Minimal-time mean field games

TL;DR

This paper models a crowd motion mean field game where agents aim to exit a domain in minimal time, considering congestion effects, and establishes existence and properties of equilibria through a Lagrangian framework and PDE analysis.

Contribution

It formulates a novel mean field game model with congestion-dependent speed limits and proves existence of equilibria using a fixed point approach and PDE techniques.

Findings

Existence of Lagrangian equilibria established.

Equilibria characterized by coupled PDE system.

Simulations demonstrate equilibrium behaviors.

Abstract

This paper considers a mean field game model inspired by crowd motion where agents want to leave a given bounded domain through a part of its boundary in minimal time. Each agent is free to move in any direction, but their maximal speed is bounded in terms of the average density of agents around their position in order to take into account congestion phenomena. After a preliminary study of the corresponding minimal-time optimal control problem, we formulate the mean field game in a Lagrangian setting and prove existence of Lagrangian equilibria using a fixed point strategy. We provide a further study of equilibria under the assumption that agents may leave the domain through the whole boundary, in which case equilibria are described through a system of a continuity equation on the distribution of agents coupled with a Hamilton--Jacobi equation on the value function of the optimal control problem solved by each agent. This is possible thanks to the semiconcavity of the value function, which follows from some further regularity properties of optimal trajectories obtained through Pontryagin Maximum Principle. Simulations illustrate the behavior of equilibria in some particular situations.