A variational mean field game of controls with free final time and pairwise interactions

TL;DR

This paper introduces a mean field game model with pairwise interactions and free final time, demonstrating its variational structure, existence of equilibria, and providing numerical insights.

Contribution

It establishes a variational framework for a class of mean field games with pairwise interactions and free final time, including existence results and conditions for equilibrium equivalence.

Findings

Proved the variational structure of the game model.

Established the existence of equilibria.

Provided a numerical illustration of the model.

Abstract

This article considers a mean field game model inspired by crowd motion models in which agents aim at reaching a given target set and wish to minimize a cost consisting of an individual running cost, an individual cost depending on the arrival time at the target set, and an interaction running cost, which takes the form of pairwise interactions with other agents through both positions and velocities. We subsume this game under a more general class of games on abstract Polish spaces with pairwise interactions, and prove that the latter games have a variational structure (in the sense that their equilibria can be characterized as critical points of some potential functional) and admit equilibria. We also discuss two a priori distinct notions of equilibria, providing a sufficient condition under which both notions coincide. The results for the games in abstract Polish spaces are applied to our mean field game model, and a numerical illustration concludes the paper.