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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Appert-Rolland, C., Degond, P., Pettre, J., Theraulaz, G.: A hierarchy of macroscopic crowd models based on behavioral heuristics. Preprint (2013a)
Appert-Rolland, C., Degond, P., Pettre, J., Theraulaz, G.: Macroscopic pedestrian models based on synthetic vision. Preprint (2013b)
Aumann, R.: Existence of competitive equilibria in markets with a continuum of traders. Econometrica 32, 39–50 (1964)
Blanchet, A.: Variational methods applied to biology and economics. Dissertation for the Habilitation, University Toulouse 1 Capitole, December (2012)
Blanchet, A., Carlier, G.: Optimal transport and Cournot–Nash equilibria. Preprint (2012)
Blanchet, A., Mossay, P., Santambrogio, F.: Existence and uniqueness of equilibrium for a spatial model of social interactions. Preprint (2012)
Cardaliaguet, P.: Notes on Mean Field Games (from P.-L. Lions’ lectures at Collège de France) (2012)
Degond, P., Motsch, S.: Continuum limit of self-driven particles with orientation interaction. Math. Models Methods Appl. Sci. 18(Suppl.), 1193–1215 (2008)
Degond, P., Liu, J.-G., Mieussens, L.: Macroscopic fluid models with localized kinetic upscaling effects. SIAM J. Multiscale Model. Simul. 5, 940–979 (2006)
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)
Degond, P., Liu, J.-G., Ringhofer, C.: Evolution of the distribution of wealth in economic neighborhood by local Nash equilibrium closure (2013b, submitted)
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)
E, W.: Principles of Multiscale Modeling. Cambridge University Press, Cambridge (2011)
Frouvelle, A., Liu, J.G.: Dynamics in a kinetic model of oriented particles with phase transition. SIAM J. Math. Anal. 44, 791–826 (2012)
Green, E.J., Porter, R.H.: Noncooperative collusion under imperfect price information. Econometrica 52, 87–100 (1984)
Hsu, E.P.: Stochastic Analysis on Manifolds. Graduate Series in Mathematics. American Mathematical Society, Providence (2002)
Huang, K.: Statistical Mechanics, 2nd edn. Wiley, New York (1987)
Jordan, R., Kinderlehrer, D., Otto, F.: Free energy and the Fokker-Planck equation. Physica D 107, 265–271 (1997)
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)
Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2, 229–260 (2007)
Lazear, E.P., Rosen, S.: Rank-order tournaments as optimum labor contracts. Preprint, National Bureau of Economic Research Cambridge, USA (1979)
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)
Mas-Colell, A.: On a theorem of Schmeidler. J. Math. Econ. 13, 201–206 (1984)
Monderer, D., Shapley, L.S.: Potential games. Games Econ. Behav. 14, 124–143 (1996)
Nash, J.F.: Equilibrim points in n-person games. Proc. Natl. Acad. Sci. USA 36, 48–49 (1950)
Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. Int. J. Game Theory 2, 65–67 (1973)
Schmeidler, D.: Equilibrium points of nonatomic games. J. Stat. Phys. 7, 295–300 (1973)
Schultz, T.W.: Investment in human capital. Am. Econ. Rev. 51, 1–17 (1961)
Shapiro, N.Z., Shapley, L.S.: Values of large games, I: a limit theorem. Math. Oper. Res. 3, 1–9 (1978)
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)
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)
Villani, C.: Topics in Optimal Transportation. AMS Graduate Studies in Mathematics, vol. 58. AMS, Providence (2003)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Bertozzi.
Rights and permissions
About this article
Cite this article
Degond, P., Liu, JG. & Ringhofer, C. Large-Scale Dynamics of Mean-Field Games Driven by Local Nash Equilibria. J Nonlinear Sci 24, 93–115 (2014). https://doi.org/10.1007/s00332-013-9185-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-013-9185-2
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