Article Contents
Article Contents

# A review of dynamic Stackelberg game models

• * Corresponding author: Tao Li
• 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.

Mathematics Subject Classification: Primary:91A25, 91A15, 91A23;Secondary:91A10.

 Citation:

• Figure 1.  Optimal Policies with Promotion

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

Table 2.  Summary of Model Descriptions

 Section Dynamics L's decisions F's decisions Solution $^{1}$ 2.2.1 Seasonal Production rate, Price Price OLSE 2.2.2 Seasonal Production rate, Price Price OLSE 2.2.3 Seasonal Production rate, Price Price OLSE 2.2.4 General Price, Production rate Price OLSE 2.2.5 General Price Order quantity OLSE 2.2.6 General Price Order quantity OLSE 2.2.7 General Price Price OLSE 2.2.8 General Labor, investment Labor, investment OLSE 2.2.9 Bass-type Price Price OLSE 2.2.10 Bass type Price Price OLSE 2.3.1 NA dynamics Participation rate, Ad. effort Ad. effort FSE 2.3.2 NA dynamics Ad. effort, Price Ad. effort, Price FSE 2.3.3 NA dynamics Participation rate, Ad. effort Ad. effort, Price FSE 2.3.4 NA dynamics Ad. effort, (coop), Price Ad. effort, Price FSE 2.3.5 NA dynamics Ad. effort, Incentive Shelf-space FSE 2.3.6 NA dynamics Ad. effort Ad. effort FSE 2.3.7 Lanchester type Ad. effort Ad. effort FSE 2.3.8 Sethi 1983 Participation rate, Price Ad. effort, Price FSE 2.3.9 Sethi 1983 Participation&Ad. rate Participation&Ad. rate FSE 2.5.1 Inventory Price Order Quantity, Price OLSE, FSE 2.5.2 Production Cost Price Order quantity FSE 2.5.3 Inv., Prod. Cost Price, Production quantity Price, Order quantity FSE 1 The symbol OLSE = Open-loop Stackelberg equilibrium and FSE = Feedback Stackelberg equilibrium.
•  K. Anand , R. Anupindi  and  Y. Bassok , Strategic inventories in vertical contracts, Management Science, 54 (2008) , 1792-1804. A. Bagchi, Stackelberg Differential Games in Economic Models Springer-Verlag, New York, 1994. doi: 10.1007/BFb0009151. 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. T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory 2nd ed. , SIAM, Philadelphia, PA, 1999. 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. 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. 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. 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. 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. 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. P. D. Berger , Vertical cooperative advertising ventures, Journal of Marketing Research, 9 (1972) , 309-312.  doi: 10.2307/3149542. 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. 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. K. R. Deal , Optimizing advertising expenditures in a dynamic duopoly, Operations Research, 27 (1979) , 682-692.  doi: 10.1287/opre.27.4.682. 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. 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. V. S. Desai , Marketing-production decisions under independent and integrated channel structure, Annals of Operations Research, 34 (1992) , 275-306.  doi: 10.1007/BF02098183. 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. 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. 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. G. M. Erickson , Empirical analysis of closed-loop duopoly advertising strategies, Management Science, 38 (1992) , 1732-1749.  doi: 10.1287/mnsc.38.12.1732. 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. 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. 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. 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. 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. 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. 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. 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. 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. S. Jorgensen , S. P. Sigue  and  G. Zaccour , Dynamic cooperative advertising in a channel, Journal of Retailing, 76 (2000) , 71-92. 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. 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. 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. 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. 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. K. Kogan and C. S. Tapiero, Co-investment in Supply Chain Infrastructure working paper, Bar Ilan University, Israel, 2007. 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. G. Leitmann , On generalized Stackelberg strategies, J. Optimization Theory and Applications, 26 (1978) , 637-643.  doi: 10.1007/BF00933155. 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. 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. J. D. C. Little , Aggregate advertising models: The state of the art, Operations Research, 27 (1979) , 629-667. 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. 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. 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. 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. D. Pekelman , Simultaneous price production in channels, Marketing Science, 7 (1974) , 335-355. 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. 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. S. P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics 2$^{nd}$ edition, Springer, New York, 2000. 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. 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. H. V. Stackelberg, The Theory of the Market Economy translated by Peacock A. T. , William Hodge and Co. , London, 1952. 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. 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. 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. 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. )

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