Conservative difference schemes for the computation of mean-field equilibria

Abstract

The numerical methods are presented for solving economic problems formulated in the Mean Field Game (MFG) form. The mean-field equilibrium (i.e., the Nash equilibrium for an infinite number of players) leads to the coupled system of two parabolic partial differential equations: the Hamilton-Jacobi-Bellman-Isaacs equation and the Fokker-Planck-Kolmogorov one. The description is focused on the discrete approximation of these equations and on the application of the MFG theory directly at discrete level. This approach results in an efficient algorithm for finding the corresponding grid control function. Contrary to difference schemes with directed differences used by other authors, here the semi-Lagrangian approximation is applied which improves some properties of a discrete problem of this type. This implies the fast convergence of an iterative algorithm for the monotone minimization of the cost functional.