Ultimate Fate of Constrained Voters

TL;DR

This paper models the long-term opinion dynamics in a population with three opinion groups, deriving exact probabilities and times for reaching consensus or frozen states, with implications for social and cultural models.

Contribution

It provides an exact analytical solution for the probabilities and times of different final states in a constrained opinion dynamics model.

Findings

Exact probability of consensus or frozen states as a function of initial conditions

Mean time to reach the final state

Implications for the Axelrod cultural dissemination model

Abstract

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.