The Myth of the Folk Theorem

A well-known result in game theory known as “the Folk Theorem” 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 {−1, 0,−1}), 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 “no regret” algorithms, can be rational (in general games with three or more players) in the sense that, when one’s opponents use such a strategy, it is not in general a best reply to follow suit.