Homogenization and Mean-Field Approximation for Multi-Player Games

TL;DR

This paper develops a quantitative homogenization approach for multi-player games with heterogeneous populations, showing how to approximate equilibria using mean-field models and optimizing sub-population partitions.

Contribution

It introduces a method to approximate multi-player game equilibria via mean-field games, including explicit bounds and an optimal partitioning framework.

Findings

Explicit bounds for approximation errors based on population size and heterogeneity

A method to compute approximate equilibria using auxiliary mean-field games

Optimal sub-population partitioning formulated as a mixed-integer program

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 $\epsilon$-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.