Dynamic Games and Applications

, Volume 7, Issue 4, pp 657–682 | Cite as

Two Numerical Approaches to Stationary Mean-Field Games

  • Noha Almulla
  • Rita Ferreira
  • Diogo Gomes


Here, we consider numerical methods for stationary mean-field games (MFG) and investigate two classes of algorithms. The first one is a gradient-flow method based on the variational characterization of certain MFG. The second one uses monotonicity properties of MFG. We illustrate our methods with various examples, including one-dimensional periodic MFG, congestion problems, and higher-dimensional models.


Mean-field games Monotone schemes Numerical methods 


  1. 1.
    Achdou Y (2013) Finite difference methods for mean field games. In: Hamilton-Jacobi equations: approximations, numerical analysis and applications. Lecture notes in mathematics, vol 2074. Springer, Heidelberg, pp. 1–47. doi: 10.1007/978-3-642-36433-4_1
  2. 2.
    Achdou Y, Camilli F, Capuzzo-Dolcetta I (2012) Mean field games: numerical methods for the planning problem. SIAM J Control Optim 50(1):77–109MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Achdou Y, Capuzzo-Dolcetta I (2010) Mean field games: numerical methods. SIAM J Numer Anal 48(3):1136–1162MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Achdou Y, Perez V (2012) Iterative strategies for solving linearized discrete mean field games systems. Netw Heterog Media 7(2):197–217MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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):161–186. doi: 10.1137/15M1015455 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barles G, Souganidis PE (1991) Convergence of approximation schemes for fully nonlinear second order equations. Asymptot Anal 4(3):271–283MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bensoussan A, Frehse J, Yam P (2013) Mean field games and mean field type control theory. Springer Briefs in Mathematics. Springer, New YorkCrossRefzbMATHGoogle Scholar
  8. 8.
    Cacace S, Camilli F (2016) Ergodic problems for Hamilton-Jacobi equations: yet another but efficient numerical method (preprint)Google Scholar
  9. 9.
    Camilli F, Festa A, Schieborn D (2013) An approximation scheme for a Hamilton–Jacobi equation defined on a network. Appl Numer Math 73:33–47MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Camilli F, Silva F (2012) A semi-discrete approximation for a first order mean field game problem. Netw Heterog Media 7(2):263–277MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cardaliaguet P (2011) Notes on mean-field gamesGoogle Scholar
  12. 12.
    Cardaliaguet P, Graber PJ (2015) Mean field games systems of first order. ESAIM Control Optim Calc Var 21(3):690–722MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cardaliaguet P, Graber PJ, Porretta A, Tonon D (2015) Second order mean field games with degenerate diffusion and local coupling. NoDEA Nonlinear Differ Equ Appl 22(5):1287–1317MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Carlini E, Silva FJ (2014) A fully discrete semi-Lagrangian scheme for a first order mean field game problem. SIAM J Numer Anal 52(1):45–67MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Evans LC (1998) Partial differential equations. Graduate Studies in Mathematics. American Mathematical Society, ProvidenceGoogle Scholar
  16. 16.
    Evans LC (2003) Some new PDE methods for weak KAM theory. Calc Var Partial Differ Equ 17(2):159–177MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Evans LC (2009) Further PDE methods for weak KAM theory. Calc Var Partial Differ Equ 35(4):435–462MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ferreira R, Gomes D (2016) Existence of weak solutions for stationary mean-field games through weak solutions (preprint)Google Scholar
  19. 19.
    Gomes D, Iturriaga R, Sánchez-Morgado H, Yu Y (2010) Mather measures selected by an approximation scheme. Proc Am Math Soc 138(10):3591–3601MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gomes D, Nurbekyan L, Prazeres M (2016) Explicit solutions of one-dimensional, first-order, stationary mean-field games with congestion. Proceeding of IEEE-CDC (to appear)Google Scholar
  21. 21.
    Gomes D, Nurbekyan L, Prazeres M (2016) One-dimensional, stationary mean-field games with a local coupling (preprint)Google Scholar
  22. 22.
    Gomes D, Patrizi S, Voskanyan V (2014) On the existence of classical solutions for stationary extended mean field games. Nonlinear Anal 99:49–79MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gomes D, Pimentel E (2016) Local regularity for mean-field games in the whole space. Minimax Theory Appl 1(1):65–82MathSciNetzbMATHGoogle Scholar
  24. 24.
    Gomes D, Pimentel E, Sánchez-Morgado H (2016) Time-dependent mean-field games in the superquadratic case. ESAIM Control Optim Calc Var 22(2):562–580. doi: 10.1051/cocv/2015029
  25. 25.
    Gomes D, Pimentel E, Sánchez-Morgado H (2015) Time-dependent mean-field games in the subquadratic case. Commun Partial Differ Equ 40(1):40–76MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Gomes D, Pimentel E, Voskanyan V (2016) Regularity theory for mean-field game systems. Springer, Berlin, p x \(+\) 118. doi: 10.1007/978-3-319-38934-9
  27. 27.
    Gomes D, Pires GE, Sánchez-Morgado H (2012) A-priori estimates for stationary mean-field games. Netw Heterog Media 7(2):303–314MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Gomes D, Sánchez Morgado H (2014) A stochastic Evans-Aronsson problem. Trans Am Math Soc 366(2):903–929MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Gomes D, Velho RM, Wolfram M-T (2014) Dual two-state mean-field games. In: Proceedings CDC 2014Google Scholar
  30. 30.
    Gomes D, Velho RM, Wolfram M-T (2014) Socio-economic applications of finite state mean field games. Philos Trans R Soc Lond Ser A Math Phys Eng Sci 372(2028):20130405, 18MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Gomes D, Voskanyan V (2015) Short-time existence of solutions for mean-field games with congestion. J Lond Math Soc 92(3):778–799. doi: 10.1112/jlms/jdv052
  32. 32.
    Gomes DA, Mitake H (2015) Existence for stationary mean-field games with congestion and quadratic Hamiltonians. NoDEA Nonlinear Differ Equ Appl 22(6):1897–1910MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Gomes DA, Patrizi S (2015) Obstacle mean-field game problem. Interfaces Free Bound 17(1):55–68MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Gomes DA, Pimentel E (2015) Time-dependent mean-field games with logarithmic nonlinearities. SIAM J Math Anal 47(5):3798–3812MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Gomes DA, Saúde J (2014) Mean field games models—a brief survey. Dyn Games Appl 4(2):110–154MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Guéant O (2012) Mean field games equations with quadratic Hamiltonian: a specific approach. Math Models Methods Appl Sci 22(9):1250022, 37MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Guéant O (2012) Mean field games with a quadratic Hamiltonian: a constructive scheme. In: Advances in dynamic games, vol 12 of Ann Int Soc Dyn Games. Birkhäuser/Springer, New York, pp 229–241Google Scholar
  38. 38.
    Guéant O (2012) New numerical methods for mean field games with quadratic costs. Netw Heterog Media 7(2):315–336MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Huang M, Caines PE, Malhamé RP (2007) Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized \(\epsilon \)-Nash equilibria. IEEE Trans Automat Control 52(9):1560–1571MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Huang M, Malhamé RP, Caines PE (2006) Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle. Commun Inf Syst 6(3):221–251MathSciNetzbMATHGoogle Scholar
  41. 41.
    Lasry J-M, Lions P-L (2006) Jeux à champ moyen. I. Le cas stationnaire. C R Math Acad Sci Paris 343(9):619–625MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    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):679–684MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Lasry J-M, Lions P-L (2007) Mean field games. Jpn J Math 2(1):229–260MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Mészáros AR, Silva FJ (2015) A variational approach to second order mean field games with density constraints: the stationary case. J Math Pures Appl (9) 104(6):1135–1159MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Oberman AM (2006) Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton–Jacobi equations and free boundary problems. SIAM J Numer Anal 44(2):879–895 (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Pimentel E, Voskanyan V (2016) Regularity for second-order stationary mean-field games. Indiana Univ Math J (to appear)Google Scholar
  47. 47.
    Porretta A (2015) Weak solutions to Fokker–Planck equations and mean field games. Arch Ration Mech Anal 216(1):1–62MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Porretta Alessio (2014) On the planning problem for the mean field games system. Dyn Games Appl 4(2):231–256MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Santambrogio F (2012) A modest proposal for MFG with density constraints. Netw Heterog Media 7(2):337–347MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Voskanyan VK (2013) Some estimates for stationary extended mean field games. Dokl Nats Akad Nauk Armen 113(1):30–36MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.College of ScienceUniversity of DammamDammamSaudi Arabia
  2. 2.CEMSE Division and KAUST SRI, Center for Uncertainty Quantification in Computational Science and EngineeringKing Abdullah University of Science and Technology (KAUST)ThuwalSaudi Arabia

Personalised recommendations