Extended mean-field games

We introduce a new class of coupled forward-backward in time systems consisting of a forward Hamilton–Jacobi and a backward quasilinear transport equation, which we call extended mean-field games system. This new class of equations strictly contains the classical mean-field games system with no common noise and its homogenization limit, and optimal transportation-type control problems. We also identify a new and meaningful ‘‘monotonicity’’-type condition that yields well-posedeness. The same condition yields uniqueness in the Hilbertian setting for the master equation without common noise as well as the hyperbolic system describing finite-state mean-field games.