Optimal Controls in Spatial Advertising Diffusion Models

We start with a brief review of a descriptive multi population (multi market) advertising diffusion model, where word of mouth communication takes place both within single markets and between individuals from different markets. Assuming that a monopolist for one market who seeks to maximize profits has naive expectations about the states of the neighbouring markets, we derive an optimal advertising effort for this particular monopolist. Even if the according advertising strategy is not truly optimal in the sense that it does not take into account the variations in the neighbouring markets, in numerical analyses it has so far always proven to be better than constant or random advertising efforts. The technique proposed here may hence be seen as a fair trade-off between accuracy and practicability for multi state optimal control problems as the one presented in this paper.