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.

利用基于 M3FG（major-minor MFGs）的学习算法，我们提出了一种新的离散时间版本的 M3FG，能够解决具有强影响力的主要玩家的问题，并在三个实例问题中验证了该理论结果的实际效果，从而为一类广泛可解的博弈问题建立了学习框架。