A complete folk theorem for finitely repeated games
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A complete folk theorem for finitely repeated games Ghislain-Herman Demeze-Jouatsa1 Accepted: 26 August 2020 © The Author(s) 2020
Abstract This paper analyzes the set of pure strategy subgame perfect Nash equilibria of any finitely repeated game with complete information and perfect monitoring. The main result is a complete characterization of the limit set, as the time horizon increases, of the set of pure strategy subgame perfect Nash equilibrium payoff vectors of the finitely repeated game. This model includes the special case of observable mixed strategies. Keywords Finitely repeated games · Pure strategy · Observable mixed strategies · Subgame perfect Nash equilibrium · Limit perfect folk theorem JEL Classification C72 and C73
1 Introduction This paper provides a full characterization of the limit set, as the time horizon increases, of the set of pure strategy subgame perfect Nash equilibrium payoff vectors of any finitely repeated game. The obtained characterization is in terms of appropriate notions of feasible and individually rational payoff vectors of the stage-game. These notions are based on Smith’s (1995) notion of Nash decomposition and appropriately generalize the classic notion of feasible payoff vectors as well as the notion of effective minimax
I acknowledges DAAD and the DFG (Deutsche Forschungsgemeinschaft/German Research Foundation) via Grant Ri 1128-9-1 (Open Research Area in the Social Sciences, Ambiguity in Dynamic Environments) for funding support and thank Christoph Kuzmics, Frank Riedel, Lones Smith, Michael Greinecker, Karl Schlag, Roland Pongou, Tondji-Jean Baptiste and Olivier Gossner for useful comments. I also thank seminar and conference participants at Bielefeld University, Cardiff University, University of Yaoundé I, Stony Brook University, Lisbon School of Economics and Management, and University of Graz for positive inputs. I thank two anonymous reviewers and an associated editor for their comments and suggestions.
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Ghislain-Herman Demeze-Jouatsa [email protected] Center for Mathematical Economics, Bielefeld University, Bielefeld, Germany
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payoff defined by Wen (1994). The main theorem nests earlier results of Benoit and Krishna (1985), Smith (1995), and Demeze-Jouatsa and Wilson (2019). Whether non-Nash outcomes of the stage-game can be sustained via subgame perfect Nash equilibria of the finitely repeated game depends on whether players can be incentivized to abandon their short term interests and to follow some collusive paths that have greater long-run average payoffs. There are two extreme cases. On the one hand, in any finite repetition of a stage-game that has a unique Nash equilibrium payoff vector such as the prisoners’ dilemma, only the stage-game Nash equilibrium payoff vector is sustainable by subgame perfect Nash equilibria of finite repetitions of that stage-game. On the other hand, for stage-games in which all players receive different Nash equilibrium payoffs such as the battle of sexes, the limit perfect folk the
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