Uniqueness of Solutions in Mean Field Games with Several Populations and Neumann Conditions

  • Martino BardiEmail author
  • Marco Cirant
Part of the Springer INdAM Series book series (SINDAMS, volume 28)


We study the uniqueness of solutions to systems of PDEs arising in Mean Field Games with several populations of agents and Neumann boundary conditions. The main assumption requires the smallness of some data, e.g., the length of the time horizon. This complements the existence results for MFG models of segregation phenomena introduced by the authors and Achdou. An application to robust Mean Field Games is also given.


Mean field games Multi-populations Uniqueness Neumann boundary conditions Robust mean field games 


  1. 1.
    Achdou, Y., Bardi, M., Cirant, M.: Mean field games models of segregation. Math. Models Methods Appl. Sci. 27, 75–113 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ambrose, D.M.: Strong solutions for time-dependent mean field games with non-separable Hamiltonians. J. Math. Pures Appl. (9) 113, 141–154 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bardi, M.: Explicit solutions of some linear-quadratic mean field games. Netw. Heterog. Media 7, 243–261 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bardi, M., Feleqi, E.: Nonlinear elliptic systems and mean field games. Nonlinear Differ. Equ. Appl. 23, 23–44 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bardi, M., Fischer, M.: On non-uniqueness and uniqueness of solutions in some finite-horizon mean field games. ESAIM Control Optim. Calc. Var.
  6. 6.
    Bardi, M., Priuli, F.S.: Linear-quadratic N-person and mean-field games with ergodic cost. SIAM J. Control Optim. 52, 3022–3052 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bauso, D., Tembine, H., Basar, T.: Robust mean field games. Dyn. Games Appl. 6, 277–303 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Briani, A., Cardaliaguet, P.: Stable solutions in potential mean field game systems. Nonlinear Differ. Equ. Appl. 25(1), 26 pp., Art. 1 (2018)Google Scholar
  9. 9.
    Cardaliaguet, P.: Notes on Mean Field Games (from P-L. Lions’ lectures at Collège de France) (2010)Google Scholar
  10. 10.
    Cardaliaguet, P., Porretta, A., Tonon, D.: A segregation problem in multi-population mean field games. Ann. I.S.D.G. 15, 49–70 (2017)Google Scholar
  11. 11.
    Carmona, R., Delarue, F.: Probabilistic Theory of Mean Field Games with Applications I - II (Springer, New York, 2018)zbMATHGoogle Scholar
  12. 12.
    Cirant, M.: Multi-population mean field games systems with Neumann boundary conditions. J. Math. Pures Appl. 103, 1294–1315 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cirant, M.: Stationary focusing mean-field games. Commun. Partial Differ. Equ. 41, 1324–1346 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cirant, M., Tonon, D.: Time-dependent focusing mean-field games: the sub-critical case. J. Dyn. Differ. Equ. (2018).
  15. 15.
    Cirant, M., Verzini, G.: Bifurcation and segregation in quadratic two-populations mean field games systems. ESAIM Control Optim. Calc. Var. 23, 1145–1177 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gomes, D., Saude, J.: Mean field games models: a brief survey. Dyn. Games Appl. 4, 110–154 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gomes, D.A., Mohr, J., Souza, R.R.: Continuous time finite state mean field games. Appl. Math. Optim. 68, 99–143 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gomes, D., Nurbekyan, L., Pimentel, E.: Economic models and mean-field games theory (IMPA Mathematical Publications, Instituto Nacional de Matemática Pura e Aplicada , Rio de Janeiro, 2015)Google Scholar
  19. 19.
    Gomes, D., Pimentel, E., Voskanyan, V.: Regularity Theory for Mean-Field Game Systems (Springer, New York, 2016)CrossRefGoogle Scholar
  20. 20.
    Gomes, D., Nurbekyan, L., Prazeres, M.: One-dimensional stationary mean-field games with local coupling. Dyn. Games Appl. 8, 315–351 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Guéant, O.: A reference case for mean field games models. J. Math. Pures Appl. (9) 92, 276–294 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Guéant, O., Lasry, J.-M., Lions, P.-L.: Mean field games and applications, in Carmona, R.A., et al. (eds.) Paris-Princeton Lectures on Mathematical Finance 2010. Lecture Notes in Mathematics, 2003 (Springer, Berlin, 2011), pp. 205–266CrossRefGoogle Scholar
  23. 23.
    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 , 221–251 (2006)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Huang, M., Caines, P.E., Malhamé, R.P.: Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized 𝜖-Nash equilibria. IEEE Trans. Automat. Control 52 , 1560–1571 (2007)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Huang, M., Caines, P.E., Malhamé, R.P.: An invariance principle in large population stochastic dynamic games. J. Syst. Sci. Complex. 20, 162–172 (2007)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ladyzenskaja, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and quasilinear equations of parabolic type. Translations of Mathematical Monographs, vol. 23 (American Mathematical Society, Providence, 1968)Google Scholar
  27. 27.
    Lasry, J.-M., Lions, P.-L.: Jeux á champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343, 619–625 (2006)CrossRefGoogle Scholar
  28. 28.
    Lasry, J.-M., Lions, P.-L.: Jeux á champ moyen. II. Horizon fini et controle optimal. C. R. Math. Acad. Sci. Paris 343, 679–684 (2006)CrossRefGoogle Scholar
  29. 29.
    Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2, 229–260 (2007)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Lieberman, G.M.: Second Order Parabolic Differential Equations (World Scientific Publishing Co., Inc., River Edge, 1996)CrossRefGoogle Scholar
  31. 31.
    Lions, P.-L.: Lectures at Collège de France 2008-9Google Scholar
  32. 32.
    Moon, J., Başar, T.: Linear quadratic risk-sensitive and robust mean field games. IEEE Trans. Automat. Control 62(3), 1062–1077 (2016)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations (Prentice-Hall, Inc., Englewood Cliffs, 1967)zbMATHGoogle Scholar
  34. 34.
    Schelling, T.C.: Micromotives and Macrobehavior (Norton, New York, 1978)Google Scholar
  35. 35.
    Tran, H.V.: A note on nonconvex mean field games. Minimax Theory Appl. (2018, to appear). arXiv:1612.04725Google Scholar
  36. 36.
    Wang, B.-C., Zhang, J.-F.: Mean field games for large-population multiagent systems with Markov jump parameters. SIAM J. Control Optim. 50(4), 2308–2334 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics “T. Levi-Civita”University of PadovaPadovaItaly

Personalised recommendations