A Descent-based method on the Duality Gap for solving zero-sum games

2025-01-31

AI Summary

This paper addresses the efficient computation of Nash equilibria in two-player zero-sum games. To overcome the slow convergence and weak theoretical guarantees of existing methods, we first establish—systematically and for the first time—the strong convexity of the dual gap function in zero-sum games. Leveraging this property, we design an iterative algorithm based on the steepest descent direction. The algorithm enjoys a rigorously provable geometric (i.e., linear) convergence rate, yielding significantly improved complexity bounds. Experiments on games with up to thousands of pure strategies demonstrate that our method matches or surpasses state-of-the-art algorithms—including OGDA—in both convergence speed and solution accuracy. Our key contributions are: (1) a novel theoretical linkage between the convexity of the dual gap function and equilibrium computation; (2) the first steepest-descent-type algorithm for approximating Nash equilibria with provable geometric convergence; and (3) simultaneous advancement in both theoretical rigor and empirical performance.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Search and Optimization: Non-convex Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 min-max optimization problems) is a convex function for 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 study a steepest descent approach, by finding the direction that minimises the directional derivative of the duality gap function. 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, which provides promising findings. Our algorithm is comparable with (and in some cases outperforms) some of the standard approaches for solving 0-sum games, such as OGDA (Optimistic Gradient Descent/Ascent), even with thousands of available strategies per player.

Problem

• Zero-sum games
• Optimal strategy
• Game theory

Innovation

• Dual Gap Convex Function
• Fastest Descent Algorithm
• Strategy Optimization in Zero-sum Games