Mean Field Game Theory with a Partially Observed Major Agent

Abstract

Mean field game (MFG) theory where there is a major agent and many minor agents (MM-MFG) has been formulated for both the linear quadratic Gaussian (LQG) case and for the case of nonlinear state dynamics and nonlinear cost functions. In this framework, even asymptotically (as the population size $N$ approaches infinity), and in contrast to the situation without major agents, the mean field term becomes stochastic due to the stochastic evolution of the state of the major agent; furthermore, the best response control actions of the minor agents depend on the state of the major agent as well as the stochastic mean field. In a decentralized environment, one is led to consider the situation where the agents are provided only with partial information on the major agent's state; in this work such a scenario is considered for systems with nonlinear dynamics and cost functions, and an $\epsilon$-Nash MFG theory is developed for this MM-MFG setup. The approach to the problem of partially observed MM-MFG systems adopted in this work is to follow the procedure of constructing the associated completely observed system via the application of nonlinear filtering theory; consequently, as a first step, nonlinear filtering equations are obtained for partially observed stochastic dynamical systems whose state equations contain a measure term corresponding to the distribution of the solution of a state process. Stochastic control theory for systems with random parameters is next generalized to the partially observed case by lifting the analysis to the infinite-dimensional domain. To achieve this, the Itô--Kunita lemma is first generalized to processes taking values in a subset of $L^1$ consisting of the space of solutions of the conditional density process generated by the filtering equations. The existence and uniqueness of solutions to the MFG system of equations is next established by a fixed point argument in the Wasserstein space of random probability measures in which the robustness property of nonlinear filtering theory is used. Finally, the $\epsilon$-Nash property of such a solution is analyzed in this setting where the state consists of finite- and infinite-dimensional (density) valued stochastic processes.