A probabilistic approach to mean field games with major and minor players. (English) Zbl 1342.93121

Summary: We propose a new approach to mean field games with major and minor players. Our formulation involves a two player game where the optimization of the representative minor player is standard while the major player faces an optimization over conditional McKean-Vlasov stochastic differential equations. The definition of this limiting game is justified by proving that its solution provides approximate Nash equilibria for large finite player games. This proof depends upon the generalization of standard results on the propagation of chaos to conditional dynamics. Because it is of independent interest, we prove this generalization in full detail. Using a conditional form of the Pontryagin stochastic maximum principle (proven in the appendix), we reduce the solution of the mean field game to a forward-backward system of stochastic differential equations of the conditional McKean-Vlasov type, which we solve in the linear quadratic setting. We use this class of models to show that Nash equilibriums in our formulation can be different from those originally found in the literature.


93E20 Optimal stochastic control
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
49K45 Optimality conditions for problems involving randomness
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[1] Aldous, D. J. (1985). Exchangeability and related topics. In École D’été de Probabilités de Saint-Flour , XIII- 1983. Lecture Notes in Math. 1117 1-198. Springer, Berlin. · Zbl 0562.60042
[2] Baghery, F. and Øksendal, B. (2007). A maximum principle for stochastic control with partial information. Stoch. Anal. Appl. 25 705-717. · Zbl 1140.93046
[3] Bensoussan, A., Chau, M. H. M. and Yam, S. C. P. (2014). Mean field games with a dominating player. Available at . arXiv:1404.4148 · Zbl 1348.49031
[4] Cardaliaguet, P. (2010). Notes on mean field games. Technical report.
[5] Carmona, R. and Delarue, F. (2013). Probabilistic analysis of mean-field games. SIAM J. Control Optim. 51 2705-2734. · Zbl 1275.93065
[6] Carmona, R. and Delarue, F. (2013). Mean field forward-backward stochastic differential equations. Electron. Commun. Probab. 18 no. 68, 15. · Zbl 1297.93182
[7] Carmona, R. and Delarue, F. (2015). Forward-backward stochastic differential equations and controlled McKean-Vlasov dynamics. Ann. Probab. 43 2647-2700. · Zbl 1322.93103
[8] Carmona, R., Delarue, F. and Lacker, D. (2014). Probabilistic analysis of mean field games with a common noise. Technical report, Princeton Univ. · Zbl 1422.91083
[9] Carmona, R., Fouque, J.-P. and Sun, L.-H. (2015). Mean field games and systemic risk. Commun. Math. Sci. 13 911-933. · Zbl 1337.91031
[10] Carmona, R. and Lacker, D. (2015). A probabilistic weak formulation of mean field games and applications. Ann. Appl. Probab. 25 1189-1231. · Zbl 1332.60100
[11] Çınlar, E. (2011). Probability and Stochastics. Graduate Texts in Mathematics 261 . Springer, New York. · Zbl 1226.60001
[12] Crisan, D., Kurtz, T. and Lee, Y. (2012). Conditional distributions, exchangeable particle systems, and stochastic partial differential equations. Technical report. · Zbl 1306.60086
[13] Delarue, F. (2002). On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case. Stochastic Process. Appl. 99 209-286. · Zbl 1058.60042
[14] Huang, M. (2010). Large-population LQG games involving a major player: The Nash equilvanece principle. SIAM J. Control Optim. 48 3318-3353. · Zbl 1200.91020
[15] Huang, M., Malhamé, R. P. and Caines, P. E. (2006). Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6 221-251. · Zbl 1136.91349
[16] Jourdain, B., Méléard, S. and Woyczynski, W. A. (2008). Nonlinear SDEs driven by Lévy processes and related PDEs. ALEA Lat. Am. J. Probab. Math. Stat. 4 1-29. · Zbl 1162.60327
[17] Lasry, J.-M. and Lions, P.-L. (2006). Jeux à champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343 619-625. · Zbl 1153.91009
[18] Lasry, J.-M. and Lions, P.-L. (2006). Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343 679-684. · Zbl 1153.91010
[19] Lasry, J.-M. and Lions, P.-L. (2007). Mean field games. Jpn. J. Math. 2 229-260. · Zbl 1156.91321
[20] Ma, J., Wu, Z., Zhang, D. and Zhang, J. (2011). On well-posedness of forward-backward SDEs-A unified approach. Technical report. · Zbl 1319.60132
[21] Ma, J. and Yong, J. (1999). Forward-Backward Stochastic Differential Equations and Their Applications. Lecture Notes in Math. 1702 . Springer, Berlin. · Zbl 0927.60004
[22] Nguyen, S. and Huang, M. (2012). Mean field LQG games with mass behavior responsive to a major player. In In 51 th IEEE Conference on Decision and Control . Maui, HI.
[23] Nguyen, S. L. and Huang, M. (2012). Linear-quadratic-Gaussian mixed games with continuum-parametrized minor players. SIAM J. Control Optim. 50 2907-2937. · Zbl 1262.91020
[24] Nourian, M. and Caines, P. E. (2013). \(\varepsilon\)-Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents. SIAM J. Control Optim. 51 3302-3331. · Zbl 1275.93067
[25] Rachev, S. T. and Rüschendorf, L. (1998). Mass Transportation Problems. Vol. II : Applications . Springer, New York. · Zbl 0990.60500
[26] Sznitman, A. S. (1989). Topics in propagation of chaos. In Ecole de Probabilités de Saint Flour , XIX- 1989 (D. L. Burkholder et al., eds.). Lecture Notes in Math. 1464 165-251.
[27] Villani, C. (2009). Optimal Transport : Old and New. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 338 . Springer, Berlin. · Zbl 1156.53003
[28] Yor, M. (1977). Sur les théorie du filtrage et de la prédiction. In Séminaire de Probabilités. XI ( Univ. Strasbourg , Strasbourg , 1975 / 1976). Lecture Notes in Math. 581 257-297. Springer, Berlin.
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