A Dynamic Collective Choice Model with an Advertiser

This paper studies a dynamic collective choice model in the presence of an advertiser, where a large number of consumers are choosing between two alternatives. Their choices are influenced by the group’s aggregate choice and an advertising effect. The latter is produced by an advertiser making investments to convince as many consumers as possible to choose a specific alternative. In schools, for example, teenagers’ decisions to smoke are considerably affected by their peers’ decisions, as well as the ministry of health campaigns against smoking. We model the problem as a Stackelberg dynamic game, where the advertiser makes its investment decision first, and then the consumers choose one of the alternatives. On the methodological side, we use the theory of mean field games to solve the game for a continuum of consumers. This allows us to describe the consumers’ individual and aggregate behaviors, and the advertiser’s optimal investment strategies. When the consumers have sufficiently diverse a priori opinions toward the alternatives, we show that a unique Nash equilibrium exists between them, which predicts the distribution of choices over the alternatives, and the advertiser can always make optimal investments. For a certain uniform distribution of a priori opinions, we give an explicit form of the advertiser’s optimal investment strategy and of the consumers’ optimal choices.