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Control and nash games with mean field effect

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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.

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References

  1. Bensoussan, A., Bulicěk, M. and Frehse, J., Existence and Compactness for weak solutions to Bellman systems with critical growth, Discrete Contin. Dyn. Syst., Ser. B, 17(6), 2012, 1–21.

    Article  MathSciNet  Google Scholar 

  2. Bensoussan, A. and Frehse, J., Regularity results for nonlinear elliptic systems and applications, Appl. Math. Sci., 151, Springer-Verlag, Berlin, 2002.

    Google Scholar 

  3. Bensoussan, A. and Frehse, J., Ergodic Bellman systems for stochastic games, Markus Feestricht Volume Differential Equations, Dynamical Systems and Control Sciences, Lecture Notes in Pure and Appl. Math., K. D. Elworty, W., Norrie Everitt, E. Bruce Lee, M. Dekker (eds.), Vol. 152, 1993, 411–421.

  4. Bensoussan, A. and Frehse, J., Ergodic Bellman systems for stochastic games in arbitrary dimension, Proc. Royal Soc., London, Math. Phy. Sci. A, 449, 1995, 65–67.

    Article  MathSciNet  MATH  Google Scholar 

  5. Bensoussan, A. and Frehse, J., Smooth solutions of systems of quasilinear parabolic equations, ESAIM: Control, Optimization and Calculus of Variations, 8, 2002, 169–193.

    Article  MathSciNet  MATH  Google Scholar 

  6. Bensoussan, A., Frehse, J. and Vogelgesang, J., Systems of Bellman equations to stochastic differential games with noncompact coupling, Discrete Contin. Dyn. Syst., Ser. A, 274, 2010, 1375–1390.

    MathSciNet  Google Scholar 

  7. Bensoussan, A., Frehse, J. and Vogelgesang, J., Nash and stackelberg differential games, Chin. Ann. Math., 33B(3), 2012, 317–332.

    Article  MathSciNet  Google Scholar 

  8. Bulicěk, M. and Frehse, J., On nonlinear elliptic Bellman systems for a class of stochastic differential games in arbitrary dimension, Math. Models Methods Appl. Sci., 21(1), 2011, 215–240.

    Article  MathSciNet  MATH  Google Scholar 

  9. Guéant, O., Lasry, J. M. and Lions, P. L., Mean field games and applications, Paris-Princeton Lectures on Mathematical Sciences 2010, A. R. Carmona, et al. (eds.), 2011, 205–266.

    Chapter  Google Scholar 

  10. Huang, M., Caines, P. E. and 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(9), 2007, 1560–1571.

    Article  MathSciNet  Google Scholar 

  11. Huang, M., Caines, P. E. and Malhamé, R. P., An invariance principle in large population stochastic dynamic games, J. Syst. Sci. Comp., 20(2), 2007, 162–172.

    Article  Google Scholar 

  12. Ladyzhenskaya, O. A. and Uraltseva, N. N., Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.

    MATH  Google Scholar 

  13. Lasry, J. M. and Lions, P. L., Jeux champ moyen I-Le cas stationnaire, Comptes Rendus de l’Académie des Sciences, Series I, 343, 2006, 619–625.

    MathSciNet  MATH  Google Scholar 

  14. Lasry, J. M. and Lions, P. L., Jeux à champ moyen II-Horizn fini et contrôle optimal, Comptes Rendus de l’Académie des Sciences, Series I, 343, 2006, 679–684.

    MathSciNet  MATH  Google Scholar 

  15. Lasry, J. M. and Lions, P. L., Mean field games, Japanese J. Math., 2(1), 2007, 229–260.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. International Center for Decision and Risk Analysis, School of Management, University of Texas-Dallas, Dallas, France

    Alain Bensoussan

  2. School of Business, the Hong Kong Polytechnic University, Hong Kong, China

    Alain Bensoussan

  3. Graduate Department of Financial Engineering, Ajou University, Ajou, Korea

    Alain Bensoussan

  4. Institute for Applied Mathematics, University of Bonn, Bonn, Germany

    Jens Frehse

Authors
  1. Alain Bensoussan
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  2. Jens Frehse
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Corresponding author

Correspondence to Alain Bensoussan.

Additional information

In honor of the scientific heritage of Jacques-Louis Lions

Project supported by the WCU World Class University program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (No. R31-20007) and the Research Grants Council of HKSAR (No. PolyU 5001/11P).

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Bensoussan, A., Frehse, J. Control and nash games with mean field effect. Chin. Ann. Math. Ser. B 34, 161–192 (2013). https://doi.org/10.1007/s11401-013-0767-y

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  • DOI: https://doi.org/10.1007/s11401-013-0767-y

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