A well-known result in game theory known as \u201cthe Folk Theorem\u201d 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 (approximate) Nash equilibrium for a three-player infinitely repeated game is computationally intractable (even when all payoffs are in {\u22121, 0,\u22121}), unless all of PPAD can be solved in randomized polynomial time. This is done by showing that finding Nash equilibria of (k + 1)-player infinitely-repeated games is as hard as finding Nash equilibria of k-player one-shot games, for which PPAD-hardness is known (Daskalakis, Goldberg and Papadimitriou, 2006; Chen, Deng and Teng, 2006; Chen, Teng and Valiant, 2007). This also explains why no computationally-efficient learning dynamics, such as the \u201cno regret\u201d algorithms, can be rational (in general games with three or more players) in the sense that, when one\u2019s opponents use such a strategy, it is not in general a best reply to follow suit.<\/p>\n","protected":false},"excerpt":{"rendered":"
A well-known result in game theory known as \u201cthe Folk Theorem\u201d 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 (approximate) Nash equilibrium for a three-player infinitely repeated game is computationally intractable (even when all payoffs are in {\u22121, […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-317225","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-locale-en_us"],"msr_publishername":"ACM Press","msr_edition":"STOC '08 Proceedings of the 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