First-order, stationary mean-field games with congestion

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 density. Our main contribution is a new variational formulation for MFGs with congestion. This formulation was not previously known, and, thanks to it, we prove the existence and uniqueness of solutions. Finally, we consider applications to numerical methods.

Comparing the best-reply strategy and mean-field games: The stationary case

“…They then perturb the solutions to prove existence in the case of a local dependence on the distribution. Other typical methods of proof use continuation methods [21,23,26], Schauder's fixed point theorem [5,17] or variational approaches through energy minimisation problems [14,20]. In our proof we exploit the linear-quadratic nature of the control.…”

“…Due to assumption (A 2 ), which states that the running cost h is an increasing function of density, we are in the setting of monotone stationary MFGs. Existence and uniqueness of such MFGs has been studied extensively by Gomes and collaborators in a number of papers e.g [20,25,28]. Although the setting in these papers focusses on domains with periodic boundary conditions, it is worth mentioning the types of techniques used and how they compare to the method in this paper.…”

Optimal control of a first order Fokker-Planck equation with reaction term and density constraints

“…m ≤ m at any point (t, s). The variational approach shows good results when applied to MFC [1] and MFG with "soft congestion" in a stationary framework [20], as well as to MFG problems dealing with "hard congestion" constraints. This has been first investigated in [40] where the density of the population does not exceed a given threshold, then in [35] where stationary second order MFG are considered.…”

Sharp semi-concavity in a non-autonomous control problem and $$\pmb {L^p}$$ estimates in an optimal-exit MFG

“…The function k in our model is intended to represent congestion, i.e., the difficulty of moving in high-density areas. Several mean field games with congestion have been previously considered [3,36,38,41,44,63], their common feature being to model congestion as a penalization in the cost function of each agent when passing through crowded regions. The penalization term is usually chosen as a negative power of the density, which introduces a singularity in the Hamilton-Jacobi equation of the corresponding optimal control problem.…”