Mean Field Analysis of Mutual Insurance Market

A mutual insurance company (MIC) is a type of consumer cooperative owned by its policyholders. By purchasing insurance from an MIC, policyholders effectively become member-owners of the company and are entitled to a share of the surplus, which is determined by their own collective claims and premium contributions. This sharing mechanism creates an interactive environment in which individual insurance strategies are influenced by the actions of others. Given that mutual insurers account for nearly one-third of the global insurance market, the analysis of members' behavior under such a sharing mechanism is of both practical and theoretical importance. This article presents a first dynamic study of members' behavior in the prevalent mutual insurance market under the large-population limit. With members' wealth processes depending on the law of the insurance strategies, we model the surplus-sharing mechanism using an extended mean field game (MFG) framework and address the fundamental question of how strategic interactions in this setting influence individual decisions. Mathematically, we establish the global-in-time existence and uniqueness of the mean field forward-backward stochastic differential equation (MF-FBSDE) characterizing the Nash equilibrium strategy, employing techniques to accommodate realistic insurance constraints. Computationally, we develop a modified deep BSDE algorithm capable of solving the extended MFG problem with an additional fixed-point structure on the control. Utilizing this scheme, we examine how structural features of the MIC's design, such as the composition of risk classes and surplus-sharing proportions, reshape members' decisions and wealth through collective interactions, underscoring the central role of these mechanisms in MICs.