A straightforward proof of the critical value in the Hegselmann-Krause model: up to one-half

TL;DR

This paper provides a simple proof that in the Hegselmann-Krause model, the critical confidence threshold for consensus is up to one-half, showing that consensus probability approaches one for thresholds above or equal to this value.

Contribution

It offers a straightforward proof establishing the critical value in the Hegselmann-Krause model as up to one-half, simplifying previous complex analyses.

Findings

Critical value in the model is up to one-half.

Consensus probability tends to one for thresholds ≥ 0.5.

Proof applies to uniformly distributed initial opinions.

Abstract

In the Hegselmann-Krause model, an agent updates its opinion by averaging with others whose opinions differ by at most a given confidence threshold. With agents' initial opinions uniformly distributed on the unit interval, we provide a straightforward proof that establishes the critical value is up to one-half. This implies that the probability of consensus approaches one as the number of agents tends to infinity for confidence thresholds larger than or equal to one-half.