Computational Economics

, Volume 53, Issue 2, pp 479–505 | Cite as

Opinion Formation with Imperfect Agents as an Evolutionary Process

  • Matjaž Steinbacher
  • Mitja SteinbacherEmail author


We develop and simulate an interaction-based model of continuous opinion formation under bounded confidence to identify conditions and understand circumstances that lead a society into either a consensus, multiple opinion classes or perpetual opinion dynamics. The society is modeled as a social network and random meetings are presumed. When only regular agents are present, we have shown that the small world networks may bring the society very close to consensus for even small threshold levels, but require higher tolerance than the complete network to reach consensus. We have identified the conditions under which the process with stubborn agents generates long-run consensus, permanent disagreement or permanent fluctuation in opinions. There cannot be a persistent fluctuation in opinions in the environment of regular agents nor in the presence of a single group of stubborn agents. In the runs with a single group of stubborn extremists, we have identified the Popper paradox despite the existence of a tolerance span in which the proportion of extremism decreases as the tolerance level increases. Further, in a highly tolerant society with two competing extremist groups, they have no supporters among the regular agents whose opinions are oscillating around the center of the opinion space. The influence of inconsistent agents is persistent and induces a perpetual opinion dynamics. The model is non-equilibrium and emerging, while consensus, if attainable, can be reached in a finite time.


Opinion formation Continuous opinions Consensus Social networks Bounded confidence Stubborn agents Insincere agents 



The algorithm is written in C++ (compiled for 64-bit Visual Studio 12). The source code is available at The authors would like to thank seminar participants at the Fifth World Congress of the Game Theory Society (GAMES 2016), Maastricht, the Netherlands, July 24–28, 2016, and the 22nd Annual Workshop on Economic Science with Heterogeneous Interacting Agents (WEHIA 2017), Milan, Italy, June 12–14, 2017, for helpful comments and suggestions. We also thank two anonymous referees who made a number of helpful comments and suggestions. All errors remain the responsibility of the authors.


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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Faculty of Business StudiesLjubljanaSlovenia
  2. 2.Kiel Institute for the World EconomyKielGermany

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