The Truth Behind the Myth of the Folk Theorem

TL;DR

This paper demonstrates that computing approximate Nash equilibria in repeated games is feasible in polynomial time under cryptographic assumptions, challenging previous beliefs about its intractability.

Contribution

The paper introduces a polynomial-time algorithm for computing approximate Nash equilibria in repeated games using cryptographic assumptions, extending to graphical games and defining a computational subgame-perfect equilibrium.

Findings

Polynomial-time algorithm for approximate Nash equilibria

Applicability to graphical games with constant degree

Efficient computation of computational subgame-perfect equilibrium

Abstract

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 model---modeling 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).