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Homogenization and Mean-Field Approximation for Multi-Player Games

Rama Cont, Anran Hu

TL;DR

The paper extends mean-field game theory to heterogeneous populations by introducing a homogenization framework that partitions agents into near-homogeneous sub-populations and models their interactions via a multi-population mean-field game. It derives non-asymptotic bounds linking ε-Nash equilibria of the original N-player game to equilibria of the MP-MFG, decomposing errors into mean-field and heterogeneity contributions and revealing a fundamental trade-off. The authors propose optimal partitioning methods, including K-means clustering for large populations and a mixed-integer convex program to jointly determine the number and composition of groups, with a detailed case study on uniformly distributed agent features. Overall, the work provides constructive, quantitative tools to approximate heterogeneous N-player games with tractable MP-MFGs, enabling informed decisions about partitioning and model fidelity in large-scale strategic systems.

Abstract

We investigate how the framework of mean-field games may be used to investigate strategic interactions in large heterogeneous populations. We consider strategic interactions in a population of players which may be partitioned into near-homogeneous sub-populations subject to peer group effects and interactions across groups. We prove a quantitative homogenization result for multi-player games in this setting: we show that $ε$-Nash equilibria of a general multi-player game with heterogeneity may be computed in terms of the Nash equilibria of an auxiliary multi-population mean-field game. We provide explicit and non-asymptotic bounds for the distance from optimality in terms of the number of players and the deviations from homogeneity in sub-populations. The best mean-field approximation corresponds to an optimal partition into sub-populations, which may be formulated as the solution of a mixed-integer program.

Homogenization and Mean-Field Approximation for Multi-Player Games

TL;DR

The paper extends mean-field game theory to heterogeneous populations by introducing a homogenization framework that partitions agents into near-homogeneous sub-populations and models their interactions via a multi-population mean-field game. It derives non-asymptotic bounds linking ε-Nash equilibria of the original N-player game to equilibria of the MP-MFG, decomposing errors into mean-field and heterogeneity contributions and revealing a fundamental trade-off. The authors propose optimal partitioning methods, including K-means clustering for large populations and a mixed-integer convex program to jointly determine the number and composition of groups, with a detailed case study on uniformly distributed agent features. Overall, the work provides constructive, quantitative tools to approximate heterogeneous N-player games with tractable MP-MFGs, enabling informed decisions about partitioning and model fidelity in large-scale strategic systems.

Abstract

We investigate how the framework of mean-field games may be used to investigate strategic interactions in large heterogeneous populations. We consider strategic interactions in a population of players which may be partitioned into near-homogeneous sub-populations subject to peer group effects and interactions across groups. We prove a quantitative homogenization result for multi-player games in this setting: we show that -Nash equilibria of a general multi-player game with heterogeneity may be computed in terms of the Nash equilibria of an auxiliary multi-population mean-field game. We provide explicit and non-asymptotic bounds for the distance from optimality in terms of the number of players and the deviations from homogeneity in sub-populations. The best mean-field approximation corresponds to an optimal partition into sub-populations, which may be formulated as the solution of a mixed-integer program.

Paper Structure

This paper contains 30 sections, 9 theorems, 98 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related literature.
  4. Mean-field games as approximations of $N$-player games
  5. Dynamic $N$-player games
  6. Mean-field games (MFGs) and multi-population MFGs
  7. Example: Equilibrium pricing
  8. Homogenization of dynamic multi-player games
  9. $N$-player games of mean-field type.
  10. Example revisited.
  11. Quantitative estimates of the homogenization error
  12. Main results
  13. Mean-field approximation for $N$-player games of mean-field type.
  14. The contribution of population heterogeneity
  15. How accurate is the representative agent model?
  16. ...and 15 more sections

Key Result

Theorem 1

Let the mean-field strategy profile $\boldsymbol{\bar{\pi}}=\{\bar{\pi}_t^k\}_{t\in\mathcal{T},k\in[K]}$ be an $(N_1/N,\dots,N_K/N)$-weighted $\epsilon$-NE of $G^{\texttt{MF}}$. Let $\boldsymbol{\pi}=\{\pi_t^i\}_{t\in\mathcal{T},i\in[N]}$ with $\pi_t^i:=\bar{\pi}_t^k$ for $i\in\mathcal{I}_k$, $k\in[ is the mean-field approximation error, with $C(T,W_{\max})>0$ depending only on the horizon $T$ and

Figures (1)

  • Figure 1: Decomposition of homogenization error.

Theorems & Definitions (27)

  • Definition 2.1: Nash equilibrium (NE) of an $N$-player game
  • Definition 2.2: $\epsilon$-Nash equilibrium of an $N$-player game
  • Definition 2.3: NE and $\epsilon$-NE of an MP-MFG
  • Remark 1
  • Theorem 1
  • Remark 2: Mean-field approximation error $\epsilon_{\texttt{MF}}$
  • Remark 3: Heterogeneity error $\epsilon_{\texttt{heter}}$
  • Theorem 2
  • Corollary 3
  • proof : Sketch of proof
  • ...and 17 more