Mean Field Game Master Equations with Anti-monotonicity Conditions

Abstract

It is well known that the monotonicity condition, either in Lasry-Lions sense or in displacement sense, is crucial for the global well-posedness of mean field game master equations, as well as for the uniqueness of mean field equilibria and solutions to mean field game systems.In the literature, the monotonicity conditions are always taken in a fixed direction. In this paper we propose a new type of monotonicity condition in the opposite direction, which we call the anti-monotonicity condition, and establish the global well-posedness for mean field game master equations with nonseparable Hamiltonians. Our anti-monotonicity condition allows our data to violate both the Lasry-Lions monotonicity and the displacement monotonicity conditions.

Mean Field Games of Controls: Propagation of Monotonicities

“…Note that H depends on ν, while V depends on µ = π 1# ν. We also remark that the Hamiltonian in [20,34] is −H. To introduce the master equation, which we will do in the next section, we need the following fixed point.…”

“…Our ultimate goal is to establish the global wellposedness of the master equation (3.6). We shall adopt the strategy in [20,34], which consists of three steps:…”

“…See, e.g., Bensoussan-Graber-Yam [6] and Gangbo-Meszaros-Mou-Zhang [20]. The anti-monotonicity, recently introduced by the authors [34], takes the following form…”

“…In [20,34] we made a simple but crucial observation: the propagation of a monotonicity is crucial for the global wellposedness of the (standard) MFG master equations. That is, provided the terminal condition G satisfies one of the above three types of monotonicity conditions, if one can show a priori that any classical solution V of the master equation satisfies the same type of monotonicity for all time t, then one can establish the global wellposedness of the master equation, which in turn will imply the uniqueness of mean field equilibria and the convergence from the N -player game to the MFG.…”

“…Our goal is to extend all these results to MFGCs, but in this paper we focus only on the propagation of these three types of monotonicities. That is, we shall follow the approach in [20,34] to find sufficient conditions on the Hamiltonian H (or alternatively on b and f ) so that the monotonicity of G can be propagated along V (t, •, •), provided the master equation has a classical solution V . We shall leave the global wellposedness of the master equations and the convergence of the N -player games to an accompanying paper.…”