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Opinion Formation with Imperfect Agents as an Evolutionary Process

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Abstract

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.

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Notes

  1. Not only are individuals’ opinions different, in some instances they are even contrasting (Hayek 1945).

  2. This assumption is grounded on the premise that opinions are not true or false in the way the logical statements are but are essentially subjective and agents’ private information. Therefore, no such objective metric exists, which would allow an agent to check the correctness of his opinion. Hence, everyone is right (or wrong) in his own way.

  3. Stubborn agents cannot be persuaded but can persuade others (Katz and Lazarsfeld 1955).

  4. Stubborn agents may be seen as opinion leaders, while insincere agents could also be referred to as misbehaving or faulty agents.

  5. Network-based models of opinion formation have largely developed within two broad categories. In the discrete opinion models, an individual has to choose from a given number of alternatives (voter model, Ising model, etc.). In continuous opinion models, the opinion is represented by a real number for which communication can bring some unique opinions which were non-existent before. This is not possible in discrete models. See Galam (2008), Castellano et al. (2009), Acemoglu and Ozdaglar (2011) and Young and Zamir (2015) for an overview of the recent models. We are concerned with continuous opinions.

  6. Even though the models of opinion dynamics usually presume an honest individual, this sort of uncertainty, i.e. misbehaving parts and component failures, has been regularly used in explaining the behavior of distributed systems in some other disciplines (Fischer et al. 1985; Lamport et al. 1982; LeBlanc et al. 2012; Marti et al. 2000; Pasqualetti et al. 2012; Ren and Beard 2005).

  7. The network is a mathematical representation of a society. For a thorough overview of the social network concepts, we refer the reader to Wasserman and Faust (1994) and Jackson (2010).

  8. This excludes the cases in which John would intentionally call for Peter, asking him for the advice, as well as the cases in which he would want to avoid certain counterparts with adverse opinions or undesired characteristics Gentzkow and Shapiro (2011).

  9. The idea of bounded confidence suggests that people ignore individuals with whom they largely disagree.

  10. The maximum number of opinion classes corresponds to the number of agents. Initially, we presume a pluralistic society.

  11. The same is true if \(x_{i,t}, x_{j,t} < x^{*}\).

  12. Uniform distribution of initial opinions was confirmed with the Kolmogorov–Smirnov test (p value  =  0.1108); \(\overline{x}_0 = 0.5\).

  13. We should note that unique opinions in single repetitions do not mean that these opinions are always the same.

  14. The small world network can be described as the set of small and connected communities with high local clustering in which agents have equal average topological features. Degree distribution in the scale free network exhibits a power law, which means that there are a few agents with many more links than others (Barabási and Albert 1999).

  15. Node degree determines the average size of the neighborhood (premise 2). We examine the cases with \(k=10, 20, 30, 50\). Names of the networks: SW10, SW20, SW30 and SW50 denote the average node degree of the corresponding small world network. In all small world networks, each link is randomly rewired with probability of 5%.

  16. Stubborn agents should not be confused with extremists, per se. Stubbornness relate to the willingness to change the opinion, while the extremism relates to the (extreme) position of an opinion on the opinion scale. Extremists are agents with opinions close to zero or one, while stubborn agents are those with \(\epsilon =0\). However, we have hinted that extremists may also be more convinced individuals, reflecting the behavior of the stubborn agents.

  17. The assumption of sincere agents is certainly valid for some individuals but, generally, it does not match the conclusions from behavioral science which provides plenty of evidence that real-world agents are all but perfect [see Hirshleifer (2001) for an overview].

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Acknowledgements

The algorithm is written in C++ (compiled for 64-bit Visual Studio 12). The source code is available at https://github.com/pixlifai/opiform. 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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  1. Faculty of Business Studies, Ciril-Metodov trg 9, 1000, Ljubljana, Slovenia

    Matjaž Steinbacher & Mitja Steinbacher

  2. Kiel Institute for the World Economy, Kiellinie 66, 24105, Kiel, Germany

    Matjaž Steinbacher

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  1. Matjaž Steinbacher
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Correspondence to Mitja Steinbacher.

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Steinbacher, M., Steinbacher, M. Opinion Formation with Imperfect Agents as an Evolutionary Process. Comput Econ 53, 479–505 (2019). https://doi.org/10.1007/s10614-017-9751-z

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