  We study the effects of free will and massive opinion of multi-agents in a
majority rule model wherein the competition of the two types of opinions is
taken into account. To address this issue, we consider two specific models
(model I and model II) involving different opinion-updating dynamics. During
the opinion-updating process, the agents either interact with their neighbors
under a majority rule with probability $1-q$, or make their own decisions with
free will (model I) or according to the massive opinion (model II) with
probability $q$. We investigate the difference of the average numbers of the
two opinions as a function of $q$ in the steady state. We find that the
location of the order-disorder phase transition point may be shifted according
to the involved dynamics, giving rise to either smooth or harsh conditions to
achieve an ordered state. For the practical case with a finite population size,
we conclude that there always exists a threshold for $q$ below which a full
consensus phase emerges. Our analytical estimations are in good agreement with
simulation results.
