A variational formulation of a Multi-Population Mean Field Games with non-local interactions

TL;DR

This paper introduces a variational approach to multi-population mean field games with non-local interactions, providing a weak solution framework and a numerical Sinkhorn-like scheme to analyze different interaction behaviors.

Contribution

It develops a novel Eulerian variational formulation for multi-population MFGs with non-local interactions and proposes a numerical method for solving the resulting system.

Findings

Numerical simulations show different behaviors for repulsive and attractive interactions.

The variational formulation yields weak solutions despite non-convexity.

The Sinkhorn-like scheme effectively solves the MFG system numerically.

Abstract

We propose a MFG model with quadratic Hamiltonian involving $N$ populations. This results in a system of $N$ Hamilton-Jacobi-Bellman and $N$ Fokker-Planck equations with non-local interactions. As in the classical case we introduce an Eulerian variational formulation which, despite the non convexity of the interaction, still gives a weak solution to the MFG model. The problem can be reformulated in Lagrangian terms and solved numerically by a Sinkhorn-like scheme. We present numerical results based on this approach, these simulations exhibit different behaviours depending on the nature (repulsive or attractive) of the non-local interaction.