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

arXiv: https://arxiv.org/abs/1507.02019 · HTML: https://mlsanity.com/paper/1507.02019/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jul 2015.

## Built on by (most cited first, 1 of 1)

- [Kernel-based potential mean-field games with unbiased random Fourier U-statistics](https://mlsanity.com/paper/2605.29371/) (2026)

## Abstract (arXiv)

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.