First Order Mean Field Games with Density Constraints: Pressure Equals Price

Cardaliaguet, Pierre; Mészáros, Alpár R.; Santambrogio, Filippo

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 equations à la Brenier. By this observation we manage to obtain a minimal regularity, which allows us to write optimality conditions at the level of single-agent trajectories and to define a weak notion of Nash equilibrium for our model.

Citation

Cardaliaguet, P., Mészáros, A. R., & Santambrogio, F. (2016). First Order Mean Field Games with Density Constraints: Pressure Equals Price. SIAM Journal on Control and Optimization, 54(5), 2672-2709. https://doi.org/10.1137/15m1029849

• Journal Article Type: Article
• Acceptance Date: Jul 20, 2016
• Online Publication Date: Oct 11, 2016
• Publication Date: 2016
• Deposit Date: Oct 1, 2019
• Publicly Available Date: Feb 28, 2020
• Journal: SIAM Journal on Control and Optimization
• Print ISSN: 0363-0129
• Electronic ISSN: 1095-7138
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 54
• Issue: 5
• Pages: 2672-2709
• DOI: https://doi.org/10.1137/15m1029849
• Related Public URLs: https://arxiv.org/abs/1507.02019