Learning Discrete-Time Major-Minor Mean Field Games

TL;DR

This paper introduces a discrete-time major-minor mean field game model that incorporates influential major players and develops a learning algorithm with theoretical guarantees, expanding the applicability of mean field game analysis.

Contribution

It proposes a novel discrete-time major-minor MFG framework with a fictitious play learning algorithm, handling stochastic mean fields and major players, with proven convergence and approximation guarantees.

Findings

The M3FG model is well-posed starting from finite games.

The fictitious play algorithm converges and approximates equilibria.

Empirical results confirm theoretical predictions in example problems.

Abstract

Recent techniques based on Mean Field Games (MFGs) allow the scalable analysis of multi-player games with many similar, rational agents. However, standard MFGs remain limited to homogeneous players that weakly influence each other, and cannot model major players that strongly influence other players, severely limiting the class of problems that can be handled. We propose a novel discrete time version of major-minor MFGs (M3FGs), along with a learning algorithm based on fictitious play and partitioning the probability simplex. Importantly, M3FGs generalize MFGs with common noise and can handle not only random exogeneous environment states but also major players. A key challenge is that the mean field is stochastic and not deterministic as in standard MFGs. Our theoretical investigation verifies both the M3FG model and its algorithmic solution, showing firstly the well-posedness of the M3FG model starting from a finite game of interest, and secondly convergence and approximation guarantees of the fictitious play algorithm. Then, we empirically verify the obtained theoretical results, ablating some of the theoretical assumptions made, and show successful equilibrium learning in three example problems. Overall, we establish a learning framework for a novel and broad class of tractable games.