1Repeated Games with Incomplete InformationEE228A Project by Alex K.Fall 2003Bayesian Collective Choice Problem23Can we achieve this in Repeated Game?Friedman’s TheoremJames W. Friedman “A Non-cooperative Equilibrium for Supergames”, Review of Economic Studies 38/1 (1971)4Repeated games: what can happen?1. Arbitrator and players do not learn2. Arbitrator does not learn but the players do3. Arbitrator does learn but the players do not4. Arbitrator and players do learnArbitrator and players do not learnMakes sense to treat the repeated game as series of independent one-shot games, in which we look for equilibrium5Response Plan Equilibrium6Arbitrator does not learn but the players doThe answer to this problem is given byEhud Kalai, Ehud Lehrer“Rational Learning Leads to Nash Equilibrium”Econometrica 61/5 (1993)Arbitrator does learnPatience – what is it?• Patient Arbitrator: if outcome functions c(θ)=c’(θ) after certain T, then they are treated as equal• Patient Player: discounting factor δ→17Arbitrator does learnIdea 1: the arbitrator has to force the players to reveal their true types. The game starts with the sequence of incentive compatible rounds. The length of the sequence depends on max{δi}n. In the infinite game the arbitrator can make this sequence as long as needed. After that the arbitrator makes Pareto optimal choice using the types that were revealed by the players.Arbitrator does learnIdea 2: let each player be a dictator for dinumber of rounds• θ and θ’ are distinguishable types for the player i if c(θ)>c(θ’) and c’(θ)<c’(θ’)• di= number of pair-wise distinguishable types for the i-th player• Arbitrator has all these pairs enumerated and at every dictatorship round of the player i this player is offered only two choice alternatives – each giving him better payoff if he were of the corresponding type• After all the dictatorships there is a period of conflict outcomes, which duration depends on max{δi}n• After that the arbitrator makes Pareto optimal choice using the types that were revealed by the players during their dictatorships8Open QuestionWhich Pareto optimal allocation should the arbitrator choose?What if everybody learns, but the players are patient and the arbitrator is not?The repeated game turns into the sequence of one-shot games9References1.1.J.W. FriedmanJ.W. Friedman “A Non-cooperative Equilibrium for Supergames”, Review of Economic Studies 38/1 (1971)2.2.D. D. FudenbergFudenberg, E. , E. MaskinMaskin “The Folk Theorem in Repeated Games with Discounting or with Incomplete Information”, Econometrica 54/3 (1986)3.3.E. E. KalaiKalai, J. Ledyard, J. Ledyard “Repeated Implementation”, Journal of Economic Theory 83 (1998)4.4.E. E. KalaiKalai, E. Lehrer, E. Lehrer “Rational Learning Leads to Nash Equilibrium”, Econometrica 61/5 (1993)5.5.R.B. MyersonR.B. Myerson “Incentive Compatibility and the Bargaining Problem”, Econometrica 47/1 (1979)6.6.R.B. MyersonR.B. Myerson “Bayesian Equilibrium and Incentive Compatibility: Introduction”, Social Goals and Social Organization, 19857.7.R.W. RosenthalR.W. Rosenthal “Arbitration of Two-Party Disputes Under Uncertainty”, Review of Economic Studies 45/3
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