Advertisement

Journal of Nonlinear Science

, Volume 24, Issue 1, pp 93–115 | Cite as

Large-Scale Dynamics of Mean-Field Games Driven by Local Nash Equilibria

  • Pierre DegondEmail author
  • Jian-Guo Liu
  • Christian Ringhofer
Article

Abstract

We introduce a new mean field kinetic model for systems of rational agents interacting in a game-theoretical framework. This model is inspired from non-cooperative anonymous games with a continuum of players and Mean-Field Games. The large time behavior of the system is given by a macroscopic closure with a Nash equilibrium serving as the local thermodynamic equilibrium. An application of the presented theory to a social model (herding behavior) is discussed.

Keywords

Non-cooperative non-atomic anonymous games Continuum of players Rational agent Best-reply scheme Local Nash equilibrium Potential games Social herding behavior Coarse graining procedure Macroscopic closure Kinetic theory 

Mathematics Subject Classification

91A10 91A13 91A40 82C40 82C21 

Notes

Acknowledgements

This work has been supported by KI-Net NSF RNMS grant No. 1107291. JGL and CR are grateful for the opportunity to stay and work at the Institut de Mathématiques de Toulouse in fall 2012, under sponsorship from Centre National de la Recherche Scientifique and University Paul-Sabatier. The authors wish to thank A. Blanchet from University Toulouse 1 Capitole for enlightening discussions.

References

  1. Appert-Rolland, C., Degond, P., Pettre, J., Theraulaz, G.: A hierarchy of macroscopic crowd models based on behavioral heuristics. Preprint (2013a) Google Scholar
  2. Appert-Rolland, C., Degond, P., Pettre, J., Theraulaz, G.: Macroscopic pedestrian models based on synthetic vision. Preprint (2013b) Google Scholar
  3. Aumann, R.: Existence of competitive equilibria in markets with a continuum of traders. Econometrica 32, 39–50 (1964) CrossRefzbMATHMathSciNetGoogle Scholar
  4. Blanchet, A.: Variational methods applied to biology and economics. Dissertation for the Habilitation, University Toulouse 1 Capitole, December (2012) Google Scholar
  5. Blanchet, A., Carlier, G.: Optimal transport and Cournot–Nash equilibria. Preprint (2012) Google Scholar
  6. Blanchet, A., Mossay, P., Santambrogio, F.: Existence and uniqueness of equilibrium for a spatial model of social interactions. Preprint (2012) Google Scholar
  7. Cardaliaguet, P.: Notes on Mean Field Games (from P.-L. Lions’ lectures at Collège de France) (2012) Google Scholar
  8. Degond, P., Motsch, S.: Continuum limit of self-driven particles with orientation interaction. Math. Models Methods Appl. Sci. 18(Suppl.), 1193–1215 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  9. Degond, P., Liu, J.-G., Mieussens, L.: Macroscopic fluid models with localized kinetic upscaling effects. SIAM J. Multiscale Model. Simul. 5, 940–979 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  10. Degond, P., Frouvelle, A., Liu, J.-G.: Macroscopic limits and phase transition in a system of self-propelled particles. J. Nonlinear Sci. 23, 427–456 (2013a) CrossRefzbMATHMathSciNetGoogle Scholar
  11. Degond, P., Liu, J.-G., Ringhofer, C.: Evolution of the distribution of wealth in economic neighborhood by local Nash equilibrium closure (2013b, submitted) Google Scholar
  12. Degond, P., Liu, J.-G., Motsch, S., Panferov, V.: Hydrodynamic models of self-organized dynamics: derivation and existence theory. Methods Appl. Anal. (2013c, to appear) Google Scholar
  13. E, W.: Principles of Multiscale Modeling. Cambridge University Press, Cambridge (2011) zbMATHGoogle Scholar
  14. Frouvelle, A., Liu, J.G.: Dynamics in a kinetic model of oriented particles with phase transition. SIAM J. Math. Anal. 44, 791–826 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  15. Green, E.J., Porter, R.H.: Noncooperative collusion under imperfect price information. Econometrica 52, 87–100 (1984) CrossRefzbMATHGoogle Scholar
  16. Hsu, E.P.: Stochastic Analysis on Manifolds. Graduate Series in Mathematics. American Mathematical Society, Providence (2002) zbMATHGoogle Scholar
  17. Huang, K.: Statistical Mechanics, 2nd edn. Wiley, New York (1987) zbMATHGoogle Scholar
  18. Jordan, R., Kinderlehrer, D., Otto, F.: Free energy and the Fokker-Planck equation. Physica D 107, 265–271 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  19. Konishi, H., Le Breton, M., Weber, S.: Pure strategy Nash equilibrium in a group formation game with positive externalities. Games Econ. Behav. 21, 161–182 (1997) CrossRefzbMATHGoogle Scholar
  20. Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2, 229–260 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  21. Lazear, E.P., Rosen, S.: Rank-order tournaments as optimum labor contracts. Preprint, National Bureau of Economic Research Cambridge, USA (1979) Google Scholar
  22. Lighthill, M.J., Whitham, J.B.: On kinematic waves. I: flow movement in long rivers. II: a theory of traffic flow on long crowded roads. Proc. R. Soc. A 229, 281–345 (1955) CrossRefzbMATHMathSciNetGoogle Scholar
  23. Mas-Colell, A.: On a theorem of Schmeidler. J. Math. Econ. 13, 201–206 (1984) CrossRefzbMATHMathSciNetGoogle Scholar
  24. Monderer, D., Shapley, L.S.: Potential games. Games Econ. Behav. 14, 124–143 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  25. Nash, J.F.: Equilibrim points in n-person games. Proc. Natl. Acad. Sci. USA 36, 48–49 (1950) CrossRefzbMATHMathSciNetGoogle Scholar
  26. Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. Int. J. Game Theory 2, 65–67 (1973) CrossRefzbMATHGoogle Scholar
  27. Schmeidler, D.: Equilibrium points of nonatomic games. J. Stat. Phys. 7, 295–300 (1973) CrossRefzbMATHMathSciNetGoogle Scholar
  28. Schultz, T.W.: Investment in human capital. Am. Econ. Rev. 51, 1–17 (1961) Google Scholar
  29. Shapiro, N.Z., Shapley, L.S.: Values of large games, I: a limit theorem. Math. Oper. Res. 3, 1–9 (1978) CrossRefzbMATHMathSciNetGoogle Scholar
  30. Sznitman, A.-S.: Topics in propagation of chaos. In: École d’été de Probabilités de Saint-Flour XIX. Lecture Notes in Math., vol. 1464, pp. 165–251. Springer, Berlin (1991) Google Scholar
  31. Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226–1229 (1995) CrossRefGoogle Scholar
  32. Villani, C.: Topics in Optimal Transportation. AMS Graduate Studies in Mathematics, vol. 58. AMS, Providence (2003) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Pierre Degond
    • 1
    • 2
    Email author
  • Jian-Guo Liu
    • 3
  • Christian Ringhofer
    • 4
  1. 1.UPS, INSA, UT1, UTM, Institut de Mathématiques de ToulouseUniversité de ToulouseToulouseFrance
  2. 2.Institut de Mathématiques de Toulouse UMR 5219CNRSToulouseFrance
  3. 3.Department of Physics and Department of MathematicsDuke UniversityDurhamUSA
  4. 4.School of Mathematics and Statistical SciencesArizona State UniversityTempeUSA

Personalised recommendations