Linear quadratic differential games with mixed leadership: The open-loop solution

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2013, 3(1): 95-108. doi: 10.3934/naco.2013.3.95

Linear quadratic differential games with mixed leadership: The open-loop solution

Alain Bensoussan ^1, , Shaokuan Chen ^1, and Suresh P. Sethi ^1,

1.	Naveen Jindal School of Management, The University of Texas at Dallas, Richardson, TX 75080-3021, United States, United States, United States

Received October 2011 Revised November 2012 Published January 2013

Abstract
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This paper is concerned with open-loop Stackelberg equilibria of two-player linear-quadratic differential games with mixed leadership. We prove that, under some appropriate assumptions on the coefficients, there exists a unique Stackelberg solution to such a differential game. Moreover, by means of the close interrelationship between the Riccati equations and the set of equations satisfied by the optimal open-loop control, we provide sufficient conditions to guarantee the existence and uniqueness of solutions to the associated Riccati equations with mixed-boundary conditions. As a result, the players' open-loop strategies can be represented in terms of the system state.

Keywords: Stackelberg equilibria, Differential games, mixed leadership, Riccati equations, two-point boundary value problems..

Mathematics Subject Classification: Primary: 91A10, 91A23; Secondary: 34K1.

Citation: 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

References:

[1]	T. Başar, On the relative leadership property of Stackelberg strategies, J. Optimization Theory and Applications, 11 (1973), 655-661. doi: 10.1007/BF00935564.
[2]	T. Başar and A. Haurie, Feedback equilibria in differential games with structural and modal uncertainties, in "Advances in Large Scale Systems" (eds. J. B. Cruz, Jr.), JAE Press Inc., (1984), 163-201.
[3]	T. Başar, A. Haurie and G. Ricci, On the dominance of capitalists' leadership in a feedback Stackelberg solution of a differential game model of capitalism, J. Economic Dynamics and Control, 9 (1985), 101-125. doi: 10.1016/0165-1889(85)90026-0.
[4]	T. Başar and G. J. Olsder, "Dynamic Noncooperative Game Theory," 2^nd edition, Academic Press, New York, 1995.
[5]	T. Başar, A. Bensoussan and S. P. Sethi, Differential games with mixed leadership: the open-loop solution, Applied Mathematics and Computation, 217 (2010), 972-979. doi: 10.1016/j.amc.2010.01.048.
[6]	A. Bensoussan, S. Chen and S. P. Sethi, Feedback Stackelberg solutions of infinite-horizon stochastic differential games, forthcoming.
[7]	A. Bensoussan, S. Chen and S. P. Sethi, The maximum principle for global solutions of stochastic Stackelberg differential games, working paper.
[8]	G. F. Cachon, Supply chain coordination with contracts, in "Handbooks in OR and MS Vol. 11, SCM: Design, Coordination and Cooperation" (eds. A. G. De Kok and S. C. Graves), Elsevier, (2003), 227-339.
[9]	A. Chutani and S. P. Sethi, Cooperative advertising in a dynamic retail market oligopoly, Dynamic Games and Applications, 2012, forthcoming. doi: 10.1007/s13235-012-0053-8.
[10]	A. Chutani and S. P. Sethi, Optimal advertising and pricing in a dynamic durable goods supply chain, Journal of Optimization Theory and Applications, 154 (2012), 615-643. doi: 10.1007/s10957-012-0034-5.
[11]	E. Dockner, S. Jøgensen, N. V. Long and G. Sorger, "Differential Games in Economics and Management Science," Cambridge University Press, Cambridge, UK, 2000. doi: 10.1017/CBO9780511805127.
[12]	X. He, A. Krishnamoorthy, A. Prasad and S. P. Sethi, Retail competition and cooperative advertising, Operations Research Letters, 39 (2011), 11-16. doi: 10.1016/j.orl.2010.10.006.
[13]	X. He, A. Krishnamoorthy, A. Prasad and S. P. Sethi, Co-Op advertising in dynamic retail oligopolies, Decision Sciences, 43 (2012), 73-105. doi: 10.1111/j.1540-5915.2011.00336.x.
[14]	X. He, A. Prasad and S. P. Sethi, Cooperative advertising and pricing in a dynamic stochastic supply chain: feedback stackelberg strategies, Production and Operations Management, 18 (2009), 78-94.
[15]	X. He, A. Prasad, S. P. Sethi and G. J. Gutierrez, A survey of Stackelberg differential game models in supply chain and marketing channels, J. Systems Science and Systems Engineering, 16 (2007), 385-413. doi: 10.1007/s11518-007-5058-2.
[16]	A. Krishnamoorthy, A. Prasad and S. P. Sethi, Optimal pricing and advertising in a durable-good duopoly, European Journal of Operations Research, 200 (2010), 486-497. doi: 10.1016/j.ejor.2009.01.003.
[17]	G. Leitmann, On generalized Stackelberg strategies, J. Optimization Theory and Applications, 26 (1978), 637-643. doi: 10.1007/BF00933155.
[18]	E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Relat. Fields, 114 (1999), 123-150. doi: 10.1007/s004409970001.
[19]	A. Prasad, S. P. Sethi and P. A. Naik, Understanding the impact of churn in dynamic oligopoly markets, Automatica, 48 (2012), 2882-2887. doi: 10.1016/j.automatica.2012.08.031.
[20]	M. Simaan and J. B. Cruz, Jr., On the Stackelberg strategy in nonzero-sum games, J. Optimization Theory and Applications, 11 (1973), 533-555. doi: 10.1007/BF00935665.
[21]	M. Simaan and J. B. Cruz, Jr., Additional aspects of the Stackelberg strategy in nonzero-sum games, J. Optimization Theory and Applications, 11 (1973b), 613-626. doi: 10.1007/BF00935561.
[22]	H. von Stackelberg, "Marktform und Gleichgewicht," Springer, Vienna, 1934 (An English translation appeared in "The Theory of the Market Economy," Oxford University Press, New York, 1952).
[23]	S. Tang, General linear quadratic optimal stochastic control problems with random coefficients: linear stochastic Hamilton systems and backward stochastic Riccati equations, SIAM J. Control Optim., 42 (2003), 53-75. doi: 10.1137/S0363012901387550.
[24]	J. Yong, Linear forward-backward stochastic differential equations with random coefficients, Probab. Theory Relat. Fields, 135 (2006), 53-83. doi: 10.1007/s00440-005-0452-5.

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References:

[1]	T. Başar, On the relative leadership property of Stackelberg strategies, J. Optimization Theory and Applications, 11 (1973), 655-661. doi: 10.1007/BF00935564.
[2]	T. Başar and A. Haurie, Feedback equilibria in differential games with structural and modal uncertainties, in "Advances in Large Scale Systems" (eds. J. B. Cruz, Jr.), JAE Press Inc., (1984), 163-201.
[3]	T. Başar, A. Haurie and G. Ricci, On the dominance of capitalists' leadership in a feedback Stackelberg solution of a differential game model of capitalism, J. Economic Dynamics and Control, 9 (1985), 101-125. doi: 10.1016/0165-1889(85)90026-0.
[4]	T. Başar and G. J. Olsder, "Dynamic Noncooperative Game Theory," 2^nd edition, Academic Press, New York, 1995.
[5]	T. Başar, A. Bensoussan and S. P. Sethi, Differential games with mixed leadership: the open-loop solution, Applied Mathematics and Computation, 217 (2010), 972-979. doi: 10.1016/j.amc.2010.01.048.
[6]	A. Bensoussan, S. Chen and S. P. Sethi, Feedback Stackelberg solutions of infinite-horizon stochastic differential games, forthcoming.
[7]	A. Bensoussan, S. Chen and S. P. Sethi, The maximum principle for global solutions of stochastic Stackelberg differential games, working paper.
[8]	G. F. Cachon, Supply chain coordination with contracts, in "Handbooks in OR and MS Vol. 11, SCM: Design, Coordination and Cooperation" (eds. A. G. De Kok and S. C. Graves), Elsevier, (2003), 227-339.
[9]	A. Chutani and S. P. Sethi, Cooperative advertising in a dynamic retail market oligopoly, Dynamic Games and Applications, 2012, forthcoming. doi: 10.1007/s13235-012-0053-8.
[10]	A. Chutani and S. P. Sethi, Optimal advertising and pricing in a dynamic durable goods supply chain, Journal of Optimization Theory and Applications, 154 (2012), 615-643. doi: 10.1007/s10957-012-0034-5.
[11]	E. Dockner, S. Jøgensen, N. V. Long and G. Sorger, "Differential Games in Economics and Management Science," Cambridge University Press, Cambridge, UK, 2000. doi: 10.1017/CBO9780511805127.
[12]	X. He, A. Krishnamoorthy, A. Prasad and S. P. Sethi, Retail competition and cooperative advertising, Operations Research Letters, 39 (2011), 11-16. doi: 10.1016/j.orl.2010.10.006.
[13]	X. He, A. Krishnamoorthy, A. Prasad and S. P. Sethi, Co-Op advertising in dynamic retail oligopolies, Decision Sciences, 43 (2012), 73-105. doi: 10.1111/j.1540-5915.2011.00336.x.
[14]	X. He, A. Prasad and S. P. Sethi, Cooperative advertising and pricing in a dynamic stochastic supply chain: feedback stackelberg strategies, Production and Operations Management, 18 (2009), 78-94.
[15]	X. He, A. Prasad, S. P. Sethi and G. J. Gutierrez, A survey of Stackelberg differential game models in supply chain and marketing channels, J. Systems Science and Systems Engineering, 16 (2007), 385-413. doi: 10.1007/s11518-007-5058-2.
[16]	A. Krishnamoorthy, A. Prasad and S. P. Sethi, Optimal pricing and advertising in a durable-good duopoly, European Journal of Operations Research, 200 (2010), 486-497. doi: 10.1016/j.ejor.2009.01.003.
[17]	G. Leitmann, On generalized Stackelberg strategies, J. Optimization Theory and Applications, 26 (1978), 637-643. doi: 10.1007/BF00933155.
[18]	E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Relat. Fields, 114 (1999), 123-150. doi: 10.1007/s004409970001.
[19]	A. Prasad, S. P. Sethi and P. A. Naik, Understanding the impact of churn in dynamic oligopoly markets, Automatica, 48 (2012), 2882-2887. doi: 10.1016/j.automatica.2012.08.031.
[20]	M. Simaan and J. B. Cruz, Jr., On the Stackelberg strategy in nonzero-sum games, J. Optimization Theory and Applications, 11 (1973), 533-555. doi: 10.1007/BF00935665.
[21]	M. Simaan and J. B. Cruz, Jr., Additional aspects of the Stackelberg strategy in nonzero-sum games, J. Optimization Theory and Applications, 11 (1973b), 613-626. doi: 10.1007/BF00935561.
[22]	H. von Stackelberg, "Marktform und Gleichgewicht," Springer, Vienna, 1934 (An English translation appeared in "The Theory of the Market Economy," Oxford University Press, New York, 1952).
[23]	S. Tang, General linear quadratic optimal stochastic control problems with random coefficients: linear stochastic Hamilton systems and backward stochastic Riccati equations, SIAM J. Control Optim., 42 (2003), 53-75. doi: 10.1137/S0363012901387550.
[24]	J. Yong, Linear forward-backward stochastic differential equations with random coefficients, Probab. Theory Relat. Fields, 135 (2006), 53-83. doi: 10.1007/s00440-005-0452-5.

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