American Institute of Mathematical Sciences

Advanced
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

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 & 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.  doi: 10.1007/BF00935564.  Google Scholar

[2]

T. Başar and A. Haurie, Feedback equilibria in differential games with structural and modal uncertainties,, in, (1984), 163.   Google Scholar

[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.  doi: 10.1016/0165-1889(85)90026-0.  Google Scholar

[4]

T. Başar and G. J. Olsder, "Dynamic Noncooperative Game Theory,", 2nd edition, (1995).   Google Scholar

[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.  doi: 10.1016/j.amc.2010.01.048.  Google Scholar

[6]

A. Bensoussan, S. Chen and S. P. Sethi, Feedback Stackelberg solutions of infinite-horizon stochastic differential games,, forthcoming., ().   Google Scholar

[7]

A. Bensoussan, S. Chen and S. P. Sethi, The maximum principle for global solutions of stochastic Stackelberg differential games,, working paper., ().   Google Scholar

[8]

G. F. Cachon, Supply chain coordination with contracts,, in, (2003), 227.   Google Scholar

[9]

A. Chutani and S. P. Sethi, Cooperative advertising in a dynamic retail market oligopoly,, Dynamic Games and Applications, (2012).  doi: 10.1007/s13235-012-0053-8.  Google Scholar

[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.  doi: 10.1007/s10957-012-0034-5.  Google Scholar

[11]

E. Dockner, S. Jøgensen, N. V. Long and G. Sorger, "Differential Games in Economics and Management Science,", Cambridge University Press, (2000).  doi: 10.1017/CBO9780511805127.  Google Scholar

[12]

X. He, A. Krishnamoorthy, A. Prasad and S. P. Sethi, Retail competition and cooperative advertising,, Operations Research Letters, 39 (2011), 11.  doi: 10.1016/j.orl.2010.10.006.  Google Scholar

[13]

X. He, A. Krishnamoorthy, A. Prasad and S. P. Sethi, Co-Op advertising in dynamic retail oligopolies,, Decision Sciences, 43 (2012), 73.  doi: 10.1111/j.1540-5915.2011.00336.x.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s11518-007-5058-2.  Google Scholar

[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.  doi: 10.1016/j.ejor.2009.01.003.  Google Scholar

[17]

G. Leitmann, On generalized Stackelberg strategies,, J. Optimization Theory and Applications, 26 (1978), 637.  doi: 10.1007/BF00933155.  Google Scholar

[18]

E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs,, Probab. Theory Relat. Fields, 114 (1999), 123.  doi: 10.1007/s004409970001.  Google Scholar

[19]

A. Prasad, S. P. Sethi and P. A. Naik, Understanding the impact of churn in dynamic oligopoly markets,, Automatica, 48 (2012), 2882.  doi: 10.1016/j.automatica.2012.08.031.  Google Scholar

[20]

M. Simaan and J. B. Cruz, Jr., On the Stackelberg strategy in nonzero-sum games,, J. Optimization Theory and Applications, 11 (1973), 533.  doi: 10.1007/BF00935665.  Google Scholar

[21]

M. Simaan and J. B. Cruz, Jr., Additional aspects of the Stackelberg strategy in nonzero-sum games,, J. Optimization Theory and Applications, 11 (): 613.  doi: 10.1007/BF00935561.  Google Scholar

[22]

H. von Stackelberg, "Marktform und Gleichgewicht,", Springer, (1934).   Google Scholar

[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.  doi: 10.1137/S0363012901387550.  Google Scholar

[24]

J. Yong, Linear forward-backward stochastic differential equations with random coefficients,, Probab. Theory Relat. Fields, 135 (2006), 53.  doi: 10.1007/s00440-005-0452-5.  Google Scholar

show all references

References:
[1]

T. Başar, On the relative leadership property of Stackelberg strategies,, J. Optimization Theory and Applications, 11 (1973), 655.  doi: 10.1007/BF00935564.  Google Scholar

[2]

T. Başar and A. Haurie, Feedback equilibria in differential games with structural and modal uncertainties,, in, (1984), 163.   Google Scholar

[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.  doi: 10.1016/0165-1889(85)90026-0.  Google Scholar

[4]

T. Başar and G. J. Olsder, "Dynamic Noncooperative Game Theory,", 2nd edition, (1995).   Google Scholar

[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.  doi: 10.1016/j.amc.2010.01.048.  Google Scholar

[6]

A. Bensoussan, S. Chen and S. P. Sethi, Feedback Stackelberg solutions of infinite-horizon stochastic differential games,, forthcoming., ().   Google Scholar

[7]

A. Bensoussan, S. Chen and S. P. Sethi, The maximum principle for global solutions of stochastic Stackelberg differential games,, working paper., ().   Google Scholar

[8]

G. F. Cachon, Supply chain coordination with contracts,, in, (2003), 227.   Google Scholar

[9]

A. Chutani and S. P. Sethi, Cooperative advertising in a dynamic retail market oligopoly,, Dynamic Games and Applications, (2012).  doi: 10.1007/s13235-012-0053-8.  Google Scholar

[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.  doi: 10.1007/s10957-012-0034-5.  Google Scholar

[11]

E. Dockner, S. Jøgensen, N. V. Long and G. Sorger, "Differential Games in Economics and Management Science,", Cambridge University Press, (2000).  doi: 10.1017/CBO9780511805127.  Google Scholar

[12]

X. He, A. Krishnamoorthy, A. Prasad and S. P. Sethi, Retail competition and cooperative advertising,, Operations Research Letters, 39 (2011), 11.  doi: 10.1016/j.orl.2010.10.006.  Google Scholar

[13]

X. He, A. Krishnamoorthy, A. Prasad and S. P. Sethi, Co-Op advertising in dynamic retail oligopolies,, Decision Sciences, 43 (2012), 73.  doi: 10.1111/j.1540-5915.2011.00336.x.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s11518-007-5058-2.  Google Scholar

[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.  doi: 10.1016/j.ejor.2009.01.003.  Google Scholar

[17]

G. Leitmann, On generalized Stackelberg strategies,, J. Optimization Theory and Applications, 26 (1978), 637.  doi: 10.1007/BF00933155.  Google Scholar

[18]

E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs,, Probab. Theory Relat. Fields, 114 (1999), 123.  doi: 10.1007/s004409970001.  Google Scholar

[19]

A. Prasad, S. P. Sethi and P. A. Naik, Understanding the impact of churn in dynamic oligopoly markets,, Automatica, 48 (2012), 2882.  doi: 10.1016/j.automatica.2012.08.031.  Google Scholar

[20]

M. Simaan and J. B. Cruz, Jr., On the Stackelberg strategy in nonzero-sum games,, J. Optimization Theory and Applications, 11 (1973), 533.  doi: 10.1007/BF00935665.  Google Scholar

[21]

M. Simaan and J. B. Cruz, Jr., Additional aspects of the Stackelberg strategy in nonzero-sum games,, J. Optimization Theory and Applications, 11 (): 613.  doi: 10.1007/BF00935561.  Google Scholar

[22]

H. von Stackelberg, "Marktform und Gleichgewicht,", Springer, (1934).   Google Scholar

[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.  doi: 10.1137/S0363012901387550.  Google Scholar

[24]

J. Yong, Linear forward-backward stochastic differential equations with random coefficients,, Probab. Theory Relat. Fields, 135 (2006), 53.  doi: 10.1007/s00440-005-0452-5.  Google Scholar

[1]

Feliz Minhós, A. I. Santos. Higher order two-point boundary value problems with asymmetric growth. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 127-137. doi: 10.3934/dcdss.2008.1.127
[2]

Jerry L. Bona, Hongqiu Chen, Shu-Ming Sun, Bing-Yu Zhang. Comparison of quarter-plane and two-point boundary value problems: The KdV-equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 465-495. doi: 10.3934/dcdsb.2007.7.465
[3]

Xiao-Yu Zhang, Qing Fang. A sixth order numerical method for a class of nonlinear two-point boundary value problems. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 31-43. doi: 10.3934/naco.2012.2.31
[4]

Jerry Bona, Hongqiu Chen, Shu Ming Sun, B.-Y. Zhang. Comparison of quarter-plane and two-point boundary value problems: the BBM-equation. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 921-940. doi: 10.3934/dcds.2005.13.921
[5]

Wenming Zou. Multiple solutions results for two-point boundary value problem with resonance. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 485-496. doi: 10.3934/dcds.1998.4.485
[6]

Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839
[7]

K. Q. Lan, G. C. Yang. Optimal constants for two point boundary value problems. Conference Publications, 2007, 2007 (Special) : 624-633. doi: 10.3934/proc.2007.2007.624
[8]

Jeremiah Birrell. A posteriori error bounds for two point boundary value problems: A green's function approach. Journal of Computational Dynamics, 2015, 2 (2) : 143-164. doi: 10.3934/jcd.2015001
[9]

Wen-Chiao Cheng. Two-point pre-image entropy. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 107-119. doi: 10.3934/dcds.2007.17.107
[10]

Hermen Jan Hupkes, Emmanuelle Augeraud-Véron. Well-posedness of initial value problems for functional differential and algebraic equations of mixed type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 737-765. doi: 10.3934/dcds.2011.30.737
[11]

Felix Sadyrbaev, Inara Yermachenko. Multiple solutions of nonlinear boundary value problems for two-dimensional differential systems. Conference Publications, 2009, 2009 (Special) : 659-668. doi: 10.3934/proc.2009.2009.659
[12]

Alexei Korolev, Gennady Ougolnitsky. Optimal resource allocation in the difference and differential Stackelberg games on marketing networks. Journal of Dynamics & Games, 2020, 7 (2) : 141-162. doi: 10.3934/jdg.2020009
[13]

Chan-Gyun Kim, Yong-Hoon Lee. A bifurcation result for two point boundary value problem with a strong singularity. Conference Publications, 2011, 2011 (Special) : 834-843. doi: 10.3934/proc.2011.2011.834
[14]

M.J. Lopez-Herrero. The existence of weak solutions for a general class of mixed boundary value problems. Conference Publications, 2011, 2011 (Special) : 1015-1024. doi: 10.3934/proc.2011.2011.1015
[15]

Santiago Cano-Casanova. Coercivity of elliptic mixed boundary value problems in annulus of $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3819-3839. doi: 10.3934/dcds.2012.32.3819
[16]

J. R. L. Webb. Remarks on positive solutions of some three point boundary value problems. Conference Publications, 2003, 2003 (Special) : 905-915. doi: 10.3934/proc.2003.2003.905
[17]

K. Q. Lan. Properties of kernels and eigenvalues for three point boundary value problems. Conference Publications, 2005, 2005 (Special) : 546-555. doi: 10.3934/proc.2005.2005.546
[18]

Wenying Feng. Solutions and positive solutions for some three-point boundary value problems. Conference Publications, 2003, 2003 (Special) : 263-272. doi: 10.3934/proc.2003.2003.263
[19]

John R. Graef, Shapour Heidarkhani, Lingju Kong. Existence of nontrivial solutions to systems of multi-point boundary value problems. Conference Publications, 2013, 2013 (special) : 273-281. doi: 10.3934/proc.2013.2013.273
[20]

Lingju Kong, Qingkai Kong. Existence of nodal solutions of multi-point boundary value problems. Conference Publications, 2009, 2009 (Special) : 457-465. doi: 10.3934/proc.2009.2009.457

 Impact Factor: 

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences