Inapproximability of NP-Complete Variants of Nash Equilibrium

TL;DR

This paper investigates the computational hardness of finding approximate Nash equilibria with certain constraints, showing many variants are NP-hard or as hard as the Hidden Clique problem, highlighting the complexity of equilibrium computation.

Contribution

It demonstrates that constrained variants of approximate Nash equilibrium are NP-hard and relates their difficulty to the Hidden Clique problem, contrasting with unconstrained cases.

Findings

Finding better than 1/2-approximate equilibrium is as hard as Hidden Clique.

Optimal Nash equilibrium is NP-hard, unlike general Nash which is in PPAD.

Approximate variants of related problems are also NP-hard or as hard as Hidden Clique.

Abstract

In recent work of Hazan and Krauthgamer (SICOMP 2011), it was shown that finding an $\eps$-approximate Nash equilibrium with near-optimal value in a two-player game is as hard as finding a hidden clique of size $O(\log n)$ in the random graph $G(n,1/2)$. This raises the question of whether a similar intractability holds for approximate Nash equilibrium without such constraints. We give evidence that the constraint of near-optimal value makes the problem distinctly harder: a simple algorithm finds an optimal 1/2-approximate equilibrium, while finding strictly better than 1/2-approximate equilibria is as hard as the Hidden Clique problem. This is in contrast to the unconstrained problem where more sophisticated algorithms, achieving better approximations, are known. Unlike general 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 $\eta$ (for any $\eta>0$, even when the best Nash equilibrium has value $1-\eta$), finding a second Nash equilibrium, and finding a Nash equilibrium with small support. Finally, we consider the complexity of approximate pure Bayes Nash equilibria in two-player games. Here we show that for general Bayesian games the problem is NP-hard. For the special case where the distribution over types is uniform, we give a quasi-polynomial time algorithm matched by a hardness result based on the Hidden Clique problem.