Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Real Preferences Under Arbitrary Norms

Joshua Zeitlin, Corinna Coupette

TL;DR

Real Preferences Under Arbitrary Norms studies when a preference profile over $m$ alternatives by $n$ voters admits a rank-preserving embedding into a normed space. It extends rank-embeddability from Euclidean and Manhattan norms to all $p$-norms using two constructive approaches: an alternative-rank embedding yielding a $(\mathbb{R}^n, \|\cdot\|_p)$ realization for $d\ge n$, and a median-based embedding yielding a $(\mathbb{R}^{m-1}, \|\cdot\|_p)$ realization for $d\ge m-1$, with $p=1$ treated via a dedicated max-rank construction. A norm-independent result shows that two voter-type profiles can be realized in $(\mathbb{R}^2, \|\cdot\|)$ for any norm, together with a geometric lemma that underpins the higher-level constructions. The work culminates in a conjecture that, in general, rank-embeddability should hold for any norm in dimensions $d\ge\min\{n,m-1\}$, outlining avenues for extending to polynomial norms and exploring tightness and higher-dimensional generalizations. By broadening the scope of spatial-preference modeling beyond Euclidean settings, the paper lays groundwork for norm-aware design choices in voting, facility location, and recommender systems.

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.

Real Preferences Under Arbitrary Norms

TL;DR

Real Preferences Under Arbitrary Norms studies when a preference profile over alternatives by voters admits a rank-preserving embedding into a normed space. It extends rank-embeddability from Euclidean and Manhattan norms to all -norms using two constructive approaches: an alternative-rank embedding yielding a realization for , and a median-based embedding yielding a realization for , with treated via a dedicated max-rank construction. A norm-independent result shows that two voter-type profiles can be realized in for any norm, together with a geometric lemma that underpins the higher-level constructions. The work culminates in a conjecture that, in general, rank-embeddability should hold for any norm in dimensions , outlining avenues for extending to polynomial norms and exploring tightness and higher-dimensional generalizations. By broadening the scope of spatial-preference modeling beyond Euclidean settings, the paper lays groundwork for norm-aware design choices in voting, facility location, and recommender systems.

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 voters and alternatives (preference profile) can be embedded into -dimensional Euclidean space for under the Euclidean norm and the Manhattan norm. We show that this holds for all -norms and establish that any pair of rankings can be embedded into under arbitrary norms, significantly expanding the reach of spatial preference models.

Paper Structure

This paper contains 6 sections, 9 theorems, 53 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Rank-Preserving Embeddings under TEXT-Norms
  3. Required Dimensionality Depending on the Number of Voters
  4. Dimensionality Depending on the Number of Alternatives
  5. Norm-Independent Rank-Preserving Embeddings into TEXT
  6. Discussion and Conclusion

Key Result

Theorem 1

Given $m$ alternatives $A$ and $n$ voters $V$ with preferences over these alternatives, a preference profile $\mathcal{P}_{A,V}$ rank-embeds into $(\mathbb{R}^d,\|\cdot\|_p)$, for all $1\leq p\leq \infty$, if $d\geq \min\{n,m-1\}$.

Figures (4)

  • Figure 1: Rank-preserving AR embedding of the preference profile presented in \ref{['ex1']}, using $c = 10$. The line segments connecting the voters to the alternatives are colored from dark to light in decreasing order of their length, highlighting that the preferences are in fact ordered by distances (which we can also verify arithmetically).
  • Figure 2: In $\mathbb{R}^2$, the pairs of balls with equal radii under the $3$-norm around $\mathbf{e}_1 = (1,0)$ (blue) and $\mathbf{e}_2 = (0,1)$ (green) intersect on a line (red).
  • Figure 3: Median-based embedding of the alternatives for $m = 3$, with the corresponding $2$-dimensional hyperplane shaded in blue and the medians between any pair of alternatives drawn as dotted red lines. With this setup, for example, voters with preference $a_3\succ a_2 \succ a_1$ can be placed in the red-shaded region of the hyperplane.
  • Figure 4: Geometric intuition for \ref{['lemma3']}. Let the unit ball of our norm be given by a symmetric convex body like the green ellipse centered around $\mathbf{v}_2$. Given points $\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}_3$ in $\mathbb{R}^2$, with $\mathbf{p}_1,\mathbf{p}_2$ already placed in the plane with $\mathbf{v}_1$ and $\mathbf{v}_2$, we seek to place $\mathbf{p}_3$ such that $\mathbf{v}_1$ ranks $\mathbf{p}_3$ last but $\mathbf{v}_2$ ranks $\mathbf{p}_3$ between $\mathbf{p}_1$ and $\mathbf{p}_2$. Here, the red arcs represent the balls around $\mathbf{v}_2$ that touch $\mathbf{p}_1$ and $\mathbf{p}_2$, creating the relevant annulus around $\mathbf{v}_2$. To ensure that $\mathbf{p}_3$ lies between $\mathbf{p}_1$ and $\mathbf{p}_2$ in the ranking of $\mathbf{v}_2$, we need to place the point inside this annulus. However, we also need to guarantee that $\mathbf{p}_3$ is ranked last for $\mathbf{v}_1$. As demonstrated in the drawing, we can achieve this by placing $\mathbf{p}_3$ outside the blue balls, yet between the red arcs. \ref{['lemma3']} ensures that such a point always exists for any norm.

Theorems & Definitions (24)

  • Definition 1: Norm
  • Definition 2: $p$-Norm
  • Definition 3: Rank-preserving embedding, rank embeddability
  • Theorem 1: Rank embeddability under $p$-norms
  • Definition 4: Alternative-Rank Embedding [AR Embedding]
  • Example 1
  • Proposition 1: Dimensionality depending on $n$
  • proof
  • Proposition 2: Asymptotics of $c$ as a function of $\frac{1}{p-1}$
  • proof
  • ...and 14 more