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

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

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.
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]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[2]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[3]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[4]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[5]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052

[6]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[7]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[8]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[9]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[10]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[11]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051

[12]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[13]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[14]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267

[15]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[16]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[17]

Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465

[18]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[19]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071

[20]

Nicolas Rougerie. On two properties of the Fisher information. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020049

 Impact Factor: 

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (8)

[Back to Top]