How Many Freemasons Are There? The Consensus Voting Mechanism in Metric Spaces

TL;DR

This paper analyzes how a social group's size evolves under consensus voting in metric spaces, providing bounds on expected group size over time with geometric techniques.

Contribution

It introduces geometric methods to bound group size evolution in consensus voting within metric spaces, especially for 2D spaces like the unit ball and square.

Findings

Expected group size in the unit ball is Θ(T^{1/8})

Expected group size in the unit square is between Ω(ln T) and O(ln T · ln ln T)

Develops techniques for bounding group size in dynamic consensus scenarios

Abstract

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 $\Theta(T^{1/8})$. In sharp contrast, for the unit square the expected cardinality is at least $\Omega(\ln T)$ but at most $O(\ln T \cdot \ln\ln T )$.