Lagrangian discretization of variational mean field games

TL;DR

This paper presents a novel finite-trajectory discretization method for variational mean field games with congestion, utilizing semi-discrete optimal transport to ensure convergence to the true solution.

Contribution

It introduces a new discretization approach for mean field games with congestion, combining variational principles with semi-discrete optimal transport techniques.

Findings

Convergence of discrete trajectories to the mean field game solution.

Efficient computation of the Moreau envelope using semi-discrete optimal transport.

Validation of the method under specific discretization parameter conditions.

Abstract

In this article, we introduce a method to approximate solutions of some variational mean field game problems with congestion, by finite sets of player trajectories. These trajectories are obtained by solving a minimization problem similar to the initial variational problem. In this discretized problem, congestion is penalized by a Moreau envelop with the 2-Wasserstein distance. Study of this envelop as well as efficient computation of its values and variations is done using semi-discrete optimal transport. We show convergence of the discrete sets of trajectories toward a solution of the mean field game, under some conditions on the parameters of the discretization.