A Lanchester-Type Dynamic Game of Advertising and Pricing

  • Steffen Jørgensen
  • Simon SiguéEmail author
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 280)


The paper studies a differential game played by two competing firms over a finite time horizon. As the game progresses, the firms observe the position of the game, i.e., the current time and the current market shares. Each firm uses pricing and advertising in order to influence market shares. We suggest a generalization of the Lanchester market share dynamics such that the rates at which firms attract market share from each other are determined not only by their advertising efforts but also by the consumer prices charged in the market. A full characterization of Nash equilibrium price and advertising strategies is obtained.


Price and advertising competition Duopoly Differential game Markovian Nash equilibrium 


  1. Dockner, E. J., Jørgensen, S., Long, N. V., & Sorger, G. (2000). Differential games in economics and management science. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  2. Erickson, G. M. (1985). A model of advertising competition. Journal of Marketing Research, 22, 297–304.CrossRefGoogle Scholar
  3. Erickson, G. M. (1993). Offensive and defensive marketing: Closed-loop duopoly strategies. Marketing Letters, 4, 285–295.CrossRefGoogle Scholar
  4. Erickson, G. M. (2009). A differential game model of the marketing-operations interface. European Journal of Operational Research, 197(1), 374–388.CrossRefGoogle Scholar
  5. Fruchter, G. (2009). Signaling quality: Dynamic price-advertising model. Journal of Optimization Theory and Applications, 3(4), 301–314.Google Scholar
  6. Fruchter, G., & Messinger, P. E. (2003). Optimal management of fringe entry over time. Journal of Economic Dynamics & Control, 28(3), 445–466.CrossRefGoogle Scholar
  7. Haurie, A., Krawczyk, J. B., & Zaccour, G. (2012). Games and dynamic games. Singapore: World Scientific.CrossRefGoogle Scholar
  8. Huang, J., Leng, M., & Liang, L. (2012). Recent developments in dynamic advertising research. European Journal of Operational Research, 220, 591–609.CrossRefGoogle Scholar
  9. Jarrar, R., Martín-Herrán, G., & Zaccour, G. (2004). Markov perfect equilibrium advertising strategies of a Lanchester duopoly model: A technical note. Management Science, 50(7), 995–1000.CrossRefGoogle Scholar
  10. Jørgensen, S., & Sigué, S. P. (2015). Defensive, offensive, and generic advertising in a Lanchester model with market expansion. Dynamic Games and Applications, 4(5), 523–539.CrossRefGoogle Scholar
  11. Kress, M., Caulkins, J. P., Feichtinger, G., Grass, D., & Seidl, A. (2018). Lanchester model for three-way combat. European Journal of Operational Research, 264, 46–54.CrossRefGoogle Scholar
  12. Krishnamoorthy, A., Prasad, A., & Sethi, S. P. (2010). Optimal pricing and advertising in a durable-good duopoly. European Journal of Operational Research, 200(2), 486–497.CrossRefGoogle Scholar
  13. Martín-Herrán, G., McQuitty, S., & Sigué, S. P. (2012). Offensive versus defensive marketing: What is the optimal spending allocation. International Journal of Research in Marketing, 29, 210–219.CrossRefGoogle Scholar
  14. Nair, A., & Narasimhan, R. (2006). Dynamics of competing with quality- and advertising-based goodwill. European Journal of Operational Research, 175(1), 462–474.CrossRefGoogle Scholar
  15. Sorger, G. (1989). Competitive dynamic advertising: A modification of the Case game. Journal of Economic Dynamics & Control, 13, 55–80.CrossRefGoogle Scholar
  16. Teng, J. T., & Thompson, G. L. (1984). Optimal pricing and advertising policies for new product oligopoly models. Marketing Science, 3(2), 148–168.CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Business and EconomicsUniversity of Southern DenmarkOdenseDenmark
  2. 2.Faculty of BusinessAthabasca UniversityAthabascaCanada

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