Feedback advertising strategies in a two-firm differential game: a numerical investigation

Abstract

We present a procedure for finding feedback strategies numerically in a two-firm differential game (DG). To motivate the discussion, we start with the single-firm optimal control formulation of the problem with a linear revenue term and a nonlinear (quadratic) advertising rate cost term. We find the optimal control to maximise the firm's market share. When two firms compete for the market share, the optimal control model generalises to a differential game formulation with two differential equations (DEs) representing each firm's market share. This formulation extends earlier models with the inclusion of a new competition effect term. We then compute the equilibrium feedback advertising policies of the differential game for each firm as functions of time and the firms' current market shares by solving a system of six DEs. Our analyses are illustrated with several examples including comparative statics and discussions of the managerial implications.

Keywords:

• Game theory
• differential game
• equilibrium feedback advertising policies
• numerical solution of differential equations

Acknowledgments

The author is grateful to the referee for his/her very useful suggestions that have improved the exposition of the paper. The author acknowledges Mr. Onur Yildirim's comments on the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Research supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Notes on contributors

Mahmut Parlar

Mahmut Parlar is Distinguished Business Research Professor and Professor of Management Science at McMaster University's DeGroote School of Business. He holds a B.Sc. in mathematics (1973) and an M.Sc. in operations research/statistics (1975) both from the Middle East Technical University, Ankara, Turkey, and a Ph.D. in management sciences (1979) from the University of Waterloo, Canada. Mahmut Parlar's research program is focused in the areas of stochastic modelling, dynamic optimization and game theory applications in supply chain management. His work in these and related areas has been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) research grants without interruptions since 1980 and have appeared in the form of more than 110 articles published in journals such as Operations Research, Management Science, Production and Operations Management, Naval Research Logistics, IIE Transactions, Journal of Operations Management, Transportation Science, Applied Mathematical Finance and others. Professor Parlar is the author of a book entitled “Interactive Operations Research with Maple” published in 2000 by Birkhäuser, Boston.