Inapproximability of NP-Complete Variants of Nash Equilibrium

In recent work of Hazan and Krauthgamer (SICOMP 2011), it was shown that finding an ε-approximate Nash equilibrium with near-optimal value in a two-player game is as hard as finding a hidden clique of size O(logn) in the random graph G(n,1/2). This raises the question of whether a similar intractability holds for approximate Nash equilibrium without side constraints such as high value. We give evidence that asking for near-optimal value makes the problem distinctly harder: a simple algorithm finds a 1/2-approximate equilibrium of optimal value, but getting below 1/2 is as hard as the Hidden Clique problem. This is in contrast to the basic problem (finding a Nash equilibrium with no optimization criteria) where more sophisticated algorithms, achieving better approximations, are known. Unlike basic Nash equilibrium, which is in PPAD, optimal (maximum value) Nash equilibrium is NP-hard. We proceed to show that optimal Nash equilibrium is just one of several known NP-hard problems related to Nash equilibrium, all of which have approximate variants which are as hard as finding a planted clique. In particular, we show this for approximate variants of the following problems: finding a Nash equilibrium with value greater than η (for any fixed η > 0, even when the optimal Nash equilibrium has value 1− η), finding a second Nash equilibrium, and finding a Nash equilibrium with small support. ∗A preliminary version of this work appeared in the Proceedings of APPROX 2011 [2]. The results in the current version are essentially the same, but several minor flaws in the proofs have been fixed. Work done while the authors were at the University of Toronto, supported by NSERC Discovery Grants. MB was also supported by a Sloan Research Fellowship. ACM Classification: F.2 AMS Classification: 68Q17, 68W25, 91A05