Homogenization and Mean-Field Approximation for Multi-Player Games

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 tradeoff 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.