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 ε-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 [38], with an approximation guarantee of (0.3393+δ), 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 δ > 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 [38], and enrich the pool of strategies from which we build the strategy profiles that we output in certain bottleneck cases of the algorithm.

自从双矩阵博弈中纳什均衡的著名 PPAD 完备性结果以来，一系列研究都集中在计算 ε 近似纳什均衡的多项式时间算法上。在多项式时间内找到我们可以拥有的最佳近似保证一直是解决近似平衡复杂性的基本且重要的追求。尽管付出了巨大的努力，Tsaknakis 和 Spirakis [38] 的算法（具有 (0.3393+δ) 的近似保证）仍然是过去 15 年中最先进的算法。在本文中，我们提出了 Tsaknakis-Spirakis 算法的新改进，产生了一种多项式时间算法，可以计算任何常数的 \((\frac{1}{3}+\delta)\)-纳什均衡δ > 0。