played a β times. a a A γ β Outcome 1 a is played 1 a β times. Outcome 2 a is played 2 a β times. Outcome A a is played A a β times. Outcome 1 a is played 1 a β times. Outcome 2 a is played 2 a β times. Outcome A a is played A a β times.
Page 5 of 38 Nash Equilibrium If the limit-of-means criterion is used, the payoff of player i in each period, and hence the entire game , is ( ) a β i i γ a A w u a . The payoff profile is hence ( ) ( ) ( ) a a β β i i N i γ γ a A a A i N w w u a u a . a a A γ β Outcome 1 a is played 1 a β times. Outcome 2 a is played 2 a β times. Outcome A a is played A a β times. Outcome 1 a is played 1 a β times. Outcome 2 a is played 2 a β times. Outcome A a is played A a β times.

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Page 6 of 38 Feasible Payoff Profiles ( ) ( ) ( ) a a β β i i N i γ γ a A a A i N w w u a u a . A vector ( ) a β γ a A w u a , where a a A γ β and 0 a β for all a , is called a feasible payoff profile of ,( ),( ) i i G N A u . Note that a feasible payoff profile is not necessarily a payoff profile of ,( ),( ) i i G N A u .
Page 7 of 38 Nash Equilibrium ( ) ( ) ( ) a a β β i i N i γ γ a A a A i N w w u a u a But how can we enforce this feasible payoff profile? The feasible payoff profile is enforceable if no player can increase his payoff by deviating! a a A γ β Outcome 1 a is played 1 a β times. Outcome 2 a is played 2 a β times. Outcome A a is played A a β times. Outcome 1 a is played 1 a β times. Outcome 2 a is played 2 a β times.

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L5 - Infinitely Repeated Games We recall that an infinitely...

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