gameL7 - Repeated Games A repeated game(say infinitely...

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Repeated Games A repeated game (say, in f nitely repeated prisoner’s dilemma) is a special case of an extensive game. The additional structure of the same game being repeated allows for new results. “Folk theorems” show that any payo f s that are feasible and enforcable (individually ra- tional) can be achieved in equilibrium. Repeated interaction allows for socially bene f cial out- comes, essentially substituting for the ability to make binding agreements. One interpretation: if interaction is repeated, then socially bene f cial outcomes that cannot be sustained by players with short-term objectives can be sustained by players with long-term objectives.
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Proofs are constructive. We can interpret the equilibrium path as a “social norm,” which is supported by the threat of punishment. ButdoesGameTheo rylosea l lp red ict ivepower? There are folk theorems for f nitely repeated games as well, but for many games the in f nitely repeated game is very di f erent from the f nitely repeated game. If the prisoner’s dilemma is repeated 1,000,000 times, the only Nash equilibrium outcome is to defect in every period. The game unravels. Which is the more appropriate model of human behavior, if people have f nite lifetimes but do not perceive the distant future to be relevant?
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Assume throughout a compact action space and contin- uous preferences. De f nition 137.1: Let G = h N, ( A i ) , ( u i ) i be a strategic game, and let A = × i N A i .A n in f nitely repeated game of G is an extensive game with perfect information and simultaneous moves D N,H,P, ( º i ) E in which we have 1. H = { } [ t =1 A t A (where A is the set of in f nite sequences of action pro f les), 2. P ( h )= N for every non-terminal history, h H , 3. º i is a preference relation on A that satis f es the following notion of weak separability: if ( a t ) A , a A , a 0 A ,and u i ( a ) >u i ( a 0 ) ,thenwehavefo r all t , ( a 1 ,...,a t 1 ,a,a t +1 ,... ) º i ( a 1 , ..., a t 1 ,a 0 t +1 ) .
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Note that a strategy for player i ass ignsanact ionin A i for every f nite sequence of outcomes in G . What additional structure should we impose on º i ?F o r in f nite games, you cannot simply add up the payo f sre - ce ivedateachstage . 1. Discounting. There is a discount factor δ (0 , 1) such that ( a t ) º i ( b t ) if and only if X t =1 δ t 1 u i ( a t ) X t =1 δ t 1 u i ( b t ) . With discounting, we normalize payo f s so that a payo f of v each period yields an overall payo f of v . (1 δ ) X t =1 δ t 1 u i ( a t )
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2. Limit of Means. ( a t ) is strictly preferred to ( b t ) if and only if there exists ε> 0 such that P T t =1 [ u i ( a t ) u i ( b t )] T holds for all but a f nite number of periods T . This criterion treats periods symmetrically, and any one period has a negligible e f ect on the overall payo f .W h e n the limiting average payo f exists, the payo f is lim T →∞ Ã P T t =1 u i ( a t ) T !
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This note was uploaded on 07/17/2008 for the course ECON 817 taught by Professor Peck during the Fall '07 term at Ohio State.

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gameL7 - Repeated Games A repeated game(say infinitely...

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