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<p>We propose a MFG model with quadratic Hamiltonian involving <i>N</i> populations. This results in a system of <i>N</i> Hamilton-Jacobi-Bellman and <i>N</i> 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 behaviors depending on the nature (repulsive or attractive) of the non-local interaction.</p>

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A variational formulation of a multi-population mean field games with non-local interactions

  • 期刊论文

  • 发表日期：

    2026年8月5日

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  • Luigi De Pascale,
  • Luca Nenna

摘要

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 behaviors depending on the nature (repulsive or attractive) of the non-local interaction.