On the Weighted Top-Difference Distance: Axioms, Aggregation, and Approximation

TL;DR

This work introduces and axiomatizes a broad family of top-difference ranking distances, $d^oldsymbol{eta}_oldsymbol{eta}$, parameterized by a menu-weight measure $oldsymbol{eta}$ and a weighting measure $oldsymbol{ ext{μ}}$ over subsets, unifying Kendall-type metrics with top-difference variants. It provides a full axiomatic characterization, clarifies neutrality conditions, and analyzes social-choice properties when these distances induce median voting rules, including majority, Pareto, and Condorcet implications. Computational contributions include a polynomial-time reformulation, generalized Diaconis-Graham inequalities for approximation, and a PTAS based on truncation to top positions, enabling scalable rank aggregation under a flexible, top-weighted framework. The framework subsumes known distances as special cases and yields tractable algorithms for exact and approximate consensus rankings, with extensions to nonneutral settings and potential for strategy-proofness and preference elicitation studies. Overall, the paper advances both theory and computation for distance-based rank aggregation with top-weight emphasis, offering practical tools for fair and efficient consensus construction in voting, recommendations, and information retrieval contexts.

Abstract

We study a family of distance functions on rankings that allow for asymmetric treatments of alternatives and consider the distinct relevance of the top and bottom positions for ordered lists. We provide a full axiomatic characterization of our distance. In doing so, we retrieve new characterizations of existing axioms and show how to effectively weaken them for our purposes. This analysis highlights the generality of our distance as it embeds many (semi)metrics previously proposed in the literature. Subsequently, we show that, notwithstanding its level of generality, our distance is still readily applicable. We apply it to preference aggregation, studying the features of the associated median voting rule. It is shown how the derived preference function satisfies many desirable features in the context of voting rules, ranging from fairness to majority and Pareto-related properties. We show how to compute consensus rankings exactly, and provide generalized Diaconis-Graham inequalities that can be leveraged to obtain approximation algorithms. Finally, we propose some truncation ideas for our distances inspired by Lu and Boutilier (2010). These can be leveraged to devise a Polynomial-Time-Approximation Scheme for the corresponding rank aggregation problem.

Figures (2)

• Figure 1: The permutahedron on $[4]$.
• Figure 2: Here we illustrate the three permutations in the case with $a<a'$ and $b>b'$.

Theorems & Definitions (99)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Theorem 5
• Remark 6
• Theorem 7
• Theorem 8
• Theorem 9
• Proposition 10