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

• Main Authors: Suzuki, Mashbat, Vetta, Adrian
• Format: Journal Article
• Language: English
• Published: 25-05-2020
• Subjects: Computer Science - Computer Science and Game Theory
• DOI: 10.48550/arxiv.2005.12505

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 $O(\ln T \cdot \ln\ln T )$.