American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence and stabilization of a Kirchhoff moving string with a delay in the boundary or in the internal feedback
  • EECT Home
  • This Issue
  • Next Article
    Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable
December  2018, 7(4): 587-597. doi: 10.3934/eect.2018028

Some partially observed multi-agent linear exponential quadratic stochastic differential games

Tyrone E. Duncan

Mathematics Department, Snow Hall, 1460 Jayhawk Blvd, Lawrence, KS 66045, USA

Received  January 2018 Revised  August 2018 Published  September 2018

Fund Project: The author was supported by NSF grant DMS 1411412, AFOSR grant FA9550-17-1-0073, and ARO grant W911NF-14-10390.

Some multi-agent stochastic differential games described by a stochastic linear system driven by a Brownian motion and having an exponential quadratic payoff for the agents are formulated and solved. The agents have either complete observations or partial observations of the system state. The agents act independently of one another and the explicit optimal feedback control strategies form a Nash equilibrium. In the partially observed problem the observations are the same for all agents which occurs in broadcast situations. The optimal control strategies and optimal payoffs are given explicitly. The method of solution for both problems does not require solving either Hamilton-Jacobi-Isaacs equations or backward stochastic differential equations.

Keywords: Stochastic differential games, multi-agent games, partially observed games, risk sensitive games, exponential quadratic payoffs.
Mathematics Subject Classification: Primary: 91A15, 49N70; Secondary: 91A25.
Citation: Tyrone E. Duncan. Some partially observed multi-agent linear exponential quadratic stochastic differential games. Evolution Equations & Control Theory, 2018, 7 (4) : 587-597. doi: 10.3934/eect.2018028
References:
[1]

M. Bardi and F. S. Priuli, Linear-quadratic n-person and mean-field games with ergodic cost, SIAM J. Control Optim., 52 (2014), 3022-3052.  doi: 10.1137/140951795.  Google Scholar

[2]

T. Basar and P. Bernhard, H∞-Optimal Control and Related Minimax Design Problems, Birkhauser, Boston, 1995. doi: 10.1007/978-0-8176-4757-5.  Google Scholar

[3]

T. Basar and G. Olsder, Dynamic Noncooperative Game Theory, SIAM, 2nd Ed. 1999.  Google Scholar

[4]

A. Bensoussan and J. H. van Schuppen, Optimal control of partially observable stochastic systems with an exponential-of-integral performance index, SIAM J. Control Optim., 23 (1995), 599-613.  doi: 10.1137/0323038.  Google Scholar

[5]

R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations, SIAM J. Control Optim., 47 (2008), 444-475.  doi: 10.1137/060671954.  Google Scholar

[6]

T. E. Duncan, Evaluation of likelihood functions, Information and Control, 13 (1968), 62-74.  doi: 10.1016/S0019-9958(68)90795-X.  Google Scholar

[7]

T. E. Duncan, Linear-exponential-quadratic Gaussian control, IEEE Trans. Autom. Control, 58 (2013), 2910-2911.  doi: 10.1109/TAC.2013.2257610.  Google Scholar

[8]

T. E. Duncan, Some linear-quadratic stochastic differential games for equations in a Hilbert space with fractional Brownian motions, Discrete Cont. Dyn. Systems Ser. A, 35 (2015), 5435-5445.  doi: 10.3934/dcds.2015.35.5435.  Google Scholar

[9]

T. E. Duncan, Linear exponential quadratic stochastic differential games, IEEE Trans. Autom. Control, 61 (2016), 2550-2552.  doi: 10.1109/TAC.2015.2510983.  Google Scholar

[10]

T. E. Duncan and B. Pasik-Duncan, A solvable stochastic differential game in the two-sphere, Proc. 52nd IEEE Conf. Decision and Control, 7833–7837, Firenze, 2013. Google Scholar

[11]

T. E. Duncan and B. Pasik-Duncan, Some results on optimal control for a partially observed linear stochastic system with an exponential quadratic cost, Proc. IFAC World Congress, Cape Town, 47 (2014), 8695-8698. doi: 10.3182/20140824-6-ZA-1003.00522.  Google Scholar

[12]

R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models, Springer, 1995.  Google Scholar

[13]

W. H. Fleming and D. Hernandez-Hernandez, On the value of stochastic differential games, Commun. Stoch. Anal., 5 (2011), 341-351.   Google Scholar

[14]

W. H. Fleming and P. E. Souganidis, On the existence of value functions of two player, zero sum stochastic differential games, Indiana Math. J., 38 (1989), 293-314.  doi: 10.1512/iumj.1989.38.38015.  Google Scholar

[15]

S. Hamadene, J. P. Lepeltier and S. Peng, BSDEs with continuous coefficients and stochastic differential games, Backward Stochastic Differential Equations (N. El Karoui et al., eds.), Pitman Res. Notes Math, 364 (1997), 115-128.  Google Scholar

[16]

R. Isaacs, Differential Games, J. Wiley, New York 1965. Google Scholar

[17]

D. H. Jacobson, Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games, IEEE Trans. Autom. Control, AC-18 (1973), 124-131.   Google Scholar

[18]

M. L. Kleptsyna, A. Le Breton and M. Viot, On the linear-exponential filtering problem for general Gaussian processes, SIAM J. control Optim., 47 (2008), 2886-2911.  doi: 10.1137/070705908.  Google Scholar

[19]

J. B. Moore, R. J. Elliott and S. Dey, Risk sensitive generalizations of minimum variance estimation and control, J. Math. Syst., Estimation, Contr., (electronic), 7 (1997), 15 pp.  Google Scholar

[20]

J. Nash, Non-cooperative games, Ann. Math., 54 (1951), 286-295.  doi: 10.2307/1969529.  Google Scholar

[21]

J. L. Speyer, J. Deyst and D. H. Jacobson, Optimization of stochastic linear systems with additive measurement and process noise using exponential performance criteria, IEEE Trans. Autom. Control, AC-19 (1974), 358-366.   Google Scholar

show all references

References:
[1]

M. Bardi and F. S. Priuli, Linear-quadratic n-person and mean-field games with ergodic cost, SIAM J. Control Optim., 52 (2014), 3022-3052.  doi: 10.1137/140951795.  Google Scholar

[2]

T. Basar and P. Bernhard, H∞-Optimal Control and Related Minimax Design Problems, Birkhauser, Boston, 1995. doi: 10.1007/978-0-8176-4757-5.  Google Scholar

[3]

T. Basar and G. Olsder, Dynamic Noncooperative Game Theory, SIAM, 2nd Ed. 1999.  Google Scholar

[4]

A. Bensoussan and J. H. van Schuppen, Optimal control of partially observable stochastic systems with an exponential-of-integral performance index, SIAM J. Control Optim., 23 (1995), 599-613.  doi: 10.1137/0323038.  Google Scholar

[5]

R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations, SIAM J. Control Optim., 47 (2008), 444-475.  doi: 10.1137/060671954.  Google Scholar

[6]

T. E. Duncan, Evaluation of likelihood functions, Information and Control, 13 (1968), 62-74.  doi: 10.1016/S0019-9958(68)90795-X.  Google Scholar

[7]

T. E. Duncan, Linear-exponential-quadratic Gaussian control, IEEE Trans. Autom. Control, 58 (2013), 2910-2911.  doi: 10.1109/TAC.2013.2257610.  Google Scholar

[8]

T. E. Duncan, Some linear-quadratic stochastic differential games for equations in a Hilbert space with fractional Brownian motions, Discrete Cont. Dyn. Systems Ser. A, 35 (2015), 5435-5445.  doi: 10.3934/dcds.2015.35.5435.  Google Scholar

[9]

T. E. Duncan, Linear exponential quadratic stochastic differential games, IEEE Trans. Autom. Control, 61 (2016), 2550-2552.  doi: 10.1109/TAC.2015.2510983.  Google Scholar

[10]

T. E. Duncan and B. Pasik-Duncan, A solvable stochastic differential game in the two-sphere, Proc. 52nd IEEE Conf. Decision and Control, 7833–7837, Firenze, 2013. Google Scholar

[11]

T. E. Duncan and B. Pasik-Duncan, Some results on optimal control for a partially observed linear stochastic system with an exponential quadratic cost, Proc. IFAC World Congress, Cape Town, 47 (2014), 8695-8698. doi: 10.3182/20140824-6-ZA-1003.00522.  Google Scholar

[12]

R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models, Springer, 1995.  Google Scholar

[13]

W. H. Fleming and D. Hernandez-Hernandez, On the value of stochastic differential games, Commun. Stoch. Anal., 5 (2011), 341-351.   Google Scholar

[14]

W. H. Fleming and P. E. Souganidis, On the existence of value functions of two player, zero sum stochastic differential games, Indiana Math. J., 38 (1989), 293-314.  doi: 10.1512/iumj.1989.38.38015.  Google Scholar

[15]

S. Hamadene, J. P. Lepeltier and S. Peng, BSDEs with continuous coefficients and stochastic differential games, Backward Stochastic Differential Equations (N. El Karoui et al., eds.), Pitman Res. Notes Math, 364 (1997), 115-128.  Google Scholar

[16]

R. Isaacs, Differential Games, J. Wiley, New York 1965. Google Scholar

[17]

D. H. Jacobson, Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games, IEEE Trans. Autom. Control, AC-18 (1973), 124-131.   Google Scholar

[18]

M. L. Kleptsyna, A. Le Breton and M. Viot, On the linear-exponential filtering problem for general Gaussian processes, SIAM J. control Optim., 47 (2008), 2886-2911.  doi: 10.1137/070705908.  Google Scholar

[19]

J. B. Moore, R. J. Elliott and S. Dey, Risk sensitive generalizations of minimum variance estimation and control, J. Math. Syst., Estimation, Contr., (electronic), 7 (1997), 15 pp.  Google Scholar

[20]

J. Nash, Non-cooperative games, Ann. Math., 54 (1951), 286-295.  doi: 10.2307/1969529.  Google Scholar

[21]

J. L. Speyer, J. Deyst and D. H. Jacobson, Optimization of stochastic linear systems with additive measurement and process noise using exponential performance criteria, IEEE Trans. Autom. Control, AC-19 (1974), 358-366.   Google Scholar

[1]

Beatris Adriana Escobedo-Trujillo, José Daniel López-Barrientos. Nonzero-sum stochastic differential games with additive structure and average payoffs. Journal of Dynamics & Games, 2014, 1 (4) : 555-578. doi: 10.3934/jdg.2014.1.555
[2]

Beatris Adriana Escobedo-Trujillo, Alejandro Alaffita-Hernández, Raquiel López-Martínez. Constrained stochastic differential games with additive structure: Average and discount payoffs. Journal of Dynamics & Games, 2018, 5 (2) : 109-141. doi: 10.3934/jdg.2018008
[3]

Beatris A. Escobedo-Trujillo. Discount-sensitive equilibria in zero-sum stochastic differential games. Journal of Dynamics & Games, 2016, 3 (1) : 25-50. doi: 10.3934/jdg.2016002
[4]

Jingzhen Liu, Ka-Fai Cedric Yiu. Optimal stochastic differential games with VaR constraints. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1889-1907. doi: 10.3934/dcdsb.2013.18.1889
[5]

Alain Bensoussan, Jens Frehse, Christine Grün. Stochastic differential games with a varying number of players. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1719-1736. doi: 10.3934/cpaa.2014.13.1719
[6]

Matthew Bourque, T. E. S. Raghavan. Policy improvement for perfect information additive reward and additive transition stochastic games with discounted and average payoffs. Journal of Dynamics & Games, 2014, 1 (3) : 347-361. doi: 10.3934/jdg.2014.1.347
[7]

Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435
[8]

Libin Mou, Jiongmin Yong. Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method. Journal of Industrial & Management Optimization, 2006, 2 (1) : 95-117. doi: 10.3934/jimo.2006.2.95
[9]

Alain Bensoussan, Shaokuan Chen, Suresh P. Sethi. Linear quadratic differential games with mixed leadership: The open-loop solution. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 95-108. doi: 10.3934/naco.2013.3.95
[10]

John A. Morgan. Interception in differential pursuit/evasion games. Journal of Dynamics & Games, 2016, 3 (4) : 335-354. doi: 10.3934/jdg.2016018
[11]

Alain Bensoussan, Jens Frehse, Jens Vogelgesang. Systems of Bellman equations to stochastic differential games with non-compact coupling. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1375-1389. doi: 10.3934/dcds.2010.27.1375
[12]

Qingmeng Wei, Zhiyong Yu. Time-inconsistent recursive zero-sum stochastic differential games. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1051-1079. doi: 10.3934/mcrf.2018045
[13]

Xiangxiang Huang, Xianping Guo, Jianping Peng. A probability criterion for zero-sum stochastic games. Journal of Dynamics & Games, 2017, 4 (4) : 369-383. doi: 10.3934/jdg.2017020
[14]

Leon Petrosyan, David Yeung. Shapley value for differential network games: Theory and application. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2020021
[15]

Ellina Grigorieva, Evgenii Khailov. Hierarchical differential games between manufacturer and retailer. Conference Publications, 2009, 2009 (Special) : 300-314. doi: 10.3934/proc.2009.2009.300
[16]

Alan Beggs. Learning in monotone bayesian games. Journal of Dynamics & Games, 2015, 2 (2) : 117-140. doi: 10.3934/jdg.2015.2.117
[17]

Konstantin Avrachenkov, Giovanni Neglia, Vikas Vikram Singh. Network formation games with teams. Journal of Dynamics & Games, 2016, 3 (4) : 303-318. doi: 10.3934/jdg.2016016
[18]

Carlos Hervés-Beloso, Emma Moreno-García. Market games and walrasian equilibria. Journal of Dynamics & Games, 2020, 7 (1) : 65-77. doi: 10.3934/jdg.2020004
[19]

Hassan Najafi Alishah, Pedro Duarte. Hamiltonian evolutionary games. Journal of Dynamics & Games, 2015, 2 (1) : 33-49. doi: 10.3934/jdg.2015.2.33
[20]

Serap Ergün, Osman Palanci, Sirma Zeynep Alparslan Gök, Şule Nizamoğlu, Gerhard Wilhelm Weber. Sequencing grey games. Journal of Dynamics & Games, 2020, 7 (1) : 21-35. doi: 10.3934/jdg.2020002

2019 Impact Factor: 0.953

Tools

Metrics

  • PDF downloads (144)
  • HTML views (348)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences