Bounded Confidence Revisited: What We Overlooked, Underestimated, and Got Wrong

Abstract

In the bounded confidence model (BC-model) (Hegselmann & Krause 2002), period by period, all agents average over all opinions that are no further away from their actual opinion than a given distance ϵ, i.e., their 'bound of confidence'. With the benefit of hindsight, it is clear that we completely overlooked a crucial feature of our model back in 2002. That is for increasing values of ϵ, our analysis suggested smooth transitions in model behaviour. However, the transitions are in fact wild, chaotic and non-monotonic-as described by Jan Lorenz ( 2006). The most dramatic example of these effects is a consensus that breaks down for larger values of ϵ. The core of this article is a fundamentally new approach to the analysis of the BC-model. This new approach makes the non-monotonicities unmissable. To understand this approach, we start with the question: how many different BC processes can we initiate with any given start distribution? The answer to this question is almost certainly for all possible start distributions and certainly in all cases analysed here, it is always a finite number of ϵ-values that make a difference for the processes we start. Moreover, there is an algorithm that finds, for any start distribution, the complete list of ϵ-values that make a difference. Using this list, we can then go directly through all the possible BC-processes given the start distribution. We can therefore check them for non-monotonicity of any kind, and will be able to find them all. This good news comes however with bad news. That is the algorithm that inevitably and without exception finds all the ϵ-values that matter requires exact arithmetics, without any rounding and without even the slightest rounding error. As a consequence, we have to abandon the usual floating-point arithmetic used in today's computers and programming languages. What we need to use instead is absolutely exact fractional arithmetic with integers of arbitrary length. This numerical approach is feasible on all modern computers. The new analytical approach and results are likely to have implications for many applications of the BC-model.

A novel framework to classify opinion dynamics of mobile agents under the bounded confidence model

“…Lorenz (2007) further highlights that Hegselmann and Krause (2002) (group-based interactions) in comparison to the DW model (peer-based interactions) have no minor clusters. Gargiulo and Huet (2010); Kozma and Barrat (2008) demonstrate the DW model while also implementing group-based interactions while permitting rewiring and these outliers disappear, similar to the static Hegselmann and Krause (2002) model. Therefore, in this article, we propose a quantifiable measurement of the occurrence of isolated agents to be considered along with the formation of communities.…”

“…One of the most well-studied opinion models is the Bounded Confidence (BC) models proposed in Krause (1997); Deffuant et al (2000); Hegselmann and Krause (2002) using a threshold for influence. Hegselmann and Krause (2002) propose a model with a simultaneous update rather than the pairwise sequential interactions implemented in Deffuant et al (2000).…”

“…One of the most well-studied opinion models is the Bounded Confidence (BC) models proposed in Krause (1997); Deffuant et al (2000); Hegselmann and Krause (2002) using a threshold for influence. Hegselmann and Krause (2002) propose a model with a simultaneous update rather than the pairwise sequential interactions implemented in Deffuant et al (2000). The need to calculate opinion averages of large groups of agents makes computer simulations of the Hegselmann and Krause (2002) model rather lengthy as compared to Deffuant et al (2000) model (Castellano et al 2009).…”

“…Hegselmann and Krause (2002) propose a model with a simultaneous update rather than the pairwise sequential interactions implemented in Deffuant et al (2000). The need to calculate opinion averages of large groups of agents makes computer simulations of the Hegselmann and Krause (2002) model rather lengthy as compared to Deffuant et al (2000) model (Castellano et al 2009).…”

On the Impact of Social Media Recommendations on Consensus of Discrete Opinions

“…Several extensions have been recently proposed to the seminal models by DeGroot and by Friedkin and Johnsen (and their discrete counterparts), by considering only limited interaction by agents (Fotakis et al, 2016(Fotakis et al, , 2018, or an evolving environment (Hegselmann and Krause, 2002;Bhawalkar et al, 2013;Bilò et al, 2018;Ferraioli and Ventre, 2017;Auletta et al, 2019a), or both repulsive and attractive interaction (Auletta et al, 2016;Acar et al, 2017). Despite their larger adherence with many real world aspects, however none of these variants has received the same level of interest as the models by DeGroot and by Friedkin and Johnsen.…”