Mean Field Games of Controls: Finite Difference Approximations

We consider a class of mean field games in which the agents interact through both their states and controls, and we focus on situations in which a generic agent tries to adjust her speed (control) to an average speed (the average is made in a neighborhood in the state space). In such cases, the monotonicity assumptions that are frequently made in the theory of mean field games do not hold, and uniqueness cannot be expected in general. Such model lead to systems of forward-backward nonlinear nonlocal parabolic equations; the latter are supplemented with various kinds of boundary conditions, in particular Neumann-like boundary conditions stemming from reflection conditions on the underlying controled stochastic processes. The present work deals with numerical approximations of the above mentioned systems. After describing the finite difference scheme, we propose an iterative method for solving the systems of nonlinear equations that arise in the discrete setting; it combines a continuation method, Newton iterations and inner loops of a bigradient like solver. The numerical method is used for simulating two examples. We also make experiments on the behaviour of the iterative algorithm when the parameters of the model vary. The theory of mean field games, (MFGs for short), aims at studying deterministic or stochastic differential games (Nash equilibria) as the number of agents tends to infinity. It supposes that the rational agents are indistinguishable and individually have a negligible influence on the game, and that each individual strategy is influenced by some averages of quantities depending on the states (or the controls as in the present work) of the other agents. MFGs have been introduced in the pioneering works of J-M. Lasry and P-L. Lions [17, 18, 19]. Independently and at approximately the same time, the notion of mean field games arose in the engineering literature, see the works of M.Y. Huang, P.E. Caines and R.Malham{\'e} [14, 15]. The present work deals with numerical approximations of mean field games in which the agents interact through both their states and controls; it follows a more theoretical work by the second author, [16], which is devoted to the mathematical analysis of the related systems of nonlocal partial differential equations. There is not much literature on MFGs in which the agents also interact through their controls, see [13, 12, 8, 10, 7, 16]. To stress the fact that the latter situation is considered, we will sometimes use