Real Preferences Under Arbitrary Norms

2025-08-12

AI Summary

This paper addresses the problem of embedding ordinal preferences—such as ranked ballots—into metric spaces endowed with numerical utilities, to support applications in voting analysis, facility location, and recommender systems. We propose a general framework for embedding into $mathbb{R}^d$ equipped with arbitrary $p$-norms ($p geq 1$). Our method constructs embeddings that preserve pairwise Kendall–Tau distances between rankings. We establish three key theoretical results: (i) any set of ordinal rankings admits an isometric embedding into Euclidean space ($ell_2$); (ii) for any two rankings, a two-dimensional isometric embedding always exists; and (iii) more generally, $n$ voters and $m$ alternatives can be isometrically embedded into $mathbb{R}^d$ for any $d geq min{n, m-1}$. These results significantly extend prior work restricted to $ell_1$ or $ell_2$ norms, unifying and broadening the theoretical foundations and applicability of spatial preference models.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Knowledge Representation and Reasoning: Preferences
• Game Theory and Economic Paradigms: Social Choice / Voting

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Translating ordinal rankings into numerical utilities under norms
• Embedding preference profiles into metric spaces using p-norms
• Extending spatial preference models to arbitrary distance norms

Innovation

• Extends embedding to all p-norms universally
• Enables 2D representation under arbitrary norms
• Generalizes spatial preference model applicability