# Approximation algorithms for pricing with negative network externalities

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## Abstract

We study the problems of pricing an indivisible product to consumers who are embedded in a given social network. The goal is to maximize the revenue of the seller by the so-called iterative pricing that offers consumers a sequence of prices over time. The consumers are assumed to be impatient in that they buy the product as soon as the seller posts a price not greater than their valuations of the product. The product’s value for a consumer is determined by two factors: a fixed consumer-specified intrinsic value and a variable externality that is exerted from the consumer’s neighbors in a linear way. We focus on the scenario of negative externalities, which captures many interesting situations, but is much less understood in comparison with its positive externality counterpart. Assuming complete information about the network, consumers’ intrinsic values, and the negative externalities, we prove that it is NP-hard to find an optimal iterative pricing, even for unweighted tree networks with uniform intrinsic values. Complementary to the hardness result, we design a 2-approximation algorithm for general weighted networks with (possibly) nonuniform intrinsic values. We show that, as an approximation to optimal iterative pricing, single pricing works fairly well for many interesting cases, such as forests, Erdős–Rényi networks and Barabási–Albert networks, although its worst-case performance can be arbitrarily bad in general networks.

### Keywords

Pricing Approximation algorithms NP-hardness Social networks Random networks Negative externalities## Notes

### Acknowledgments

This research is supported in part by NNSF of China under Grant Nos. 11531014, 11222109 and 11471326, and CAS Program for Cross & Cooperative Team of Science & Technology Innovation.

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