Mean field games of controls and moderate interactions.

Mean Field Game theory, a quickly growing field emerging around 2007, seeks to answer qualitative and quantitative questions about high population competitive dynamics. A typical Mean Field Game is comprised of a large pool of identical, weak, rational agents, each seeking to maximize a payout (or, equivalently, minimize a cost). When an agent selects a strategy, it alters the state of the playing field for all other agents, that is, the actions of all agents must be factored into the strategy choice of any agent. The main object of interest is the Nash Equilibria of the game: a strategy choice that, if chosen by all players, no single player may improve their outcome by adjusting their own strategy. The question of these Nash Equilibria can be cast as solutions of a coupled system of partial differential equations: a backward in time Hamilton Jacobi Equation that encapsulates the strategy costs, the unknown being the value function u(x,t), and a forward in time Fokker-Planck type equation governing the evolution m(x,t): the probability distribution of player states. We look at two adaptations of the traditional Mean Field Game. First, a Potential Mean Field Game of Controls, in which agents have knowledge of not only the current states of other agents, but also their current adjustment to strategy. While the potential structure allows for the use of convex analysis techniques, we introduce a new, spatially non-local coupling component that depends on both the distribution of player states and the feedback. We also consider a Mean Field Game of Moderate Interactions, in which agents are more intensely affected by those in their immediate vicinity. This augments the standard Mean Field Game by introducing a local coupling term that has interaction with the gradient of the value function. We establish, in each case, well posedness results under generic assumptions on the data. In the conclusion we remark on the possible generalizations and further directions to take each work.