Feedback competitive advertising strategies with a general objective function

We introduce a general objective function, which incorporates competitive situations, such as conservative, punitive, and predatory advertising. Linking together the particular situations into a two-parameter family of max–min problems, and using the Lanchester model to describe the dynamics of the market, a bilinear–quadratic differential game is obtained. For this game, we find saddle-point feed-back time-invariant advertising strategies and show when these strategies are Nash equilibrium strategies. In an empirical application involving duopolistic competition in the cola market, we find evidence of a punitive motivation for the advertising strategies.