Euclidean preferences in the plane under $\ell_1$, $\ell_2$ and $\ell_\infty$ norms

TL;DR

This paper investigates Euclidean preferences in the plane under different norms, establishing bounds on preference profile sizes, last-ranked candidates, and providing new characterizations and proofs for these preferences.

Contribution

It extends known results to higher dimensions and different norms, offering new bounds, characterizations, and simplified proofs for Euclidean preference profiles.

Findings

Maximum size of preference profiles under $\, ext{ell}_1$ and $ ext{ell}_ ext{infty}$ norms is 19 for four candidates.

At most four candidates can be ranked last in two-dimensional Euclidean preferences under $\, ext{ell}_1$ or $ ext{ell}_ ext{infty}$ norms.

Maximal size of two-dimensional Euclidean preference profiles on m candidates is in $\, heta(m^4)$, similar to $\, ext{ell}_2$ norm.

Abstract

We present various results about Euclidean preferences in the plane under $\ell_1$, $\ell_2$ and $\ell_{\infty}$ norms. When there are four candidates, we show that the maximal size (in terms of the number of pairwise distinct preferences) of Euclidean preference profiles in the plane under norm $\ell_1$ or $\ell_{\infty}$ is 19. Whatever the number of candidates, we prove that at most four distinct candidates can be ranked in last position of a two-dimensional Euclidean preference profile under norm $\ell_1$ or $\ell_\infty$, which generalizes the case of one-dimensional Euclidean preferences (for which it is well known that at most two candidates can be ranked last). We generalize this result to $2^d$ (resp. $2d$) for $\ell_1$ (resp. $\ell_\infty$) for $d$-dimensional Euclidean preferences. We also establish that the maximal size of a two-dimensional Euclidean preference profile on $m$ candidates under norm $\ell_1$ is in $\Theta(m^4)$, i.e., the same order of magnitude as under norm $\ell_2$. Finally, we provide a new proof that two-dimensional Euclidean preference profiles under norm $\ell_2$ for four candidates can be characterized by three voter-maximal two-dimensional Euclidean profiles. This proof is a simpler alternative to that proposed by Kamiya et al. in Ranking patterns of unfolding models of codimension one, Advances in Applied Mathematics 47(2):379-400.