OR Spectrum

, Volume 35, Issue 4, pp 1059–1084 | Cite as

A dynamic game with monopolist manufacturer and price-competing duopolist retailers

  • Fouad El OuardighiEmail author
  • Steffen Jørgensen
  • Federico Pasin
Regular Article


The paper considers a dynamic game with a single manufacturer who supplies two retailers. The manufacturer determines his production rate of a specific product, the rate of quality improvement efforts as well as the rate of advertising for the product. Each retailer controls her purchasing rate and the consumer sales price. Payments from a retailer to the manufacturer are determined by a wholesale price or a revenue-sharing scheme. The retailers operate in the same consumer market in which they compete in prices for the consumer demand. Nash equilibrium conditions are derived and numerical methods are employed to characterize equilibrium behavior of the players in a differential game of fixed and finite duration.


Operations management Marketing Supply chain management Differential games 


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  1. Anupindi R, Bassok Y (1999) Centralization of stocks: retailers versus manufacturer. Manag Sci 45: 178–191CrossRefGoogle Scholar
  2. Başar T, Olsder GJ (1999) Dynamic noncooperative game theory: SIAM series in classics in applied mathematics, 3rd edn. SIAM, PhiladelphiaGoogle Scholar
  3. Bensoussan A, Hurst EG Jr, Näslund B (1974) Management applications of modern control theory. North-Holland, New YorkGoogle Scholar
  4. Bernstein F, Federgruen A (2003) Pricing and replenishment strategies in a distribution system with competing retailers. Operat Res 51: 409–426CrossRefGoogle Scholar
  5. Bernstein F, Federgruen A (2005) Decentralized supply chains with competing retailers under demand uncertainty. Manag Sci 51: 18–29CrossRefGoogle Scholar
  6. Bresnahan TF, Reiss PC (1985) Dealer and manufacturer margins. Rand J Econ 16: 253–268CrossRefGoogle Scholar
  7. Cachon G (2003) Supply chain coordination with contracts. In: Graves S, de Kok T (eds) Handbooks in operations research and management science: supply chain management. Elsevier, AmsterdamGoogle Scholar
  8. Cachon GP, Lariviere MA (2005) Supply chain coordination with revenue sharing contracts. Strengths Limitat Manag Sci 51: 30–44Google Scholar
  9. Chand S, Moskowitz H, Novak A, Rekhi I, Sorger G (1996) Capacity allocation for dynamic process improvement with quality and demand considerations. Operat Res 44: 964–975CrossRefGoogle Scholar
  10. Chen F, Federgruen A, Zheng Y (2001) Coordination mechanisms for a distribution system with one supplier and multiple retailers. Manag Sci 47: 693–708CrossRefGoogle Scholar
  11. Choi SC (1996) Price competition in a duopoly common retailer channel. J Retail 72(2): 117–134CrossRefGoogle Scholar
  12. Coughlan AT, Wernerfelt B (1989) On credible delegation by oligopolists: a discussion of distribution channel managements. Manag Sci 35(2): 226–239CrossRefGoogle Scholar
  13. Dana J, Spier K (2001) Revenue sharing and vertical control in the video rental industry. J Indus Econ 59: 223–245Google Scholar
  14. Dockner E, Feichtinger G, Jørgensen S (1985) Tractable classes of nonzero-sum open-loop Nash differential games: theory and examples. J Optimiz Theory Appl 45: 179–197CrossRefGoogle Scholar
  15. Dockner E, Jørgensen S, Van Long N, Sorger G (2000) Differential games in economics and management science. Cambridge University Press, CambridgeGoogle Scholar
  16. El Ouardighi F, Jørgensen S, Pasin F (2008) A dynamic game model of operations and marketing management in a supply chain. Int Game Theory Rev 10: 373–397CrossRefGoogle Scholar
  17. El Ouardighi F, Kim B (2010) Supply quality management with wholesale price and revenue-sharing contracts under horizontal competition. Eur J Operat Res 206: 329–340CrossRefGoogle Scholar
  18. El Ouardighi F, Pasin F (2006) Quality improvement and goodwill accumulation in a dynamic duopoly. Eur J Operat Res 175: 1021–1032CrossRefGoogle Scholar
  19. Erickson G (2003) Dynamic models of advertising competition: international series in quantitative marketing, 2nd edn. Kluwer Academic Publishers, BostonGoogle Scholar
  20. Erickson GM (2011) A differential game model of the marketing-operations interface. Eur J Operat Res 211: 394–402CrossRefGoogle Scholar
  21. Erickson GM (2012) Transfer pricing in a dynamic marketing-operations interface. Eur J Operat Res 216: 326–333CrossRefGoogle Scholar
  22. Feichtinger F (1982) Optimal pricing in a diffusion model with concave price-dependent market potential. Operat Res Lett 1: 236–240CrossRefGoogle Scholar
  23. Feichtinger G, Hartl R (1985) Optimal pricing and production in an inventory model. Eur J Operat Res 19: 45–56CrossRefGoogle Scholar
  24. Gaimon C (1988) Simultaneous and dynamic price, production, inventory and capacity decisions. Eur J Operat Res 35: 426–441CrossRefGoogle Scholar
  25. Garvin D (1988) Managing quality. Free Press, New YorkGoogle Scholar
  26. Ittner CD, Nagar V, Rajan MV (2001) An empirical examination of dynamic quality-based leaning models. Manag Sci 47: 563–578CrossRefGoogle Scholar
  27. Jørgensen S (1986) Optimal production, purchasing and pricing: a differential games approach. Eur J Operat Res 24: 64–76CrossRefGoogle Scholar
  28. Jørgensen S, Zaccour G (2004) Differential games in marketing. Kluwer Academic Publishers, BostonCrossRefGoogle Scholar
  29. Kogan K, Tapiero CS (2007) Supply chain games: operations management and risk valuation. Springer, New YorkCrossRefGoogle Scholar
  30. Lariviere MA, Porteus EL (2001) Selling to the newsvendor: an analysis of price-only contracts. Manufact Serv Operat Manag 3: 293–305CrossRefGoogle Scholar
  31. Léonard D (1981) The signs of the co-state variables and sufficiency conditions in a class of optimal control problems. Econ Lett 8: 321–325CrossRefGoogle Scholar
  32. Lippman S, McCardle K (1997) The competitive newsboy. Operat Res 45: 54–65CrossRefGoogle Scholar
  33. Lourenço HR, Martin OC, Stützle T (2002) Iterated local search. In: Glover F, Kochenberger G (eds) Handbook of Metaheuristics. Kluwer Academic Publishers, BostonGoogle Scholar
  34. O’Brien DP, Shaffer G (1993) On the dampening-of-competition effect of exclusive dealing. J Indus Econ 41: 215–221CrossRefGoogle Scholar
  35. Perakis G, Roels G (2007) The price of anarchy in supply chains: quantifying the efficiency of price-only contracts. Manag Sci 53: 1249–1268CrossRefGoogle Scholar
  36. Press WH, Teukolsky AA, Vetterling WT, Flannery BP (2002) Numerical recipes in C++: the art of scientific computing. Cambridge University Press, CambridgeGoogle Scholar
  37. Tsay AA, Agrawal N (2004) Modeling conflict and coordination in multi-channel distribution systems: a review. In: Simchi-Levi D, Wu SD, Shen Z-J (eds) Handbook of quantitative supply chain analysis: modeling in the e-business era. Springer, BerlinGoogle Scholar
  38. Weng ZK (1995) Channel coordination and quantity discounts. Manag Sci 41: 1509–1522CrossRefGoogle Scholar
  39. Zaccour G (2008) On the coordination of dynamic marketing channels and two-part tariffs. Automatica 44: 1233–1239CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Fouad El Ouardighi
    • 1
    Email author
  • Steffen Jørgensen
    • 2
  • Federico Pasin
    • 3
  1. 1.Department of Operations ManagementESSEC Business SchoolCergy PontoiseFrance
  2. 2.Department of Business and EconomicsUniversity of Southern DenmarkOdenseDenmark
  3. 3.Department of Logistics and Operations ManagementHEC MontrealMontréalCanada

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