First-order, stationary mean-field games with congestion. (August 2018)

• Record Type: Journal Article
• Authors: Evangelista, David; Ferreira, Rita; Gomes, Diogo A.; Nurbekyan, Levon; Voskanyan, Vardan
• Is Part Of: Nonlinear analysis. Volume 173(2018)
• Page Start: 37
• Page End: 74
• Publication Date: 2018-08
• Subjects: 35J47 -- 35A01; Mean-field game -- Congestion -- Calculus of variations; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2018.03.011
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Mean-field games (MFGs) are models for large populations of competing rational agents that seek to optimize a suitable functional. In the case of congestion, this functional takes into account the difficulty of moving in high-density areas. Here, we study stationary MFGs with congestion with quadratic or power-like Hamiltonians. First, using explicit examples, we illustrate two main difficulties: the lack of classical solutions and the existence of areas with vanishing densities. Our main contribution is a new variational formulation for MFGs with congestion. With this formulation, we prove the existence and uniqueness of solutions. Finally, we consider applications to numerical methods.