  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.
