# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2022105
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## 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

*Corresponding author: Yu Li

Received  December 2021 Revised  May 2022 Early access June 2022

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.

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.

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##### 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.
$c_e = 0.5, 1.0, 1.5, 2.0$ from left to right and from up to down
The optimal polices under promotion
Zoom in at $t_s$ and $t_f$
The optimal polices under promotion
Zoom in at $t_s$ and $t_f$

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