  Motivated by the dynamics of cultural change and diversity, we generalize the
three-species constrained voter model on a complete graph introduced in [J.
Phys. A 37, 8479 (2004)]. In this opinion dynamics model, a population of size
N is composed of "leftists" and "rightists" that interact with "centrists": 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
the 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 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 when |q|~1/N and 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. Our findings
are corroborated by stochastic simulations.
