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We determine the ultimate fate of individual opinions in a socially-interacting population of leftists, centrists, and rightists. In an elemental interaction between agents, a centrist and a leftist can become both centrists or both become leftists with equal rates (and similarly for a centrist and a rightist). However leftists and rightists do not interact. This interaction step between pairs of agents is applied repeatedly until the system can no longer evolve. In the mean-field limit, we determine the exact probability that the system reaches consensus (either leftist, rightist, or centrist) or a frozen mixture of leftists and rightists as a function of the initial composition of the population. We also determine the mean time until the final state is reached. Some implications of our results for the ultimate fate in a limit of the Axelrod model are discussed.
Vázquez et al. (Wed,) studied this question.