Abstract
The goal of the paper is to introduce a formulation of the mean field game with major and minor players as a fixed point on a space of controls. This approach emphasizes naturally the role played by McKean–Vlasov dynamics in some of the players’ optimization problems. We apply this approach to linear quadratic models for which we recover the existing solutions for open loop equilibria, and we show that we can also provide solutions for closed loop versions of the game. Finally, we implement numerically our theoretical results on a simple model of flocking.
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References
Bensoussan, A., Chau, M., Yam, S.: Mean field games with a dominating player. Tech. Rep. (2013). doi:10.1007/s00245-015-9309-1
Carmona, R., Delarue, E.: Probabilistic Theory of Mean Field Games: vol. I, Mean Field FBSDEs, Control, and Games, Stochastic Analysis and Applications. Springer, New York (2017)
Carmona, R., Delarue, E.: Probabilistic Theory of Mean Field Games: vol. II, Mean Field Games with Common Noise and Master Equations, Stochastic Analysis and Applications. Springer, New York (2017)
Carmona, R., Zhu, G.: A probabilistic approach to mean field games with major and minor players. Ann. Appl. Probab. 26, 1535–1580 (2014)
Cucker, F., Smale, S.: Emergent behavior in flocks. IEEE Trans. Autom. Control 52, 852–862 (2007)
Huang, M.: Large-population lqg games involving a major player: the nash equivalence principle. SIAM J. Control Optim. 48, 3318–3353 (2010)
Jaimungal, S., Nourian, M.: Mean-field game strategies for a major-minor agent optimal execution problem. Tech. Rep. University of Toronto, Toronto (2015)
Nourian, M., Caines, P., Malhamé, R.: Mean field analysis of controlled Cucker-Smale type flocking: linear analysis and perturbation equations. In: Proceedings of the 18th IFAC World Congress, Milan, pp. 4471–4476 (2011)
Nguyen, S., Huang, M.: Linear-quadratic-Gaussian mixed games with continuum-parametrized minor players. SIAM J. Control Optim. 50(5), 2907–2937 (2015)
Nguyen, S., Huang, M.: Mean field LQG games with mass behavior responsive to a major player. In: 51th IEEE Conference on Decision and Control (2012)
Nourian, M., Caines, P.: \(\epsilon \)-nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents. Tech. Rep. (2013). doi:10.1137/120889496
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Carmona, R., Wang, P. An Alternative Approach to Mean Field Game with Major and Minor Players, and Applications to Herders Impacts. Appl Math Optim 76, 5–27 (2017). https://doi.org/10.1007/s00245-017-9430-4
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DOI: https://doi.org/10.1007/s00245-017-9430-4