Finding Approximate Nash Equilibria of Bimatrix Games via Payoff Queries

Abstract: We study the deterministic and randomized query complexity of finding approximate equilibria in bimatrix games. We show that the deterministic query complexity of finding an $\epsilon$-Nash equilibrium when $\epsilon &lt; \frac{1}{2}$ is $\Omega(k<sup>2)$, even in zero-one constant-sum games. In combination with previous results \cite{FGGS13}, this provides a complete characterization of the deterministic query complexity of approximate Nash equilibria. We also study randomized querying algorithms. We give a randomized algorithm for finding a $(\frac{3 - \sqrt{5}}{2} + \epsilon)$-Nash equilibrium using $O(\frac{k \cdot \log k}{\epsilon<sup>2})$ payoff queries, which shows that the $\frac{1}{2}$ barrier for deterministic algorithms can be broken by randomization. For well-supported Nash equilibria (WSNE), we first give a randomized algorithm for finding an $\epsilon$-WSNE of a zero-sum bimatrix game using $O(\frac{k \cdot \log k}{\epsilon<sup>4})$ payoff queries, and we then use this to obtain a randomized algorithm for finding a $(\frac{2}{3} + \epsilon)$-WSNE in a general bimatrix game using $O(\frac{k \cdot \log k}{\epsilon<sup>4})$ payoff queries. Finally, we initiate the study of lower bounds against randomized algorithms in the context of bimatrix games, by showing that randomized algorithms require $\Omega(k<sup>2)$ payoff queries in order to find a $\frac{1}{6k}$-Nash equilibrium, even in zero-one constant-sum games. In particular, this rules out query-efficient randomized algorithms for finding exact Nash equilibria.