## Annals of Applied Probability

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

#### 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
Revised: May 2015
First available in Project Euclid: 14 June 2016

https://projecteuclid.org/euclid.aoap/1465905011

Digital Object Identifier
doi:10.1214/15-AAP1125

Mathematical Reviews number (MathSciNet)
MR3513598

Zentralblatt MATH identifier
1342.93121

#### 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.
• [2] Baghery, F. and Øksendal, B. (2007). A maximum principle for stochastic control with partial information. Stoch. Anal. Appl. 25 705–717.
• [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.
• [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.
• [6] Carmona, R. and Delarue, F. (2013). Mean field forward–backward stochastic differential equations. Electron. Commun. Probab. 18 no. 68, 15.
• [7] Carmona, R. and Delarue, F. (2015). Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics. Ann. Probab. 43 2647–2700.
• [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.
• [10] Carmona, R. and Lacker, D. (2015). A probabilistic weak formulation of mean field games and applications. Ann. Appl. Probab. 25 1189–1231.
• [11] Çınlar, E. (2011). Probability and Stochastics. Graduate Texts in Mathematics 261. Springer, New York.
• [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.
• [14] Huang, M. (2010). Large-population LQG games involving a major player: The Nash equilvanece principle. SIAM J. Control Optim. 48 3318–3353.
• [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.
• [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.
• [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.
• [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.
• [19] Lasry, J.-M. and Lions, P.-L. (2007). Mean field games. Jpn. J. Math. 2 229–260.
• [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.
• [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.
• [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.
• [25] Rachev, S. T. and Rüschendorf, L. (1998). Mass Transportation Problems. Vol. II: Applications. Springer, New York.
• [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.
• [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.