Real Preferences Under Arbitrary Norms

TL;DR

This paper demonstrates that all preference profiles can be embedded into two-dimensional space under any p-norm, broadening the applicability of spatial models for preferences beyond Euclidean and Manhattan norms.

Contribution

It proves that any pair of rankings can be embedded into R^2 under arbitrary norms, extending previous results limited to Euclidean and Manhattan norms.

Findings

All preference profiles can be embedded into R^2 under any p-norm.

Embedding is possible for all pairs of rankings, not just specific cases.

Expands the theoretical foundation for spatial preference models.

Abstract

Whether the goal is to analyze voting behavior, locate facilities, or recommend products, the problem of translating between (ordinal) rankings and (numerical) utilities arises naturally in many contexts. This task is commonly approached by representing both the individuals doing the ranking (voters) and the items to be ranked (alternatives) in a shared metric space, where ordinal preferences are translated into relationships between pairwise distances. Prior work has established that any collection of rankings with $n$ voters and $m$ alternatives (preference profile) can be embedded into $d$-dimensional Euclidean space for $d \geq \min\{n,m-1\}$ under the Euclidean norm and the Manhattan norm. We show that this holds for all $p$-norms and establish that any pair of rankings can be embedded into $R^2$ under arbitrary norms, significantly expanding the reach of spatial preference models.