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Lecture16-new

# Lecture16-new - 6.254 Game Theory with Engineering...

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6.254 : Game Theory with Engineering Applications Lecture 16: Repeated Games – II Asu Ozdaglar MIT April 13, 2010 1

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Game Theory: Lecture 16 Introduction Outline Repeated Games – perfect monitoring Folk Theorems Repeated Games – imperfect monitoring Price-trigger strategies Reference: Fudenberg and Tirole, Section 5.1 and 5.5. 2
Game Theory: Lecture 16 Repeated Games Repeated Games By repeated games , we refer to a situation in which the same stage game (strategic form game) is played at each date for some duration of T periods. More formally, imagine that I players playing a strategic form game G = hI , ( A i ) i ∈I , ( g i ) i ∈I i for T periods (infinitely-repeated game if T = ). At each period, the outcomes of all past periods are observed by all players perfect monitoring The notation a = { a t } t = 0 denotes the (infinite) sequence of action profiles. The payoff to player i for the entire repeated game is then u i ( a ) = ( 1 - δ ) t = 0 δ t g i ( a t i , a t - i ) where δ [ 0, 1 ) . We have seen that grim trigger strategies can sustain “cooperation” in infinitely repeated games. This motivated the question what payoffs are achievable in equilibrium when players are sufficiently patient (i.e., when δ 1). 3

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Game Theory: Lecture 16 Folk Theorems Folk Theorems for Infinitely Repeated Games We started last time talking about folk theorems which study equilibrium payoffs that can be obtained in infinitely repeated games. Recall that we use v i to denote the minmax payoff of player i , i.e., v i = min α - i max α i g i ( α i , α - i ) . The strategy profile m i - i denotes the minmax strategy of opponents of player i and m i i denotes the best response of player i to m i - i , i.e., g i ( m i i , m i - i ) = v i . The set V * R I denotes the set of feasible and strictly individually rational payoffs. We have seen that equilibrium payoffs of repeated games are bounded from below by minmax payoffs. 4
Game Theory: Lecture 16 Folk Theorems Folk Theorems Our first theorem is the Nash folk theorem which shows that any feasible and strictly individually rational payoff vector can be achieved if the player is sufficiently patient. Theorem (Nash Folk Theorem) If ( v 1 , . . . , v I ) is feasible and strictly individually rational, then there exists some δ < 1 such that for all δ > δ , there is a Nash equilibrium of G ( δ ) with payoffs ( v 1 , · · · , v I ) . The Nash folk theorem states that essentially any payoff can be obtained as a Nash Equilibrium when players are patient enough. However, the corresponding strategies involve this non-forgiving punishments, which may be very costly for the punisher to carry out (i.e., they represent non-credible threats).

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