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Active Particles, Volume 1 pp 141–171Cite as

Birkhäuser

Variational Mean Field Games

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Abstract

This paper is a brief presentation of those mean field games with congestion penalization which have a variational structure, starting from the deterministic dynamical framework. The stochastic framework (i.e., with diffusion) is also presented in both the stationary and dynamic cases. The variational problems relevant to MFG are described via Eulerian and Lagrangian languages, and the connection with equilibria is explained by means of convex duality and of optimality conditions. The convex structure of the problem also allows for efficient numerical treatment, based on augmented Lagrangian algorithms, and some new simulations are shown at the end of the paper.

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Acknowledgements

The authors acknowledge the support of the ANR project ISOTACE (ANR-12-MONU-0013). The third author also acknowledges the support of the iCODE project “Strategic Crowds,” funded by IDEX Paris-Saclay.

Author information

Authors and Affiliations

  1. INRIA-MOKAPLAN, 2 rue Simone Iff, 75012, Paris, France

    Jean-David Benamou

  2. Ceremade Univ. Paris Dauphine and INRIA-MOKAPLAN, Paris, France

    Guillaume Carlier

  3. Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405, Orsay Cedex, France

    Filippo Santambrogio

Authors
  1. Jean-David Benamou
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  2. Guillaume Carlier
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  3. Filippo Santambrogio
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Corresponding author

Correspondence to Filippo Santambrogio .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, King Abdulaziz University, Jeddah, Saudi Arabia

    Nicola Bellomo

  2. Department of Mathematics, Imperial College London, London, United Kingdom

    Pierre Degond

  3. CSCAMM, University of Maryland, College Park, USA

    Eitan Tadmor

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Benamou, JD., Carlier, G., Santambrogio, F. (2017). Variational Mean Field Games. In: Bellomo, N., Degond, P., Tadmor, E. (eds) Active Particles, Volume 1 . Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-49996-3_4

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