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Particle system approximation of Nash equilibria in large games

Ludovic Tangpi, Nizar Touzi

TL;DR

This work develops a probabilistic framework to approximate Nash equilibria in symmetric $N$-player games via mean field games, using a McKean-Vlasov Langevin dynamics and its particle system. It establishes the existence and ergodicity of invariant measures $m^\sigma$ for the McKean-Vlasov dynamics for all $\sigma>0$, and shows tightness and convergence to a mean field equilibrium $m^0$ under monotonicity (Lasry-Lions or displacement). Under strict monotonicity and mild convexity, the authors prove contractivity and uniform-in-time propagation of chaos, yielding quantitative convergence of the $N$-player Nash equilibria to the MFE as $N o ext{large}$ and $\sigma o0$, with concentration bounds and explicit rates. Importantly, the framework avoids small-interaction assumptions by leveraging monotonicity, offering concrete error bounds (Fournier-Guilin rates) for finite-population approximations with potential applications to scalable multi-agent optimization and large-scale economic or engineering systems.

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.

Particle system approximation of Nash equilibria in large games

TL;DR

This work develops a probabilistic framework to approximate Nash equilibria in symmetric -player games via mean field games, using a McKean-Vlasov Langevin dynamics and its particle system. It establishes the existence and ergodicity of invariant measures for the McKean-Vlasov dynamics for all , and shows tightness and convergence to a mean field equilibrium under monotonicity (Lasry-Lions or displacement). Under strict monotonicity and mild convexity, the authors prove contractivity and uniform-in-time propagation of chaos, yielding quantitative convergence of the -player Nash equilibria to the MFE as and , with concentration bounds and explicit rates. Importantly, the framework avoids small-interaction assumptions by leveraging monotonicity, offering concrete error bounds (Fournier-Guilin rates) for finite-population approximations with potential applications to scalable multi-agent optimization and large-scale economic or engineering systems.

Abstract

We develop a probabilistic framework to approximate Nash equilibria in symmetric -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.
Paper Structure (14 sections, 8 theorems, 94 equations)

This paper contains 14 sections, 8 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Description of the main results
  3. Related Literature
  4. Outline of the rest of the paper
  5. The general framework
  6. Notations
  7. Finite population game
  8. Symmetric cost functions and mean field games
  9. Symmetric finite population game and mean field limit
  10. Assumptions and examples
  11. Main results
  12. Mean field games and Ergodicity of Langevin dynamics
  13. Particle system: contractivity and propagation of chaos
  14. Approximation of equilibria

Key Result

Proposition 2.3

Assume that nablaFLip and eq:convex.group hold and that the family $\{F_i\}_{i=1,\dots,N}$ is strictly displacement monotone. Let $\mathbf{x}^*$ be a Nash equilibrium and assume that $U$ satisfies nablaULip and is $\ell_U-$convexi.e. the map $x\longmapsto U(x)-\frac{1}{2}\ell_U|x|^2$ is convex. for where the two limits in the second claim commute.

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Example 2.4
  • Remark 2.5
  • Remark 3.1
  • Definition 3.2
  • Definition 3.5
  • Remark 3.6
  • ...and 20 more