Particle system approximation of Nash equilibria in large games

TL;DR

This paper introduces a probabilistic particle system approach to approximate Nash equilibria in large symmetric games, leveraging mean field game analysis and monotonicity conditions for convergence.

Contribution

It develops a novel framework using McKean-Vlasov Langevin dynamics to approximate Nash equilibria without small interaction assumptions, emphasizing monotonicity.

Findings

Proves convergence of particle systems to mean field game solutions.

Establishes uniform-in-time propagation of chaos under monotonicity.

Shows that monotonicity ensures convergence without functional inequalities.

Abstract

We develop a probabilistic framework to approximate Nash equilibria in symmetric $N$-player games in the large population regime, via the analysis of associated mean field games (MFGs). The approximation is achieved through the analysis of a McKean-Vlasov type Langevin dynamics and their associated particle systems, with convergence to the MFG solution established in the limit of vanishing temperature parameter. Relying on displacement monotonicity or Lasry-Lions monotonicity of the cost function, we prove contractility of the McKean-Vlasov process and uniform-in-time propagation of chaos for the particle system. Our results contribute to the general theory of interacting diffusions by showing that monotonicity can ensure convergence without requiring small interaction assumptions or functional inequalities.