Small One-Dimensional Euclidean Preference Profiles

文献摘要

We characterize one-dimensional Euclidean preference profiles with a small number of alternatives and voters. In particular, we show the following. Every preference profile with up to two voters is one-dimensional Euclidean if and only if it is single-peaked. 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.

We characterize one-dimensional Euclidean preference profiles with a small number of alternatives and voters. In particular, we show the following.

Every preference profile with up to two voters is one-dimensional Euclidean if and only if it is single-peaked.
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.