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