Figure 1.
$ c_e = 0.5, 1.0, 1.5, 2.0 $ from left to right and from up to down
-
Previous Article
Optimal pricing strategy in a dual-channel supply chain: A two-period game analysis
- JIMO Home
- This Issue
-
Next Article
Equilibrium decisions on pricing and innovation that impact reference price dynamics
doi: 10.3934/jimo.2022105
Online First
Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.
Readers can access Online First articles via the “Online First” tab for the selected journal.
A new feedback form of open-loop Stackelberg strategy in a general linear-quadratic differential game
| 1. | Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin, China |
| 2. | School of Mathematical Sciences, Sunway University, Malaysia |
In this paper, we consider a general form of linear-quadratic Stackelberg deterministic differential game model, which consists of one leader and one follower. Each of their utility functions includes all possible squared terms, cross terms and single terms of states and controls of the two players, and constant terms. The time-consistent state feedback form of Stackelberg equilibrium strategy is obtained. Its explicit expression is in terms of the solutions of three decoupled symmetric Riccati differential equations. These decoupled symmetric Riccati differential equations are independent of the state and can be solved backward in time one by one. The proposed model and theory are applied to some classical Stackelberg games.
Keywords:
Leader-follower,
Stackelberg strategy,
linear-quadratic differential game,
Riccati differential equation,
supply chain management.
Mathematics Subject Classification:
Primary: 91A65, 49N10; Secondary: 90B06.
Citation:
Yu Li, Kok Lay Teo, Shuhua Zhang.
A new feedback form of open-loop Stackelberg strategy in a general linear-quadratic differential game. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022105
![]()
References:
| [1] |
H. Abou-Kandil and P. Bertrand,
Analytical solution for an open-loop Stackelberg game, IEEE Transactions on Automatic Control, 30 (1985), 1222-1224.
doi: 10.1109/TAC.1985.1103886. |
| [2] |
H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati Equations in Control and Systems Theory, Birkhäuser, 2003.
doi: 10.1007/978-3-0348-8081-7. |
| [3] |
J. F. Bard, Practical Bilevel Optimization: Algorithms and Applications, 30, Springer Science & Business Media, 2013.
doi: 10.1007/978-1-4757-2836-1. |
| [4] |
A. Bensoussan, S. Chen and S. P. Sethi,
The maximum principle for global solutions of stochastic Stackelberg differential games, SIAM Journal on Control and Optimization, 53 (2015), 1956-1981.
doi: 10.1137/140958906. |
| [5] |
J. A. E. Bryson and Y.-C. Ho, Applied Optimal Control: Optimization, Estimation and Control, Taylor & Francis, 1975. |
| [6] |
R. Caldentey and M. Haugh, A Cournot-Stackelberg model of supply contracts with financial hedging and identical retailers, Foundations and Trends® in Technology, Information and Operations Management, 11 (2017), 124–143.
doi: 10.1287/opre.1080.0521. |
| [7] |
B. Colson, P. Marcotte and G. Savard,
An overview of bilevel optimization, Annals of Operations Research, 153 (2007), 235-256.
doi: 10.1007/s10479-007-0176-2. |
| [8] |
G. Freiling, G. Jank and and H. Abou-Kandil,
Discrete time Riccati equations in open loop Nash and Stackelberg games, European Journal of Control, 5 (1999), 56-66.
doi: 10.1016/S0947-3580(99)70139-1. |
| [9] |
G. Freiling, G. Jank and S. R. Lee,
Existence and uniqueness of open-loop Stackelberg equilibria in linear-quadratic differential games, Journal of Optimization Theory & Applications, 110 (2001), 515-544.
doi: 10.1023/A:1017532210579. |
| [10] |
X. He, A. Prasad, S. P. Sethi and G. J. Gutierrez,
A survey of Stackelberg differential game models in supply and marketing channels, Journal of Systems Science and Systems Engineering, 16 (2007), 385-413.
|
| [11] |
S. Jørgensen and G. Zaccour,
Developments in differential game theory and numerical methods: economic and management applications, Computational Management Science, 4 (2007), 159-181.
doi: 10.1007/s10287-006-0032-x. |
| [12] |
M. Jungers,
On linear-quadratic Stackelberg games with time preference rates, IEEE Transactions on Automatic Control, 53 (2008), 621-625.
doi: 10.1109/TAC.2008.917649. |
| [13] |
K. Kogan and C. S. Tapiero, Supply Chain Games: Operations Management and Risk Valuation, Springer US, 2007.
doi: 10.1007/978-0-387-72776-9. |
| [14] |
D. Korzhyk, V. Conitzer and R. Parr, Solving Stackelberg games with uncertain observability, in The 10th International Conference on Autonomous Agents and Multiagent Systems-Volume 3, International Foundation for Autonomous Agents and Multiagent Systems, 2011, 1013–1020. |
| [15] |
D. Korzhyk, Z. Yin, C. Kiekintveld, V. Conitzer and M. Tambe,
Stackelberg vs. Nash in security games: An extended investigation of interchangeability, equivalence, and uniqueness, Journal of Artificial Intelligence Research, 41 (2011), 297-327.
doi: 10.1613/jair.3269. |
| [16] |
T. Li and S. P. Sethi,
A review of dynamic Stackelberg game models, Discrete and Continuous Dynamical Systems - Series B, 22 (2016), 125-159.
doi: 10.3934/dcdsb.2017007. |
| [17] |
K. Madani,
Game theory and water resources, Journal of Hydrology, 381 (2010), 225-238.
doi: 10.1016/j.jhydrol.2009.11.045. |
| [18] |
M. Simaan and J. B. Cruz,
On the Stackelberg strategy in nonzero-sum games, Journal of Optimization Theory & Applications, 11 (1973), 533-555.
doi: 10.1007/BF00935665. |
| [19] |
H. V. Stackelberg, Marktform und Gleichgewicht, Springer, Vienna, 1934. |
| [20] |
G. Tecuceanu and C. Popeea,
The closed loop implementation of the open loop Stackelberg solution in the linear quadratic problems, Journal of Applied Mathematics and Mechanics, 78 (1998), 1097-1100.
|
| [21] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, 1991. |
| [22] |
J. Xu and H. Zhang,
Sufficient and necessary open-loop Stackelberg strategy for two-player game with time delay, IEEE Transactions on Cybernetics, 46 (2016), 438-449.
doi: 10.1109/TCYB.2015.2403262. |
| [23] |
J. Xu, H. Zhang and T. Chai,
Necessary and sufficient condition for two-player Stackelberg strategy, IEEE Transactions on Automatic Control, 60 (2015), 1356-1361.
doi: 10.1109/TAC.2014.2346460. |
| [24] |
Y. Xu, Stackelberg equilibriums of open-loop differential games, in Proceedings of the 26th Chinese Control Conference, 2007. |
| [25] |
J. Yong,
A leader-follower stochastic linear quadratic differential game, SIAM Journal on Control & Optimization, 41 (2002), 1015-1041.
doi: 10.1137/S0363012901391925. |
show all references
References:
| [1] |
H. Abou-Kandil and P. Bertrand,
Analytical solution for an open-loop Stackelberg game, IEEE Transactions on Automatic Control, 30 (1985), 1222-1224.
doi: 10.1109/TAC.1985.1103886. |
| [2] |
H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati Equations in Control and Systems Theory, Birkhäuser, 2003.
doi: 10.1007/978-3-0348-8081-7. |
| [3] |
J. F. Bard, Practical Bilevel Optimization: Algorithms and Applications, 30, Springer Science & Business Media, 2013.
doi: 10.1007/978-1-4757-2836-1. |
| [4] |
A. Bensoussan, S. Chen and S. P. Sethi,
The maximum principle for global solutions of stochastic Stackelberg differential games, SIAM Journal on Control and Optimization, 53 (2015), 1956-1981.
doi: 10.1137/140958906. |
| [5] |
J. A. E. Bryson and Y.-C. Ho, Applied Optimal Control: Optimization, Estimation and Control, Taylor & Francis, 1975. |
| [6] |
R. Caldentey and M. Haugh, A Cournot-Stackelberg model of supply contracts with financial hedging and identical retailers, Foundations and Trends® in Technology, Information and Operations Management, 11 (2017), 124–143.
doi: 10.1287/opre.1080.0521. |
| [7] |
B. Colson, P. Marcotte and G. Savard,
An overview of bilevel optimization, Annals of Operations Research, 153 (2007), 235-256.
doi: 10.1007/s10479-007-0176-2. |
| [8] |
G. Freiling, G. Jank and and H. Abou-Kandil,
Discrete time Riccati equations in open loop Nash and Stackelberg games, European Journal of Control, 5 (1999), 56-66.
doi: 10.1016/S0947-3580(99)70139-1. |
| [9] |
G. Freiling, G. Jank and S. R. Lee,
Existence and uniqueness of open-loop Stackelberg equilibria in linear-quadratic differential games, Journal of Optimization Theory & Applications, 110 (2001), 515-544.
doi: 10.1023/A:1017532210579. |
| [10] |
X. He, A. Prasad, S. P. Sethi and G. J. Gutierrez,
A survey of Stackelberg differential game models in supply and marketing channels, Journal of Systems Science and Systems Engineering, 16 (2007), 385-413.
|
| [11] |
S. Jørgensen and G. Zaccour,
Developments in differential game theory and numerical methods: economic and management applications, Computational Management Science, 4 (2007), 159-181.
doi: 10.1007/s10287-006-0032-x. |
| [12] |
M. Jungers,
On linear-quadratic Stackelberg games with time preference rates, IEEE Transactions on Automatic Control, 53 (2008), 621-625.
doi: 10.1109/TAC.2008.917649. |
| [13] |
K. Kogan and C. S. Tapiero, Supply Chain Games: Operations Management and Risk Valuation, Springer US, 2007.
doi: 10.1007/978-0-387-72776-9. |
| [14] |
D. Korzhyk, V. Conitzer and R. Parr, Solving Stackelberg games with uncertain observability, in The 10th International Conference on Autonomous Agents and Multiagent Systems-Volume 3, International Foundation for Autonomous Agents and Multiagent Systems, 2011, 1013–1020. |
| [15] |
D. Korzhyk, Z. Yin, C. Kiekintveld, V. Conitzer and M. Tambe,
Stackelberg vs. Nash in security games: An extended investigation of interchangeability, equivalence, and uniqueness, Journal of Artificial Intelligence Research, 41 (2011), 297-327.
doi: 10.1613/jair.3269. |
| [16] |
T. Li and S. P. Sethi,
A review of dynamic Stackelberg game models, Discrete and Continuous Dynamical Systems - Series B, 22 (2016), 125-159.
doi: 10.3934/dcdsb.2017007. |
| [17] |
K. Madani,
Game theory and water resources, Journal of Hydrology, 381 (2010), 225-238.
doi: 10.1016/j.jhydrol.2009.11.045. |
| [18] |
M. Simaan and J. B. Cruz,
On the Stackelberg strategy in nonzero-sum games, Journal of Optimization Theory & Applications, 11 (1973), 533-555.
doi: 10.1007/BF00935665. |
| [19] |
H. V. Stackelberg, Marktform und Gleichgewicht, Springer, Vienna, 1934. |
| [20] |
G. Tecuceanu and C. Popeea,
The closed loop implementation of the open loop Stackelberg solution in the linear quadratic problems, Journal of Applied Mathematics and Mechanics, 78 (1998), 1097-1100.
|
| [21] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, 1991. |
| [22] |
J. Xu and H. Zhang,
Sufficient and necessary open-loop Stackelberg strategy for two-player game with time delay, IEEE Transactions on Cybernetics, 46 (2016), 438-449.
doi: 10.1109/TCYB.2015.2403262. |
| [23] |
J. Xu, H. Zhang and T. Chai,
Necessary and sufficient condition for two-player Stackelberg strategy, IEEE Transactions on Automatic Control, 60 (2015), 1356-1361.
doi: 10.1109/TAC.2014.2346460. |
| [24] |
Y. Xu, Stackelberg equilibriums of open-loop differential games, in Proceedings of the 26th Chinese Control Conference, 2007. |
| [25] |
J. Yong,
A leader-follower stochastic linear quadratic differential game, SIAM Journal on Control & Optimization, 41 (2002), 1015-1041.
doi: 10.1137/S0363012901391925. |
| [1] |
Weijun Meng, Jingtao Shi. A linear quadratic stochastic Stackelberg differential game with time delay. Mathematical Control and Related Fields, 2022, 12 (3) : 581-609. doi: 10.3934/mcrf.2021035 |
| [2] |
Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control and Related Fields, 2022, 12 (2) : 371-404. doi: 10.3934/mcrf.2021026 |
| [3] |
Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001 |
| [4] |
Yeong-Cheng Liou, Siegfried Schaible, Jen-Chih Yao. Supply chain inventory management via a Stackelberg equilibrium. Journal of Industrial and Management Optimization, 2006, 2 (1) : 81-94. doi: 10.3934/jimo.2006.2.81 |
| [5] |
Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040 |
| [6] |
Kai Du, Jianhui Huang, Zhen Wu. Linear quadratic mean-field-game of backward stochastic differential systems. Mathematical Control and Related Fields, 2018, 8 (3&4) : 653-678. doi: 10.3934/mcrf.2018028 |
| [7] |
Xingwu Chen, Jaume Llibre, Weinian Zhang. Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3953-3965. doi: 10.3934/dcdsb.2017203 |
| [8] |
Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435 |
| [9] |
Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control and Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97 |
| [10] |
Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control and Related Fields, 2021, 11 (4) : 797-828. doi: 10.3934/mcrf.2020047 |
| [11] |
Lianju Sun, Ziyou Gao, Yiju Wang. A Stackelberg game management model of the urban public transport. Journal of Industrial and Management Optimization, 2012, 8 (2) : 507-520. doi: 10.3934/jimo.2012.8.507 |
| [12] |
Xihong Yan. An augmented Lagrangian-based parallel splitting method for a one-leader-two-follower game. Journal of Industrial and Management Optimization, 2016, 12 (3) : 879-890. doi: 10.3934/jimo.2016.12.879 |
| [13] |
David W. K. Yeung, Yingxuan Zhang, Hongtao Bai, Sardar M. N. Islam. Collaborative environmental management for transboundary air pollution problems: A differential levies game. Journal of Industrial and Management Optimization, 2021, 17 (2) : 517-531. doi: 10.3934/jimo.2019121 |
| [14] |
Haijiao Li, Kuan Yang, Guoqing Zhang. Optimal pricing strategy in a dual-channel supply chain: A two-period game analysis. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022072 |
| [15] |
Sara Bernardi, Gissell Estrada-Rodriguez, Heiko Gimperlein, Kevin J. Painter. Macroscopic descriptions of follower-leader systems. Kinetic and Related Models, 2021, 14 (6) : 981-1002. doi: 10.3934/krm.2021035 |
| [16] |
Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations and Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017 |
| [17] |
Shigeaki Koike, Hiroaki Morimoto, Shigeru Sakaguchi. A linear-quadratic control problem with discretionary stopping. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 261-277. doi: 10.3934/dcdsb.2007.8.261 |
| [18] |
Russell Johnson, Carmen Núñez. Remarks on linear-quadratic dissipative control systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 889-914. doi: 10.3934/dcdsb.2015.20.889 |
| [19] |
Alain Bensoussan, Shaokuan Chen, Suresh P. Sethi. Linear quadratic differential games with mixed leadership: The open-loop solution. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 95-108. doi: 10.3934/naco.2013.3.95 |
| [20] |
Isaac A. García, Jaume Giné, Jaume Llibre. Liénard and Riccati differential equations related via Lie Algebras. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 485-494. doi: 10.3934/dcdsb.2008.10.485 |
2021 Impact Factor: 1.411
Tools
Metrics
Other articles
by authors
[Back to Top]