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Finding Approximate Nash Equilibria of Bimatrix Games via Payoff Queries

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 $ε$-Nash equilibrium when $ε< \frac{1}{2}$ is $Ω(k^2)$, 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 $(\frac{3 - \sqrt{5}}{2} + ε)$-Nash equilibrium using $O(\frac{k \cdot \log k}{ε^2})$ 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 $ε$-WSNE of a zero-sum bimatrix game using $O(\frac{k \cdot \log k}{ε^4})$ payoff queries, and we then use this to obtain a randomized algorithm for finding a $(\frac{2}{3} + ε)$-WSNE in a general bimatrix game using $O(\frac{k \cdot \log k}{ε^4})$ 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 $Ω(k^2)$ 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.