  A novel framework is presented that combines Mean Field Game (MFG) theory and
Hybrid Optimal Control (HOC) theory to obtain a unique $\epsilon$-Nash
equilibrium for a non-cooperative game with switching and stopping times. We
consider the case where there exists one major agent with a significant
influence on the system together with a large number of minor agents
constituting two subpopulations, each agent with individually asymptotically
negligible effect on the whole system. Each agent has stochastic linear
dynamics with quadratic costs, and the agents are coupled in their dynamics and
costs by the average state of minor agents (i.e. the empirical mean field). It
is shown that for a class of Hybrid LQG MFGs, the optimal switching and
stopping times are state-invariant and only depend on the dynamical parameters
of each agent. Accordingly, a hybrid systems formulation of the game is
presented via the indexing by discrete events: (i) the switching of the major
agent between alternative dynamics or (ii) the termination of the agents'
trajectories in one or both of the subpopulations of minor agents. Optimal
switchings and stopping time strategies together with best response control
actions for, respectively, the major agent and all minor agents are established
with respect to their individual cost criteria by an application of Hybrid LQG
MFG theory.
