Daskalakis C., Mehta A., Papadimitriou C. (2006) A Note on Approximate Nash Equilibria. In: Spirakis P., Mavronicolas M., Kontogiannis S. (eds) Internet and Network Economics. WINE 2006. Lecture Notes in Computer Science, vol 4286. Springer, Berlin, Heidelberg
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.