A Lanchester-Type Dynamic Game of Advertising and Pricing

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.