The Truth Behind the Myth of the Folk Theorem

We study the problem of computing an $\epsilon$-Nash equilibrium in repeated games. Earlier work by Borgs et al. [2010] suggests that this problem is intractable. We show that if we make a slight change to their modelmodeling the players as polynomial-time Turing machines that maintain state and make some standard cryptographic hardness assumptions (the existence of public-key encryption), the problem can actually be solved in polynomial time. Our algorithm works not only for games with a finite number of players, but also for constant-degree graphical games. As Nash equilibrium is a weak solution concept for extensive form games, we additionally define and study an appropriate notion of a subgame-perfect equilibrium for computationally bounded players, and show how to efficiently find such an equilibrium in repeated games (again, making standard cryptographic hardness assumptions).