Voting in agreeable societies

TL;DR

This paper explores the mathematical conditions under which a majority can find common ground in voting scenarios, using convex set theory to analyze social agreement and the influence of political spectrum shapes.

Contribution

It extends classical convex set theorems like Helly's theorem to model preferences in voting, providing new tools for analyzing agreement in societies.

Findings

Extensions of Helly's theorem applicable to preference sets

Conditions for majority agreement in convex preference models

Insights into how political spectrum shapes affect voting outcomes

Abstract

When can a majority of voters find common ground, that is, a position they all agree upon? How does the shape of the political spectrum influence the outcome? When mathematical objects have a social interpretation, the associated theorems have social applications. In this article we give examples of situations where sets model preferences and develop extensions of classical theorems about convex sets, such as Helly's theorem, that can be used in the analysis of voting in "agreeable" societies.