The myth of the Folk Theorem

• Published in: Games and economic behavior Vol. 70; no. 1; pp. 34 - 43
• Main Authors: Borgs, Christian, Chayes, Jennifer, Immorlica, Nicole, Kalai, Adam Tauman, Mirrokni, Vahab, Papadimitriou, Christos
• Format: Journal Article
• Language: English
• Published: Duluth Elsevier Inc 01.09.2010; Elsevier; Academic Press
• Series: Games and Economic Behavior
• Subjects: Applied general equilibrium models; Computable general equilibrium models; Computational methods; Equilibrium models; Game theory; Games; Nash equilibrium; Repeated games; Repeated games Computable general equilibrium models; Studies; C68; Repeated games; Computable general equilibrium models; C73
• ISSN: 0899-8256; 1090-2473
• DOI: 10.1016/j.geb.2009.04.016

The Folk Theorem for repeated games suggests that finding Nash equilibria in repeated games should be easier than in one-shot games. In contrast, we show that the problem of finding any Nash equilibrium for a three-player infinitely-repeated game is as hard as it is in two-player one-shot games. More specifically, for any two-player game, we give a simple construction of a three-player game whose Nash equilibria (even under repetition) correspond to those of the one-shot two-player game. Combined with recent computational hardness results for one-shot two-player normal-form games ( Daskalakis et al., 2006; Chen et al., 2006; Chen et al., 2007), this gives our main result: the problem of finding an (epsilon) Nash equilibrium in a given n × n × n game (even when all payoffs are in { − 1 , 0 , 1 } ) is PPAD-hard (under randomized reductions).