A Linear State Game of Advertising à la Vidale-Wolfe

We revisit the tradition of advertising models stemming from Vidale and Wolfe (1957), to illustrate the possibility of building up a game delivering a (degenerate) feedback equilibrium under open-loop rules. To this aim, we reformulate the state equation of the generic firm in such a way that its own advertising effort and the rivals' reaction to it enter the state dynamics additively. This approach amounts to envisaging situations where advertising has an essentially predatory/defensive nature, as it is not designed to modify the natural growth rate of a firm's sales or market share. This modelling strategy gives the game a state-linear structure, which also delivers an Arrovian result concerning the relationship between the aggregate advertising effort and industry structure.