$\epsilon$-Nash Mean Field Game Theory for Nonlinear Stochastic Dynamical Systems with Major and Minor Agents

Abstract

This paper studies large population dynamic games involving nonlinear stochastic dynamical systems with agents of the following mixed types: (i) a major agent and (ii) a population of $N$ minor agents where $N$ is very large. The major and minor agents are coupled via both (i) their individual nonlinear stochastic dynamics and (ii) their individual finite time horizon nonlinear cost functions. This problem is analyzed by the so-called $\epsilon$-Nash mean field game theory. A distinct feature of the mixed agent mean field game problem is that even asymptotically (as the population size $N$ approaches infinity) the noise process of the major agent causes random fluctuation of the mean field behavior of the minor agents. To deal with this, the overall asymptotic ($N \rightarrow \infty$) mean field game problem is decomposed into (i) two nonstandard stochastic optimal control problems with random coefficient processes which yield forward adapted stochastic best response control processes determined from the solution of (backward in time) stochastic Hamilton--Jacobi--Bellman (SHJB) equations and (ii) two stochastic coefficient McKean--Vlasov (SMV) equations which characterize the state of the major agent and the measure determining the mean field behavior of the minor agents. This yields a stochastic mean field game (SMFG) system which is in contrast to the deterministic mean field game systems of standard MFG problems with only minor agents. Existence and uniqueness of the solutions to SMFG systems (SHJB and SMV equations) is established by a fixed point argument in the Wasserstein space of random probability measures. In the case where minor agents are coupled to the major agent only through their cost functions, the $\epsilon_N$-Nash equilibrium property of the SMFG best responses is shown for a finite $N$ population system where $\epsilon_N=O(1/\sqrt N)$.