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

In the bounded confidence model (BC-model) (Hegselmann and Krause 2002), period by period, each agents averages 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 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.