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Linear-quadratic mean field Stackelberg games with state and control delays. (English) Zbl 1372.91021

Summary: In this article, we consider a linear-quadratic mean field game between a leader (dominating player) and a group of followers (agents) under the Stackelberg game setting as proposed in [A. Bensoussan et al., Appl. Math. Optim. 74, No. 1, 91–128 (2016; Zbl 1348.49031)], so that the evolution of each individual follower is now also subjected to delay effects from both his/her state and control variables, as well as those of the leader. The overall Stackelberg game is solved by tackling three subproblems hierarchically. Their resolution corresponds to the establishment of the existence and uniqueness of the solutions of three different forward-backward stochastic functional differential equations, which we manage by applying the unified continuation method as first developed in, for example, [Y. Hu and S. Peng, Probab. Theory Relat. Fields 103, No. 2, 273–283 (1995; Zbl 0831.60065)] and [X. Xu, “Fully coupled forward-backward stochastic functional differential equations and applications to quadratic optimal control”, Preprint, arXiv:1310.6846]. In particular, by first regarding the mean field term and the delay influence of the leader as exogenous, we use the adjoint equation approach to solve the optimal control of each follower. Next, we utilize the fixed point property to get the desired mean field equilibrium, with which we propose a time independent sufficient condition that warrants its existence and uniqueness. Finally, we solve the optimal control of the leader and conclude that its presence would not interfere with the original existence of the equilibrium of the community. Our present setting introduces a much different challenge than that in the former work [Bensoussan et al., loc. cit.], which stems from the delay effects from each follower’s state and control, which result in a more complicated system of infinite dimensional forward-backward stochastic functional differential equations. To find the optimal control of the leader, we need to develop a thorough understanding of a special linear operator resulting from the delay effects of the leader’s control on the dynamics of each follower, which hinges on the invertibility of another certain functional operator. This was not considered in [Bensoussan et al., loc. cit.].

MSC:

91A65 Hierarchical games (including Stackelberg games)
91A07 Games with infinitely many players
34K50 Stochastic functional-differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93E20 Optimal stochastic control
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[1] D. Andersson and B. Djehiche, {\it A maximum principle for SDEs of mean-field type}, Appl. Math. Optim., 63 (2011), pp. 341-356. · Zbl 1215.49034
[2] M. Bardi, {\it Explicit solutions of some linear-quadratic mean field games}, Netw. Heterog. Media, 7 (2012), pp. 243-261. · Zbl 1268.91017
[3] A. Bensoussan, M. H. M. Chau, and S. C. P. Yam, {\it Mean field Stackelberg games: Aggregation of delayed instructions}, SIAM J. Control Optim., 53 (2015), pp. 2237-2266, . · Zbl 1320.91028
[4] A. Bensoussan, M. Chau, and S. Yam, {\it Mean field games with a dominating player}, Appl. Math. Optim., 74 (2016), pp. 91-128. · Zbl 1348.49031
[5] A. Bensoussan, J. Frehse, and P. Yam, {\it Mean Field Games and Mean Field Type Control Theory}, Springer Briefs in Math., Springer-Verlag, 2013. · Zbl 1287.93002
[6] A. Bensoussan, J. Sung, S. Yam, and S. Yung, {\it Linear-quadratic mean field games}, J. Optim. Theory Appl., 169 (2016), pp. 496-529. · Zbl 1343.91010
[7] R. Buckdahn, B. Djehiche, J. Li, and S. Peng, {\it Mean-field backward stochastic differential equations: A limit approach}, Ann. Probab., 37 (2009), pp. 1524-1565. · Zbl 1176.60042
[8] P. Cardaliaguet, {\it Notes on Mean Field Games}, Tech. report, Univ. Paris, Dauphine, 2010.
[9] R. Carmona, {\it Minerva Lecture: Master Equations, Games with Common Noise, and with Major and Minor Players}, , 2016.
[10] R. Carmona and F. Delarue, {\it Probabilistic analysis of mean-field games}, SIAM J. Control Optim., 51 (2013), pp. 2705-2734, . · Zbl 1275.93065
[11] R. Carmona, J.-P. Fouque, S. M. Mousavi, and L.-H. Sun, {\it Systemic Risk and Stochastic Games with Delay}, preprint, , 2016. · Zbl 1418.91062
[12] F. Cucker and S. Smale, {\it Emergent behavior in flocks}, IEEE Trans. Automat. Control, 52 (2007), pp. 852-862. · Zbl 1366.91116
[13] M. C. Delfour, {\it The linear-quadratic optimal control problem with delays in state and control variables: A state space approach}, SIAM J. Control Optim., 24 (1986), pp. 835-883, . · Zbl 0606.93037
[14] S. Federico, B. Goldys, and F. Gozzi, {\it HJB equations for the optimal control of differential equations with delays and state constraints, I: Regularity of viscosity solutions}, SIAM J. Control Optim., 48 (2010), pp. 4910-4937, . · Zbl 1208.49048
[15] S. Federico, B. Goldys, and F. Gozzi, {\it HJB equations for the optimal control of differential equations with delays and state constraints, II: Verification and optimal feedbacks}, SIAM J. Control Optim., 49 (2011), pp. 2378-2414, . · Zbl 1242.49058
[16] S. Federico and E. Tacconi, {\it Dynamic programming for optimal control problems with delays in the control variable}, SIAM J. Control Optim., 52 (2014), pp. 1203-1236, . · Zbl 1293.49060
[17] D. Firoozi and P. Caines, {\it Mean field game epsilon-Nash equilibria for partially observed optimal execution problems in finance}, in Proceedings of the 55th IEEE Conference on Decision and Control, IEEE, 2016, pp. 268-275.
[18] J. Garnier, G. Papanicolaou, and T.-W. Yang, {\it Large deviations for a mean field model of systemic risk}, SIAM J. Financial Math., 4 (2013), pp. 151-184, . · Zbl 1283.60044
[19] F. Gozzi and C. Marinelli, {\it Stochastic optimal control of delay equations arising in advertising models}, in Stochastic Partial Differential Equations and Applications - VII, G. Da Prato and L. Tubaro, eds., Chapman and Hall/CRC, 2005, pp. 133-148. · Zbl 1107.93035
[20] F. Gozzi and F. Masiero, {\it Stochastic Optimal Control with Delay in the Control: Solution through Partial Smoothing}, preprint, , 2015. · Zbl 1375.93140
[21] O. Guéant, J. Lasry, and P. Lions, {\it Mean field games and applications}, in Paris-Princeton Lectures on Mathematical Finance 2010, Springer, 2011, pp. 205-266. · Zbl 1205.91027
[22] S. Hadd, {\it An evolution equation approach to nonautonomous linear systems with state, input, and output delays}, SIAM J. Control Optim., 45 (2006), pp. 246-272, . · Zbl 1119.34058
[23] Y. Hu and S. Peng, {\it Solution of forward-backward stochastic differential equations}, Probab. Theory Related Fields, 103 (1995), pp. 273-283. · Zbl 0831.60065
[24] J. Huang, X. Li, and T. Wang, {\it Mean-field linear-quadratic-Gaussian (LQG) games for stochastic integral systems}, IEEE Trans. Automat. Control, 61 (2016), pp. 2670-2675, . · Zbl 1359.91008
[25] M. Huang, P. Caines, and R. Malhamé, {\it Individual and mass behaviour in large population stochastic wireless power control problems: Centralized and Nash equilibrium solutions}, in Proceedings of the 42nd IEEE Conference on Decision and Control, vol. 1, IEEE, 2003, pp. 98-103.
[26] M. Huang, R. Malhamé, and P. Caines, {\it Large population stochastic dynamic games: Closed-loop Mckean-Vlasov systems and the Nash certainty equivalence principle}, Commun. Inf. Syst., 6 (2006), pp. 221-252. · Zbl 1136.91349
[27] A. Ichikawa, {\it Quadratic control of evolution equations with delays in control}, SIAM J. Control Optim., 20 (1982), pp. 645-668, . · Zbl 0495.49006
[28] J. Lasry and P. Lions, {\it Jeux à champ moyen. I-Le cas stationnaire}, C. R. Math. Acad. Sci. Paris, 343 (2006), pp. 619-625. · Zbl 1153.91009
[29] J. Lasry and P. Lions, {\it Jeux à champ moyen. II-Horizon fini et contrôle optimal}, C. R. Math. Acad. Sci. Paris, 343 (2006), pp. 679-684. · Zbl 1153.91010
[30] J. Lasry and P. Lions, {\it Mean field games}, Jpn. J. Math., 2 (2007), pp. 229-260. · Zbl 1156.91321
[31] T. Meyer-Brandis, B. Øksendal, and X. Zhou, {\it A mean-field stochastic maximum principle via Malliavin calculus}, Stochastics, 84 (2012), pp. 643-666. · Zbl 1252.49039
[32] M. Huang, {\it Large-population LQG games involving a major player: The Nash certainty equivalence principle}, SIAM J. Control Optim., 48 (2010), pp. 3318-3353, . · Zbl 1200.91020
[33] M. Nourian and P. E. Caines, {\it epsilon-Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents}, SIAM J. Control Optim., 51 (2013), pp. 3302-3331, . · Zbl 1275.93067
[34] M. Nourian, P. Caines, and R. Malhamé, {\it Mean field analysis of controlled Cucker-Smale type flocking: Linear analysis and perturbation equations}, IFAC Proc. Vol., 44 (2011), pp. 4471-4476.
[35] B. Øksendal, A. Sulem, and T. Zhang, {\it Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations}, Adv. Appl. Prob., 43 (2011), pp. 572-596. · Zbl 1217.93183
[36] W. Rudin, {\it Functional Analysis}, McGraw-Hill, 1991. · Zbl 0867.46001
[37] A. J. Pritchard and D. Salamon, {\it The linear-quadratic control problem for retarded systems with delays in control and observation}, IMA J. Math. Control Inform., 2 (1985), pp. 335-362. · Zbl 0646.34078
[38] H. von Stackelberg, {\it Marktform und gleichgewicht}, Springer, Vienna, 1934. · Zbl 1405.91003
[39] R. Vinter, {\it Control of linear hereditary systems with control and output delays}, Ann. N.Y. Acad. Sci., 410 (1983), pp. 121-128.
[40] R. B. Vinter and R. H. Kwong, {\it The infinite time quadratic control problem for linear systems with state and control delays: An evolution equation approach}, SIAM J. Control Optim., 19 (1981), pp. 139-153, . · Zbl 0465.93043
[41] X. Xu, {\it Fully Coupled Forward-Backward Stochastic Functional Differential Equations and Applications to Quadratic Optimal Control}, preprint, , 2013.
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