Inapproximability Results for Approximate Nash Equilibria

We study the problem of finding approximate Nash equilibria that satisfy certain conditions, such as providing good social welfare. In particular, we study the problem $\epsilon$-NE $\delta$-SW: find an $\epsilon$-approximate Nash equilibrium ($\epsilon$-NE) that is within $\delta$ of the best social welfare achievable by an $\epsilon$-NE. Our main result is that, if the exponential-time hypothesis (ETH) is true, then solving $\left(\frac{1}{8} - \mathrm{O}(\delta)\right)$-NE $\mathrm{O}(\delta)$-SW for an $n\times n$ bimatrix game requires $n^{\mathrm{\widetilde \Omega}(\log n)}$ time. Building on this result, we show similar conditional running time lower bounds on a number of decision problems for approximate Nash equilibria that do not involve social welfare, including maximizing or minimizing a certain player's payoff, or finding approximate equilibria contained in a given pair of supports. We show quasi-polynomial lower bounds for these problems assuming that ETH holds, where these lower bounds apply to $\epsilon$-Nash equilibria for all $\epsilon < \frac{1}{8}$. The hardness of these other decision problems has so far only been studied in the context of exact equilibria.