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The Squared Kemeny Rule for Averaging Rankings

Patrick Lederer, Dominik Peters, Tomasz Wąs

TL;DR

This work argues that Kemeny’s median-based rank aggregation can overly suppress low-weight criteria in weighted settings. It proposes the Squared Kemeny rule, defined by minimizing the weighted sum of squared swap distances, as a natural mean-like alternative that respects input weights. The authors provide a full axiomatic characterization—showing SqK is the unique SPF satisfying neutrality, reinforcement, continuity, and 2RP—and establish both proportionality guarantees and general performance bounds. They also address computation (NP-hardness with ILP formulations and approximation algorithms) and validate the approach via simulations and a city-ranking case study, highlighting SqK’s practical appeal for proportional and multi-criteria ranking tasks.

Abstract

For the problem of aggregating several rankings into one ranking, Kemeny (1959) proposed two methods: the median rule which selects the ranking with the smallest total swap distance to the input rankings, and the mean rule which minimizes the squared swap distances to the input rankings. The median rule has been extensively studied since and is now known simply as Kemeny's rule. It exhibits majoritarian properties, so for example if more than half of the input rankings are the same, then the output of the rule is the same ranking. We observe that this behavior is undesirable in many rank aggregation settings. For example, when we rank objects by different criteria (quality, price, etc.) and want to aggregate them with specified weights for the criteria, then a criterion with weight 51% should have 51% influence on the output instead of 100%. We show that the Squared Kemeny rule (i.e., the mean rule) behaves this way, by establishing a bound on the distance of the output ranking to any input rankings, as a function of their weights. Furthermore, we give an axiomatic characterization of the Squared Kemeny rule, which mirrors the existing characterization of the Kemeny rule but replaces the majoritarian Condorcet axiom by a proportionality axiom. Finally, we discuss the computation of the rule and show its behavior in a simulation study.

The Squared Kemeny Rule for Averaging Rankings

TL;DR

This work argues that Kemeny’s median-based rank aggregation can overly suppress low-weight criteria in weighted settings. It proposes the Squared Kemeny rule, defined by minimizing the weighted sum of squared swap distances, as a natural mean-like alternative that respects input weights. The authors provide a full axiomatic characterization—showing SqK is the unique SPF satisfying neutrality, reinforcement, continuity, and 2RP—and establish both proportionality guarantees and general performance bounds. They also address computation (NP-hardness with ILP formulations and approximation algorithms) and validate the approach via simulations and a city-ranking case study, highlighting SqK’s practical appeal for proportional and multi-criteria ranking tasks.

Abstract

For the problem of aggregating several rankings into one ranking, Kemeny (1959) proposed two methods: the median rule which selects the ranking with the smallest total swap distance to the input rankings, and the mean rule which minimizes the squared swap distances to the input rankings. The median rule has been extensively studied since and is now known simply as Kemeny's rule. It exhibits majoritarian properties, so for example if more than half of the input rankings are the same, then the output of the rule is the same ranking. We observe that this behavior is undesirable in many rank aggregation settings. For example, when we rank objects by different criteria (quality, price, etc.) and want to aggregate them with specified weights for the criteria, then a criterion with weight 51% should have 51% influence on the output instead of 100%. We show that the Squared Kemeny rule (i.e., the mean rule) behaves this way, by establishing a bound on the distance of the output ranking to any input rankings, as a function of their weights. Furthermore, we give an axiomatic characterization of the Squared Kemeny rule, which mirrors the existing characterization of the Kemeny rule but replaces the majoritarian Condorcet axiom by a proportionality axiom. Finally, we discuss the computation of the rule and show its behavior in a simulation study.
Paper Structure (34 sections, 24 theorems, 21 equations, 12 figures, 2 tables)

This paper contains 34 sections, 24 theorems, 21 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Model
  3. Axiomatic Analysis
  4. 2-Rankings-Proportionality and Single-Crossing Profiles
  5. 2-Rankings-Proportionality.
  6. Single-Crossing-Proportionality.
  7. Characterization of the Squared Kemeny Rule
  8. Neutrality.
  9. Reinforcement.
  10. Continuity.
  11. Efficiency, Participation, and Strategyproofness
  12. Efficiency.
  13. Participation.
  14. Strategyproofness.
  15. Proportionality Guarantees
  16. ...and 19 more sections

Key Result

theorem \ref{thm:characterization}

An SPF satisfies neutrality, reinforcement, continuity, and 2RP if and only if it is the Squared Kemeny rule.

Figures (12)

  • Figure 1: Sort options on booking.com.
  • Figure 2: Squared Kemeny when mixing two criteria.
  • Figure 3: The maximum swap distance (normalized to $[0, 1]$) between the output of the Kemeny or Squared Kemeny rule to an input ranking, as a function of the weight $\alpha$ of the input ranking, for $m = 6$ alternatives.
  • Figure 4: Euclidean profiles with Kemeny (red) and Squared Kemeny (green) ranking locations shown
  • Figure 5: A single-crossing profile, on which Squared Kemeny outputs $\succ_4$ (the average of the voter locations) and Kemeny outputs $\succ_2$ (the median).
  • ...and 7 more figures

Theorems & Definitions (24)

  • theorem \ref{thm:characterization}
  • theorem \ref{thm:characterization}
  • theorem \ref{thm:characterization}
  • theorem \ref{thm:characterization}
  • proposition 1
  • theorem \ref{thm:characterization}
  • theorem \ref{thm:characterization}
  • theorem \ref{thm:characterization}
  • theorem \ref{thm:characterization}
  • lemma 1
  • ...and 14 more