{'fraction_non_alphanumeric': 0.01850453172205438, 'fraction_numerical': 0.00037764350453172205, 'mean_word_length': 5.146171693735499, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Optimal Pricing with Positive Network Effects: The Big Benefits of Just a Little Discrimination

We study the revenue-optimal pricing strategies of a monopolist selling a divisible good (service) that exhibits positive externalities to consumers embedded in a social network. The positive externalities in our model mean that a consumer's usage level depends directly on the usage of his neighbors in the network. Optimal pricing may therefore involve offering different prices based on consumers' position. We first consider a setting where the monopolist can offer individualized prices and derive an explicit characterization of the optimal price for each consumer as a function of his network position. We show that such a policy amounts to the solution to a quadratic program and that it is optimal for the seller to charge consumers a price that is proportional to a measure of social network importance called Bonacich centrality. We next study a constrained policy whereby the seller can choose no more than k distinct prices. While the problem is tractable for uniform pricing, i.e. k=1, we show that the problem is generally NP-complete and that there is no polynomial-time approximation scheme of the optimal solution. Next, we consider a relaxation of the k-price problem that is polynomial-time solvable and provide lower bounds on the revenue to the fully discriminatory policy. We show that the latter revenue gap is naturally decreasing in k and is proportional to the Bonacich centrality and to the variance of the network’s degree distribution. The results suggest that knowledge of the externalities is more important to optimal pricing as a network's density and the variance of its degree distribution increases. Put another way, a seller's capability to discriminate between consumers is more important for settings in which consumers' relative influence on each other (or, equally, consumers' susceptibility to influence) from externalities varies greatly. Significantly, we also show that allowing for even a modest number of prices provides a good approximation to the fully discriminatory policy. Finally, we consider seeding strategies as a special case of k-pricing where the good is given to a subset of consumers for free. Seeding always represents a large improvement over uniform pricing, is a good approximation of the unconstrained k-price policy, and the benefits of seeding are greatest for high-variance networks. While the number of seeds varies substantively depending on whether influence is unilateral or bilateral, the revenue extracted remains largely unaffected by the degree of reciprocity of influence.