# L4 - Repeated Games Consider the game of Iterated Prisoners...

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Page 1 of 27 Repeated Games Consider the game of Iterated Prisoner’s Dilemma , in which players repeatedly engage in the Prisoner’s Dilemma (the ‘ constituent game ’). ( C , C ) (3, 3) ( D , C ) (5, 0) ( D , C ) (5, 0) ( D , D ) (1, 1) Suppose the prisoners encounter for T round. What are Nash equilibria? Subgame perfect equilibria?

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Page 2 of 27 Iterated Prisoner’s Dilemma ( C , C ) (3, 3) ( D , C ) (5, 0) ( D , C ) (5, 0) ( D , D ) (1, 1) Suppose the prisoners encounter for T round. Are Nash equilibria ‘good?’ Are Subgame perfect equilibria ‘good?’ What should be the best for both?
Page 3 of 27 Iterated Prisoner’s Dilemma ( C , C ) (3, 3) ( D , C ) (5, 0) ( D , C ) (5, 0) ( D , D ) (1, 1) What if the game is infinitely repeated?

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Page 4 of 27 Repeated Games D EFINITION . Let ,( ),( ) ii G N A be a strategic game; let  i N i AA . An infinitely repeated game of G is an extensive game with perfect information and simultaneous moves * , , ,( ) i N H P in which      1 { } ( ) t t H A A (where is the initial history and A is the set of infinite sequences 1 () t t a of action profiles in G )
Page 5 of 27 Note:      1 { } ( ) t t H A A A history is a sequence of outcomes (action profiles), of the constituent game. is the initial history t A is the set of histories of length t . A history  12 ( ) ( , , , ) t t t a a a a A is a sequence of outcomes: i a is the outcome of round i . A is the set of infinit e sequences (‘ terminal history ’) 1 () t t a of action profiles

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Page 6 of 27 () P h N for each nonterminal history hH * i is a preference relation on A that extends the preference relation i in the constituent game, in the sense that it satisfies the following condition of weak separability : if 1 2 3 ( , , , ) a a a A is a terminal history, aA and are act ion profiles (or ‘ outcome ’) of a constituent game, and i aa then 1 1 1 * 1 1 1 ( , , , , , ) ( , , , , , ) t t t t i a a a a a a a a for all values of t .
Page 7 of 27 We further assume a payoff function i u : * ( ) ( ) tt i ab iff ( ) ( ) ii u a u b .

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Page 8 of 27 Preference Relations If i aa then 1 1 1 * 1 1 1 ( , , , , , ) ( , , , , , ) t t t t i a a a a a a a a
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## This note was uploaded on 04/23/2010 for the course CSC CSC5350 taught by Professor Leunghofung during the Winter '09 term at CUHK.

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L4 - Repeated Games Consider the game of Iterated Prisoners...

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