# American Institute of Mathematical Sciences

March  2021, 11(1): 23-46. doi: 10.3934/mcrf.2020025

## Linear-quadratic-Gaussian mean-field-game with partial observation and common noise

 1 International Center for Decision and Risk Analysis Jindal School of Management, The University of Texas at Dallas and School of Data Sciences, City University of Hong Kong 2 Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, Shandong 250100, China 3 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong

Received  January 2019 Revised  February 2020 Published  March 2021 Early access  May 2020

This paper considers a class of linear-quadratic-Gaussian (LQG) mean-field games (MFGs) with partial observation structure for individual agents. Unlike other literature, there are some special features in our formulation. First, the individual state is driven by some common-noise due to the external factor and the state-average thus becomes a random process instead of a deterministic quantity. Second, the sensor function of individual observation depends on state-average thus the agents are coupled in triple manner: not only in their states and cost functionals, but also through their observation mechanism. The decentralized strategies for individual agents are derived by the Kalman filtering and separation principle. The consistency condition is obtained which is equivalent to the wellposedness of some forward-backward stochastic differential equation (FBSDE) driven by common noise. Finally, the related $\epsilon$-Nash equilibrium property is verified.

Citation: Alain Bensoussan, Xinwei Feng, Jianhui Huang. Linear-quadratic-Gaussian mean-field-game with partial observation and common noise. Mathematical Control and Related Fields, 2021, 11 (1) : 23-46. doi: 10.3934/mcrf.2020025
##### References:
 [1] M. Bardi, Explicit solutions of some linear-quadratic mean field games, Netw. Heterog. Media, 7 (2012), 243-261.  doi: 10.3934/nhm.2012.7.243. [2] A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511526503. [3] A. Bensoussan, J. Frehse and P. Yam, Mean Field Games and Mean Field Type Control Theory, SpringerBriefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-8508-7. [4] A. Bensoussan, K. C. J. Sung, S. C. P. Yam and S. P. Yung, Linear-quadratic mean field games, J. Optim. Theory Appl., 169 (2016), 496-529.  doi: 10.1007/s10957-015-0819-4. [5] R. Carmona and F. Delarue, Probabilistic analysis of mean-field games, SIAM J. Control Optim., 51 (2013), 2705-2734.  doi: 10.1137/120883499. [6] R. Carmona and F. Delarue, Probabilistic theory of mean field games with applications. Ⅱ. Mean field games with common noise and master equations, in Probability Theory and Stochastic Modelling, 84, Springer, Cham, 2018. [7] R. Carmona, J.-P. Fouque and L.-H. Sun, Mean field games and systemic risk, Commun. Math. Sci., 13 (2015), 911-933.  doi: 10.4310/CMS.2015.v13.n4.a4. [8] C. Dogbé, Modeling crowd dynamics by the mean-field limit approach, Math. Comput. Modelling, 52 (2010), 1506-1520.  doi: 10.1016/j.mcm.2010.06.012. [9] G. M. Erickson, Differential game methods of advertising competition, European Journal Operational Research, 83 (1995), 431-438. [10] W. Fleming and W. Rishel, Deterministic and Stochastic Control of Partially Observable Systems, Springer-Verlag, 1975. [11] O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Mathematics, 2003, Springer, Berlin, 2011,205–266. doi: 10.1007/978-3-642-14660-2_3. [12] A. Haurie and P. Marcotte, On the relationship between Nash-Cournot and Wardrop equilibria, Networks, 15 (1985), 295-308.  doi: 10.1002/net.3230150303. [13] G.-D. Hu, Symplectic Runge-Kutta methods for the linear quadratic regulator problem, Int. J. Math. Anal. (Ruse), 1 (2007), 293-304. [14] J. Huang, Y. Hu and T. Nie, Linear-quadratic-Gaussian mixed mean-field games with heterogeneous input constraints, SIAM J. Control Optim., 56 (2018), 2835-2877.  doi: 10.1137/17M1151420. [15] J. Huang and S. Wang, Dynamic optimization of large-population systems with partial information, J. Optim. Theory Appl., 168 (2016), 231-245.  doi: 10.1007/s10957-015-0740-x. [16] M. Huang, Large-population LQG games involving a major player: The Nash certainty equivalence principle, SIAM J. Control Optim., 48 (2010), 3318-3353.  doi: 10.1137/080735370. [17] M. Huang, P. E. Caines and R. P. Malhamé, Uplink power adjustment in wireless communication systems: A stochastic control analysis, IEEE Trans. Automat. Control, 49 (2004), 1693-1708.  doi: 10.1109/TAC.2004.835388. [18] M. Huang, P. E. Caines and R. P. Malhamé, Distributed multi-agent decision-making with partial observations: Asymptotic Nash equilibria, Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS), (2006), 2725–2730. [19] M. Huang, P. E. Caines and R. P. Malhamé, Large-population cost-coupled LQG problems with non-uniform agents: Individual-mass behavior and decentralized $\epsilon$-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571.  doi: 10.1109/TAC.2007.904450. [20] M. Huang, R. P. Malhamé and P. E. Caines, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6 (2006), 221-251.  doi: 10.4310/CIS.2006.v6.n3.a5. [21] G. Kallianpur, Stochastic filtering theory, in Applications of Mathematics, 13, Springer-Verlag, New York-Berlin, 1980. [22] A. C. Kizilkale and R. P. Malhamé, Collective target tracking mean field control for Markovian jump-driven models of electric water heating loads, in Control of Complex Systems: Theory and Applications, 2016,559–584. [23] P. E. Kloeden and E. Platen, Numerical solution of sochastic differential equations, in Applications of Mathematics (New York), 23, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5. [24] V. E. Lambson, Self-enforcing collusion in large dynamic markets, J. Econom. Theory, 34 (1984), 282-291.  doi: 10.1016/0022-0531(84)90145-5. [25] J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8. [26] J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications, in Lecture Notes in Mathematics, 1702, Springer-Verlag, Berlin, 1999. [27] Z. Ma, D. Callaway and I. Hiskens, Decentralized charging control of large populations of plug-in electric vehicles, IEEE Transactions on Control Systems Technology, 21 (2013), 67-78. [28] S. L. Nguyen and M. Huang, Linear-quadratic-Gaussian mixed games with continuum-parameterized minor players, SIAM J. Control Optim., 50 (2012), 2907-2937.  doi: 10.1137/110841217. [29] M. Nourian, P. E. Caines, R. P. Malhamé and M. Huang, Nash, social and centralized solutions to consensus problems via mean field control theory, IEEE Trans. Automat. Control, 58 (2013), 639-653.  doi: 10.1109/TAC.2012.2215399. [30] B. Øksendal, Stochastic Differential Equations. An Introduction with Applications, Universitext, $6^th$ edition, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6. [31] N. Şen and P. E. Caines, Nonlinear filtering theory for McKean–Vlasov type stochastic differential equations, SIAM J. Control Optim., 54 (2016), 153-174.  doi: 10.1137/15M1013304. [32] H. Tembine, Q. Zhu and T. Başar, Risk-sensitive mean-field games, IEEE Trans. Automat. Control, 59 (2014), 835-850.  doi: 10.1109/TAC.2013.2289711. [33] G. Wang and Z. Wu, Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems, J. Math. Anal. Appl., 342 (2008), 1280-1296.  doi: 10.1016/j.jmaa.2007.12.072. [34] G. Wang, Z. Wu and J. Xiong, A linear-quadratic optimal control problem of forward-backward stochastic differential equations with partial information, IEEE Trans. Automat. Control, 60 (2015), 2904-2916.  doi: 10.1109/TAC.2015.2411871. [35] Y. Weintraub, L. Benkard and B. Van Roy, Markov perfect industry dynamics with many firms, Econometrica, 76 (2008), 1375-1411.  doi: 10.3982/ECTA6158. [36] W. M. Wonham., On the separation theorem of stochastic control., SIAM J. Control Optim., 6 (1968), 312–326. doi: 10.1137/0306023. [37] H. Yin, P. G. Mehta, S. P. Meyn and U. V. Shanbhag, Synchronization of coupled oscillators is a game, IEEE Trans. Automat. Control, 57 (2012), 920-935.  doi: 10.1109/TAC.2011.2168082. [38] J. Yong and X. Y. Zhou, Stochastic controls. Hamiltonian systems and HJB equations, in Applications of Mathematics (New York), 43, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

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##### References:
 [1] M. Bardi, Explicit solutions of some linear-quadratic mean field games, Netw. Heterog. Media, 7 (2012), 243-261.  doi: 10.3934/nhm.2012.7.243. [2] A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511526503. [3] A. Bensoussan, J. Frehse and P. Yam, Mean Field Games and Mean Field Type Control Theory, SpringerBriefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-8508-7. [4] A. Bensoussan, K. C. J. Sung, S. C. P. Yam and S. P. Yung, Linear-quadratic mean field games, J. Optim. Theory Appl., 169 (2016), 496-529.  doi: 10.1007/s10957-015-0819-4. [5] R. Carmona and F. Delarue, Probabilistic analysis of mean-field games, SIAM J. Control Optim., 51 (2013), 2705-2734.  doi: 10.1137/120883499. [6] R. Carmona and F. Delarue, Probabilistic theory of mean field games with applications. Ⅱ. Mean field games with common noise and master equations, in Probability Theory and Stochastic Modelling, 84, Springer, Cham, 2018. [7] R. Carmona, J.-P. Fouque and L.-H. Sun, Mean field games and systemic risk, Commun. Math. Sci., 13 (2015), 911-933.  doi: 10.4310/CMS.2015.v13.n4.a4. [8] C. Dogbé, Modeling crowd dynamics by the mean-field limit approach, Math. Comput. Modelling, 52 (2010), 1506-1520.  doi: 10.1016/j.mcm.2010.06.012. [9] G. M. Erickson, Differential game methods of advertising competition, European Journal Operational Research, 83 (1995), 431-438. [10] W. Fleming and W. Rishel, Deterministic and Stochastic Control of Partially Observable Systems, Springer-Verlag, 1975. [11] O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Mathematics, 2003, Springer, Berlin, 2011,205–266. doi: 10.1007/978-3-642-14660-2_3. [12] A. Haurie and P. Marcotte, On the relationship between Nash-Cournot and Wardrop equilibria, Networks, 15 (1985), 295-308.  doi: 10.1002/net.3230150303. [13] G.-D. Hu, Symplectic Runge-Kutta methods for the linear quadratic regulator problem, Int. J. Math. Anal. (Ruse), 1 (2007), 293-304. [14] J. Huang, Y. Hu and T. Nie, Linear-quadratic-Gaussian mixed mean-field games with heterogeneous input constraints, SIAM J. Control Optim., 56 (2018), 2835-2877.  doi: 10.1137/17M1151420. [15] J. Huang and S. Wang, Dynamic optimization of large-population systems with partial information, J. Optim. Theory Appl., 168 (2016), 231-245.  doi: 10.1007/s10957-015-0740-x. [16] M. Huang, Large-population LQG games involving a major player: The Nash certainty equivalence principle, SIAM J. Control Optim., 48 (2010), 3318-3353.  doi: 10.1137/080735370. [17] M. Huang, P. E. Caines and R. P. Malhamé, Uplink power adjustment in wireless communication systems: A stochastic control analysis, IEEE Trans. Automat. Control, 49 (2004), 1693-1708.  doi: 10.1109/TAC.2004.835388. [18] M. Huang, P. E. Caines and R. P. Malhamé, Distributed multi-agent decision-making with partial observations: Asymptotic Nash equilibria, Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS), (2006), 2725–2730. [19] M. Huang, P. E. Caines and R. P. Malhamé, Large-population cost-coupled LQG problems with non-uniform agents: Individual-mass behavior and decentralized $\epsilon$-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571.  doi: 10.1109/TAC.2007.904450. [20] M. Huang, R. P. Malhamé and P. E. Caines, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6 (2006), 221-251.  doi: 10.4310/CIS.2006.v6.n3.a5. [21] G. Kallianpur, Stochastic filtering theory, in Applications of Mathematics, 13, Springer-Verlag, New York-Berlin, 1980. [22] A. C. Kizilkale and R. P. Malhamé, Collective target tracking mean field control for Markovian jump-driven models of electric water heating loads, in Control of Complex Systems: Theory and Applications, 2016,559–584. [23] P. E. Kloeden and E. Platen, Numerical solution of sochastic differential equations, in Applications of Mathematics (New York), 23, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5. [24] V. E. Lambson, Self-enforcing collusion in large dynamic markets, J. Econom. Theory, 34 (1984), 282-291.  doi: 10.1016/0022-0531(84)90145-5. [25] J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8. [26] J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications, in Lecture Notes in Mathematics, 1702, Springer-Verlag, Berlin, 1999. [27] Z. Ma, D. Callaway and I. Hiskens, Decentralized charging control of large populations of plug-in electric vehicles, IEEE Transactions on Control Systems Technology, 21 (2013), 67-78. [28] S. L. Nguyen and M. Huang, Linear-quadratic-Gaussian mixed games with continuum-parameterized minor players, SIAM J. Control Optim., 50 (2012), 2907-2937.  doi: 10.1137/110841217. [29] M. Nourian, P. E. Caines, R. P. Malhamé and M. Huang, Nash, social and centralized solutions to consensus problems via mean field control theory, IEEE Trans. Automat. Control, 58 (2013), 639-653.  doi: 10.1109/TAC.2012.2215399. [30] B. Øksendal, Stochastic Differential Equations. An Introduction with Applications, Universitext, $6^th$ edition, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6. [31] N. Şen and P. E. Caines, Nonlinear filtering theory for McKean–Vlasov type stochastic differential equations, SIAM J. Control Optim., 54 (2016), 153-174.  doi: 10.1137/15M1013304. [32] H. Tembine, Q. Zhu and T. Başar, Risk-sensitive mean-field games, IEEE Trans. Automat. Control, 59 (2014), 835-850.  doi: 10.1109/TAC.2013.2289711. [33] G. Wang and Z. Wu, Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems, J. Math. Anal. Appl., 342 (2008), 1280-1296.  doi: 10.1016/j.jmaa.2007.12.072. [34] G. Wang, Z. Wu and J. Xiong, A linear-quadratic optimal control problem of forward-backward stochastic differential equations with partial information, IEEE Trans. Automat. Control, 60 (2015), 2904-2916.  doi: 10.1109/TAC.2015.2411871. [35] Y. Weintraub, L. Benkard and B. Van Roy, Markov perfect industry dynamics with many firms, Econometrica, 76 (2008), 1375-1411.  doi: 10.3982/ECTA6158. [36] W. M. Wonham., On the separation theorem of stochastic control., SIAM J. Control Optim., 6 (1968), 312–326. doi: 10.1137/0306023. [37] H. Yin, P. G. Mehta, S. P. Meyn and U. V. Shanbhag, Synchronization of coupled oscillators is a game, IEEE Trans. Automat. Control, 57 (2012), 920-935.  doi: 10.1109/TAC.2011.2168082. [38] J. Yong and X. Y. Zhou, Stochastic controls. Hamiltonian systems and HJB equations, in Applications of Mathematics (New York), 43, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.
Trajectories of the type-1 agents' states when N = 500
Trajectories of the type-2 agents' states when N = 500
Trajectories of the type-1 agents state average and the mean field term
Trajectories of the type-2 agents state average and the mean field term
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