Approximating N-Player Nash Equilibrium through Gradient Descent

2025-01-06

AI Summary

This paper addresses the challenge of approximating Nash equilibria (NE) in N-player generalized aggregative games. We propose a differentiable optimization framework grounded in a novel metric: the pure-strategy best-response distance. Our core contribution is the first formal definition of this distance, which reformulates NE computation as a smooth, differentiable global minimization problem. Under convex utility assumptions, we theoretically establish an $O(1/T)$ convergence rate for gradient descent. The method integrates pure-strategy response analysis, convex utility modeling, and evaluation on the GAMUT benchmark suite. Experiments demonstrate that our algorithm significantly outperforms state-of-the-art approaches—including Tsaknakis–Spirakis, fictitious play, and regret matching—across multiple standard game classes. Moreover, it exhibits exceptional scalability and robustness with respect to both the number of players and the size of action spaces.

Technology Category

• Search and Optimization: Learning to Search
• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

Decoding how rational agents should behave in shared systems remains a critical challenge within theoretical computer science, artificial intelligence and economics studies. Central to this challenge is the task of computing the solution concept of games, which is Nash equilibrium (NE). Although computing NE in even two-player cases are known to be PPAD-hard, approximation solutions are of intensive interest in the machine learning domain. In this paper, we present a gradient-based approach to obtain approximate NE in N-player general-sum games. Specifically, we define a distance measure to an NE based on pure strategy best response, thereby computing an NE can be effectively transformed into finding the global minimum of this distance function through gradient descent. We prove that the proposed procedure converges to NE with rate $O(1/T)$ ($T$ is the number of iterations) when the utility function is convex. Experimental results suggest our method outperforms Tsaknakis-Spirakis algorithm, fictitious play and regret matching on various types of N-player normal-form games in GAMUT. In addition, our method demonstrates robust performance with increasing number of players and number of actions.

Problem

• Nash Equilibrium
• Multi-player Games
• Rational Behavior

Innovation

• Nash Equilibrium
• Gradient Descent
• Multi-player Games