Opinion Formation Games with Aggregation and Negative Influence

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10504)


We study continuous opinion formation games with aggregation aspects. In many domains, expressed opinions of people are not only affected by local interaction and personal beliefs, but also by influences that stem from global properties of the opinions in the society. To capture the interplay of such global and local effects, we propose a model of opinion formation games with aggregation, where we concentrate on the average public opinion as a natural way to represent a global trend in the society. While the average alone does not have good strategic properties as an aggregation rule, we show that with a reasonable influence of the average public opinion, the good properties of opinion formation models are preserved. More formally, we prove that a unique equilibrium exists in average-oriented opinion formation games. Simultaneous best-response dynamics converge to within distance \(\varepsilon \) of equilibrium in \(O(n^2 \ln (n/\varepsilon ))\) rounds, even in a model with outdated information on the average public opinion. For the Price of Anarchy, we show a small bound of \(9/8 + o(1)\), almost matching the tight bound for games without aggregation. Moreover, some of the results apply to a general class of opinion formation games with negative influences, and we extend our results to the case where expressed opinions come from a restricted domain.


Formulate Game Rules Best-response Dynamics Opinion Formation Process Intrinsic Belief Nice Algorithmic Properties 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Altafini, C.: Consensus problems on networks with antagonistic interactions. IEEE Trans. Autom. Control 58(4), 935–946 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Auletta, V., Caragiannis, I., Ferraioli, D., Galdi, C., Persiano, G.: Generalized discrete preference games. In: Proceedings of 25th International Joint Conference on Artificial Intelligence (IJCAI 2016), pp. 53–59 (2016)Google Scholar
  3. 3.
    Barberà, S.: An introduction to strategy proof social choice functions. Soc. Choice Welf. 18, 619–653 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bhawalkar, K., Gollapudi, S., Munagala, K.: Coevolutionary opinion formation games. In: Proceedings of 45th ACM Symposium on Theory of Computing (STOC 2013), pp. 41–50 (2013)Google Scholar
  5. 5.
    Bilò, V., Fanelli, A., Moscardelli, L.: Opinion formation games with dynamic social influences. In: Cai, Y., Vetta, A. (eds.) WINE 2016. LNCS, vol. 10123, pp. 444–458. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-54110-4_31CrossRefGoogle Scholar
  6. 6.
    Bindel, D., Kleinberg, J.M., Oren, S.: How bad is forming your own opinion? In: Proceeding of 52nd IEEE Symposium on Foundations of Computer Science (FOCS 2011), pp. 57–66 (2011)Google Scholar
  7. 7.
    Chazelle, B., Wang, C.: Inertial Hegselmann-Krause systems. IEEE Trans. Autom. Control (2017, to appear)Google Scholar
  8. 8.
    Chen, P.-A., Chen, Y.-L., Lu, C.-J.: 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
  9. 9.
    DeGroot, M.H.: Reaching a consensus. J. Am. Stat. Assoc. 69, 118–121 (1974)CrossRefGoogle Scholar
  10. 10.
    Ferraioli, D., Goldberg, P.W., Ventre, C.: Decentralized dynamics for finite opinion games. In: Serna, M. (ed.) SAGT 2012. LNCS, pp. 144–155. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-33996-7_13CrossRefGoogle Scholar
  11. 11.
    Fotouhi, B., Rabbat, M.G.: The effect of exogenous inputs and defiant agents on opinion dynamics with local and global interactions. IEEE J. Sel. Top. Sig. Process. 7(2), 347–357 (2013)CrossRefGoogle Scholar
  12. 12.
    Friedkin, N.E., Johnsen, E.C.: Social influence and opinions. J. Math. Sociol. 15(3–4), 193–205 (1990)CrossRefGoogle Scholar
  13. 13.
    Ghaderi, J., Srikant, R.: Opinion dynamics in social networks with stubborn agents: equilibrium and convergence rate. Automatica 50, 3209–3215 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Golub, B., Jackson, M.O.: Naïve learning in social networks and the wisdom of crowds. Am. Econ. J.: Microecon. 2(1), 112–149 (2010)Google Scholar
  15. 15.
    Hegselmann, R., Krause, U.: Opinion dynamics and bounded confidence models, analysis, and simulation. J. Artif. Soc. Soc. Simul. 5, 2 (2002)Google Scholar
  16. 16.
    Jackson, M.O.: Social and Economic Networks. Princeton University Press, Princeton (2008)CrossRefGoogle Scholar
  17. 17.
    Moulin, H.: On strategy-proofness and single-peakedness. Publ. Choice 35, 437–455 (1980)CrossRefGoogle Scholar
  18. 18.
    Rosen, J.B.: Existence and uniqueness of equilibrium points in concave \(n\)-person games. Econometrica 33, 520–534 (1965)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Roughgarden, T., Schoppmann, F.: Local smoothness and the price of anarchy in splittable congestion games. J. Econ. Theory 156, 317–342 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Yildiz, E., 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

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Electrical and Systems EngineeringUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.School of Electrical and Computer EngineeringNTU AthensZografouGreece
  3. 3.Institut Für InformatikGoethe-Universität Frankfurt/MainFrankfurtGermany

Personalised recommendations