A mean field game model for the evolution of cities

César Barilla; Guillaume Carlier; Jean-Michel Lasry

doi:10.3934/jdg.2021017

Article Contents

2021, Volume 8, Issue 3: 299-329. Doi: 10.3934/jdg.2021017

This issue Previous Article A graph cellular automaton with relation-based neighbourhood describing the impact of peer influence on the consumption of marijuana among college-aged youths Next Article

A mean field game model for the evolution of cities

1.
Department of Economics, Columbia University, New York, USA
2.
CEREMADE, Université Paris Dauphine, PSL Research University and INRIA-Paris, MOKAPLAN, Paris, France
3.
CEREMADE, Université Paris Dauphine, PSL Research University, Paris, France

^* Corresponding author: Guillaume Carlier
^* Corresponding author: Guillaume Carlier

Received: November 30, 2020

Early access: May 2021

Published: July 2021

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Abstract

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.

Keywords:

Mean field games,
convex duality,
optimal transport,
labour market equilibrium,
Iterative Proportional Fitting procedure (IPFP).

Mathematics Subject Classification: Primary: 49L12, 91A14; Secondary: 65K10.

Citation:

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