On the optimal mixing problem of approximate Nash equilibria in bimatrix games

This paper introduces the optimal mixing problem, a natural extension of the computation of approximate Nash Equilibria (NE) in bimatrix games. The problem focuses on determining the optimal convex combination of given strategies that minimizes the approximation (i.e., regret) in NE computation. We develop algorithms for the exact and approximate optimal mixing problems and present new complexity results that bridge both practical and theoretical aspects of NE computation. Practically, our algorithms can be used to enhance and integrate arbitrary existing constant-approximate NE algorithms, offering a powerful tool for the design of approximate NE algorithms. Theoretically, these algorithms allow us to explore the implications of support restrictions on approximate NE and derive the upper-bound separations between approximate NE and exact NE. Consequently, this work contributes to theoretical understandings of the computational complexity of approximate NE under various constraints and practical improvements in multi- agent reinforcement learning (MARL) and other fields where NE computation is involved.