American Institute of Mathematical Sciences

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January  2017, 22(1): 125-159. doi: 10.3934/dcdsb.2017007

A review of dynamic Stackelberg game models

Tao Li 1,, and  Suresh P. Sethi 2,

1. 

Santa Clara University, 500 El Camino Real, Santa Clara, CA 95053, USA
2. 

The University of Texas at Dallas, 800 W Campbell Rd, Richardson, TX 75080, USA

* Corresponding author: Tao Li

Received  January 2016 Revised  June 2016 Published  December 2016

Dynamic Stackelberg game models have been used to study sequential decision making in noncooperative games in various fields. In this paper we give relevant dynamic Stackelberg game models, and review their applications to operations management and marketing channels. A common feature of these applications is the specification of the game structure: a decentralized channel consists of a manufacturer and independent retailers, and a sequential decision process with a state dynamics. In operations management, Stackelberg games have been used to study inventory issues, such as wholesale and retail pricing strategies, outsourcing, and learning effects in dynamic environments. The underlying demand typically has a growing trend or seasonal variation. In marketing, dynamic Stackelberg games have been used to model cooperative advertising programs, store brand and national brand advertising strategies, shelf space allocation, and pricing and advertising decisions. The demand dynamics are usually extensions of the classic advertising capital models or sales-advertising response models. We begin each section by introducing the relevant dynamic Stackelberg game formulation along with the definition of the equilibrium used, and then review the models and results appearing in the literature.

Keywords: Dynamic game, Stackelberg game, Nash game, non-cooperative game, operations management, marketing, open-loop equilibrium, feedback equilibrium, maximum principle, time consistency.
Mathematics Subject Classification: Primary:91A25, 91A15, 91A23;Secondary:91A1.
Citation: Tao Li, Suresh P. Sethi. A review of dynamic Stackelberg game models. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 125-159. doi: 10.3934/dcdsb.2017007
References:
[1]

K. Anand, R. Anupindi and Y. Bassok, Strategic inventories in vertical contracts, Management Science, 54 (2008), 1792-1804. Google Scholar

[2]

A. Bagchi, Stackelberg Differential Games in Economic Models Springer-Verlag, New York, 1994. doi: 10.1007/BFb0009151. Google Scholar

[3]

T. Başar and A. Haurie, Feedback equilibria in differential games with structural and modal uncertainties, in Advances in Large Scale Systems (eds. Jose B. Cruz Jr.), JAE Press Inc., Connecticut, 1 (1984), 163-201. Google Scholar

[4]

T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory 2nd ed. , SIAM, Philadelphia, PA, 1999. Google Scholar

[5]

F. M. Bass, A new product growth for model consumer durables, Chapter: Mathematical Models in Marketing, 132 (1976), 351-353. doi: 10.1007/978-3-642-51565-1_107. Google Scholar

[6]

A. Bensoussan, S. Chen, A. Chutani and S. P. Sethi, Feedback Stackelberg-Nash equilibria in mixed leadership games with an application to cooperative advertising, working paper, The University of Texas at Dallas, 2016.Google Scholar

[7]

A. Bensoussan, S. Chen and S. P. Sethi, Linear quadratic differential games with mixed leadership: The open-loop solution, Numerical Algebra, Control and Optimization, 3 (2013), 95-108. doi: 10.3934/naco.2013.3.95. Google Scholar

[8]

A. Bensoussan, S. Chen and S. P. Sethi, Feedback Stackelberg solutions of infinite-horizon stochastic differential games, in Models and Methods in Economics and Management Science (eds. F. El Ouardighi and K. Kogan), International Series in Operations Research and Management Science 198, Springer, (2014), 3-15. doi: 10.1007/978-3-319-00669-7_1. Google Scholar

[9]

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. Google Scholar

[10]

M. Bergen and G. John, Understanding cooperative advertising participation rates in conventional channels, Journal of Marketing Research, 34 (1997), 357-369. doi: 10.2307/3151898. Google Scholar

[11]

P. D. Berger, Vertical cooperative advertising ventures, Journal of Marketing Research, 9 (1972), 309-312. doi: 10.2307/3149542. Google Scholar

[12]

M. Breton, R. Jarrar and G. Zaccour, A note on feedback sequential equilibria in a Lanchester model with empirical application, Management Science, 52 (2006), 804-811. doi: 10.1287/mnsc.1050.0475. Google Scholar

[13]

P. Chintagunta and D. Jain, A dynamic model of channel member strategies for marketing expenditures, Marketing Science, 11 (1992), 168-188. doi: 10.1287/mksc.11.2.168. Google Scholar

[14]

K. R. Deal, Optimizing advertising expenditures in a dynamic duopoly, Operations Research, 27 (1979), 682-692. doi: 10.1287/opre.27.4.682. Google Scholar

[15]

K. R. Deal, S. P. Sethi and G. L. Thompson, A bilinear-quadratic differential game in advertising, in Control Theory in Mathematical Economics (eds. P.-T. Liu and J. G. Sutinen), Marcel Dekker, Inc., New York, NY., 47 (1979), 91-109. Google Scholar

[16]

N. A. Derzko, S. P. Sethi and G. L. Thompson, Necessary and sufficient conditions for optimal control of quasilinear partial differential systems, Journal of Optimal Theory and Applications, 43 (1984), 89-101. doi: 10.1007/BF00934748. Google Scholar

[17]

V. S. Desai, Marketing-production decisions under independent and integrated channel structure, Annals of Operations Research, 34 (1992), 275-306. doi: 10.1007/BF02098183. Google Scholar

[18]

V. S. Desai, Interactions between members of a marketing-production channel under seasonal demand, European Journal of Operational Research, 90 (1996), 115-141. doi: 10.1016/0377-2217(94)00308-4. Google Scholar

[19]

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

[20]

J. Eliashberg and R. Steinberg, Marketing-production decisions in an industrial channel of distribution, Management Science, 33 (1987), 981-1000. doi: 10.1287/mnsc.33.8.981. Google Scholar

[21]

G. M. Erickson, Empirical analysis of closed-loop duopoly advertising strategies, Management Science, 38 (1992), 1732-1749. doi: 10.1287/mnsc.38.12.1732. Google Scholar

[22]

W. H. Fleming and R. W. Rishel, Deterministic and stochastic optimal control, in Stochastic Modelling and Applied Probability (eds. P. W. Glynn and Y. Le Jan), V. 1, Springer-Verlag, New York, NY, 1975. Google Scholar

[23]

G. E. Fruchter and S. Kalish, Closed-loop advertising strategies in a duopoly, Management Science, 43 (1997), 54-63. doi: 10.1287/mnsc.43.1.54. Google Scholar

[24]

J. V. Gray, B. Tomlin and A. V. Roth, Outsourcing to a powerful contract manufacturer: The effect of learning-by-doing, Production and Operations Management, 18 (2009), 487-505. doi: 10.1111/j.1937-5956.2009.01024.x. Google Scholar

[25]

G. J. Gutierrez and X. He, Life-cycle channel coordination issues in launching an innovative durable product, Production and Operations Management, 20 (2011), 268-279. doi: 10.1111/j.1937-5956.2010.01197.x. Google Scholar

[26]

X. He, A. Prasad, S. P. Sethi and G. J. Gutierrez, A survey of differential game models in supply and marketing channels, Journal of Systems Science and Systems Engineering, 16 (2007), 385-413. Google Scholar

[27]

X. He and S. P. Sethi, Dynamic slotting and pricing decisions in a durable product supply chain, Journal of Optimization Theory and Applications, 137 (2008), 363-379. doi: 10.1007/s10957-007-9330-x. Google Scholar

[28]

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. doi: 10.1109/PICMET.2008.4599783. Google Scholar

[29]

Z. Huang, S. X. Li and V. Mahajan, An analysis of manufacturer-retailer supply chain coordination in cooperative advertising, Decision Sciences, 33 (2002), 469-494. doi: 10.1111/j.1540-5915.2002.tb01652.x. Google Scholar

[30]

R. Jarrar, G. Martin-Herran and G. Zaccour, Markov perfect equilibrium advertising strategies of Lanchester duopoly model: A technical note, Management Science, 50 (2004), 995-1000. doi: 10.1287/mnsc.1040.0249. Google Scholar

[31]

S. Jorgensen, S. P. Sigue and G. Zaccour, Dynamic cooperative advertising in a channel, Journal of Retailing, 76 (2000), 71-92. Google Scholar

[32]

S. Jorgensen, S. P. Sigue and G. Zaccour, Stackelberg leadership in a marketing channel, International Game Theory Review, 3 (2001), 13-26. doi: 10.1142/S0219198901000282. Google Scholar

[33]

S. Jorgensen, S. Taboubi and G. Zaccour, Retail promotions with negative brand image effects: Is cooperation possible?, European Journal of Operational Research, 150 (2003), 395-405. doi: 10.1016/S0377-2217(02)00641-0. Google Scholar

[34]

S. Jorgensen, S. Taboubi and G. Zaccour, Incentives for retailer promotion in a marketing channel, Annals of the International Society of dynamic Games, 8 (2006), 365-378. doi: 10.1007/0-8176-4501-2_19. Google Scholar

[35]

S. Karray and G. Zaccour, A differential game of advertising for national and store brands, in Dynamic Games: Theory and Applicatoins (eds. A. Haurie and G. Zaccour), Springer, New York, NY, 10 (2005), 213-229. doi: 10.1007/0-387-24602-9_11. Google Scholar

[36]

K. Kogan and C. S. Tapiero, Supply Chain Games: Operations Management and Risk Valuation Springer, New York, NY, 2007. doi: 10.1007/978-0-387-72776-9. Google Scholar

[37]

K. Kogan and C. S. Tapiero, Co-investment in Supply Chain Infrastructure working paper, Bar Ilan University, Israel, 2007.Google Scholar

[38]

H. L. Lee, V. Padmanabhan, T. A. Taylor and S. Whang, Price protection in the personal computer industry, Management Science, 46 (2000), 467-482. doi: 10.1287/mnsc.46.4.467.12058. Google Scholar

[39]

G. Leitmann, On generalized Stackelberg strategies, J. Optimization Theory and Applications, 26 (1978), 637-643. doi: 10.1007/BF00933155. Google Scholar

[40]

T. Li, S. P. Sethi and X. He, Dynamic pricing, production, and channel coordination with stochastic learning, Production and Operations Management, 24 (2015), 857-882. doi: 10.1111/poms.12320. Google Scholar

[41]

C. T. Linh and Y. Hong, Channel coordination through a revenue sharing contract in a two-period newsboy problem, European Journal of Operational Research, 198 (2009), 822-829. doi: 10.1016/j.ejor.2008.10.019. Google Scholar

[42]

J. D. C. Little, Aggregate advertising models: The state of the art, Operations Research, 27 (1979), 629-667. Google Scholar

[43]

X. Lu, J. S. Song and A. Regan, Rebate, returns and price protection policies in channel coordination, IIE Transactions, 39 (2007), 111-124. doi: 10.1080/07408170600710408. Google Scholar

[44]

G. Martin-Herran and S. Taboubi, Incentive strategies for shelf-space allocation in duopolies, in Dynamic Games Theory and Applications (eds. A. Haurie and G. Zaccour), Springer, New York, NY, 10 (2005), 231-253. doi: 10.1007/0-387-24602-9_12. Google Scholar

[45]

M. Nerlove and K. J. Arrow, Optimal advertising policy under dynamic conditions, Mathematical Models in Marketing, 132 (1976), 167-168. doi: 10.1007/978-3-642-51565-1_54. Google Scholar

[46]

G. P. Papavassilopoulos and J. B. Cruz, Nonclassical control problems and Stackelberg games, IEEE Transactions on Automatic Control, 24 (1979), 155-166. doi: 10.1109/TAC.1979.1101986. Google Scholar

[47]

D. Pekelman, Simultaneous price production in channels, Marketing Science, 7 (1974), 335-355. Google Scholar

[48]

S. J. Rubio, On coincidence of feedback Nash equilibria and Stackelberg equilibria in economic applications of differential games, Journal of Optimization Theory and Applications, 128 (2006), 203-221. doi: 10.1007/s10957-005-7565-y. Google Scholar

[49]

S. P. Sethi, Deterministic and stochastic optimization of a dynamic advertising model, Optimal Control Applications and Methods, 4 (1983), 179-184. doi: 10.1002/oca.4660040207. Google Scholar

[50]

S. P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics 2$^{nd}$ edition, Springer, New York, 2000. Google Scholar

[51]

M. Simaan and J. B. Cruz, On the Stackelberg strategy in nonzero-sum games, J. Optimization Theory and Applications, 11 (1973), 533-555. doi: 10.1007/BF00935665. Google Scholar

[52]

M. Simaan and J. B. Cruz, Additional aspects of the Stackelberg strategy in nonzero-sum games, J. Optimization Theory and Applications, 11 (1973), 613-626. doi: 10.1007/BF00935561. Google Scholar

[53]

H. V. Stackelberg, The Theory of the Market Economy translated by Peacock A. T. , William Hodge and Co. , London, 1952.Google Scholar

[54]

T. A. Taylor, Channel coordination under price protection, midlife returns, and end-of-life returns in dynamic markets, Management Science, 47 (2001), 1220-1234. doi: 10.1287/mnsc.47.9.1220.9786. Google Scholar

[55]

J. T. Teng and G. L. Thompson, Oligopoly models for optimal advertising when production costs obey a learning curve, Management Science, 29 (1983), 1087-1101. doi: 10.1287/mnsc.29.9.1087. Google Scholar

[56]

M. L. Vidale and H. B. Wolfe, An operations research study of sales response to advertising, Operations Research, 5 (1957), 370-381. doi: 10.1287/opre.5.3.370. Google Scholar

[57]

H. Von Stackelberg, Marktform und Gleichgewicht Springer, Vienna, 1934. (An English translation appeared in The Theory of the Market Economy Oxford University Press, Oxford, England, 1952. )Google Scholar

show all references

References:
[1]

K. Anand, R. Anupindi and Y. Bassok, Strategic inventories in vertical contracts, Management Science, 54 (2008), 1792-1804. Google Scholar

[2]

A. Bagchi, Stackelberg Differential Games in Economic Models Springer-Verlag, New York, 1994. doi: 10.1007/BFb0009151. Google Scholar

[3]

T. Başar and A. Haurie, Feedback equilibria in differential games with structural and modal uncertainties, in Advances in Large Scale Systems (eds. Jose B. Cruz Jr.), JAE Press Inc., Connecticut, 1 (1984), 163-201. Google Scholar

[4]

T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory 2nd ed. , SIAM, Philadelphia, PA, 1999. Google Scholar

[5]

F. M. Bass, A new product growth for model consumer durables, Chapter: Mathematical Models in Marketing, 132 (1976), 351-353. doi: 10.1007/978-3-642-51565-1_107. Google Scholar

[6]

A. Bensoussan, S. Chen, A. Chutani and S. P. Sethi, Feedback Stackelberg-Nash equilibria in mixed leadership games with an application to cooperative advertising, working paper, The University of Texas at Dallas, 2016.Google Scholar

[7]

A. Bensoussan, S. Chen and S. P. Sethi, Linear quadratic differential games with mixed leadership: The open-loop solution, Numerical Algebra, Control and Optimization, 3 (2013), 95-108. doi: 10.3934/naco.2013.3.95. Google Scholar

[8]

A. Bensoussan, S. Chen and S. P. Sethi, Feedback Stackelberg solutions of infinite-horizon stochastic differential games, in Models and Methods in Economics and Management Science (eds. F. El Ouardighi and K. Kogan), International Series in Operations Research and Management Science 198, Springer, (2014), 3-15. doi: 10.1007/978-3-319-00669-7_1. Google Scholar

[9]

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. Google Scholar

[10]

M. Bergen and G. John, Understanding cooperative advertising participation rates in conventional channels, Journal of Marketing Research, 34 (1997), 357-369. doi: 10.2307/3151898. Google Scholar

[11]

P. D. Berger, Vertical cooperative advertising ventures, Journal of Marketing Research, 9 (1972), 309-312. doi: 10.2307/3149542. Google Scholar

[12]

M. Breton, R. Jarrar and G. Zaccour, A note on feedback sequential equilibria in a Lanchester model with empirical application, Management Science, 52 (2006), 804-811. doi: 10.1287/mnsc.1050.0475. Google Scholar

[13]

P. Chintagunta and D. Jain, A dynamic model of channel member strategies for marketing expenditures, Marketing Science, 11 (1992), 168-188. doi: 10.1287/mksc.11.2.168. Google Scholar

[14]

K. R. Deal, Optimizing advertising expenditures in a dynamic duopoly, Operations Research, 27 (1979), 682-692. doi: 10.1287/opre.27.4.682. Google Scholar

[15]

K. R. Deal, S. P. Sethi and G. L. Thompson, A bilinear-quadratic differential game in advertising, in Control Theory in Mathematical Economics (eds. P.-T. Liu and J. G. Sutinen), Marcel Dekker, Inc., New York, NY., 47 (1979), 91-109. Google Scholar

[16]

N. A. Derzko, S. P. Sethi and G. L. Thompson, Necessary and sufficient conditions for optimal control of quasilinear partial differential systems, Journal of Optimal Theory and Applications, 43 (1984), 89-101. doi: 10.1007/BF00934748. Google Scholar

[17]

V. S. Desai, Marketing-production decisions under independent and integrated channel structure, Annals of Operations Research, 34 (1992), 275-306. doi: 10.1007/BF02098183. Google Scholar

[18]

V. S. Desai, Interactions between members of a marketing-production channel under seasonal demand, European Journal of Operational Research, 90 (1996), 115-141. doi: 10.1016/0377-2217(94)00308-4. Google Scholar

[19]

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

[20]

J. Eliashberg and R. Steinberg, Marketing-production decisions in an industrial channel of distribution, Management Science, 33 (1987), 981-1000. doi: 10.1287/mnsc.33.8.981. Google Scholar

[21]

G. M. Erickson, Empirical analysis of closed-loop duopoly advertising strategies, Management Science, 38 (1992), 1732-1749. doi: 10.1287/mnsc.38.12.1732. Google Scholar

[22]

W. H. Fleming and R. W. Rishel, Deterministic and stochastic optimal control, in Stochastic Modelling and Applied Probability (eds. P. W. Glynn and Y. Le Jan), V. 1, Springer-Verlag, New York, NY, 1975. Google Scholar

[23]

G. E. Fruchter and S. Kalish, Closed-loop advertising strategies in a duopoly, Management Science, 43 (1997), 54-63. doi: 10.1287/mnsc.43.1.54. Google Scholar

[24]

J. V. Gray, B. Tomlin and A. V. Roth, Outsourcing to a powerful contract manufacturer: The effect of learning-by-doing, Production and Operations Management, 18 (2009), 487-505. doi: 10.1111/j.1937-5956.2009.01024.x. Google Scholar

[25]

G. J. Gutierrez and X. He, Life-cycle channel coordination issues in launching an innovative durable product, Production and Operations Management, 20 (2011), 268-279. doi: 10.1111/j.1937-5956.2010.01197.x. Google Scholar

[26]

X. He, A. Prasad, S. P. Sethi and G. J. Gutierrez, A survey of differential game models in supply and marketing channels, Journal of Systems Science and Systems Engineering, 16 (2007), 385-413. Google Scholar

[27]

X. He and S. P. Sethi, Dynamic slotting and pricing decisions in a durable product supply chain, Journal of Optimization Theory and Applications, 137 (2008), 363-379. doi: 10.1007/s10957-007-9330-x. Google Scholar

[28]

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. doi: 10.1109/PICMET.2008.4599783. Google Scholar

[29]

Z. Huang, S. X. Li and V. Mahajan, An analysis of manufacturer-retailer supply chain coordination in cooperative advertising, Decision Sciences, 33 (2002), 469-494. doi: 10.1111/j.1540-5915.2002.tb01652.x. Google Scholar

[30]

R. Jarrar, G. Martin-Herran and G. Zaccour, Markov perfect equilibrium advertising strategies of Lanchester duopoly model: A technical note, Management Science, 50 (2004), 995-1000. doi: 10.1287/mnsc.1040.0249. Google Scholar

[31]

S. Jorgensen, S. P. Sigue and G. Zaccour, Dynamic cooperative advertising in a channel, Journal of Retailing, 76 (2000), 71-92. Google Scholar

[32]

S. Jorgensen, S. P. Sigue and G. Zaccour, Stackelberg leadership in a marketing channel, International Game Theory Review, 3 (2001), 13-26. doi: 10.1142/S0219198901000282. Google Scholar

[33]

S. Jorgensen, S. Taboubi and G. Zaccour, Retail promotions with negative brand image effects: Is cooperation possible?, European Journal of Operational Research, 150 (2003), 395-405. doi: 10.1016/S0377-2217(02)00641-0. Google Scholar

[34]

S. Jorgensen, S. Taboubi and G. Zaccour, Incentives for retailer promotion in a marketing channel, Annals of the International Society of dynamic Games, 8 (2006), 365-378. doi: 10.1007/0-8176-4501-2_19. Google Scholar

[35]

S. Karray and G. Zaccour, A differential game of advertising for national and store brands, in Dynamic Games: Theory and Applicatoins (eds. A. Haurie and G. Zaccour), Springer, New York, NY, 10 (2005), 213-229. doi: 10.1007/0-387-24602-9_11. Google Scholar

[36]

K. Kogan and C. S. Tapiero, Supply Chain Games: Operations Management and Risk Valuation Springer, New York, NY, 2007. doi: 10.1007/978-0-387-72776-9. Google Scholar

[37]

K. Kogan and C. S. Tapiero, Co-investment in Supply Chain Infrastructure working paper, Bar Ilan University, Israel, 2007.Google Scholar

[38]

H. L. Lee, V. Padmanabhan, T. A. Taylor and S. Whang, Price protection in the personal computer industry, Management Science, 46 (2000), 467-482. doi: 10.1287/mnsc.46.4.467.12058. Google Scholar

[39]

G. Leitmann, On generalized Stackelberg strategies, J. Optimization Theory and Applications, 26 (1978), 637-643. doi: 10.1007/BF00933155. Google Scholar

[40]

T. Li, S. P. Sethi and X. He, Dynamic pricing, production, and channel coordination with stochastic learning, Production and Operations Management, 24 (2015), 857-882. doi: 10.1111/poms.12320. Google Scholar

[41]

C. T. Linh and Y. Hong, Channel coordination through a revenue sharing contract in a two-period newsboy problem, European Journal of Operational Research, 198 (2009), 822-829. doi: 10.1016/j.ejor.2008.10.019. Google Scholar

[42]

J. D. C. Little, Aggregate advertising models: The state of the art, Operations Research, 27 (1979), 629-667. Google Scholar

[43]

X. Lu, J. S. Song and A. Regan, Rebate, returns and price protection policies in channel coordination, IIE Transactions, 39 (2007), 111-124. doi: 10.1080/07408170600710408. Google Scholar

[44]

G. Martin-Herran and S. Taboubi, Incentive strategies for shelf-space allocation in duopolies, in Dynamic Games Theory and Applications (eds. A. Haurie and G. Zaccour), Springer, New York, NY, 10 (2005), 231-253. doi: 10.1007/0-387-24602-9_12. Google Scholar

[45]

M. Nerlove and K. J. Arrow, Optimal advertising policy under dynamic conditions, Mathematical Models in Marketing, 132 (1976), 167-168. doi: 10.1007/978-3-642-51565-1_54. Google Scholar

[46]

G. P. Papavassilopoulos and J. B. Cruz, Nonclassical control problems and Stackelberg games, IEEE Transactions on Automatic Control, 24 (1979), 155-166. doi: 10.1109/TAC.1979.1101986. Google Scholar

[47]

D. Pekelman, Simultaneous price production in channels, Marketing Science, 7 (1974), 335-355. Google Scholar

[48]

S. J. Rubio, On coincidence of feedback Nash equilibria and Stackelberg equilibria in economic applications of differential games, Journal of Optimization Theory and Applications, 128 (2006), 203-221. doi: 10.1007/s10957-005-7565-y. Google Scholar

[49]

S. P. Sethi, Deterministic and stochastic optimization of a dynamic advertising model, Optimal Control Applications and Methods, 4 (1983), 179-184. doi: 10.1002/oca.4660040207. Google Scholar

[50]

S. P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics 2$^{nd}$ edition, Springer, New York, 2000. Google Scholar

[51]

M. Simaan and J. B. Cruz, On the Stackelberg strategy in nonzero-sum games, J. Optimization Theory and Applications, 11 (1973), 533-555. doi: 10.1007/BF00935665. Google Scholar

[52]

M. Simaan and J. B. Cruz, Additional aspects of the Stackelberg strategy in nonzero-sum games, J. Optimization Theory and Applications, 11 (1973), 613-626. doi: 10.1007/BF00935561. Google Scholar

[53]

H. V. Stackelberg, The Theory of the Market Economy translated by Peacock A. T. , William Hodge and Co. , London, 1952.Google Scholar

[54]

T. A. Taylor, Channel coordination under price protection, midlife returns, and end-of-life returns in dynamic markets, Management Science, 47 (2001), 1220-1234. doi: 10.1287/mnsc.47.9.1220.9786. Google Scholar

[55]

J. T. Teng and G. L. Thompson, Oligopoly models for optimal advertising when production costs obey a learning curve, Management Science, 29 (1983), 1087-1101. doi: 10.1287/mnsc.29.9.1087. Google Scholar

[56]

M. L. Vidale and H. B. Wolfe, An operations research study of sales response to advertising, Operations Research, 5 (1957), 370-381. doi: 10.1287/opre.5.3.370. Google Scholar

[57]

H. Von Stackelberg, Marktform und Gleichgewicht Springer, Vienna, 1934. (An English translation appeared in The Theory of the Market Economy Oxford University Press, Oxford, England, 1952. )Google Scholar

Figure 1.  Optimal Policies with Promotion
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Table 1.  Notations
$*$ Optimal/ equilibrium levels $\hat{c}_{i}$ Unit advertising cost
$i$ Denotes the $i^{th}$ player $f$ Production function, $\dot{x}$
$A$ , $\hat{A}$ Advertising level $h_{i}$ Unit inventory/ backlog cost
$B$ , $\hat{B}$ Local advertising $h_{i}^{+}$ Unit inventory holding cost
$C_{i}$ Cost of production/advertising $h_{i}^{-}$ Unit backlog cost
$D$ Demand, Revenue rate $m_{i}$ Margin
$G$ , $G_{i}$ Goodwill $p$ , $p_{i}$ Retail price
$H_{i}$ , $\bar{H}_{i}$ Current -value Hamiltonians $q$ Manufacturer's share of revenue
$I_{i}$ Inventory level $r_{1}$ , $r_{2}$ Effectiveness of advertising
$J_{i}$ Objective functional $t_{1}^{d}$ , $t_{2}^{d}$ Time parameters
$K$ Infrastructure capital $t_{s}$ , $t_{f}$ Start & end of promotional period
$L_{i}$ Labor force $u$ Leader's control variable
$M$ Market size $v$ ,Follower's control variable
$N$ Number of firms $v^0$ Optimal response of the follower
$Q_{i}$ Production rate, Processing rate $w$ Wholesale price
$\bar{Q}_{i}$ Capacity limit $x$ State variable; Sales rate
$S$ Shelf space $\alpha$ External market influence
$S_{i}$ Salvage value, Unit salvage value $\alpha_{i}$ Demand parameters
$T$ Planning horizon $\beta$ Internal market influence
$r$ Optimal response of the follower $\delta$ Decay rate
$V_{i}$ Value function $\theta$ , $\hat{\theta}$ M's share of R's advertising cost
$X$ Cumulative sales $\lambda_{i}$ Adjoint variable, Shadow price
$a$ ,Market potential/Advertising effectiveness $\pi_{i}$ Instantaneous profit rate
$a_{i}$ , $b_{i}$ , $d$ , $e_{i}$ Problem parameters $\rho$ Discount rate
$a_{_{l}}$ , $a_{_{s}}$ , $b_{_{l}}$ , $b_{_{s}}$ Advertising effectiveness $\phi$ , $\psi$ Adjoint variable, Shadow price
$b$ Price sensitivity/Advertising effectiveness $\omega_{i}$ Coefficient of incentive strategy
$c_{i}$ Unit production/advertising cost $\mathcal {U}$ , $\mathcal {V}$ Feasible set of controls
$q_i$ Order quantity $\Lambda$ , $\gamma_i$ Learning efficiency
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$*$ Optimal/ equilibrium levels $\hat{c}_{i}$ Unit advertising cost
$i$ Denotes the $i^{th}$ player $f$ Production function, $\dot{x}$
$A$ , $\hat{A}$ Advertising level $h_{i}$ Unit inventory/ backlog cost
$B$ , $\hat{B}$ Local advertising $h_{i}^{+}$ Unit inventory holding cost
$C_{i}$ Cost of production/advertising $h_{i}^{-}$ Unit backlog cost
$D$ Demand, Revenue rate $m_{i}$ Margin
$G$ , $G_{i}$ Goodwill $p$ , $p_{i}$ Retail price
$H_{i}$ , $\bar{H}_{i}$ Current -value Hamiltonians $q$ Manufacturer's share of revenue
$I_{i}$ Inventory level $r_{1}$ , $r_{2}$ Effectiveness of advertising
$J_{i}$ Objective functional $t_{1}^{d}$ , $t_{2}^{d}$ Time parameters
$K$ Infrastructure capital $t_{s}$ , $t_{f}$ Start & end of promotional period
$L_{i}$ Labor force $u$ Leader's control variable
$M$ Market size $v$ ,Follower's control variable
$N$ Number of firms $v^0$ Optimal response of the follower
$Q_{i}$ Production rate, Processing rate $w$ Wholesale price
$\bar{Q}_{i}$ Capacity limit $x$ State variable; Sales rate
$S$ Shelf space $\alpha$ External market influence
$S_{i}$ Salvage value, Unit salvage value $\alpha_{i}$ Demand parameters
$T$ Planning horizon $\beta$ Internal market influence
$r$ Optimal response of the follower $\delta$ Decay rate
$V_{i}$ Value function $\theta$ , $\hat{\theta}$ M's share of R's advertising cost
$X$ Cumulative sales $\lambda_{i}$ Adjoint variable, Shadow price
$a$ ,Market potential/Advertising effectiveness $\pi_{i}$ Instantaneous profit rate
$a_{i}$ , $b_{i}$ , $d$ , $e_{i}$ Problem parameters $\rho$ Discount rate
$a_{_{l}}$ , $a_{_{s}}$ , $b_{_{l}}$ , $b_{_{s}}$ Advertising effectiveness $\phi$ , $\psi$ Adjoint variable, Shadow price
$b$ Price sensitivity/Advertising effectiveness $\omega_{i}$ Coefficient of incentive strategy
$c_{i}$ Unit production/advertising cost $\mathcal {U}$ , $\mathcal {V}$ Feasible set of controls
$q_i$ Order quantity $\Lambda$ , $\gamma_i$ Learning efficiency
Table 2.  Summary of Model Descriptions
SectionDynamicsL's decisionsF's decisionsSolution $^{1}$
2.2.1SeasonalProduction rate, PricePriceOLSE
2.2.2SeasonalProduction rate, PricePriceOLSE
2.2.3SeasonalProduction rate, PricePriceOLSE
2.2.4GeneralPrice, Production ratePriceOLSE
2.2.5GeneralPriceOrder quantityOLSE
2.2.6GeneralPriceOrder quantityOLSE
2.2.7GeneralPricePriceOLSE
2.2.8GeneralLabor, investmentLabor, investmentOLSE
2.2.9Bass-typePricePriceOLSE
2.2.10Bass typePricePriceOLSE
2.3.1NA dynamicsParticipation rate, Ad. effortAd. effortFSE
2.3.2NA dynamicsAd. effort, PriceAd. effort, PriceFSE
2.3.3NA dynamicsParticipation rate, Ad. effortAd. effort, PriceFSE
2.3.4NA dynamicsAd. effort, (coop), PriceAd. effort, PriceFSE
2.3.5NA dynamicsAd. effort, IncentiveShelf-spaceFSE
2.3.6NA dynamicsAd. effortAd. effortFSE
2.3.7Lanchester typeAd. effortAd. effortFSE
2.3.8Sethi 1983Participation rate, PriceAd. effort, PriceFSE
2.3.9Sethi 1983Participation&Ad. rateParticipation&Ad. rateFSE
2.5.1InventoryPriceOrder Quantity, PriceOLSE, FSE
2.5.2Production CostPriceOrder quantityFSE
2.5.3Inv., Prod. CostPrice, Production quantityPrice, Order quantityFSE
1 The symbol OLSE = Open-loop Stackelberg equilibrium and FSE = Feedback Stackelberg equilibrium.
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SectionDynamicsL's decisionsF's decisionsSolution $^{1}$
2.2.1SeasonalProduction rate, PricePriceOLSE
2.2.2SeasonalProduction rate, PricePriceOLSE
2.2.3SeasonalProduction rate, PricePriceOLSE
2.2.4GeneralPrice, Production ratePriceOLSE
2.2.5GeneralPriceOrder quantityOLSE
2.2.6GeneralPriceOrder quantityOLSE
2.2.7GeneralPricePriceOLSE
2.2.8GeneralLabor, investmentLabor, investmentOLSE
2.2.9Bass-typePricePriceOLSE
2.2.10Bass typePricePriceOLSE
2.3.1NA dynamicsParticipation rate, Ad. effortAd. effortFSE
2.3.2NA dynamicsAd. effort, PriceAd. effort, PriceFSE
2.3.3NA dynamicsParticipation rate, Ad. effortAd. effort, PriceFSE
2.3.4NA dynamicsAd. effort, (coop), PriceAd. effort, PriceFSE
2.3.5NA dynamicsAd. effort, IncentiveShelf-spaceFSE
2.3.6NA dynamicsAd. effortAd. effortFSE
2.3.7Lanchester typeAd. effortAd. effortFSE
2.3.8Sethi 1983Participation rate, PriceAd. effort, PriceFSE
2.3.9Sethi 1983Participation&Ad. rateParticipation&Ad. rateFSE
2.5.1InventoryPriceOrder Quantity, PriceOLSE, FSE
2.5.2Production CostPriceOrder quantityFSE
2.5.3Inv., Prod. CostPrice, Production quantityPrice, Order quantityFSE
1 The symbol OLSE = Open-loop Stackelberg equilibrium and FSE = Feedback Stackelberg equilibrium.
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