Numerical Methods for Finite-State Mean-Field Games Satisfying a Monotonicity Condition


Here, we develop numerical methods for finite-state mean-field games (MFGs) that satisfy a monotonicity condition. MFGs are determined by a system of differential equations with initial and terminal boundary conditions. These non-standard conditions make the numerical approximation of MFGs difficult. Using the monotonicity condition, we build a flow that is a contraction and whose fixed points solve both for stationary and time-dependent MFGs. We illustrate our methods with a MFG that models the paradigm-shift problem.

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  1. 1.

    Achdou, Y.: Finite difference methods for mean field games. In: Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, Lecture Notes in Mathematics, vol. 2074, pp. 1–47. Springer, Heidelberg (2013).

  2. 2.

    Achdou, Y., Camilli, F., Capuzzo-Dolcetta, I.: Mean field games: numerical methods for the planning problem. SIAM J. Control Optim. 50(1), 77–109 (2012)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Achdou, Y., Capuzzo-Dolcetta, I.: Mean field games: numerical methods. SIAM J. Numer. Anal. 48(3), 1136–1162 (2010)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Achdou, Y., Cirant, M., Bardi, M.: Mean field games models of segregation. Math. Models Methods Appl. Sci. 27(1), 75–113 (2017)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Achdou, Y., Perez, V.: Iterative strategies for solving linearized discrete mean field games systems. Netw. Heterog. Media 7(2), 197–217 (2012)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Al-Mulla, N., Ferreira, R., Gomes, D.: Two numerical approaches to stationary mean-field games. Dyn. Games Appl. 7(4), 657–682 (2016)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Basna, R., Hilbert, A., Kolokoltsov, V.N.: An epsilon-Nash equilibrium for non-linear Markov games of mean-field-type on finite spaces. Commun. Stoch. Anal. 8(4), 449–468 (2014)

    MathSciNet  Google Scholar 

  8. 8.

    Besancenot, D., Dogguy, H.: Paradigm shift: a mean-field game approach. Bull. Econ. Res. 67(3), 289–302 (2015).

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Briceño Arias, L.M., Kalise, D., Silva, F.J.: Proximal methods for stationary mean field games with local couplings. SIAM J. Control Optim. 56(2), 801–836 (2018).

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Cardaliaguet, P., Lasry, J.-M., Lions, P.-L., Porretta, A.: Long time average of mean field games. Netw. Heterog. Media 7(2), 279–301 (2012)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Cardaliaguet, P., Lasry, J.-M., Lions, P.-L., Porretta, A.: Long time average of mean field games with a nonlocal coupling. SIAM J. Control Optim. 51(5), 3558–3591 (2013)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Carlini, E., Silva, F.J.: A fully discrete semi-Lagrangian scheme for a first order mean field game problem. SIAM J. Numer. Anal. 52(1), 45–67 (2014)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Carlini, E., Silva, F.J.: A semi-Lagrangian scheme for a degenerate second order mean field game system. Discret. Contin. Dyn. Syst. 35(9), 4269–4292 (2015)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Ferreira, R., Gomes, D.: On the convergence of finite state mean-field games through \(\Gamma \)-convergence. J. Math. Anal. Appl. 418(1), 211–230 (2014)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Ferreira, R., Gomes, D.: Existence of weak solutions for stationary mean-field games through variational inequalities. Preprint (2016)

  16. 16.

    Gomes, D., Mohr, J., Souza, R.R.: Discrete time, finite state space mean field games. J. Math. Pures Appl. 93(2), 308–328 (2010)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Gomes, D., Mohr, J., Souza, R.R.: Continuous time finite state mean-field games. Appl. Math. Optim. 68(1), 99–143 (2013)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Gomes, D., Pimentel, E., Voskanyan, V.: Regularity theory for mean-field game systems. Springer Briefs in Mathematics. Springer, Cham (2016)

  19. 19.

    Gomes, D., Saúde, J.: Mean field games models—a brief survey. Dyn. Games Appl. 4(2), 110–154 (2014)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Gomes, D., Velho, R.M., Wolfram, M.-T.: Dual two-state mean-field games. In: Proceedings CDC 2014 (2014)

  21. 21.

    Gomes, D., Velho, R.M., Wolfram, M.-T.: Socio-economic applications of finite state mean field games. In: Proceedings of the Royal Society A, Bd. 372(2028(S.)) (2014)

  22. 22.

    Guéant, O.: Existence and uniqueness result for mean field games with congestion effect on graphs. Appl. Math. Optim. 72(2), 291–303 (2015).

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Guéant, O.: From infinity to one: the reduction of some mean field games to a global control problem. Preprint (2011)

  24. 24.

    Huang, M., Caines, P.E., Malhamé, R.P.: Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized \(\epsilon \)-Nash equilibria. IEEE Trans. Autom. Control 52(9), 1560–1571 (2007)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Huang, M., Malhamé, R.P., Caines, P.E.: Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–251 (2006)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Kolokoltsov, V.N., Malafeyev, O.A.: Mean-field-game model of corruption. Dyn. Games Appl. 7(1), 34–47 (2017)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343(9), 619–625 (2006)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343(10), 679–684 (2006)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Mészáros, A., Silva, F.J.: On the variational formulation of some stationary second-order mean field games systems. SIAM J. Math. Anal. 50(1), 1255–1277 (2018).

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Mészáros, A.R., Silva, F.J.: A variational approach to second order mean field games with density constraints: the stationary case. J. Math. Pures Appl. (9) 104(6), 1135–1159 (2015)

    MathSciNet  Article  Google Scholar 

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Corresponding author

Correspondence to Diogo A. Gomes.

Additional information

D. Gomes was partially supported by KAUST baseline and start-up funds and KAUST SRI, Center for Uncertainty Quantification in Computational Science and Engineering. J. Saúde was partially supported by FCT/Portugal through the CMU-Portugal Program.

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Gomes, D.A., Saúde, J. Numerical Methods for Finite-State Mean-Field Games Satisfying a Monotonicity Condition. Appl Math Optim 83, 51–82 (2021).

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  • Mean-field games
  • Finite state problems
  • Monotonicity methods

Mathematics Subject Classification

  • 91A13
  • 91A10
  • 49M30