Opinion Dynamics with Limited Information

  • Dimitris Fotakis
  • Vardis Kandiros
  • Vasilis Kontonis
  • Stratis SkoulakisEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11316)


We study opinion formation games based on the Friedkin-Johnsen (FJ) model. We are interested in simple and natural variants of the FJ model that use limited information exchange in each round and converge to the same stable point. As in the FJ model, we assume that each agent i has an intrinsic opinion \(s_i \in [0,1]\) and maintains an expressed opinion \(x_i(t) \in [0,1]\) in each round t. To model limited information exchange, we assume that each agent i meets with one random friend j at each round t and learns only \(x_j(t)\). The amount of influence j imposes on i is reflected by the probability \(p_{ij}\) with which i meets j. Then, agent i suffers a disagreement cost that is a convex combination of \((x_i(t) - s_i)^2\) and \((x_i(t) - x_j(t))^2\).

An important class of dynamics in this setting are no regret dynamics. We show an exponential gap between the convergence rate of no regret dynamics and of more general dynamics that do not ensure no regret. We prove that no regret dynamics require roughly \({\varOmega }(1/\varepsilon )\) rounds to be within distance \(\varepsilon \) from the stable point \(x^*\) of the FJ model. On the other hand, we provide an opinion update rule that does not ensure no regret and converges to \(x^*\) in \({\tilde{O}}(\log ^2(1/\varepsilon ))\) rounds. Finally, we show that the agents can adopt a simple opinion update rule that ensures no regret and converges to \(x^*\) in \(\mathrm {poly}(1/\varepsilon )\) rounds.


  1. 1.
    Abebe, R., Kleinberg, J., Parkes, D., Tsourakakis, C.E.: Opinion dynamics with varying susceptibility to persuasion. CoRR abs/1801.07863 (2018)Google Scholar
  2. 2.
    Bertsekas, D., Tsitsiklis, J.: Parallel and Distributed Computation: Numerical Methods. Athena Scientific, Belmont (1997)zbMATHGoogle Scholar
  3. 3.
    Bhawalkar, K., Gollapudi, S., Munagala, K.: Coevolutionary opinion formation games. In: Symposium on Theory of Computing Conference, STOC 2013, pp. 41–50 (2013)Google Scholar
  4. 4.
    Bilò, V., Fanelli, A., Moscardelli, L.: Opinion formation games with dynamic social influences. In: Cai, Y., Vetta, A. (eds.) WINE 2016. Lecture Notes in Computer Science, vol. 10123, pp. 444–458. Springer, Berlin (2016). Scholar
  5. 5.
    Bindel, D., Kleinberg, J., Oren, S.: How bad is forming your own opinion? In: IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, pp. 57–66 (2011)Google Scholar
  6. 6.
    Chen, P., Chen, Y., Lu, C.: Bounds on the price of anarchy for a more general class of directed graphs in opinion formation games. Oper. Res. Lett. 44(6), 808–811 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cohen, J., Héliou, A., Mertikopoulos, P.: Hedging under uncertainty: regret minimization meets exponentially fast convergence. In: Bilò, V., Flammini, M. (eds.) SAGT 2017. LNCS, vol. 10504, pp. 252–263. Springer, Cham (2017). Scholar
  8. 8.
    DeGroot, M.: Reaching a consensus. J. Am. Stat. Assoc. 69, 118–121 (1974)CrossRefGoogle Scholar
  9. 9.
    Epitropou, M., Fotakis, D., Hoefer, M., Skoulakis, S.: Opinion formation games with aggregation and negative influence. In: Bilò, V., Flammini, M. (eds.) SAGT 2017. LNCS, vol. 10504, pp. 173–185. Springer, Cham (2017). Scholar
  10. 10.
    Even-Dar, E., Mansour, Y., Nadav, U.: On the convergence of regret minimization dynamics in concave games. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, pp. 523–532 (2009)Google Scholar
  11. 11.
    Ferraioli, D., Goldberg, P., Ventre, C.: Decentralized dynamics for finite opinion games. Theor. Comput. Sci. 648(C), 96–115 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Foster, D., Vohra, R.: Calibrated learning and correlated equilibrium. Games Econ. Behav. 21(1), 40–55 (1997)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fotakis, D., Palyvos-Giannas, D., Skoulakis, S.: Opinion dynamics with local interactions. In: Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence, IJCAI 2016, pp. 279–285 (2016)Google Scholar
  14. 14.
    Freund, Y., Schapire, R.: Adaptive game playing using multiplicative weights. Games Econ. Behav. 29(1), 79–103 (1999)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Friedkin, N., Johnsen, E.: Social influence and opinions. J. Math. Sociol. 15(3–4), 193–206 (1990)CrossRefGoogle Scholar
  16. 16.
    Ghaderi, J., Srikant, R.: Opinion dynamics in social networks with stubborn agents: equilibrium and convergence rate. Automatica 50(12), 3209–3215 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gionis, A., Terzi, E., Tsaparas, P.: Opinion maximization in social networks. In: Proceedings of the 13th SIAM International Conference on Data Mining, SDM 2013, pp. 387–395 (2013)CrossRefGoogle Scholar
  18. 18.
    Hazan, E.: Introduction to online convex optimization. Found. Trends Optim. 2(3–4), 157–325 (2016)CrossRefGoogle Scholar
  19. 19.
    Hazan, E., Agarwal, A., Kale, S.: Logarithmic regret algorithms for online convex optimization. Mach. Learn. 69(2), 169–192 (2007)CrossRefGoogle Scholar
  20. 20.
    Hegselmann, R., Krause, U.: Opinion dynamics and bounded confidence models, analysis, and simulation. J. Artif. Soc. Soc. Simul. 5 (2002)Google Scholar
  21. 21.
    Héliou, A., Cohen, J., Mertikopoulos, P.: Learning with bandit feedback in potential games. In: Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, NIPS 2017, pp. 6372–6381 (2017)Google Scholar
  22. 22.
    Jackson, M.: Social and Economic Networks. Princeton University Press, Princeton (2008)zbMATHGoogle Scholar
  23. 23.
    Kleinberg, R., Piliouras, G., Tardos, É.: Multiplicative updates outperform generic no-regret learning in congestion games: extended abstract. In: Proceedings of 21st ACM Symposium on Theory of Computing (STOC 2009), pp. 533–542 (2009)Google Scholar
  24. 24.
    Kleinberg, R., Piliouras, G., Tardos, É.: Load balancing without regret in the bulletin board model. Distrib. Comput. 24(1), 21–29 (2011)CrossRefGoogle Scholar
  25. 25.
    Krackhardt, D.: A plunge into networks. Science 326(5949), 47–48 (2009). Scholar
  26. 26.
    Mertikopoulos, P., Staudigl, M.: Convergence to nash equilibrium in continuous games with noisy first-order feedback. In: 56th IEEE Annual Conference on Decision and Control, CDC 2017, pp. 5609–5614 (2017)Google Scholar
  27. 27.
    Sergiu, S.H., Mas-Colell, A.: A simple adaptive procedure leading to correlated equilibrium. Econometrica 68(5), 1127–1150 (2000)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Yildiz, M., Ozdaglar, A., Acemoglu, D., Saberi, A., Scaglione, A.: Binary opinion dynamics with stubborn agents. ACM Trans. Econ. Comput. 1(4), 19:1–19:30 (2013)CrossRefGoogle Scholar
  29. 29.
    Zinkevich, M.: Online convex programming and generalized infinitesimal gradient ascent. In: Proceedings of the Twentieth International Conference on International Conference on Machine Learning, ICML 2003, pp. 928–935. AAAI Press (2003)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Dimitris Fotakis
    • 1
    • 2
  • Vardis Kandiros
    • 2
  • Vasilis Kontonis
    • 3
  • Stratis Skoulakis
    • 2
    Email author
  1. 1.Yahoo ResearchNew YorkUSA
  2. 2.National Technical University of AthensAthensGreece
  3. 3.University of Wisconsin-MadisonMadisonUSA

Personalised recommendations