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 and including their social interactions with other customers. We consider a setting where a monopolist sells an indivisible good to consumers that are embedded in a social network. Assuming complete information about the interactions between consumers, we model the optimal pricing problem of the firm as a two-stage game. First, the firm designs prices to maximize his profits and second the consumers choose whether to purchase the item or not at the given prices so as to optimize their own payoffs. Assuming positive network externalities, we show the existence of a pure strategy Nash equilibrium for the second stage game. Using duality theory and integer programming techniques, we reformulate the problem into a single stage linear fixed-integer program (MIP). This formulation can be used by any firm as an operational pricing tool as it can easily incorporate business rules on prices and constraints on network segmentation. We derive efficient and scalable ways of optimally solving the MIP using its linear programming relaxation for two different pricing strategies chosen by the seller: discriminative pricing where the monopolist offers prices that may differ by agent and the uniform pricing where the prices are the same for all agents. Further we extend our model and results to the case when the seller offers incentives in addition to prices to solicit actions such as buyer reviews to ensure influence on other agents. Finally, we present computational insights by comparing the various pricing strategies and highlighting the benefits of incorporating social interactions. In particular, we provide instances where it is beneficial for the seller to earn negative profit on an influential agent in order to extract significant positive profits on others.