A Modest Proposal for MFG with Density Constraints

TL;DR

This paper explores a new approach to modeling congestion in Mean Field Games by replacing the traditional $L^p$ penalization with an $L^ abla$ constraint, introducing a PDE system with a pressure field.

Contribution

It proposes a novel PDE framework incorporating density constraints in Mean Field Games, inspired by recent crowd motion models, and discusses an example and open problems.

Findings

Proposed a PDE system with a pressure field for density constraints.

Analyzed an example illustrating the model's application.

Identified open problems for future research.

Abstract

We consider a typical problem in Mean Field Games: the congestion case, where in the cost that agents optimize there is a penalization for passing through zones with high density of agents, in a deterministic framework. This equilibrium problem is known to be equivalent to the optimization of a global functional including an $L^p$ norm of the density. The question arises as to produce a similar model replacing the $L^p$ penalization with an $L^\infty$ constraint, but the simplest approaches do not give meaningful definitions. Taking into account recent works about crowd motion, where the density constraint $\rho\leq 1$ was treated in terms of projections of the velocity field onto the set of admissible velocity (with a constraint on the divergence) and a pressure field was introduced, we propose a definition and write a system of PDEs including the usual Hamilton-Jacobi equation coupled with the continuity equation. For this system, we analyze an example and propose some open problems.