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Unformatted text preview: 6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation Daron Acemoglu and Asu Ozdaglar MIT November 2, 2009 1 Networks: Lecture 15 Introduction Outline The problem of cooperation Finitelyrepeated prisoners dilemma Infinitelyrepeated games and cooperation Folk theorems Cooperation in finitelyrepeated games Social preferences Reading: Osborne, Chapters 14 and 15. 2 Networks: 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 operating. Cooperate Defect Cooperate 1 , 1 1 , 2 Defect 2 , 1 , 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 Networks: 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. Key new concept: discounting . We will imagine that 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 ). 4 Networks: 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 twoperiod 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 Networks: Lecture 15 Introduction Mathematical Model More formally, imagine that I players are playing a strategic form game G = I , ( A i ) i I , ( u i ) i I for T periods. At each period, the outcomes of all past periods are observed by all players. 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 u i : A R , where A = A 1 A I . That is, u i a i t , a t is the state payoff to player i when action profile i a t = a i t , a t is played. i 6 Networks: Lecture 15 Introduction Mathematical Model (continued) t T We use the notation a = { a } t =0 to denote the sequence of action profiles. We could also define = { t } T to be the profile of mixed t =0 strategies....
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 Fall '09
 Acemoglu

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