Consensus and Polarization in a Three-State Bounded-Compromise Voter Model

Abstract: It has recently been argued that the seek for "consensus" and some form of "incompatibility" are basic mechanisms to explain the dynamics of cultural change and diversity [1]. Here, we will consider a basic, but mathematically amenable, three-state bounded compromise voter model (a constrained generalization of the classic two-state voter model [2]) that includes these ingredients. In this opinion dynamics model, a population of size N is composed of "leftists" and "rightists" that interact with "centrists" on a complete graph: a leftist and centrist can both become leftists with rate (1+q)/2 or centrists with rate (1-q)/2 (and similarly for rightists and centrists), where q denotes a selective bias towards extremism (q>0) or centrism (q<0). This system admits three absorbing fixed points and a "polarization" line along which a frozen mixture of leftists and rightists coexist. In the realm of the Fokker-Planck equation, and using a mapping onto a population genetics model, we compute the fixation probability of ending in every absorbing state and the mean times for these events. We therefore show, especially in the limit of weak bias and large population size (|q|~1/N, N>>1), how fluctuations alter the mean field predictions: polarization is likely when q>0, but there is always a finite probability to reach a consensus; the opposite happens when q<0. The findings are illustrated and corroborated by stochastic simulations. This presentation is based on the recent Ref.[3]"