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 [Richard J. Lipton, Evangelos Markakis, Aranyak Mehta, Playing large games using simple strategies, in: EC, 2003], and no approximation better than 1 4 is possible by any algorithm that examines equilibria involving fewer than log n strategies [Ingo Althöfer, On sparse approximations to randomized strategies and convex combinations, Linear Algebra and its Applications (1994) 199]. We give a simple, linear-time algorithm examining just two strategies per player and resulting in a 1 2 -approximate Nash equilibrium in any 2-player game. For the more demanding notion of approximately well supported Nash equilibrium due to [Constantinos Daskalakis, Paul W. Goldberg, Christos H. Papadimitriou, The complexity of computing a Nash equilibrium, SIAM Journal on Computing (in press) Preliminary version appeared in STOC (2006)] 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 or 1 ), and that an approximation of 5 6 is possible, contingent upon a graph-theoretic conjecture. Subsequent work extends the 1 4 impossibility result of Ingo Althöfer’s paper, as mentioned above, to 1 2 [Tomás Feder, Hamid Nazerzadeh, Amin Saberi, Approximating nash equilibria using small-support strategies, in: EC, 2007], making our 1 2 -approximate Nash equilibrium algorithm optimal among the algorithms that only consider mixed strategies of sublogarithmic size support. Moreover, techniques similar to our techniques for approximately well supported Nash equilibria are used in [Spyros Kontogiannis, Paul G. Spirakis, Efficient algorithms for constant well supported approximate equilibria in bimatrix games, in: ICALP, 2007] for obtaining an efficient algorithm for 0.658-approximately well supported Nash equilibria, unconditionally.