Small One-Dimensional Euclidean Preference Profiles

TL;DR

This paper characterizes small one-dimensional Euclidean preference profiles, showing conditions under which preferences are Euclidean based on the number of voters and alternatives, with implications for preference profile structure.

Contribution

It provides a complete characterization of small Euclidean preference profiles, linking them to single-peaked and single-crossing properties, and identifies minimal non-Euclidean profiles.

Findings

Profiles with up to two voters are Euclidean iff single-peaked.

Profiles with up to five alternatives are Euclidean iff single-peaked and single-crossing.

Smallest non-Euclidean profiles have three voters and six alternatives.

Abstract

We characterize one-dimensional Euclidean preference profiles with a small number of alternatives and voters. In particular, we show the following. 1. Every preference profile with up to two voters is one-dimensional Euclidean if and only if it is single-peaked. 2. Every preference profile with up to five alternatives is one-dimensional Euclidean if and only if it is single-peaked and single-crossing. By [12], we thus obtain that the smallest single-peaked and single-crossing preference profiles that are not one-dimensional Euclidean consist of three voters and six alternatives.