Mean Field Games for Modeling Crowd Motion

  • Yves AchdouEmail author
  • Jean-Michel Lasry
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 47)


We present a model for crowd motion based on the recent theory of mean field games. The model takes congestion effects into account. A robust and efficient numerical method is discussed. Numerical simulations are presented for two examples. The second example, in which all the agents share a common source of risk and have incomplete information, is of particular interest, because it cannot be dealt with without modeling rational anticipation.



The first author would like to affectionately dedicate this work to Yuri Kuznetsov and Olivier Pironneau for their seventieth birthdays. The first author was partially funded by the ANR projects ANR-12-MONU-0013 and ANR-12-BS01-0008-01. The two authors acknowledge the support of the Chaire “Finance et développement durable”.


  1. 1.
    Achdou Y (2013) Finite difference methods for mean field games. In: Hamilton-Jacobi Equations: approximations, numerical analysis and applications, volume 2074 of Lecture Notes in Mathematics. Springer, Heidelberg, pp 1–47Google Scholar
  2. 2.
    Achdou Y, Buera FJ, Lasry J-M, Lions P-L, Moll B (2014) Partial differential equation models in macroeconomics. Philos Trans R Soc Lond Ser A Math Phys Eng Sci 372(2028):20130397 (19 pp)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Achdou Y, Camilli F, Capuzzo-Dolcetta I (2013) Mean field games: convergence of a finite difference method. SIAM J Numer Anal 51(5):2585–2612MathSciNetCrossRefGoogle Scholar
  4. 4.
    Achdou Y, Capuzzo-Dolcetta I (2010) Mean field games: numerical methods. SIAM J Numer Anal 48(3):1136–1162MathSciNetCrossRefGoogle Scholar
  5. 5.
    Achdou Y, Han J, Lasry J-M, Lions P-L, Moll B (2015) Heterogeneous agent models in continuous time. Working papers, Princeton UniversityGoogle Scholar
  6. 6.
    Achdou Y, Perez V (2012) Iterative strategies for solving linearized discrete mean field games systems. Netw Heterog Media 7(2):197–217MathSciNetCrossRefGoogle Scholar
  7. 7.
    Achdou Y, Porretta A (2016) Convergence of a finite difference scheme to weak solutions of the system of partial differential equations arising in mean field games. SIAM J Numer Anal 54(1):161–186MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bardi M, Capuzzo-Dolcetta I (1997) Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston, MA (With appendices by M. Falcone and P. Soravia)CrossRefGoogle Scholar
  9. 9.
    Burger M, Di Francesco M, Markowich PA, Wolfram M-T (2014) Mean field games with nonlinear mobilities in pedestrian dynamics. Discrete Contin Dyn Syst Ser B 19(5):1311–1333MathSciNetCrossRefGoogle Scholar
  10. 10.
    Djehiche B, Tcheukam A, Tembine H A mean-field game of evacuation in multi-level building. IEEE Trans Automat Control. (to appear)
  11. 11.
    Faure S, Maury B (2015) Crowd motion from the granular standpoint. Math Models Methods Appl Sci 25(3):463–493MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fleming WH, Soner HM (2006) Controlled Markov processes and viscosity solutions, volume 25 of Stochastic modelling and applied probability, 2nd edn. Springer, New YorkGoogle Scholar
  13. 13.
    Hughes RL (2003) The flow of human crowds. In: Annual review of fluid mechanics, volume 35 of Annual Review of Fluid Mechanics. Annual Reviews, Palo Alto, CA, pp 169–182Google Scholar
  14. 14.
    Krusell P, Smith AA Jr (1998) Income and wealth heterogeneity in the macroeconomy. J Polit Econ 106(5):867–896CrossRefGoogle Scholar
  15. 15.
    Lachapelle A, Wolfram M-T (2011) On a mean field game approach modeling congestion and aversion in pedestrian crowds. Transp Res Part B: Methodol 45(10):1572–1589CrossRefGoogle Scholar
  16. 16.
    Lasry J-M, Lions P-L (2006) Jeux à champ moyen. I. Le cas stationnaire. C R Math Acad Sci Paris 349:619–625MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lasry J-M, Lions P-L (2006) Jeux à champ moyen. II. Horizon fini et contrôle optimal. C R Math Acad Sci Paris 343(10):679–684MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lasry J-M, Lions P-L (2007) Mean field games. Jpn J Math 2(1):229–260MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lions P-L (2007–2011) Cours du Collège de France.
  20. 20.
  21. 21.
    van der Vorst HA (1992) Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J Sci Statist Comput 13(2):631–644MathSciNetCrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Université Paris Diderot, Sorbonne Paris Cité, Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRSParisFrance
  2. 2.CEREMADEUniversité de Paris-DauphineParisFrance

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