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

Luigi De Pascale, Luca Nenna

TL;DR

The paper addresses multi-population mean field games with non-local interactions by developing a robust variational framework in both Eulerian and Lagrangian forms. It convexifies the dynamic terms through momentum variables and leverages Fenchel-Rockafellar duality to connect optimality conditions to a system of HJB-FP equations, even in the presence of non-convex interactions. A Lagrangian entropic formulation, Γ-convergence of time-discretized problems, and a Sinkhorn-type numerical scheme are developed to ensure existence, stability, and computability, with explicit schemes handling non-local terms. Numerical experiments reveal how repulsive and attractive interactions drive distinct population dynamics and demonstrate the method’s capacity to approximate first-order MFG limits as viscosity vanishes. The work provides a principled, computationally efficient approach to complex, non-local MFGs with practical implications for systems with multiple interacting populations.

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.

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

TL;DR

The paper addresses multi-population mean field games with non-local interactions by developing a robust variational framework in both Eulerian and Lagrangian forms. It convexifies the dynamic terms through momentum variables and leverages Fenchel-Rockafellar duality to connect optimality conditions to a system of HJB-FP equations, even in the presence of non-convex interactions. A Lagrangian entropic formulation, Γ-convergence of time-discretized problems, and a Sinkhorn-type numerical scheme are developed to ensure existence, stability, and computability, with explicit schemes handling non-local terms. Numerical experiments reveal how repulsive and attractive interactions drive distinct population dynamics and demonstrate the method’s capacity to approximate first-order MFG limits as viscosity vanishes. The work provides a principled, computationally efficient approach to complex, non-local MFGs with practical implications for systems with multiple interacting populations.

Abstract

We propose a MFG model with quadratic Hamiltonian involving populations. This results in a system of Hamilton-Jacobi-Bellman and 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.
Paper Structure (7 sections, 12 theorems, 127 equations, 5 figures, 1 algorithm)

This paper contains 7 sections, 12 theorems, 127 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Variational formulation
  3. Analysis of the minimizers
  4. The Entropic problem
  5. Time discretization and $\Gamma$-convergence
  6. Numerical Approximation
  7. Entropy and the De la Vallée Poussin Theorem

Key Result

Theorem 2.1

Let $\rho_t$ be an absolutely continuous curve in $\mathcal{P}_2 (\mathbb R^d)$ then for a.e. $t \in [0,T]$, there exists a vector field $v_t \in L^2_{\rho_t}$ such that and for a.e. $t$ we have $\|v_t\|_{L^2_{\rho_t}} \leq |\rho'|$, where $|\rho'|$ denotes the metric derivative of $\rho$. Conversely, if $\rho_t$ is a curve in $\mathcal{P}_2 (\mathbb R^d)$ and for a.e. $t \in [0,T]$ there exists

Figures (5)

  • Figure 5.1: Support of $\rho^1$ and $\rho^2$ for $V(x,y)=120\chi_{||x-y||<0.2}(x,y)$.
  • Figure 5.2: Support of $\rho^1$ and $\rho^2$ for $V(x,y)=\min(1000,\frac{1}{||x-y||})$.
  • Figure 5.3: Support of $\rho^1$ and $\rho^2$ for $\varepsilon=.005$ and $V(x,y)=120\chi_{||x-y||<0.2}(x,y)$.
  • Figure 5.4: Support of $\rho^1$ and $\rho^2$ for $\varepsilon=.005$ and $V(x,y)=\min(1000,\frac{1}{||x-y||})$.
  • Figure 5.5: Support of $\rho^1$, $\rho^2$ and $\rho^3$ for $V(x,y)=120\chi_{||x-y||<0.2}(x,y)$.

Theorems & Definitions (24)

  • Remark 1.1
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • Theorem 2.6
  • ...and 14 more