Piercing Numbers in Circular Societies

TL;DR

This paper investigates the piercing number in circular approval voting societies, providing bounds, probability distributions, and simulations for societies with voters' preferences modeled as arcs on a circle.

Contribution

It introduces bounds and probabilistic analysis for the piercing number in fixed-length circular societies, extending geometric approval voting models.

Findings

Bounds on piercing numbers based on voter arc length

Probability distribution of piercing numbers under uniform arc placement

Simulation results for societies with larger voter preference arcs

Abstract

In the system of approval voting, individuals vote for all candidates they find acceptable. Many approval voting situations can be modeled geometrically, and thus geometric concepts such as the piercing number have a natural interpretation. In this paper, we explore piercing numbers in the setting where voter preferences can be modeled by congruent arcs on a circle -- i.e., in fixed-length circular societies. Given a number of voters and the length of the voter preference arcs, we give bounds on the possible piercing number of the society. Further, we explore which piercing numbers are more likely. Specifically, under the assumption of uniformly distributed voter preference arcs, we determine the probability distribution of the piercing number of societies in which the length of the arcs is sufficiently small. We end with simulations that give estimated probabilities of piercing number for societies with larger voter preference arcs.