## Optimal transport coupling in multi-population mean field games : Matching equilibrium displacement and applications to urban planning

Mathematics of Modeling M2 (Sorbonne University) Masters Thesis ; Supervisors : Guillaume Carlier (CEREMADE) and Jean-Michel Lasry (CEREMADE)

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We present a multi-population mean-field game (MFG) model in which the running cost functions of each population are coupled via the potentials of an instantaneous optimal transport problem between their respective densities, with applications to geographical population dynamics and urban planning. The modeling approach is abstract and very stylized ; its applications bear no pretensions of being a realistic nor complete description of observed dynamics. Indeed, our ambition is rather to present conceptual mechanisms susceptible to drive the dynamics of such systems and to explore new tools for analyzing them. We first approach the model as a system of partial differential equations that are very close to the classical MFG system. We then present a Eulerian variational formulation, as a problem of optimal control of Fokker-Planck equations. Finally, we recast the problem in a third equivalent formulation as an entropy minimization problem in the space of paths measures (Lagrangian approach). The latter is then used to construct an algorithm for practical simulation by performing alternate maximization on the dual problem, which corresponds to Dykstra’s algorithm, often referred to as Sinkhorn algorithm or IPFP in optimal transport litterature. It is noteworthy that, although we use variational formulations, the model might not be variational – and is likely not for many interesting cost functions. However, the variational formulations can still be obtained through a fixed point trick. In applying this model to urban planning problems, we view the populations as e.g. inhabitants and firms, and the instantaneous optimal transport as instanta- neous equilibrium on the labour market, while allowing both populations to move. More generally, this framework allows to tackle any situation where you combine an instantaneous matching equilibrium condition and dynamic optimal control – the model provides a Nash equilibrium over a succession of instantaneous equilibria.