Computational Economics

, Volume 25, Issue 4, pp 381–405 | Cite as

Opinion Dynamics Driven by Various Ways of Averaging

  • Rainer Hegselmann
  • Ulrich Krause


The paper treats opinion dynamics under bounded confidence when agents employ, beside an arithmetic mean, means like a geometric mean, a power mean or a random mean in aggregating opinions. The different kinds of collective dynamics resulting from these various ways of averaging are studied and compared by simulations. Particular attention is given to the random mean which is a new concept introduced in this paper. All those concrete means are just particular cases of a partial abstract mean, which also is a new concept. This comprehensive concept of averaging opinions is investigated also analytically and it is shown in particular, that the dynamics driven by it always stabilizes in a certain pattern of opinions.


opinion dynamics bounded confidence averaging power mean random mean abstract means 


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  1. Arazy, J., Claesson, T., Janson, S. and Peetre, J. (1985). Mean and their iterations. In Proceedings of the Nineteenth Nordic Congress of Mathematics, Iceland Mathematical Society, Reykjavik, pp. 191–212.Google Scholar
  2. Axelrod, R. (1997). The dissemination of culture: A model with local convergence and global polarization. Journal of Conflict Resolution, 41, 203–226.Google Scholar
  3. Bénabou, R. (1996). Heterogeneity, stratifcation and growth: Macroeconomic implications of community structure and school finance. American Economic Review, 86, 584–609.Google Scholar
  4. Borwein J.M. and Borwein, P.B. (1987). Pi and the AGM. A Study in Analytic Number Theory and Computational Complexity, Wiley, New York.Google Scholar
  5. Ben-Naim, E., Krapivsky, P.L. and Redner, S. (2003). Bifurcations and patterns in compromise processes. Physica D, 183, 190–204.Google Scholar
  6. Bullen, P.S., Mitrinović, D.S. and Vasić, P.M. (eds.) (1988). Means and Their Inequalities. D. Reidel Publication, Dordrecht.Google Scholar
  7. Carlson, B.C. (1971). Algorithms involving arithmetic and geometric means. American Mathematical Monthly, 78, 496–505.Google Scholar
  8. Deffuant, G., Amblard, F., Weisbuch, G. and Faure, Th. (2002). How can extremism prevail? A study based on the relative agreement interaction model. Journal of Artificial Societies and Social Simulation, 5(4),
  9. Dittmer, J.C. (2001). Consensus formation under bounded confidence. Nonlinear Analysis, 47, 4615–4621.CrossRefGoogle Scholar
  10. Hardy, G.H., Littlewood, J.E. and Pólya, G. (1973). Inequalities, Cambridge University Press, Cambridge, (first published 1934)Google Scholar
  11. Hegselmann, R. (2004). Opinion Dynamics – insights by radically simplifyinga models. In: D. Gillies (ed.), Laws and Models in Science, London, pp. 1–29.Google Scholar
  12. Hegselmann R. and Krause, U. (2002). Opinion dynamics and bounded confidence: Models, analysis and simulation, Journal of Artificial Societies and Social Simulation 5(3).
  13. Ioannides, Y.M. (2002). Nonlinear neighbourhood interactions and intergenerational transmissions of human capital. In G. Bitros and Y. Katsoulacos (eds.), Essays in Economic Theory, Growtand Labour Markets: A Festschrift in Honour of Emmanuel Drandakis, Edward Elgar, Cheltenham, pp. 75–112.Google Scholar
  14. Keynes, J.M. (1973). The General Theory of Employment. In The Collected Writings of John Maynard Keynes, Vol. 14. Macmillan, London, pp. 109–123.Google Scholar
  15. Krause U. and Nesemann, T. (1999). Differenzengleichungen and Diskrete Dynamische Systeme. Teubner, Stuttgart.Google Scholar
  16. Krause, U. (2000). A discrete nonlinear and non-autonomous model of consensus formation. In S. Elaydi, G. Ladas, J. Popenda and J. Rakowski (eds.), Communication in Difference Equations, Gordon and Breach Publication, Amsterdam, pp. 227–236.Google Scholar
  17. Krause, U. (2003). Positive particle interaction. In L. Benvenuti, A. De Santis, L. Farina (eds.), Proceeding of the First Multidisciplinarty International Symposium on Positive Systems, Rome, 2003. Springer, Berlin, pp. 199–206.Google Scholar
  18. Krause, U. Collective dynamics of many-faceted agents. Preprint.
  19. Stauffer, D. (2001). Monte Carlo simulations of the Sznajd models. Journal of Artificial Societies and Social Simulation, 5(1),
  20. Stauffer D. and Meyer-Ortmanns, H. (2004). Simulation of consensus model of Deffuant et al. on a Barabási-Albert network. cond-mat/0308231, International Journal of Modern Physics C15, No. 2, in press.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity BayreuthGermany
  2. 2.Department of MathematicsUniversity BremenGermany

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