Double-interval societies

TL;DR

This paper investigates societies where voters have approval sets as two disjoint intervals on a spectrum, ensuring pairwise intersection, and determines the minimal guaranteed approval ratio, providing bounds and examples from double-n string arrangements.

Contribution

It introduces the concept of double-interval societies with pairwise intersecting approval sets and establishes lower bounds on approval ratios, connecting to arrangements derived from double-n strings.

Findings

Established a lower bound for approval ratios in double-interval societies.

Analyzed societies from double-n strings with low approval ratios.

Connected approval ratios to arrangements of symbols in double-n strings.

Abstract

Consider a society of voters, each of whom specify an approval set over a linear political spectrum. We examine double-interval societies, in which each person's approval set is represented by two disjoint closed intervals, and study this situation where the approval sets are pairwise-intersecting: every pair of voters has a point in the intersection of their approval sets. The approval ratio for a society is, loosely speaking, the popularity of the most popular position on the spectrum. We study the question: what is the minimal guaranteed approval ratio for such a society? We provide a lower bound for the approval ratio, and examine a family of societies that have rather low approval ratios. These societies arise from double-n strings: arrangements of n symbols in which each symbol appears exactly twice.