Global Convergence to Nash Equilibrium in Nonconvex General-Sum Games under the $n$-Sided PL Condition

2026-02-12

AI Summary

This work addresses the challenge of computing Nash equilibria in non-convex general-sum games, where standard gradient-based methods often fail due to the absence of convergence guarantees. To this end, the authors propose a novel solvability framework centered on an $n$-sided Polyak–Łojasiewicz (PL) condition, which unifies and extends both the classical PL condition and the notion of multi-convexity. Building upon this condition, they develop enhanced algorithms based on first-order gradient descent, block coordinate descent, and their variants, and rigorously establish their global convergence to Nash equilibria under the proposed framework. Both theoretical analysis and empirical experiments demonstrate that the approach effectively overcomes the limitations of conventional gradient methods in non-convex game settings.

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: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We consider the problem of finding a Nash equilibrium (NE) in a general-sum game, where player $i$'s objective is $f_i(x)=f_i(x_1,...,x_n)$, with $x_j\in\mathbb{R}^{d_j}$ denoting the strategy variables of player $j$. Our focus is on investigating first-order gradient-based algorithms and their variations, such as the block coordinate descent (BCD) algorithm, for tackling this problem. We introduce a set of conditions, called the $n$-sided PL condition, which extends the well-established gradient dominance condition a.k.a Polyak-{\L}ojasiewicz (PL) condition and the concept of multi-convexity. This condition, satisfied by various classes of non-convex functions, allows us to analyze the convergence of various gradient descent (GD) algorithms. Moreover, our study delves into scenarios where the standard gradient descent methods fail to converge to NE. In such cases, we propose adapted variants of GD that converge towards NE and analyze their convergence rates. Finally, we evaluate the performance of the proposed algorithms through several experiments.

Problem

• Nash equilibrium
• nonconvex general-sum games
• global convergence
• gradient-based algorithms
• n-sided PL condition

Innovation

• n-sided PL condition
• nonconvex general-sum games
• Nash equilibrium
• gradient-based algorithms
• global convergence