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Mean Field Games and Applications: Numerical Aspects

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Mean Field Games

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2281))

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Abstract

The theory of mean field games aims at studying deterministic or stochastic differential games (Nash equilibria) as the number of agents tends to infinity. Since very few mean field games have explicit or semi-explicit solutions, numerical simulations play a crucial role in obtaining quantitative information from this class of models. They may lead to systems of evolutive partial differential equations coupling a backward Bellman equation and a forward Fokker–Planck equation. In the present survey, we focus on such systems. The forward-backward structure is an important feature of this system, which makes it necessary to design unusual strategies for mathematical analysis and numerical approximation. In this survey, several aspects of a finite difference method used to approximate the previously mentioned system of PDEs are discussed, including convergence, variational aspects and algorithms for solving the resulting systems of nonlinear equations. Finally, we discuss in details two applications of mean field games to the study of crowd motion and to macroeconomics, a comparison with mean field type control, and present numerical simulations.

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Acknowledgements

The research of the first author was partially supported by the ANR (Agence Nationale de la Recherche) through MFG project ANR-16-CE40-0015-01.

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Authors and Affiliations

  1. Université de Paris, Laboratoire Jacques-Louis Lions (LJLL), Paris, France

    Yves Achdou

  2. Princeton University, Operations Research and Financial Engineering (ORFE) department, Sherrerd Hall, Princeton, NJ, USA

    Mathieu Laurière

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  1. Yves Achdou
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  2. Mathieu Laurière
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Correspondence to Yves Achdou .

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Editors and Affiliations

  1. Ceremade, UMR CNRS 7534, Université Paris-Dauphine, Paris, France

    Pierre Cardaliaguet

  2. Dipartimento di Matematica, University of Rome Tor Vergata, Roma, Italy

    Alessio Porretta

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Achdou, Y., Laurière, M. (2020). Mean Field Games and Applications: Numerical Aspects. In: Cardaliaguet, P., Porretta, A. (eds) Mean Field Games. Lecture Notes in Mathematics(), vol 2281. Springer, Cham. https://doi.org/10.1007/978-3-030-59837-2_4

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