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We study the effects of free will and massive opinion of multi-agents in amajority rule model wherein the competition of the two types of opinions istaken into account. To address this issue, we consider two specific models(model I and model II) involving different opinion-updating dynamics. Duringthe opinion-updating process, the agents either interact with their neighborsunder a majority rule with probability $1-q$, or make their own decisions withfree will (model I) or according to the massive opinion (model II) withprobability $q$. We investigate the difference of the average numbers of thetwo opinions as a function of $q$ in the steady state. We find that thelocation of the order-disorder phase transition point may be shifted accordingto the involved dynamics, giving rise to either smooth or harsh conditions toachieve 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 fullconsensus phase emerges. Our analytical estimations are in good agreement withsimulation results.

Effects of free will and massive opinion in majority rule model