Social networks are a good example of the complexity of human relationships, where a person’s daily life decisions are regularly affected by the choices of their friends. This challenges extensive research on the economics of networks, in which individual activities have global impacts on the activities of others. In short, the externalities of network activities are localised within social circles.

Aware of this, firms devise marketing, promotions, and pricing strategies accordingly. For instance, Airbnb offers coupons to those who invite friends to join its platform. However, despite significant advances in technology, firms still struggle with network pricing, specifically due to a lack of precise network information. On one hand, because of the nature of social networks, individual customers often have limited knowledge about the entire network. This means that they may have to speculate about other customers’ choices and make decisions with only local information, thus creating information heterogeneity among customers. On the other hand, firms are at an information disadvantage in terms of individual customers because they are unable to “directly observe their respective positions in the network”, leading to information asymmetry between firms and customers.

An additional pricing hurdle comes in the form of social comparison, whereby a company that might not even face information asymmetry still struggles to price directly based on the individual’s network position because the individual is able to compare their own price with those of their friends. In part, then, information asymmetry and customers’ counteractions explain why there are still coarse pricing schemes for products whose local externalities are well documented.

In response to this, a recent paper by Yang Zhang and Ying-Ju Chen investigates a firm’s optimal nonlinear pricing (where there is a nonlinear relationship between the price and quantity of goods) of products and services in social networks, in which customers possess private information about their network position and whose consumption exhibits local externality. Nonlinear pricing was chosen because, faced with information asymmetry, a firm can offer a selection (menu) that combines quantity with price, from which customers can then choose on their own, thereby revealing their network position. It also gets around the problem raised earlier of social comparison, as price discrimination is implicit. Finally, nonlinear pricing is better than fixed pricing because it is able to obtain an individual’s private information more effectively.

Basing their model on extant research, the authors consider two alternative configurations of how the payoff externalities are generated through the network formation: the out-neighbour model and the in-neighbour model – both of which they solved by an approach based on calculus of variations and positive neighbour affiliation. In the out-neighbour model, a customer’s payoff is influenced by their out-neighbours’ consumption levels. In the in-neighbour model, it is the in-neighbours’ consumption that influences a customer’s payoff.

Zhang and Chen show that optimal pricing balances the extraction of information rent because of neighbour consumption with the incentivisation of the individual’s own consumption, giving rise to either a quantity premium or a quantity discount menu. After applying their results to Erdös–Rényi graphs (a special case of the social network model used by the authors), they found that the optimal scheme does not screen network positions, thereby allowing firms to offer a uniform price for all customers. To check the robustness of their findings, the authors used two-way connections, discovering that a firm’s optimal consumption becomes linear in customer degree in a scale-free network (a network whose degree distribution follows a power). They go on to demonstrate the advantage of nonlinear pricing over linear pricing, in that it lets firms “respond more effectively to the changes of network topology and economic factors”.

The authors end by suggesting that a future avenue of research would be the incorporation of a network formation process, in order to consider dynamic pricing in transient states of the network.