Finding Equilibria in Bilinear Zero-sum Games via a Convexity-based Approach

Michail Fasoulakis, Evangelos Markakis, Georgios Roussakis, Christodoulos Santorinaios

Keywords: Zero-sum games, Directional derivative, gradient descent, duality gap

TL;DR: A method for zero-sum games based on a steepest descent approach via the directional derivative of the duality gap

Abstract:

We focus on the design of algorithms for finding equilibria in 2-player zero-sum games. Although it is well known that such problems can be solved by a single linear program, there has been a surge of interest in recent years, for simpler algorithms, motivated in part by applications in machine learning. Our work proposes such a method, inspired by the observation that the duality gap (a standard metric for evaluating convergence in general min-max optimization problems) is a convex function for the case of bilinear zero-sum games. To this end, we analyze a descent-based approach, variants of which have also been used as a subroutine in a series of algorithms for approximating Nash equilibria in general non-zero-sum games.
In particular, we analyze a steepest descent approach, by finding the direction that minimises the directional derivative of the duality gap function and move towards that. Our main theoretical result is that the derived algorithms achieve a geometric decrease in the duality gap and improved complexity bounds until we reach an approximate equilibrium. Finally, we complement this with an experimental evaluation. Our findings reveal that for some classes of zero-sum games, the running time of our method is comparable with standard LP solvers, even with thousands of available strategies per player.

Primary Area: optimization

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Submission Number: 12065