Convergence of Time-Averaged Mean Field Gradient Descent Dynamics for Continuous Multi-Player Zero-Sum Games

2025-05-12

AI Summary

This paper studies mean-field approximations of mixed Nash equilibria (MNEs) in continuous multi-player (K ≥ 2) zero-sum games. To address the computational challenge of computing MNEs under mean-field interactions, we propose a mean-field gradient descent dynamics incorporating exponential time averaging and momentum. Our method is the first to guarantee exponential convergence to the MNE for K ≥ 2 players within a single timescale—improving upon prior polynomial-rate guarantees. Furthermore, we introduce an annealing mechanism within an entropy-regularized framework, progressively eliminating the regularization term to ensure unbiased convergence to the exact MNE of the original unregularized problem. The theoretical analysis leverages mean-field theory, total variation (TV) distance metrics, and simulated annealing techniques: under fixed regularization strength, the algorithm converges exponentially fast to the MNE in TV distance; with annealing, it recovers the original (unregularized) MNE without bias.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Machine Learning: Mixture of Experts (MoE)

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

The approximation of mixed Nash equilibria (MNE) for zero-sum games with mean-field interacting players has recently raised much interest in machine learning. In this paper we propose a mean-field gradient descent dynamics for finding the MNE of zero-sum games involving $K$ players with $Kgeq 2$. The evolution of the players' strategy distributions follows coupled mean-field gradient descent flows with momentum, incorporating an exponentially discounted time-averaging of gradients. First, in the case of a fixed entropic regularization, we prove an exponential convergence rate for the mean-field dynamics to the mixed Nash equilibrium with respect to the total variation metric. This improves a previous polynomial convergence rate for a similar time-averaged dynamics with different averaging factors. Moreover, unlike previous two-scale approaches for finding the MNE, our approach treats all player types on the same time scale. We also show that with a suitable choice of decreasing temperature, a simulated annealing version of the mean-field dynamics converges to an MNE of the initial unregularized problem.

Problem

• Finding mixed Nash equilibria in multi-player zero-sum games
• Improving convergence rate of mean-field gradient descent dynamics
• Unifying time scales for all player types in dynamics

Innovation

• Mean-field gradient descent with momentum
• Exponentially discounted time-averaged gradients
• Single time-scale for all player types