Inevitability of Polarization in Geometric Opinion Exchange

Polarization and unexpected correlations between opinions on diverse topics are an object of sustained attention. However, numerous theoretical models do not seem to convincingly explain these phenomena. This paper is motivated by a recent line of work, studying models where polarization can be explained in terms of biased assimilation and geometric interplay between opinions on various topics. The agent opinions are represented as unit vectors on a multidimensional sphere and updated according to geometric rules. In contrast to previous work, we focus on the classical opinion exchange setting, where the agents update their opinions in discrete time steps, with a pair of agents interacting randomly at every step. Our findings are twofold. First, polarization appears to be ubiquitous in the class of models we study, requiring only relatively modest assumptions on the update functions, reflecting biased assimilation. Second, there is a qualitative difference between two-dimensional dynamics on the one hand, and three or more dimensions on the other. Accordingly, we prove almost sure polarization for a large class of update rules in two dimensions. Then, we prove polarization in three and more dimensions in more limited cases and try to shed light on central difficulties absent in two dimensions.

Funding: A. M. Alidou and J. Hązła were supported by the Alexander von Humboldt Foundation German research chair funding [Deutscher Akademischer Austauschdienst (German Academic Exchange Service) Projects 57610033 and 57761435]. M. Hahn-Klimroth was partly supported by the Deutsche Forschungsgemeinschaft [Project FOR 2975]. O. Scheftelowitsch was supported by the Deutsche Forschungsgemeinschaft [Grant CO 646/5].