A fully-distributed proximal-point algorithm for Nash equilibrium seeking with linear convergence rate

TL;DR

This paper introduces a fully-distributed proximal-point algorithm for Nash equilibrium seeking in multi-agent systems with partial information, achieving linear convergence and outperforming existing gradient-based methods.

Contribution

The paper presents a novel, single-layer, fixed-step distributed algorithm based on operator theory and a new preconditioning matrix for Nash equilibrium problems.

Findings

Proves linear convergence under strong monotonicity and Lipschitz conditions.

Demonstrates superior performance over existing gradient-based schemes.

Provides theoretical and numerical validation of the convergence rate.

Abstract

We address the Nash equilibrium problem in a partial-decision information scenario, where each agent can only observe the actions of some neighbors, while its cost possibly depends on the strategies of other agents. Our main contribution is the design of a fully-distributed, single-layer, fixed-step algorithm, based on a proximal best-response augmented with consensus terms. To derive our algorithm, we follow an operator-theoretic approach. First, we recast the Nash equilibrium problem as that of finding a zero of a monotone operator. Then, we demonstrate that the resulting inclusion can be solved in a fully-distributed way via a proximal-point method, thanks to the use of a novel preconditioning matrix. Under strong monotonicity and Lipschitz continuity of the game mapping, We prove linear convergence of our algorithm to a Nash equilibrium. Furthermore, we show that our method outperforms the fastest known gradient-based schemes, both in terms of guaranteed convergence rate, via theoretical analysis, and in practice, via numerical simulations.