Mean Field Games and Applications: Numerical Aspects

Abstract

The theory of mean field games aims at studying deterministic or stochastic differential games (Nash equilibria) as the number of agents tends to infinity. Since very few mean field games have explicit or semi-explicit solutions, numerical simulations play a crucial role in obtaining quantitative information from this class of models. They may lead to systems of evolutive partial differential equations coupling a backward Bellman equation and a forward Fokker-Planck equation. In the present survey, we focus on such systems. The forward-backward structure is an important feature of this system, which makes it necessary to design unusual strategies for mathematical analysis and numerical approximation. In this survey, several aspects of a finite difference method used to approximate the previously mentioned system of PDEs are discussed, including convergence, variational aspects and algorithms for solving the resulting systems of nonlinear equations. Finally, we discuss in details two applications of mean field games to the study of crowd motion and to macroeconomics, a comparison with mean field type control, and present numerical simulations.

Monotone solutions for mean field games master equations: finite state space and optimal stopping

“…We now present uniqueness results concerning (1) and (2). Those results are direct extensions of the results established in [32] and do not need any new ideas, however, as they play a crucial role in the notion of solution we introduce in the next section, we detail their proofs.…”

“…As already mentioned several times above, monotonicity plays a crucial role in the well-posedness of (1) and (2). We say that we are in the monotone regime when the following assumption is satisfied.…”

“…We now turn to the questions of existence of solutions of (1) and (2). As the next section gives a precise definition of a solution of (1) and (2), we do not focus on the sense in which solutions satisfy the PDE but rather on how we can obtain a priori estimates.…”

“…Description of the model. -We consider a MFG in finite state space which is described by F, G, λ and T as in (2). We add the possibility for a player in the state i to instantly jump to the state j by paying a cost k ij 0.…”

“…Let us also recall that a probabilistic approach of MFG have been developed, we refer to [18,28] for more details on this approach. Finally let us mention that numerical methods have been developed to compute equilibria of MFG, mostly in the forward-backward setting, more details can be found in [1,2,3]. In recent years, the study of MFG of optimal stopping or other "singular" controls have been the subject of a growing number of researches, namely because such games have natural applications in Economy.…”