  The influence of contrarians on the noisy voter model is studied at the
mean-field level. The noisy voter model is a variant of the voter model where
agents can adopt two opinions, optimistic or pessimistic, and can change them
by means of an imitation (herding) and an intrinsic (noise) mechanisms. An
ensemble of noisy voters undergoes a finite-size phase transition, upon
increasing the relative importance of the noise to the herding, form a bimodal
phase where most of the agents shear the same opinion to a unimodal phase where
almost the same fraction of agent are in opposite states. By the inclusion of
contrarians we allow for some voters to adopt the opposite opinion of other
agents (anti-herding). We first consider the case of only contrarians and show
that the only possible steady state is the unimodal one. More generally, when
voters and contrarians are present, we show that the bimodal-unimodal
transition of the noisy voter model prevails only if the number of contrarians
in the system is smaller than four, and their characteristic rates are small
enough. For the number of contrarians bigger or equal to four, the voters and
the contrarians can be seen only in the unimodal phase. Moreover, if the number
of voters and contrarians, as well as the noise and herding rates, are of the
same order, then the probability functions of the steady state are very well
approximated by the Gaussian distribution.
