How Many Freemasons Are There? The Consensus Voting Mechanism in Metric Spaces
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We study the evolution of a social group when admission to the group is determined via consensus or unanimity voting. In each time period, two candidates apply for membership and a candidate is selected if and only if all the current group members agree. We apply the spatial theory of voting where group members and candidates are located in a metric space and each member votes for its closest (most similar) candidate. Our interest focuses on the expected cardinality of the group after T time periods. To evaluate this we study the geometry inherent in dynamic consensus voting over a metric space. This allows us to develop a set of techniques for lower bounding and upper bounding the expected cardinality of a group. We specialize these methods for two-dimensional metric spaces. For the unit ball the expected cardinality of the group after T time periods is Θ(T^1/8). In sharp contrast, for the unit square the expected cardinality is at least Ω(ln T) but at most O(ln T ·lnln T ).
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