Designing price incentives in a network with social interactions

Abstract

Problem definition: The recent ubiquity of social networks allows firms to collect vast amount of data on their customers and on their social interactions. We consider a setting in which a monopolist sells an indivisible good to consumers who are embedded in a social network. Academic/practical relevance: This is an important problem as sellers can use available data to design and send targeted promotions that account for social externality effects and ultimately increase their profits. Methodology: We capture the interactions among consumers using a broad class of nonlinear utility models. This class extends the existing models by explicitly capturing externalities from subsets of agents (communities or groups) and includes several existing models as special cases (e.g., full information version of the triggering model). Assuming complete information about the interactions, we model the optimal pricing problem as a two-stage game. First, the firm designs prices to maximize profits and then consumers choose whether to purchase the item. Results: Under 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. Finally, we propose two intuitive heuristic algorithms to solve the problem for which we derive worst-case parametric performance bounds. Managerial implications: We draw interesting insights on the structure of the optimal prices and the seller’s profit. In particular, we quantify the effect on prices when using a nonlinear utility model relative to a linear model and identify settings with which it is beneficial to offer a price below cost to influential agents. Finally, we extend our model and results to the case in which the seller offers incentives (in addition to prices) to solicit actions so as to ensure network externality effects.