Journal of Combinatorial Optimization

, Volume 33, Issue 2, pp 681–712 | Cite as

Approximation algorithms for pricing with negative network externalities

  • Zhigang Cao
  • Xujin Chen
  • Xiaodong Hu
  • Changjun Wang


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.


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



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.


  1. Akhlaghpour H, Ghodsi M, Haghpanah N, Mirrokni VS, Mahini H, Nikzad A (2010) Optimal iterative pricing over social networks (extended abstract). In: Proceedings of the 6th international conference on Internet and Network Economics, WINE’10, pp 415–423Google Scholar
  2. Alon N, Mansour Y, Tenneholtz M (2013) Differential pricing with inequity aversion in social networks. In: Proceedings of the fourteenth ACM conference on Electronic Commerce, EC’13, pp 9–24Google Scholar
  3. Barabási AL, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509–512MathSciNetCrossRefMATHGoogle Scholar
  4. Bateni M, Haghpanah N, Sivan B, Zadimoghaddam M (2013) Revenue maximization with nonexcludable goods. In: Proceedings of the 9th international conference on Web and Internet Economics, WINE’13, pp 40–53Google Scholar
  5. Bhattacharya S, Kulkarni J, Munagala K, Xu X (2011) On allocations with negative externalities. In: Proceedings of the 7th international conference on Internet and Network Economics, WINE’11, pp 25–36Google Scholar
  6. Bloch F, Quérou N (2013) Pricing in social networks. Games Econ Behav 80:243–261MathSciNetCrossRefMATHGoogle Scholar
  7. Bollobás B, Riordan O, Spencer J, Tusnády G (2001) The degree sequence of a scale-free random graph process. Random Struct Algorithms 18(3):279–290MathSciNetCrossRefMATHGoogle Scholar
  8. Bramoullé Y (2007) Anti-coordination and social interactions. Games Econ Behav 58(1):30–49MathSciNetCrossRefMATHGoogle Scholar
  9. Bramoullé Y, Kranton R (2007) Public goods in networks. J Econ Theory 135(1):478–494MathSciNetCrossRefMATHGoogle Scholar
  10. Candogan O, Bimpikis K, Ozdaglar A (2012) Optimal pricing in networks with externalities. Oper Res 60(4):883–905MathSciNetCrossRefMATHGoogle Scholar
  11. Cao Z, Chen X, Hu X, Wang C (2015) Pricing in social networks with negative externalities. In: Proceedings of the 4th international conference on Computational Social Networks, CSoNet’15, pp 14–25Google Scholar
  12. Chen N (2009) On the approximability of influence in social networks. SIAM J Discrete Math 23(3):1400–1415MathSciNetCrossRefMATHGoogle Scholar
  13. Chen W, Lu P, Sun X, Tang B, Wang Y, Zhu ZA (2011) Optimal pricing in social networks with incomplete information. In: Proceedings of the 7th international conference on Internet and Network Economics, WINE’11, pp 49–60Google Scholar
  14. Deng C, Pekeč S (2011) Money for nothing: exploiting negative externalities. In: Proceedings of the 12th ACM conference on Electronic Commerce, EC’11, pp 361–370Google Scholar
  15. Farrell J, Saloner G (1985) Standardization, compatibility, and innovation. Rand J Econ 16:70–83CrossRefGoogle Scholar
  16. Feldman M, Kempe D, Lucier B, Paes Leme R (2013) Pricing public goods for private sale. In: Proceedings of the fourteenth ACM conference on Electronic Commerce, EC’13, pp 417–434Google Scholar
  17. Haghpanah N, Immorlica N, Mirrokni V, Munagala K (2013) Optimal auctions with positive network externalities. ACM Trans Econ Comput 1(2):13CrossRefGoogle Scholar
  18. Hartline J, Mirrokni V, Sundararajan M (2008) Optimal marketing strategies over social networks. In: Proceedings of the 17th international conference on World Wide Web, WWW’08, pp 189–198Google Scholar
  19. Jehiel P, Moldovanu B, Stacchetti E (1996) How (not) to sell nuclear weapons. Am Econ Rev 86:814–829Google Scholar
  20. Katz M, Shapiro C (1985) Network externalities, competition, and compatibility. Am Econ Rev 75:424–440Google Scholar
  21. Kempe D, Kleinberg J, Tardos É (2003) Maximizing the spread of influence through a social network. In: Proceedings of the ninth ACM SIGKDD international conference on Knowledge Discovery and Data Mining, pp 137–146Google Scholar
  22. Mossel E, Roch S (2007) On the submodularity of influence in social networks. In: Proceedings of the thirty-ninth annual ACM Symposium on Theory of Computing, STOC’07, pp 128–134Google Scholar
  23. Radner R, Radunskaya A, Sundararajan A (2014) Dynamic pricing of network goods with boundedly rational consumers. Proc Natl Acad Sci 111(1):99–104MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Zhigang Cao
    • 1
  • Xujin Chen
    • 1
  • Xiaodong Hu
    • 1
  • Changjun Wang
    • 2
  1. 1.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.Beijing Institute for Scientific and Engineering ComputingBeijing University of TechnologyBeijingChina

Personalised recommendations