Posts by Collection


A Mean-Field Game Model for the Evolution of Cities

with Guillaume Carlier and Jean-Michel Lasry.


We propose a (toy) MFG model for the evolution of residents and firms densities, coupled both by labour market equilibrium conditions at each time 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 so- lutions by a variational approach. In the case of quadratic Hamil- tonians, the problem can be reformulated in Lagrangian terms and solved numerically by an IPFP/Sinkhorn-like scheme as in [2]. We present numerical results based on this approach, these simulations exhibit different behaviours with either residential or business centers depending on the initial conditions and parameters.