Mean field game theory for agents with individual-state partial observations

Subject to reasonable conditions, in large population stochastic dynamics games where the agents are coupled by the system's mean field through their nonlinear dynamics and cost functions, it can be shown that a best response control action for each agent exists which (i) depends only upon the individual agent's state observations and the mean field, and (ii) achieves an ∈-Nash equilibrium for the system. In this work we formulate a class of problems where each agent has only partial observations on its individual state. The main result is that the ∈-Nash equilibrium property holds where the best response control action of each agent depends upon the conditional density of its own state generated by a nonlinear filter, together with the system's mean field. Finally, it is worthwhile comparing this MFG state estimation problem to one found in the literature where there exists a major agent whose partially observed state process is independent of the control action of any individual agent; by contrast, in this work, the partially observed state process of any agent depends upon that agent's control action.