Comments on “Opinion Dynamics Driven by Various Ways of Averaging”

The two main findings in Hegselmann and Krause (Comput Econ 25:381–405, 2005), the theorem on opinion stabilization and the corollary on consensus formation, are built on partial abstract means (PAMs) with bounded confidence sets that are assumed to be continuous. However, we show that any PAM with bounded confidence sets cannot be continuous. The discontinuity of such PAMs threatens the validity of the main findings in Hegselmann and Krause (Comput Econ 25:381–405, 2005). Moreover, the condition that the corollary on consensus formation considers necessary and sufficient, under which agents will approach a consensus, is, in fact, not a sufficient condition. To resolve these issues, we provide a sufficient condition for PAMs with bounded confidence sets under which the theorem on opinion stabilization becomes valid. We also show that under this condition the condition in the corollary on consensus formation is necessary condition for agents to approach a consensus.