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Control and Nash Games with Mean Field Effect

Chapter

Abstract

Mean field theory has raised a lot of interest in the recent years (see in particular the results of Lasry-Lions in 2006 and 2007, of Gueant-Lasry-Lions in 2011, of Huang-Caines-Malham in 2007 and many others). There are a lot of applications. In general, the applications concern approximating an infinite number of players with common behavior by a representative agent. This agent has to solve a control problem perturbed by a field equation, representing in some way the behavior of the average infinite number of agents. This approach does not lead easily to the problems of Nash equilibrium for a finite number of players, perturbed by field equations, unless one considers averaging within different groups, which has not been done in the literature, and seems quite challenging. In this paper, the authors approach similar problems with a different motivation which makes sense for control and also for differential games. Thus the systems of nonlinear partial differential equations with mean field terms, which have not been addressed in the literature so far, are considered here.

Keywords

Mean field Dynamic programming Nash games Equilibrium Calculus of variations 

Mathematics Subject Classification

49L20 

References

  1. 1.
    Bensoussan, A., Bulicěk, M., Frehse, J.: Existence and Compactness for weak solutions to Bellman systems with critical growth. Discrete Contin. Dyn. Syst., Ser. B 17(6), 1–21 (2012) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bensoussan, A., Frehse, J.: Regularity Results for Nonlinear Elliptic Systems and Applications. Appl. Math. Sci., vol. 151. Springer, Berlin (2002) CrossRefzbMATHGoogle Scholar
  3. 3.
    Bensoussan, A., Frehse, J.: Ergodic Bellman systems for stochastic games. In: Elworty, K.D., Norrie Everitt, W., Bruce Lee, E., Dekker, M. (eds.) Markus Feestricht Volume Differential Equations, Dynamical Systems and Control Sciences. Lecture Notes in Pure and Appl. Math., vol. 152, pp. 411–421 (1993) Google Scholar
  4. 4.
    Bensoussan, A., Frehse, J.: Ergodic Bellman systems for stochastic games in arbitrary dimension. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 449, 65–67 (1995) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bensoussan, A., Frehse, J.: Smooth solutions of systems of quasilinear parabolic equations. ESAIM, COCV 8, 169–193 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bensoussan, A., Frehse, J., Vogelgesang, J.: Systems of Bellman equations to stochastic differential games with noncompact coupling. Discrete Contin. Dyn. Syst., Ser. A 274, 1375–1390 (2010) MathSciNetGoogle Scholar
  7. 7.
    Bensoussan, A., Frehse, J., Vogelgesang, J.: Nash and Stackelberg differential games. Chin. Ann. Math. 33B(3), 317–332 (2012) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bulicěk, M., Frehse, J.: On nonlinear elliptic Bellman systems for a class of stochastic differential games in arbitrary dimension. Math. Models Methods Appl. Sci. 21(1), 215–240 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Guéant, O., Lasry, J.M., Lions, P.L.: Mean field games and applications. In: Carmona, A.R., et al. (eds.) Paris-Princeton Lectures on Mathematical Sciences 2010, pp. 205–266 (2011) CrossRefGoogle Scholar
  10. 10.
    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. Autom. Control 52(9), 1560–1571 (2007) CrossRefGoogle Scholar
  11. 11.
    Huang, M., Caines, P.E., Malhamé, R.P.: An invariance principle in large population stochastic dynamic games. J. Syst. Sci. Complex. 20(2), 162–172 (2007) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ladyzhenskaya, O.A., Uraltseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968) zbMATHGoogle Scholar
  13. 13.
    Lasry, J.M., Lions, P.L.: Jeux champ moyen I. Le cas stationnaire. C. R. Acad. Sci., Ser. 1 Math. 343, 619–625 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lasry, J.M., Lions, P.L.: Jeux à champ moyen II. Horizn fini et contrôle optimal. C. R. Acad. Sci., Ser. 1 Math. 343, 679–684 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lasry, J.M., Lions, P.L.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.International Center for Decision and Risk Analysis, School of ManagementUniversity of Texas-DallasRichardsonUSA
  2. 2.School of BusinessThe Hong Kong Polytechnic UniversityHong KongChina
  3. 3.Graduate Department of Financial EngineeringAjou UniversitySuwonKorea
  4. 4.Institute for Applied MathematicsUniversity of BonnBonnGermany

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