2-Dimensional Euclidean Preferences

TL;DR

This paper investigates the conditions under which preference profiles with alternatives and voters can be represented in a 2-dimensional Euclidean space, revealing tight bounds for different numbers of voters and alternatives.

Contribution

It establishes tight bounds for 2-dimensional Euclidean representations based on the number of voters and alternatives, extending previous understanding.

Findings

Profiles with up to two voters are 2-dimensional Euclidean.

Profiles with up to three alternatives are 2-dimensional Euclidean.

The property holds for up to seven alternatives with three voters.

Abstract

A preference profile with m alternatives and n voters is 2-dimensional Euclidean if both the alternatives and the voters can be placed into a 2-dimensional space such that for each pair of alternatives, every voter prefers the one which has a shorter Euclidean distance to the voter. We study how 2-dimensional Euclidean preference profiles depend on the values m and n. We find that any profile with at most two voters or at most three alternatives is 2-dimensional Euclidean while for three voters, we can show this property for up to seven alternatives. The results are tight in terms of Bogomolnaia and Laslier [2, Proposition 15(1)].