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 < frac12 is Omega(k2), even in zero-one constant-sum games. In combination with previous results citeFGGS13, 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 (frac3 - √52 + ε)-Nash equilibrium using O(frack · log kε2) payoff queries, which shows that the frac12 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(frack · log kε4) payoff queries, and we then use this to obtain a randomized algorithm for finding a (frac23 + ε)-WSNE in a general bimatrix game using O(frack · log kε4) 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(k2) payoff queries in order to find a frac16k-Nash equilibrium, even in zero-one constant-sum games. In particular, this rules out query-efficient randomized algorithms for finding exact Nash equilibria.