The planning problem in mean field games as regularized mass transport

Graber, P. Jameson; Mészáros, Alpár Richárd; Silva, Francisco; Tonon, Daniela

doi:10.1007/s00526-019-1561-9

Cited by 32 publications

(32 citation statements)

References 24 publications

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“…As for J * , using [28] as well as the definition of F and G (see for instance [20] for details in a similar variational MFG context), we have…”

Section: G(β)mentioning

confidence: 99%

A mean field game model for the evolution of cities

Barilla¹,

Carlier²

2021

JDG

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We propose a (toy) MFG model for the evolution of residents and firms densities, coupled both by labour market equilibrium conditions and competition for land use (congestion). This results in a system of two Hamilton-Jacobi-Bellman and two Fokker-Planck equations with a new form of coupling related to optimal transport. This MFG has a convex potential which enables us to find weak solutions by a variational approach. In the case of quadratic Hamiltonians, the problem can be reformulated in Lagrangian terms and solved numerically by an IPFP/Sinkhorn-like scheme as in [4]. We present numerical results based on this approach, these simulations exhibit different behaviours with either agglomeration or segregation dominating depending on the initial conditions and parameters.

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“…As for J * , using [28] as well as the definition of F and G (see for instance [20] for details in a similar variational MFG context), we have…”

Section: G(β)mentioning

confidence: 99%

A mean field game model for the evolution of cities

Barilla¹,

Carlier²

2021

JDG

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“…Note that written in (2) or (3), the Schrödinger bridge problem has the form of a first-order mean field game [10,21]. More accurately, since the boundary conditions consist of fixing the densities, it is rather an instance of the planning problem [26,33,18]. Rather formidably, system (3) can be rewritten into a simpler and more symmetric way thanks to the Hopf-Cole transformation.…”

Section: Reviewmentioning

confidence: 99%

“…Note that here the tensor (δ 2 F) −1 denotes the inverse of the Hessian δ 2 F. Applying it to (18), and inverting δ 2 F(η * ) we derive…”

Section: Symplectic Aspects Of Hopf-cole Transformationmentioning

confidence: 99%

Hopf–Cole Transformation and Schrödinger Problems

Léger¹,

Li²

2019

Lecture Notes in Computer Science

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We study generalized Hopf-Cole transformations motivated by the Schrödinger bridge problem, which can be seen as a boundary value Hamiltonian system on the Wasserstein space. We prove that generalized Hopf-Cole transformations are symplectic submersions in the Wasserstein symplectic geometry. Many examples, including a Hopf-Cole transformation for the shallow water equations, are given. Based on this transformation, energy splitting inequalities are provided.

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“…Numerical methods for this type of system has been presented in [1]. More recently, the works [6,9] studied (1) in the potential case. In the so-called potential approach, (1) is interpreted as the optimality conditions of an infinite dimensional optimal control problem.…”

Section: Introductionmentioning

confidence: 99%

Master Equation for the Finite State Space Planning Problem

Bertucci

Lions²

2021

Arch Rational Mech Anal

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We present results of existence, regularity and uniqueness of solutions of the master equation associated with the mean field planning problem in the finite state space case, in the presence of a common noise. The results hold under monotonicity assumptions, which are used crucially in the different proofs of the paper. We also make a link with the trajectories induced by the solution of the master equation and start a discussion on the case of boundary conditions. Contents 1.2. Preliminary results 2. Planning problem master equation in R d 2.1. Statement of the problem 2.2. Properties of the Yosida approximation 2.3. The limit master equation 2.4. Links with the induced trajectories 3. A comment on the case of a restricted domain 3.1. Main differences with the previous case 3.2. The half space case with vanishing conditions

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The planning problem in mean field games as regularized mass transport

Cited by 32 publications

References 24 publications

A mean field game model for the evolution of cities

A mean field game model for the evolution of cities

Hopf–Cole Transformation and Schrödinger Problems

Master Equation for the Finite State Space Planning Problem

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