A variational formulation of a Multi-Population Mean Field Games with non-local interactions
For researchers in mean field games, this extends variational methods to multi-population settings with non-local interactions, but the approach is incremental as it adapts existing techniques.
The paper proposes a multi-population mean field game model with non-local interactions and provides a variational formulation that yields weak solutions despite non-convex interactions. Numerical simulations using a Sinkhorn-like scheme demonstrate different behaviors for repulsive vs. attractive interactions.
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.