A probabilistic approach to mean field games with major and minor players

René Carmona; Xiuneng Zhu

doi:10.1214/15-AAP1125

Annals of Applied Probability

Ann. Appl. Probab.
Volume 26, Number 3 (2016), 1535-1580.

A probabilistic approach to mean field games with major and minor players

René Carmona and Xiuneng Zhu

Full-text: Open access

Enhanced PDF (383 KB)

Abstract

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 equilibriums 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.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 3 (2016), 1535-1580.

Dates
Received: September 2014
Revised: May 2015
First available in Project Euclid: 14 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1465905011

Digital Object Identifier
doi:10.1214/15-AAP1125

Mathematical Reviews number (MathSciNet)
MR3513598

Zentralblatt MATH identifier
1342.93121

Subjects
Primary: 93E20: Optimal stochastic control
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Mean Field Games stochastic control McKean–Vlasov diffusion stochastic Pontryagin principle mean-field interaction mean-field forward–backward stochastic differential equation

Citation

Carmona, René; Zhu, Xiuneng. A probabilistic approach to mean field games with major and minor players. Ann. Appl. Probab. 26 (2016), no. 3, 1535--1580. doi:10.1214/15-AAP1125. https://projecteuclid.org/euclid.aoap/1465905011

References

[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.
Mathematical Reviews (MathSciNet): MR883646
[2] Baghery, F. and Øksendal, B. (2007). A maximum principle for stochastic control with partial information. Stoch. Anal. Appl. 25 705–717.
Mathematical Reviews (MathSciNet): MR2321904
Digital Object Identifier: doi:10.1080/07362990701283128
[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.
arXiv: 1404.4148
[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.
Mathematical Reviews (MathSciNet): MR3072222
Digital Object Identifier: doi:10.1137/120883499
[6] Carmona, R. and Delarue, F. (2013). Mean field forward–backward stochastic differential equations. Electron. Commun. Probab. 18 no. 68, 15.
Mathematical Reviews (MathSciNet): MR3091726
Digital Object Identifier: doi:10.1214/ECP.v18-2446
[7] Carmona, R. and Delarue, F. (2015). Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics. Ann. Probab. 43 2647–2700.
Mathematical Reviews (MathSciNet): MR3395471
Digital Object Identifier: doi:10.1214/14-AOP946
Project Euclid: euclid.aop/1441792295
[8] Carmona, R., Delarue, F. and Lacker, D. (2014). Probabilistic analysis of mean field games with a common noise. Technical report, Princeton Univ.
[9] Carmona, R., Fouque, J.-P. and Sun, L.-H. (2015). Mean field games and systemic risk. Commun. Math. Sci. 13 911–933.
Mathematical Reviews (MathSciNet): MR3325083
Digital Object Identifier: doi:10.4310/CMS.2015.v13.n4.a4
[10] Carmona, R. and Lacker, D. (2015). A probabilistic weak formulation of mean field games and applications. Ann. Appl. Probab. 25 1189–1231.
Mathematical Reviews (MathSciNet): MR3325272
Digital Object Identifier: doi:10.1214/14-AAP1020
Project Euclid: euclid.aoap/1427124127
[11] Çınlar, E. (2011). Probability and Stochastics. Graduate Texts in Mathematics 261. Springer, New York.
Mathematical Reviews (MathSciNet): MR2767184
[12] Crisan, D., Kurtz, T. and Lee, Y. (2012). Conditional distributions, exchangeable particle systems, and stochastic partial differential equations. Technical report.
[13] Delarue, F. (2002). On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case. Stochastic Process. Appl. 99 209–286.
Mathematical Reviews (MathSciNet): MR1901154
Digital Object Identifier: doi:10.1016/S0304-4149(02)00085-6
[14] Huang, M. (2010). Large-population LQG games involving a major player: The Nash equilvanece principle. SIAM J. Control Optim. 48 3318–3353.
Mathematical Reviews (MathSciNet): MR2599921
Digital Object Identifier: doi:10.1137/080735370
[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.
Mathematical Reviews (MathSciNet): MR2346927
Digital Object Identifier: doi:10.4310/CIS.2006.v6.n2.a2
Project Euclid: euclid.cis/1183728987
[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.
Mathematical Reviews (MathSciNet): MR2383731
[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.
Mathematical Reviews (MathSciNet): MR2269875
Digital Object Identifier: doi:10.1016/j.crma.2006.09.019
[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.
Mathematical Reviews (MathSciNet): MR2271747
Digital Object Identifier: doi:10.1016/j.crma.2006.09.018
[19] Lasry, J.-M. and Lions, P.-L. (2007). Mean field games. Jpn. J. Math. 2 229–260.
Mathematical Reviews (MathSciNet): MR2295621
Digital Object Identifier: doi:10.1007/s11537-007-0657-8
[20] Ma, J., Wu, Z., Zhang, D. and Zhang, J. (2011). On well-posedness of forward–backward SDEs—A unified approach. Technical report.
[21] Ma, J. and Yong, J. (1999). Forward–Backward Stochastic Differential Equations and Their Applications. Lecture Notes in Math. 1702. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1704232
[22] Nguyen, S. and Huang, M. (2012). Mean field LQG games with mass behavior responsive to a major player. In In 51th 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.
Mathematical Reviews (MathSciNet): MR3022092
Digital Object Identifier: doi:10.1137/110841217
[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.
Mathematical Reviews (MathSciNet): MR3092738
Digital Object Identifier: doi:10.1137/120889496
[25] Rachev, S. T. and Rüschendorf, L. (1998). Mass Transportation Problems. Vol. II: Applications. Springer, New York.
Mathematical Reviews (MathSciNet): MR1619171
[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.
Mathematical Reviews (MathSciNet): MR1108185
[27] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR2459454
[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.
Mathematical Reviews (MathSciNet): MR471060

Project Euclid

mathematics and statistics online

Annals of Applied Probability

A probabilistic approach to mean field games with major and minor players

More by René Carmona

More by Xiuneng Zhu

Abstract

Article information

Citation

References

The Institute of Mathematical Statistics

New content alerts

More like this