The Complexity of Finding Nash Equilibria

The recent proof that finding a Nash equilibrium is complete for the complexity class PPAD, even in the case of two players, is outlined, evidence that the problem is intractable.

Abstract

Computing a Nash equilibrium, given a game in normal form, is a fundamental problem for Algorithmic Game Theory. The problem is essentially combinatorial, and in the case of two players it can be solved by a pivoting technique called the Lemke–Howson algorithm, which however is exponential in the worst case. We outline the recent proof that finding a Nash equilibrium is complete for the complexity class PPAD, even in the case of two players; this is evidence that the problem is intractable. We also introduce several variants of succinctly representable games, a genre important in terms of both applications and computational considerations, and discuss algorithms for correlated equilibria, a more relaxed equilibrium concept.