Designing Price Incentives in a Network with Social Interactions

The recent ubiquity of social networks allows firms to collect vast amount of data about their customers including their social interactions. We consider a setting where a monopolist sells an indivisible good to consumers that are embedded in a social network. We capture interactions among consumers using a broad class of non-linear influence models. This class includes several existing influence models as special cases (e.g., linear influence and non-linear triggering model). Assuming complete information about the interactions, we model the optimal pricing problem as a two-stage game. The firm first designs prices to maximize profits and then, the consumers choose whether or not to purchase the item. Assuming positive network externalities, we show the existence of a pure Nash equilibrium that is preferred by both the seller and the buyers. Using duality theory and integer programming techniques, we reformulate the problem into a linear mixed-integer program (MIP). We derive efficient ways to optimally solve the MIP using its linear programming relaxation for two pricing strategies: discriminative and uniform. We then draw interesting insights about the structure of the optimal prices and profit of the seller. In particular, we quantify the effect on prices when using a non-linear influence model relative to a linear utility, and also identify settings when it is beneficial for the seller to offer a price below cost for influential agents. Further, we extend our model and results to the case where the seller offers incentives in addition to prices to solicit actions to ensure influence.