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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.

APE M2 Masters Thesis (PSE) ; Supervisor : Alfred Galichon (NYU) ; Referee : Olivier Tercieux (PSE)

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Complementarities are known to jeopardize stability in many-to-one matching markets. The present work explores a modeling approach tailored for large markets characterized by dynamic allocation with temporary partnerships and search frictions – the prominent example being labour markets – with the hope to obtain less stringent stability requirement. Weaker notions of stability of the process locally in time and asymptotically in market size are defined, according to which equilibrium may be obtained even when a stable equilibrium may not exist on the complete market. A general reduction method is proposed to study stability under general constraint graphs, in particular the category of graphs produced by a sequential search process. Application of this method to a simple search process yields an overall negative results : local stability on simple structures is sufficient, but symetrically local instability can spread to the whole graph. Asymptotic results may be recovered for large markets in which tree graphs are obtained ; it remains extremely challenging to ensure stability in general settings without any assumptions on the production function. Several qualitative properties of such models are explored, notably its (lack of) convergence to a static stable allocation when there exists one and optimality. In the particular case where the usual gross substitutes assumption is satisfied, the model should converge towards the static stable equilibria. This is, however, not the case in general even when the surplus function is submodular and there exists a stable equilibrium. Well known examples of instability are also revisited. Finally, what this approach can and cannot explain highlights several intuitions that might prove fruitful for further studies of stability issues with complementarities.

** Published:**

Presentation of the paper A Mean Field Game Model for the Dynamics of Cities (slides) in the Applied Micro Theory Colloquium at Columbia University, New York.

Joint with Guillaume Carlier and Jean-Michel Lasry.

Published in *Journal of Dynamics and Games*, 2021

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.

Teaching Assistant for Prof. Ceyhun Elgin, *Columbia University*, 2020

Main instructor, *Columbia University*, 2021

Joint with Guillaume Carlier and Jean-Michel Lasry.

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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 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.