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.

翻译：近期基于平均场博弈（MFGs）的技术能够对大量相似理性智能体参与的多智能体博弈进行可扩展分析。然而，标准MFGs仅限于弱相互影响的同质化智能体，无法建模对其它智能体产生强影响的主要玩家（major players），严重限制了可处理问题的类别。我们提出了一种新颖的离散时间主次平均场博弈（M3FGs）及其基于虚拟博弈与概率单纯形分割的学习算法。重要的是，M3FGs泛化了含共同噪声的MFGs，不仅能处理随机外生环境状态，还可建模主要玩家。关键挑战在于其平均场具有随机性，而非标准MFGs中的确定性。我们的理论研究验证了M3FG模型及其算法解决方案：首先证明了从相关有限博弈出发的M3FG模型的适定性，其次给出了虚拟博弈算法的收敛性与近似保证。随后通过实验验证理论结果，消融部分理论假设，并在三个示例问题中成功学习了均衡解。总体而言，我们为新颖且广泛的可解博弈类别建立了一套学习框架。