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 is possible by any algorithm that examines 4 equilibria involving fewer than log n strategies [Alt94]. We give a simple, linear-time algorithm examining just two strategies per player and resulting in a 1-approximate Nash equilibrium in any 2-player game. For the 2 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 is possible contingent upon a 6 graph-theoretic conjecture
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