A Note on Approximate Nash Equilibria

Abstract

In view of the intractability of finding a Nash equilibrium, it is important to understand the limits of approximation in this context. A subexponential approximation scheme is known [LMM03], and no approximation better than $1\over 4$ is possible by any algorithm that examines equilibria involving fewer than logn strategies [Alt94]. We give a simple, linear-time algorithm examining just two strategies per player and resulting in a $1\over 2$-approximate Nash equilibrium in any 2-player game. For the more demanding notion of well-supported approximate equilibrium due to [DGP06] no nontrivial bound is known; we show that the problem can be reduced to the case of win-lose games (games with all utilities 0–1), and that an approximation of $5\over 6$ is possible contingent upon a graph-theoretic conjecture.