Homogenization and Mean-Field Approximation for Multi-Player Games

Abstract

We investigate how the framework of mean-field games may be used to study strategic interactions in large heterogeneous populations. Starting from a partition of the player population into groups, we introduce an intermediate finite-player game of mean-field type and derive explicit non-asymptotic bounds for the average exploitability of strategy profiles obtained by lifting strategies from the associated multi-population mean-field game. The approximation error decomposes into two components: a finite-population mean-field error, controlled by empirical-measure approximation within each group, and a heterogeneity error measuring deviations of the original players' rewards and transition dynamics from their group-level approximations. Our results apply to compact state and action spaces and allow heterogeneous deterministic initial states within each population. We further study the resulting trade-off between group size and intra-group heterogeneity. In a parametrized heterogeneous setting, the choice of partition minimizing the resulting certified upper bound can be formulated as a mixed-integer second-order cone program. In the large-population regime, this problem is shown to be related to $K$-means clustering.