Abstract
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.
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Notes
- 1.
This observation was also made in the survey by Huang et al. (2012).
- 2.
In quite many games played with the Lanchester dynamics it is assumed that the revenue rate is π iX i(t) where π i > 0 is the constant revenue per unit of market share. This formulation often simplifies the analysis considerably.
- 3.
The idea of letting market shares enter the right-hand sides of the market share dynamics as \(\sqrt {X_{i}\left ( t\right ) }\) most likely originated in Sorger (1989) and has gained some popularity in the literature.
- 4.
Erickson (1993) did not consider pricing but included defensive advertising in the attraction rates. The purpose of defensive advertising is to defend a firm’s customer base against offensive advertising done by the rival firm. In our model this is accomplished by pricing.
- 5.
- 6.
If the guess turned out to be wrong, we can make another guess.
- 7.
The above value functions also appear in Sorger (1989, p. 66).
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Jørgensen, S., Sigué, S. (2020). A Lanchester-Type Dynamic Game of Advertising and Pricing. In: Pineau, PO., Sigué, S., Taboubi, S. (eds) Games in Management Science. International Series in Operations Research & Management Science, vol 280. Springer, Cham. https://doi.org/10.1007/978-3-030-19107-8_1
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