Choosing Products in Social Networks

  • Sunil Simon
  • Krzysztof R. Apt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7695)

Abstract

We study the consequences of adopting products by agents who form a social network. To this end we use the threshold model introduced in [1], in which the nodes influenced by their neighbours can adopt one out of several alternatives, and associate with each such social network a strategic game between the agents. The possibility of not choosing any product results in two special types of (pure) Nash equilibria.

We show that such games may have no Nash equilibrium and that determining the existence of a Nash equilibrium, also of a special type, is NP-complete. The situation changes when the underlying graph of the social network is a DAG, a simple cycle, or has no source nodes. For these three classes we determine the complexity of establishing whether a (special type of) Nash equilibrium exists.

We also clarify for these categories of games the status and the complexity of the finite improvement property (FIP). Further, we introduce a new property of the uniform FIP which is satisfied when the underlying graph is a simple cycle, but determining it is co-NP-hard in the general case and also when the underlying graph has no source nodes. The latter complexity results also hold for verifying the property of being a weakly acyclic game.

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References

  1. 1.
    Apt, K.R., Markakis, E.: Diffusion in Social Networks with Competing Products. In: Persiano, G. (ed.) SAGT 2011. LNCS, vol. 6982, pp. 212–223. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Easley, D., Kleinberg, J.: Networks, Crowds, and Markets. Cambridge University Press (2010)Google Scholar
  3. 3.
    Granovetter, M.: Threshold models of collective behavior. American Journal of Sociology 83(6), 1420–1443 (1978)CrossRefGoogle Scholar
  4. 4.
    Morris, S.: Contagion. The Review of Economic Studies 67(1), 57–78 (2000)MATHCrossRefGoogle Scholar
  5. 5.
    Kempe, D., Kleinberg, J.M., Tardos, É.: Maximizing the spread of influence through a social network. In: Proceedings of the International Conference on Knowledge Discovery and Data Mining, pp. 137–146 (2003)Google Scholar
  6. 6.
    Chen, N.: On the approximability of influence in social networks. SIAM J. Discrete Math. 23(3), 1400–1415 (2009)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Immorlica, N., Kleinberg, J.M., Mahdian, M., Wexler, T.: The role of compatibility in the diffusion of technologies through social networks. In: ACM Conference on Electronic Commerce, pp. 75–83. ACM (2007)Google Scholar
  8. 8.
    Borodin, A., Filmus, Y., Oren, J.: Threshold Models for Competitive Influence in Social Networks. In: Saberi, A. (ed.) WINE 2010. LNCS, vol. 6484, pp. 539–550. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Kearns, M., Littman, M., Singh, S.: Graphical models for game theory. In: Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence (UAI 2001), pp. 253–260. Morgan Kaufmann (2001)Google Scholar
  10. 10.
    Tardos, É., Wexler, T.: Network formation games and the potential function method. In: Algorithmic Game Theory, pp. 487–516. Cambridge University Press (2007)Google Scholar
  11. 11.
    Alon, N., Feldman, M., Procaccia, A.D., Tennenholtz, M.: A note on competitive diffusion through social networks. Inf. Process. Lett. 110(6), 221–225 (2010)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Brautbar, M., Kearns, M.: A Clustering Coefficient Network Formation Game. In: Persiano, G. (ed.) SAGT 2011. LNCS, vol. 6982, pp. 224–235. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  13. 13.
    Milchtaich, I.: Congestion games with player-specific payoff functions. Games and Economic Behaviour 13, 111–124 (1996)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Monderer, D., Shapley, L.S.: Potential games. Games and Economic Behaviour 14, 124–143 (1996)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Brokkelkamp, K.R., de Vries, M.J.: Convergence of Ordered Improvement Paths in Generalized Congestion Games. In: Serna, M. (ed.) SAGT 2012. LNCS, vol. 7615, pp. 61–71. Springer, Heidelberg (2012)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sunil Simon
    • 1
  • Krzysztof R. Apt
    • 1
  1. 1.Centre for Mathematics and Computer Science (CWI)University of AmsterdamThe Netherlands

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