A survey of Stackelberg differential game models in supply and marketing channels

  • Xiuli He
  • Ashutosh Prasad
  • Suresh P. Sethi
  • Genaro J. Gutierrez


Stackelberg differential game models have been used to study sequential decision making in noncooperative games in diverse fields. In this paper, we survey recent applications of Stackelberg differential game models to the supply chain management and marketing channels literatures. A common feature of these applications is the specification of the game structure: a decentralized channel composed of a manufacturer and independent retailers, and a sequential decision procedure with demand and supply dynamics and coordination issues. In supply chain management, Stackelberg differential games have been used to investigate inventory issues, wholesale and retail pricing strategies, and outsourcing in dynamic environments. The underlying demand typically has growth dynamics or seasonal variation. In marketing, Stackelberg differential 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 classical advertising capital models or sales-advertising response models. We begin by explaining the Stackelberg differential game solution methodology and then provide a description of the models and results reported in the literature.


Stackelberg differential games supply chain management marketing channels open-loop equilibria feedback policies channel coordination optimal control 


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  1. [1]
    Bagchi, A. (1984). Stackelberg Differential Games in Economic Models. Springer-Verlag, New York, NYCrossRefzbMATHGoogle Scholar
  2. [2]
    Basar, T. & Olsder, G.J. (1999). Dynamic Noncooperative Game Theory, 2nd ed. SIAM, Philadelphia, PAzbMATHGoogle Scholar
  3. [3]
    Bass, F.M. (1969). A new product growth for model consumer durables. Management Science, 15 (5): 215–2CrossRefGoogle Scholar
  4. [4]
    Bensoussan, E.E., Hurst, Jr. & Naslund, B. (1974). Management Application of Modern Control Theory. North-Holland, AmsterdamGoogle Scholar
  5. [5]
    Bergen, M. & John, G. (1997). Understanding cooperative advertising participation rates in conventional channels. Journal of Marketing Research, 34 (3):357–369CrossRefGoogle Scholar
  6. [6]
    Berger, P.D. (1972). Vertical cooperative advertising ventures. Journal of Marketing Research, 9 (3): 309–312CrossRefGoogle Scholar
  7. [7]
    Breton, M., Jarrar, R. & Zaccour, G. (2006). A note on feedback Stackelberg equilibria in a Lanchester model with empirical application. Management Science, 52 (5): 804–811CrossRefGoogle Scholar
  8. [8]
    Chintagunta, P. & Jain, D. (1992). A dynamic model of channel member strategies for marketing expenditures. Marketing Science, 11 (2): 168–188CrossRefGoogle Scholar
  9. [9]
    Deal, K.R. (1979). Optimizing advertising expenditures in a dynamic duopoly. Operations Research, 27 (4): 682–692CrossRefzbMATHGoogle Scholar
  10. [10]
    Deal, K.R., Sethi, S.P. & Thompson, G.L. (1979). A bilinear-quadratic differential game in advertising. In: P.-T. Liu and J. G. Sutinen (eds.), Control Theory in Mathematical Economics, 91–109. Marcel Dekker, Inc., New York, NYGoogle Scholar
  11. [11]
    Derzko, N.A., Sethi, S.P. & Thompson, G.L. (1984). Necessary and sufficient conditions for optimal control of quasilinear partial differential systems. Journal of Optimal Theory and Application, 43: 89–101CrossRefMathSciNetzbMATHGoogle Scholar
  12. [12]
    Desai, V.S. (1992). Marketing-production decisions under independent and integrated channel structure. Annals of Operations Research, 34: 275–306CrossRefzbMATHGoogle Scholar
  13. [13]
    Desai, V.S. (1996). Interactions between members of a marketing-production channel under seasonal demand. European Journal of Operational Research, 90: 115–141CrossRefzbMATHGoogle Scholar
  14. [14]
    Dockner, E., Jørgensen, S., Long, N.V. & Sorger, G. (2000). Differential Games in Economics and Management Science. Cambridge University PressGoogle Scholar
  15. [15]
    Eliashberg, J. & Steinberg, R. (1987). Marketing-production decisions in an industrial channel of distribution. Management Science, 33 (8): 981–1000CrossRefzbMATHGoogle Scholar
  16. [16]
    Erickson, G.M. (1992). Empirical analysis of closed-loop duopoly advertising strategies. Management Science, 38: 1732–1749CrossRefzbMATHGoogle Scholar
  17. [17]
    Erichson, G.M. (1995). Differential game models of advertising competition. European Journal of Operational Research, 83 (3):431–438CrossRefGoogle Scholar
  18. [18]
    Erickson, G.M. (1997). Dynamic conjectural variations in a Lanchester oligopoly. Management Science, 43 (11):1603–1608CrossRefzbMATHGoogle Scholar
  19. [19]
    Feichtinger, G., Hartel, R.F. & Sethi, S.P. (1994). Dynamic optimal control models in advertising: recent developments. Management Science, 40: 29–31CrossRefGoogle Scholar
  20. [20]
    Fruchter, G.E. & Kalish, S. (1997). Closed-loop advertising strategies in a duopoly. Management Science, 43: 54–63CrossRefzbMATHGoogle Scholar
  21. [21]
    Fruchter, G.E. & Kalish, S. (1998). Dynamic promotional budgeting and media allocation. European Journal of Operational Research, 111: 15–27CrossRefzbMATHGoogle Scholar
  22. [22]
    Gutierrez, G.J. & He, X. (2007). Life-cycle channel coordination issues in launching an innovative durable product. Production and Operations Management, to appearGoogle Scholar
  23. [23]
    Harris, C. & Vickers, J. (1995). Innovation and natural resources: a dynamic game with uncertainty. RAND Journal of Economics, 26 (3): 418–430CrossRefGoogle Scholar
  24. [24]
    He, X. & Sethi, S.P. (2008). Dynamic slotting and pricing decisions in a durable product supply chain. Journal of Optimization Theory and Applications, 134 (8), to appearGoogle Scholar
  25. [25]
    He, X., Prasad, A. & Sethi, S.P. (2007). Cooperative advertising and pricing in a stochastic supply chain: feedback Stackelberg strategies. Working paper. The University of Texas at DallasGoogle Scholar
  26. [26]
    Huang, Z., Li, S.X. & Mahajan, V. (2002). An analysis of manufacturer-retailer supply chain coordination in cooperative advertising. Decision Sciences, 33 (3): 469–494CrossRefGoogle Scholar
  27. [27]
    Isaacs, R. (1965). Differential Games. Wiley, New YorkzbMATHGoogle Scholar
  28. [28]
    Jarrar, R., Martin-Herran, G. & Zaccour, G. (2004). Markov perfect equilibrium advertising strategies of Lanchester duopoly model: a technical note. Management Science, 50 (7): 995–1000CrossRefGoogle Scholar
  29. [29]
    Jørgensen, S. (1982). A survey of some differential games in advertising. Journal of Economic Dynamics and Control, Springer-Verlag, BerlinGoogle Scholar
  30. [30]
    Jørgensen, S., Sigue, S.P. & Zaccour, G. (2000). Dynamic cooperative advertising in a channel. Journal of Retailing, 76 (1): 71–92CrossRefGoogle Scholar
  31. [31]
    Jørgensen, S., Sigue, S.P. & Zaccour, G. (2001). Stackelberg leadership in a marketing channel. International Game Theory Review, 3 (1): 13–26CrossRefMathSciNetGoogle Scholar
  32. [32]
    Jørgensen, S., Taboubi, S. & Zaccour, G. (2001). Cooperative advertising in a marketing channel. Journal of Optimization Theory and Applications, 110 (1): 145–158CrossRefMathSciNetGoogle Scholar
  33. [33]
    Jørgensen, S., Taboubi, S. & Zaccour, G. (2003). Retail promotions with negative brand image effects: is cooperation possible? European Journal of Operational Research, 150: 395–405CrossRefMathSciNetGoogle Scholar
  34. [34]
    Jørgensen, S., Taboubi, S. & Zaccour, G. (2006). Incentives for retailer promotion in a marketing channel. Annals of the International Society of dynamic Games, 8:365–378CrossRefGoogle Scholar
  35. [35]
    Jorsengen, S. & Zaccour, G. (2005). Differential Games in Marketing. Springer, New York, NYGoogle Scholar
  36. [36]
    Karray, S. & Zaccour, G. (2005). A differential game of advertising for national and store brands. In: A. Haurie, G. Zaccour (eds.), Dynamic Games: Theory and Applicatoins. 213–230, Springer, New York, NYCrossRefGoogle Scholar
  37. [37]
    Kogan, K. & Tapiero, C.S. (2007a, b, c, d, e). Supply Chain Games: Operations Management and Risk Valuation. Springer, New York, NYzbMATHGoogle Scholar
  38. [38]
    Kogan, K. & Tapiero, C.S. (2007f). Co-investment in supply chain infrastructure. Working Paper. Bar Ilan University, IsraelGoogle Scholar
  39. [39]
    Little, J.D.C. (1979). Aggregate advertising models: the state of the art. Operations Research, 27 (4): 629–667CrossRefzbMATHGoogle Scholar
  40. [40]
    Martin-Herran, G. & Taboubi, S. (2005). Incentive strategies for shelf-space allocation in duopolies. In: A. Haurie, G. Zaccour (eds.), Dynamic Games Theory and Applicatoins, 231–253. Springer, New York, NYCrossRefGoogle Scholar
  41. [41]
    Nerlove, M. & Arrow, K.J. (1962). Optimal advertising policy under dynamic conditions. Economica, 39: 129–142CrossRefGoogle Scholar
  42. [42]
    Pekelman, D. (1974). Simultaneous price production in channels. Marketing Science, 7:335–355Google Scholar
  43. [43]
    Rubio, S.J. (2006). On coincidence of feedback Nash equilibria and Stackelberg equilibria in economic applications of differential games. Journal of Optimization Theory and Applications, 128 (1): 203–221CrossRefMathSciNetzbMATHGoogle Scholar
  44. [44]
    Sethi, S.P. (1983). Deterministic and stochastic optimization of a dynamic advertising model. Optimal Control Applications and Methods, 4: 179–184CrossRefMathSciNetzbMATHGoogle Scholar
  45. [45]
    Sethi, S.P. & Thompson, G.L. (2000). Optimal Control Theory: Applications to Management Science and Economics, 2nd ed. Springer, New York, NYzbMATHGoogle Scholar
  46. [46]
    Stackelberg, H.V. (1952). The Theory of the Market Economy, translated by Peacock A.T. William Hodge and Co., LondonGoogle Scholar
  47. [47]
    Teng, J.T. & Thompson, G.L. (1983). Oligopoly models for optimal advertising when production costs obey a learning curve. Management Science, 29 (9): 1087–1101CrossRefzbMATHGoogle Scholar
  48. [48]
    Thompson, G.L. & Teng, J.T. (1984). Optimal pricing and advertising policies for new product. Marketing Science, 3 (2): 148–168CrossRefGoogle Scholar
  49. [49]
    Urban, T.L. (1998). An inventory-theoretic approach to product assortment and shelf-space allocation. Journal of Retailing, 74 (1): 15–35CrossRefMathSciNetGoogle Scholar
  50. [50]
    Vidale, M.L. & Wolfe, H.B. (1957). An operations research study of sales response to advertising. Operations Research, 5: 370–381CrossRefMathSciNetGoogle Scholar

Copyright information

© Systems Engineering Society of China & Springer-Verlag 2007

Authors and Affiliations

  • Xiuli He
    • 1
  • Ashutosh Prasad
    • 1
  • Suresh P. Sethi
    • 1
  • Genaro J. Gutierrez
    • 2
  1. 1.School of ManagementThe University of Texas at DallasRichardsonUSA
  2. 2.McCombs School of BusinessThe University of Texas at AustinAustinUSA

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