Mean Field Games of Controls: Propagation of Monotonicities

Abstract

The theory of Mean Field Game of Controls considers a class of mean field games where the interaction is through the joint distribution of the state and control. It is well known that, for standard mean field games, certain monotonicity condition is crucial to guarantee the uniqueness of mean field equilibria and then the global wellposedness for master equations.In the literature, the monotonicity condition could be the Lasry-Lions monotonicity, the displacement monotonicity, or the anti-monotonicity conditions. In this paper, we investigate all these three types of monotonicity conditions for Mean Field Games of Controls and show their propagation along the solutions to the master equations with common noises. In particular, we extend the displacement monotonicity to semi-monotonicity, whose propagation result 1 is new even for standard mean field games. This is the first step towards the global wellposedness theory for master equations of Mean Field Games of Controls.

Linear-Quadratic Mean Field Games of Controls with Non-Monotone Data

“…with given k N i , 1 ≤ i ≤ N , defined in (56). The following two theorems are the well-posedness results for (60) and (61).…”

“…To our best knowledge, this is the first global well-posedness result for the master equation in the literature for mean field games of controls. There is a recent progress on (infinite dimensional) master equations for mean field games of controls in [56], where the propagation of monotonicities along classical solutions of master equations has been proved. To establish the well-posedness result, we follow the following steps.…”