Approximation of deterministic mean field games with control-affine dynamics

TL;DR

This paper studies deterministic mean field games with control-affine dynamics, proposing discrete approximations, proving convergence of equilibria, and providing numerical results for nonlinear and acceleration control scenarios.

Contribution

It introduces a method to approximate complex mean field games with control-affine dynamics using discrete models and proves convergence of solutions.

Findings

Existence of solutions for discrete mean field games.

Uniqueness under monotonicity assumptions.

Convergence of discrete equilibria to continuous solutions.

Abstract

We consider deterministic mean field games where the dynamics of a typical agent is non-linear with respect to the state variable and affine with respect to the control variable. Particular instances of the problem considered here are mean field games with control on the acceleration. We focus our attention on the approximation of such mean field games by analogous problems in discrete time and finite state space. For these approximations, we show the existence and, under an additional monotonicity assumption, uniqueness of solutions. In our main result, we establish the convergence of equilibria of the discrete mean field games problems towards equilibria of the continuous one. Finally, we provide some numerical results for two MFG problems. In the first one, the dynamics of a typical player is nonlinear with respect to the state and, in the second one, a typical player controls its acceleration.