A Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games

Since the celebrated PPAD-completeness result for Nash equilibria in bimatrix games, a long line of research has focused on polynomial-time algorithms that compute $\varepsilon$-approximate Nash equilibria. Finding the best possible approximation guarantee that we can have in polynomial time has been a fundamental and non-trivial pursuit on settling the complexity of approximate equilibria. Despite a significant amount of effort, the algorithm of Tsaknakis and Spirakis \cite{TS08}, with an approximation guarantee of $(0.3393+\delta)$, remains the state of the art over the last 15 years. In this paper, we propose a new refinement of the Tsaknakis-Spirakis algorithm, resulting in a polynomial-time algorithm that computes a $(\frac{1}{3}+\delta)$-Nash equilibrium, for any constant $\delta>0$. The main idea of our approach is to go beyond the use of convex combinations of primal and dual strategies, as defined in the optimization framework of \cite{TS08}, and enrich the pool of strategies from which we build the strategy profiles that we output in certain bottleneck cases of the algorithm.