First order Mean Field Games with density constraints: Pressure equals Price

TL;DR

This paper investigates mean field game systems with density constraints, revealing that the additional pressure term corresponds to the pressure in incompressible Euler models, and establishes a weak Nash equilibrium concept.

Contribution

It introduces a novel connection between density-constrained mean field games and pressure fields from fluid dynamics, providing a minimal regularity framework for analysis.

Findings

Pressure equals the price in saturated zones.

Weak solutions incorporate an extra pressure term.

Established a weak Nash equilibrium concept.

Abstract

In this paper we study Mean Field Game systems under density constraints as optimality conditions of two optimization problems in duality. A weak solution of the system contains an extra term, an additional price imposed on the saturated zones. We show that this price corresponds to the pressure field from the models of incompressible Euler's equations {\`a} la Brenier. By this observation we manage to obtain a minimal regularity, which allows to write optimality conditions at the level of single agent trajectories and to define a weak notion of Nash equilibrium for our model.