Major-Minor Mean Field Game of Stopping: An Entropy Regularization Approach
Xiang Yu, Jiacheng Zhang, Keyu Zhang, Zhou Zhou
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keywords: Major-minor mean field game of stopping, Markov decision processes, relaxed equilibrium, entropy regularization, linear programming, fixed point
This paper studies a discrete-time major-minor mean field game of stopping where the major player can choose either an optimal control or stopping time. We look for the relaxed equilibrium as a randomized stopping policy, which is formulated as a fixed point of a set-valued mapping, whose existence is challenging by direct arguments. To overcome the difficulties caused by the presence of a major player, we propose to study an auxiliary problem by considering entropy regularization in the major player's problem while formulating the minor players' optimal stopping problems as linear programming over occupation measures. We first show the existence of regularized equilibria as fixed points of some simplified set-valued operator using the Kakutani-Fan-Glicksberg fixed-point theorem. Next, we prove that the regularized equilibrium converges as the regularization parameter $λ$ tends to 0, and the limit corresponds to a fixed point of the original operator, thereby confirming the existence of a relaxed equilibrium in the original mean field game problem. We also extend this entropy regularization method to the mean-field game problem where the minor players choose optimal controls.
Don't relax: early stopping for convex regularization
Ecole Politechnique（巴黎综合理工学院）；Istituto Italiano di Tecnologia（意大利技术研究院）；Massachusetts Institute of Technology（麻省理工学院）；Università di Genova（热那亚大学）；Politecnico di Milano（米兰理工大学）；EPFL（洛桑联邦理工学院）
China School of Banking and Finance, University of International Business and Economics（对外经济贸易大学中国银行业与金融学院）；School of Mathematics and Statistics, University of Sydney（悉尼大学数学与统计学院）
