Abstract
We analyze optimal advertising spending in a duopolistic market where each firm's market share depends on its own and its competitor's advertising decisions, and is also subject to stochastic disturbances. We develop a differential game model of advertising in which the dynamic behavior is based on the Sethi stochastic advertising model and the Lanchester model of combat. Particularly important to note is the morphing of the sales decay term in the Sethi model into decay caused by competitive advertising and noncompetitive churn that acts to equalize market shares in the absence of advertising. We derive closed-loop Nash equilibria for symmetric as well as asymmetric competitors. For all cases, explicit solutions and comparative statics are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Vidale, M. L., and Wolfe, H. B., An Operations Research Study of Sales Response to Advertising, Operations Research, Vol. 5, pp. 370–381, 1957.
Sethi, S. P., Dynamic Optimal Control Models in Advertising:A Survey, SIAM Review, Vol. 19, pp. 685–725, 1977.
Feichtinger, G., Hartl, R. F., and Sethi, S. P., Dynamic Optimal Control Models in Advertising: Recent Developments, Management Science, Vol. 40, pp. 195–226, 1994.
CMR/TNS Media Intelligence, US Advertising Market Shows Healthy Growth: Spending Up 4. 2%in 2002; see http://www. tnsmi-cmr. com/news/2003/031003. html (March 10, 2003).
Patti, C. H., and Blasko, V. J., Budgeting Practices of Big Advertisers, Journal of Advertising Research, Vol. 21, pp. 23–29, 1981.
Blasko, V. J., and Patti, C. H., The Advertising Budgeting Practices of Industrial Marketers, Journal of Marketing, Vol. 48, pp. 104–110, 1984.
Aaker, D., and Carman, J. M., Are You Overadvertising? Journal of Advertising Research, Vol. 22, pp. 57–70, 1982.
Lodish, L. M., Abraham, M., Kalmenson, S., Livelsberger, J., Lubetkin, B., Richardson, B., and Stevens, M. E., How TV Advertising Works:A Meta-Analysis of 389 Real World Split Cable TV Advertising Experiments, Journal of Marketing Research, Vol. 32, pp. 125–139, 1995.
Chintagunta, P. K., and Vilcassim, N., An Empirical Investigation of Advertising Strategies in a Dynamic Duopoly, Management Science, Vol. 38, pp. 1230–1224, 1992.
Erickson, G. M., Empirical Analysis of Closed-Loop Duopoly Advertising Strategies, Management Science, Vol. 38, pp. 1732–1749, 1992.
Fruchter, G., and Kalish, S., Closed-Loop Advertising Strategies in a Duopoly, Management Science, Vol. 43, pp. 54–63, 1997.
Bain, J. S., Barriers to New Competition, Harvard University Press, Cambridge, Massachusetts, 1956.
Nelson, P., Advertising as Information, Journal of Political Economy, Vol. 82, pp. 729–754, 1974.
McAlister, L., and Pessemier, E., Variety Seeking Behavior:An Interdisciplinary Review, Journal of Consumer Research, Vol. 9, pp. 311–323, 1982.
Sethi, S. P., Deterministic and Stochastic Optimization of a Dynamic Advertising Model, Optimal Control Applications and Methods, Vol. 4, pp. 179–184, 1983.
Deal, K., Sethi, S. P., and Thompson, G. L., A Bilinear-Quadratic Differential Game in Advertising, Control Theory in Mathematical Economics, Edited by P. T. Lui and J. G. Sutinen, Manuel Dekker, New York, NY, pp. 91–109, 1979.
Chintagunta, P. K., and Jain, D. C., Empirical Analysis of a Dynamic Duopoly Model of Competition, Journal of Economics and Management Strategy, Vol. 4, pp. 109–131, 1995.
Horsky, D., and Mate, K., Dynamic Advertising Strategies of Competing Durable Good Producers, Marketing Science, Vol. 7, pp. 356–367, 1988.
Sorger, G., Competitive Dynamic Advertising: A Modification of the Case Game, Journal of Economics Dynamics and Control, Vol. 13, pp. 55–80, 1989.
Little, J. D. C., Aggregate Advertising Models:The State of the Art, Operations Research, Vol. 27, pp. 629–667, 1979.
Sethi, S. P., Optimal Control of the Vidale-Wolfe Advertising Model, Operations Research, Vol. 21, pp. 998–1013, 1973.
Erickson, G. M., Dynamic Models of Advertising Competition, Kluwer, Norwell, Massachusetts, 2003.
Dockner, E. J., Jorgensen, S., Long, N. V., and Sorger, G., Differential Games in Economics and Management Science, Cambridge University Press, Cambridge, UK, 2000.
Kimball, G. E., Some Industrial Application of Military Operations Research Methods, Operations Research, Vol. 5, pp. 201–204, 1957.
Gihman, I. I., and Skorohod, A. V., Stochastic Differential Equations, Springer-Verlag, New York, NY, 1972.
Gould, J. P., Diffusion Processes and Optimal Advertising Policy, Microeconomic Foundations of Employment and Inflation Theory, Edited by E. S. Phelps et al., W. W. Norton, New York, NY, pp. 338–368, 1970.
Sethi, S. P., and Thompson, G. L., Optimal Control Theory: Applications to Management Science and Economics, Kluwer, Norwell, Massachusetts, 2000.
Zwillinger, D., Handbook of Differential Equations, Academic Press, San Diego, California, 1998.
Polyanin, A. D., and Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential Equations, Chapman and Hall, CRC, Boca Raton, Florida, 2003.
Fruchter, G., Oligopoly Advertising Strategies with Market Expansion, Optimal Control Applications and Methods, Vol. 20, pp. 199–211, 1999.
Erickson, G. M., Advertising Strategies in a Dynamic Oligopoly, Journal of Marketing Research, Vol. 32, pp. 233–237, 1995.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Prasad, A., Sethi, S.P. Competitive Advertising Under Uncertainty: A Stochastic Differential Game Approach. Journal of Optimization Theory and Applications 123, 163–185 (2004). https://doi.org/10.1023/B:JOTA.0000043996.62867.20
Issue Date:
DOI: https://doi.org/10.1023/B:JOTA.0000043996.62867.20