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  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 & 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.  Google Scholar

[2] A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511526503.  Google Scholar
[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.  Google Scholar

[4]

A. BensoussanK. C. J. SungS. 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.  Google Scholar

[5]

R. Carmona and F. Delarue, Probabilistic analysis of mean-field games, SIAM J. Control Optim., 51 (2013), 2705-2734.  doi: 10.1137/120883499.  Google Scholar

[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.  Google Scholar

[7]

R. CarmonaJ.-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.  Google Scholar

[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.  Google Scholar

[9]

G. M. Erickson, Differential game methods of advertising competition, European Journal Operational Research, 83 (1995), 431-438.   Google Scholar

[10]

W. Fleming and W. Rishel, Deterministic and Stochastic Control of Partially Observable Systems, Springer-Verlag, 1975. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

G.-D. Hu, Symplectic Runge-Kutta methods for the linear quadratic regulator problem, Int. J. Math. Anal. (Ruse), 1 (2007), 293-304.   Google Scholar

[14]

J. HuangY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

M. HuangP. 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.  Google Scholar

[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. Google Scholar

[19]

M. HuangP. 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.  Google Scholar

[20]

M. HuangR. 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.  Google Scholar

[21]

G. Kallianpur, Stochastic filtering theory, in Applications of Mathematics, 13, Springer-Verlag, New York-Berlin, 1980.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications, in Lecture Notes in Mathematics, 1702, Springer-Verlag, Berlin, 1999.  Google Scholar

[27]

Z. MaD. 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.   Google Scholar

[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.  Google Scholar

[29]

M. NourianP. E. CainesR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

H. TembineQ. Zhu and T. Başar, Risk-sensitive mean-field games, IEEE Trans. Automat. Control, 59 (2014), 835-850.  doi: 10.1109/TAC.2013.2289711.  Google Scholar

[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.  Google Scholar

[34]

G. WangZ. 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.  Google Scholar

[35]

Y. WeintraubL. Benkard and B. Van Roy, Markov perfect industry dynamics with many firms, Econometrica, 76 (2008), 1375-1411.  doi: 10.3982/ECTA6158.  Google Scholar

[36]

W. M. Wonham., On the separation theorem of stochastic control., SIAM J. Control Optim., 6 (1968), 312–326. doi: 10.1137/0306023.  Google Scholar

[37]

H. YinP. G. MehtaS. 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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2] A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511526503.  Google Scholar
[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.  Google Scholar

[4]

A. BensoussanK. C. J. SungS. 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.  Google Scholar

[5]

R. Carmona and F. Delarue, Probabilistic analysis of mean-field games, SIAM J. Control Optim., 51 (2013), 2705-2734.  doi: 10.1137/120883499.  Google Scholar

[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.  Google Scholar

[7]

R. CarmonaJ.-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.  Google Scholar

[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.  Google Scholar

[9]

G. M. Erickson, Differential game methods of advertising competition, European Journal Operational Research, 83 (1995), 431-438.   Google Scholar

[10]

W. Fleming and W. Rishel, Deterministic and Stochastic Control of Partially Observable Systems, Springer-Verlag, 1975. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

G.-D. Hu, Symplectic Runge-Kutta methods for the linear quadratic regulator problem, Int. J. Math. Anal. (Ruse), 1 (2007), 293-304.   Google Scholar

[14]

J. HuangY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

M. HuangP. 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.  Google Scholar

[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. Google Scholar

[19]

M. HuangP. 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.  Google Scholar

[20]

M. HuangR. 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.  Google Scholar

[21]

G. Kallianpur, Stochastic filtering theory, in Applications of Mathematics, 13, Springer-Verlag, New York-Berlin, 1980.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications, in Lecture Notes in Mathematics, 1702, Springer-Verlag, Berlin, 1999.  Google Scholar

[27]

Z. MaD. 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.   Google Scholar

[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.  Google Scholar

[29]

M. NourianP. E. CainesR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

H. TembineQ. Zhu and T. Başar, Risk-sensitive mean-field games, IEEE Trans. Automat. Control, 59 (2014), 835-850.  doi: 10.1109/TAC.2013.2289711.  Google Scholar

[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.  Google Scholar

[34]

G. WangZ. 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.  Google Scholar

[35]

Y. WeintraubL. Benkard and B. Van Roy, Markov perfect industry dynamics with many firms, Econometrica, 76 (2008), 1375-1411.  doi: 10.3982/ECTA6158.  Google Scholar

[36]

W. M. Wonham., On the separation theorem of stochastic control., SIAM J. Control Optim., 6 (1968), 312–326. doi: 10.1137/0306023.  Google Scholar

[37]

H. YinP. G. MehtaS. 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.  Google Scholar

[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.  Google Scholar

Figure 1.  Trajectories of the type-1 agents' states when N = 500
Figure 2.  Trajectories of the type-2 agents' states when N = 500
Figure 3.  Trajectories of the type-1 agents state average and the mean field term
Figure 4.  Trajectories of the type-2 agents state average and the mean field term
[1]

Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021006

[2]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006

[3]

Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021011

[4]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[5]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[6]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

[7]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022

[8]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[9]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[10]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192

[11]

Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021008

[12]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233

[13]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089

[14]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397

[15]

Mansour Shrahili, Ravi Shanker Dubey, Ahmed Shafay. Inclusion of fading memory to Banister model of changes in physical condition. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 881-888. doi: 10.3934/dcdss.2020051

[16]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223

[17]

Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181

[18]

Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243

[19]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008

[20]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

2019 Impact Factor: 0.857

Metrics

  • PDF downloads (181)
  • HTML views (319)
  • Cited by (0)

Other articles
by authors

[Back to Top]