  This work is devoted to finding the closed-loop equilibria for a class of
mean-field games (MFGs) with infinitely many symmetric players in a common
switching environment when the cost functional is under general discount in
time. There are two key challenges in the application of the well-known
Hamilton-Jacobi-Bellman and Fokker-Planck (HJB-FP) approach to our problems:
the path-dependence due to the conditional mean-field interaction and the
time-inconsistency due to the general discounting cost. To overcome the
difficulties, a theory for a class of systems of path-dependent equilibrium
Hamilton-Jacobi-Bellman equations (HJBs) is developed. Then closed-loop
equilibrium strategies can be identified through a two-step verification
procedure. It should be noted that the closed-loop equilibrium strategies
obtained satisfy a new form of local optimality in the Nash sense. The theory
obtained extends the HJB-FP approach for classical MFGs to more general
conditional MFGs with general discounting costs.
