Particle system approximation of Nash equilibria in large games (2510.19211v1)

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.

• The paper develops a particle system approach utilizing McKean–Vlasov Langevin dynamics to approximate Nash equilibria in large symmetric games.
• It provides rigorous guarantees on contractivity and uniform-in-time propagation of chaos under key monotonicity and convexity conditions.
• The framework delivers explicit convergence rates and scalable methods for equilibrium computation in multi-agent systems.

Particle System Approximation of Nash Equilibria in Large Games

Introduction and Motivation

The paper develops a probabilistic framework for approximating Nash equilibria in symmetric $N$-player games in the large population regime, leveraging the theory of mean field games (MFGs). The central challenge addressed is the computational intractability of finding Nash equilibria in high-dimensional, non-cooperative games, especially as $N$ grows. Traditional gradient-based methods suffer from slow convergence and may fail to find true Nash equilibria in the absence of strong convexity. The authors propose a particle system approach based on McKean–Vlasov type Langevin dynamics, providing both theoretical guarantees and quantitative convergence rates under monotonicity conditions on the cost functions.

Mathematical Framework

Symmetric $N$-Player Games and Mean Field Limit

The setting considers $N$ agents, each with a cost function $F_i: (\mathbb{R}^d)^N \to \mathbb{R}$, seeking a Nash equilibrium $x^* = (x^{1,*}, \ldots, x^{N,*})$ such that

$$F_i(x^*) \leq F_i(x^{-i,*}, y), \quad \forall y \in \mathbb{R}^d, \forall i.$$

For symmetric games, the cost functions are of the form $F_i(x) = F(x^i, \mu^N_x)$, where $\mu^N_x$ is the empirical measure of the players' strategies. The mean field game (MFG) limit is defined via a cost $$F: \mathbb{R}^d \times \mathcal{P}(\mathbb{R}^d) \to \mathbb{R}$$, and the MFG equilibrium is a measure $m$ such that

$$\operatorname{supp}(m) \subset \arg\min_{x \in \mathbb{R}^d} F(x, m).$$

Particle System and McKean–Vlasov Dynamics

The core approximation is via the interacting particle system:

$$dX^i_t = -\left[ \nabla_x F(X^i_t, \mu^N_{X_t}) + \sigma \nabla U(X^i_t) \right] dt + \sqrt{2\sigma} dW^i_t,$$

where $U$ is a confining potential, $\sigma > 0$ is a temperature parameter, and $W^i_t$ are independent Brownian motions. As $N \to \infty$, the empirical measure converges to the law of the McKean–Vlasov SDE:

$$dX_t = -\left[ \nabla_x F(X_t, m_{X_t}) + \sigma \nabla U(X_t) \right] dt + \sqrt{2\sigma} dW_t.$$

Monotonicity and Contractivity

The analysis relies on two monotonicity notions for $F$:

• Lasry–Lions monotonicity: For all $m, m' \in \mathcal{P}(\mathbb{R}^d)$,

$\int [F(x, m) - F(x, m')] (m - m')(dx) \geq 0.$

• Displacement monotonicity: For all $m, m'$ and couplings $\pi \in \Pi(m, m')$,

$$\int [\nabla_x F(x, m) - \nabla_x F(x', m')] \cdot (x - x') \, \pi(dx, dx') \geq 0.$$

These conditions, together with convexity of $U$ or $F$, are shown to guarantee contractivity of the McKean–Vlasov process and uniform-in-time propagation of chaos for the particle system, even without smallness assumptions on the interaction term.

Main Results

Existence and Uniqueness of Invariant Measures

• For any $\sigma > 0$, the McKean–Vlasov SDE admits an invariant measure $m^\sigma$.
• The family $(m^\sigma)_{\sigma > 0}$ is tight, and any accumulation point as $\sigma \to 0$ is an MFG equilibrium.
• Uniqueness of the invariant measure and the MFG equilibrium is established under strict Lasry–Lions or displacement monotonicity.

Quantitative Convergence and Propagation of Chaos

• Contractivity: If $F$ is $\ell_F$-displacement semimonotone and $U$ is $\ell_U$-convex with $\ell_F + \sigma \ell_U > 0$, then

$$\sup_{t > 0} e^{2(\ell_F + \sigma \ell_U)t} W_2^2(m_{X_t}, m^\sigma) < \infty.$$

• Uniform-in-time propagation of chaos: Under additional regularity, for all $N$,

$$\sup_{t \geq 0} N \mathbb{E}[|X^{1,N}_t - X_t|^2] < \infty.$$

• Approximation error: For $F$ Lasry–Lions monotone and $\ell_F$-convex,

$$W_2(m_{X_t}, m^0) \leq C e^{-(\ell_F + \sigma \ell_U)t} + C \frac{\sigma}{\ell_F + \sigma \ell_U},$$

and for the particle system,

$$W_2(m_{X^{\sigma,N}_t}, m^0) \leq \frac{C}{N} + C e^{-(\ell_F + \sigma \ell_U)t} + C \frac{\sigma}{\ell_F + \sigma \ell_U}.$$

Approximate Nash Equilibria

• For any MFG equilibrium $m^0$, the empirical measure of $N$ i.i.d. samples from $m^0$ yields an $\varepsilon^N$-Nash equilibrium for the $N$-player game, with $\varepsilon^N \to 0$ as $N \to \infty$.
• The convergence rate of empirical measures to the MFG equilibrium is characterized in terms of the Fournier–Guillin rate, with explicit concentration inequalities.

Avoidance of Smallness Assumptions

A key claim is that monotonicity in the measure argument, together with convexity of either $F$ or $U$, suffices for contractivity and propagation of chaos, without requiring small interaction or functional inequalities. This is in contrast to much of the prior literature, which imposes smallness conditions on the Lipschitz constant of the interaction term.

Implications and Applications

Theoretical Implications

• The results provide a rigorous justification for the use of mean field approximations in large symmetric games, with explicit quantitative error bounds.
• The framework extends the applicability of particle-based methods for Nash equilibrium approximation to settings with strong interactions, provided monotonicity holds.
• The analysis clarifies the role of monotonicity in ensuring uniqueness and stability of equilibria in both finite and infinite population limits.

Practical and Computational Implications

• The particle system approach enables scalable approximation of Nash equilibria in high-dimensional, large-population games, relevant for multi-agent systems, economics, and engineering.
• The convergence rates and concentration inequalities provide guidance for selecting the number of particles $N$, simulation time $t$, and temperature parameter $\sigma$ to achieve a desired approximation accuracy.
• The method is robust to the strength of interactions, provided monotonicity is satisfied, making it suitable for a broader class of games than previously addressed.

Potential for Future Developments

• The framework can be extended to dynamic games, games with more general interaction structures, and settings with non-symmetric or heterogeneous agents.
• The results suggest new directions for the design of scalable algorithms for equilibrium computation in large-scale multi-agent reinforcement learning and distributed control.
• The avoidance of smallness assumptions opens the possibility of analyzing more realistic models in economics and social sciences, where strong interactions are prevalent.

Conclusion

This work establishes a comprehensive probabilistic framework for approximating Nash equilibria in large symmetric games via particle systems and mean field limits. By leveraging monotonicity properties, the authors provide strong theoretical guarantees for convergence, contractivity, and propagation of chaos, without restrictive smallness conditions on interactions. The results have significant implications for both the theory and practice of large-scale game-theoretic analysis, and open avenues for further research in mean field control, multi-agent learning, and beyond.