A rigorous formulation of and partial results on Lorenz's "consensus strikes back" phenomenon for the Hegselmann-Krause model

In a 2006 paper, Jan Lorenz observed a curious behaviour in numerical simulations of the Hegselmann-Krause model: Under some circumstances, making agents more closed-minded can produce a consensus from a dense configuration of opinions which otherwise leads to fragmentation. Suppose one considers initial opinions equally spaced on an interval of length $L$. As first observed by Lorenz, simulations suggest that there are three intervals $[0, L_1)$, $(L_1, L_2)$ and $(L_2, L_3)$, with $L_1 \approx 5.23$, $L_2 \approx 5.67$ and $L_3 \approx 6.84$ such that, when the number of agents is sufficiently large, consensus occurs in the first and third intervals, whereas for the second interval the system fragments into three clusters. In this paper, we prove consensus for $L \leq 5.2$ and for $L$ sufficiently close to 6. These proofs include large computations and in principle the set of $L$ for which consensus can be proven using our approach may be extended with the use of more computing power. We also prove that the set of $L$ for which consensus occurs is open. Moreover, we prove that, when consensus is assured for the equally spaced systems, this in turn implies asymptotic almost sure consensus for the same values of $L$ when initial opinions are drawn independently and uniformly at random. We thus conjecture a pair of phase transitions, making precise the formulation of Lorenz's "consensus strikes back" hypothesis. Our approach makes use of the continuous agent model introduced by Blondel, Hendrickx and Tsitsiklis. Indeed, one contribution of the paper is to provide a presentation of the relationships between the three different models with equally spaced, uniformly random and continuous agents, respectively, which is more rigorous than what can be found in the existing literature.