[Paper Review] Convergence of some Mean Field Games systems to aggregation and flocking models

Martino Bardi, Pierre Cardaliaguet|arXiv (Cornell University)|Apr 9, 2020

Mathematical Biology Tumor GrowthMathematics38 references21 citations

TL;DR

This paper establishes the convergence of two classes of Mean Field Game (MFG) systems to aggregation and flocking models as the discount rate λ→∞ and control cost vanishes. For second-order MFGs with vanishing viscosity, solutions converge to an aggregation equation; for first-order MFGs of acceleration, they converge to the kinetic Cucker-Smale model. The limit dynamics emerge as gradient descent of the running cost under the limiting agent density.

ABSTRACT

For two classes of Mean Field Game systems we study the convergence of solutions as the interest rate in the cost functional becomes very large, modeling agents caring only about a very short time-horizon, and the cost of the control becomes very cheap. The limit in both cases is a single first order integro-partial differential equation for the evolution of the mass density. The first model is a 2nd order MFG system with vanishing viscosity, and the limit is an aggregation equation. The result has an interpretation for models of collective animal behaviour and of crowd dynamics. The second class of problems are 1st order MFGs of acceleration and the limit is the kinetic equation associated to the Cucker-Smale model. The first problem is analyzed by PDE methods, whereas the second is studied by variational methods in the space of probability measures on trajectories.

Motivation & Objective

• To rigorously justify the heuristic convergence of rational, myopic agents in MFG systems to collective behavior models like aggregation and flocking.
• To analyze the limit of second-order MFG systems with vanishing viscosity and large discounting (λ→∞), showing convergence to an aggregation equation.
• To study first-order MFGs of acceleration under the same asymptotic regime, proving convergence to the kinetic Cucker-Smale model.
• To unify and extend prior heuristic results from [21] and [8] by incorporating vanishing viscosity and infinite time horizon.
• To establish convergence via PDE methods for the aggregation case and variational methods in measure space for the Cucker-Smale case.

Proposed method

• Formulate a second-order MFG system with vanishing viscosity νλ→0+ and large discount λ→∞, using a Hamilton-Jacobi-Bellman equation and a Fokker-Planck-type continuity equation.
• Prove convergence of the MFG solution (uλ, mλ) to a solution m of the aggregation equation ∂tm − div(m DxF(x,m)) = 0 via PDE techniques and compactness arguments.
• Analyze the optimal feedback −λDuλ, showing it converges a.e. to −DxF(·,m), the gradient descent of the running cost.
• For the acceleration-based MFG, use a variational formulation in the space of probability measures on trajectories, minimizing a time-integrated action functional.
• Establish tightness and relative compactness of the family of measures on trajectories, then identify the limit as a solution to the Cucker-Smale kinetic equation.
• Use Euler-Lagrange equations and integration by parts to derive the limiting dynamics, proving uniqueness of the solution via known results from [10].

Experimental results

Research questions

• RQ1 Under what conditions does a second-order Mean Field Game system with vanishing viscosity and large discount converge to an aggregation equation?
• RQ2 How does the optimal feedback control in a myopic MFG regime (λ→∞) relate to the gradient descent of the running cost in the limit?
• RQ3 Can a first-order MFG of acceleration converge to the kinetic Cucker-Smale model under vanishing control cost and large discounting?
• RQ4 What is the role of vanishing viscosity (νλ→0+) in the convergence of MFG solutions to non-diffusive collective models?
• RQ5 Is the limiting measure-valued solution to the Cucker-Smale equation unique under the proposed asymptotic regime?

Key findings

• As λ→∞ and νλ→0+, the solution mλ of the second-order MFG system converges to a solution m of the aggregation equation ∂tm − div(m DxF(x,m)) = 0.
• The optimal feedback control −λDuλ converges almost everywhere to −DxF(·,m), representing the gradient descent of the running cost under the limiting agent density.
• For the acceleration-based MFG, the limiting dynamics are described by the Cucker-Smale kinetic equation, with trajectories solving the second-order ODE ¨γ(t) = −DvF(γ(t), ˙γ(t), m(t)).
• The limiting measure m is the pushforward of the initial distribution m0 under the flow of the Cucker-Smale dynamics, uniquely determined by the initial condition.
• The convergence holds along subsequences, and due to uniqueness of the limit, the entire sequence (mλ) converges to the solution of the limiting equation.
• The results provide a rigorous justification for the heuristic derivation of aggregation and flocking models from rational MFG systems in the limit of myopic, cost-free agents.