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Frustrated Equilibrium of Asymmetric Coordinating Dynamics in a Marketing Game

  • Matthew G. Reyes
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1037)

Abstract

This paper considers a recently introduced model for socially-contingent decision-making and addresses the connection between influences on individual decision-making and the statistical, information-theoretic properties associated with such decision-making dynamics on a social network. In particular, we analytically show, on a few simple examples, the correspondence between coordinating influences and positively correlated models which in turn correspond to models with entropy that decreases monotonically in the strength of the influences. Moreover, we discuss numerical results that suggest asymmetric yet coordinating influences may converge to a frustrated equilibrium.

Keywords

Reinforcement learning Social networks Marketing Glauber dynamics Gibbs distributions Minimum conditional description length 

References

  1. 1.
    Alonzo-Sanz, R.: Self-organization in the battle of the sexes. Int. J. Mod. Phys. C 22, 1–11 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Amari, S., Nagaoka, H.: Methods of Information Geometry. Oxford University Press, Oxford (1993)zbMATHGoogle Scholar
  3. 3.
    Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Dover, Mineola (2007)zbMATHGoogle Scholar
  4. 4.
    Bell, D.E., Keeney, R.L., Little, J.D.C.: A market share theorem. J. Mark. Res. 12(2), 136–141 (1975)CrossRefGoogle Scholar
  5. 5.
    Blume, L.E.: Statistical mechanics of strategic interaction. Games Econ. Behav. 5(3), 387–424 (1993)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bramoulle, Y.: Anti-coordination and social interactions. Games Econ. Behav. 58, 30–49 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Broere, J., Buskens, V., Weesie, J., Stoof, H.: Network effects on coordination in asymmetric games. Sci. Rep. 7, 17016 (2017)CrossRefGoogle Scholar
  8. 8.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley, New York (1991)CrossRefGoogle Scholar
  9. 9.
    Georgii, H.O.: Gibbs Measures and Phase Transitions. De Grutyer, Berlin (1988)CrossRefGoogle Scholar
  10. 10.
    Gladwell, M.: The Tipping Point. Little, Brown, and Company, Boston (2000)Google Scholar
  11. 11.
    Glauber, R.J.: Time-dependent statistics of the Ising model. J. Math. Phys. 4, 294–307 (1963)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Godreche C.: Dynamics of the directed Ising chain. J. Stat. Mech. Theory Exp. P04005 (2011).  https://doi.org/10.1088/1742-5468/2011/04/P04005MathSciNetCrossRefGoogle Scholar
  13. 13.
    Griffiths, R.B.: Correlations in Ising Ferromag. I. J. Math. Phys. 8, 478–483 (1967)CrossRefGoogle Scholar
  14. 14.
    Luce, D.: Individual Choice Behavior. Dover, Mineola (1959)zbMATHGoogle Scholar
  15. 15.
    Kempe, D., Kleinberg, J., Tardos, E.: Influential nodes in a diffusion model for social networks. In: Proceedings of the 32nd International Conference on Automata, Languages, and Programming (ICALP), pp. 1127–1138 (2005)CrossRefGoogle Scholar
  16. 16.
    McFadden, D.: Conditional logit analysis of qualitative choice behavior. In: Frontiers in Econometrics. Academic Press, New York (1974)Google Scholar
  17. 17.
    Moessner, R., Sondhi, S.L.: “Ising models of quantum frustration. Phys. Rev. B 63, 224401 (2001)CrossRefGoogle Scholar
  18. 18.
    Montanari, A., Saberi, A.: The spread of innovations in social networks. PNAS 107(47), 20196–20201 (2010)CrossRefGoogle Scholar
  19. 19.
    Reyes, M.G., Neuhoff, D.L.: Entropy bounds for a Markov random subfield. In: ISIT 2009, Seoul, South Korea (2009)Google Scholar
  20. 20.
    Reyes, M.G., Neuhoff, D.L.: Minimum conditional description length estimation of Markov random fields. In: ITA Workshop, February 2016Google Scholar
  21. 21.
    Reyes, M.G.: A Marketing Game: a model for social media mining and manipulation. Accepted to Future of Information and Communication Conference, San Francisco, CA, 14–15 March 2019Google Scholar
  22. 22.
    Reyes, M.G.: Reinforcement learning in a marketing game. Accepted to Computing Conference, London, UK, 16–17 July 2019Google Scholar
  23. 23.
    Reyes, M.G., Neuhoff, D.L.: Monotonicity of entropy in positively correlated Ising trees. Accepted to ISIT, Paris, July 2019Google Scholar
  24. 24.
    Smith, J.M., Hofbauer, J.: The ‘Battle of the Sexes’: a genetic model with limit cycle behavior. Theor. Popul. Biol. 32, 1–14 (1987)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sutton, R.S., Barto, A.G.: Reinforcement Learning: An Introduction. MIT Press, Cambridge (1998)zbMATHGoogle Scholar
  26. 26.
    Wainwright, M.J.: Estimating the “Wrong” graphical model: benefits in the computation-limited setting. J. Mach. Learn. Res. 7, 1829–1859 (2006)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Wannier, G.H.: Antiferromagnetism. The triangular Ising net. Phys. Rev. 79, 357 (1950)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Watts, D.J., Dodds, P.S.: Influentials, networks, and public opinion formation. J. Consum. Res. 34, 441–458 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Matthew G. Reyes
    • 1
  1. 1.Ann ArborUSA

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