Lecture15-new

Lecture15-new - 6.254 : Game Theory with Engineering...

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Unformatted text preview: 6.254 : Game Theory with Engineering Applications Lecture 15: Repeated Games Asu Ozdaglar MIT April 1, 2010 1 Game Theory: Lecture 15 Introduction Outline Repeated Games (perfect monitoring) The problem of cooperation Finitely-repeated prisoners dilemma Infinitely-repeated games and cooperation Folk Theorems Reference: Fudenberg and Tirole, Section 5.1. 2 Game Theory: Lecture 15 Introduction Prisoners Dilemma How to sustain cooperation in the society? Recall the prisoners dilemma , which is the canonical game for understanding incentives for defecting instead of cooperating. Cooperate Defect Cooperate 1, 1- 1, 2 Defect 2,- 1 0, 0 Recall that the strategy profile ( D , D ) is the unique NE. In fact, D strictly dominates C and thus ( D , D ) is the dominant equilibrium. In society, we have many situations of this form, but we often observe some amount of cooperation. Why? 3 Game Theory: Lecture 15 Introduction Repeated Games In many strategic situations, players interact repeatedly over time. Perhaps repetition of the same game might foster cooperation. 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. Such games are also sometimes called supergames. We will assume that overall payoff is the sum of discounted payoffs at each stage. Future payoffs are discounted and are thus less valuable (e.g., money and the future is less valuable than money now because of positive interest rates; consumption in the future is less valuable than consumption now because of time preference ). We will see in this lecture how repeated play of the same strategic game introduces new (desirable) equilibria by allowing players to condition their actions on the way their opponents played in the previous periods. 4 Game Theory: Lecture 15 Introduction Discounting We will model time preferences by assuming that future payoffs are discounted proportionately ( exponentially ) at some rate [ 0, 1 ) , called the discount rate . For example, in a two-period game with stage payoffs given by u 1 and u 2 , overall payoffs will be U = u 1 + u 2 . With the interest rate interpretation, we would have = 1 1 + r , where r is the interest rate. 5 Game Theory: Lecture 15 Introduction Mathematical Model 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. At each period, the outcomes of all past periods are observed by all players perfect monitoring Let us start with the case in which T is finite, but we will be particularly interested in the case in which T = . Here A i denotes the set of actions at each stage, and g i : A R , where A = A 1 A I ....
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Lecture15-new - 6.254 : Game Theory with Engineering...

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