[Paper Review] Beyond Pairwise Comparisons in Social Choice: A Setwise Kemeny Aggregation Problem

Hugo Gilbert, Tom Portoleau+1 more|arXiv (Cornell University)|Nov 14, 2019

Game Theory and Voting Systems36 references4 citations

TL;DR

This paper introduces a k-wise generalization of the Kemeny rule for rank aggregation, where consensus rankings minimize disagreements on top choices across all subsets of alternatives of size at most k. It proposes a 2-approximation algorithm with practical performance far better than the theoretical worst case, and develops a k-wise majority graph to enable efficient preprocessing and exact computation for up to 18 candidates.

ABSTRACT

In this paper, we advocate the use of setwise contests for aggregating a set of input rankings into an output ranking. We propose a generalization of the Kemeny rule where one minimizes the number of k-wise disagreements instead of pairwise disagreements (one counts 1 disagreement each time the top choice in a subset of alternatives of cardinality at most k differs between an input ranking and the output ranking). After an algorithmic study of this k-wise Kemeny aggregation problem, we introduce a k-wise counterpart of the majority graph. This graph reveals useful to divide the aggregation problem into several sub-problems, which enables to speed up the exact computation of a consensus ranking. By introducing a k-wise counterpart of the Spearman distance, we also provide a 2-approximation algorithm for the k-wise Kemeny aggregation problem. We conclude with numerical tests.

Motivation & Objective

• To address the limitation of pairwise comparison-based aggregation rules in capturing collective preferences beyond binary comparisons.
• To formalize a new class of social welfare functions based on setwise contests (plurality scores over subsets of candidates).
• To generalize the Kemeny rule to k-wise disagreements, enabling more nuanced consensus ranking beyond pairwise comparisons.
• To develop efficient algorithms for exact and approximate computation of the k-wise Kemeny consensus ranking.
• To evaluate the practical performance of the proposed 2-approximation algorithm and its dependence on k and preference correlation.

Proposed method

• Define a k-wise disagreement count as the number of subsets S of size at most k where the top choice in S differs between an input ranking and the output ranking.
• Introduce a k-wise variant of the majority graph, where edges represent the relative plurality support for candidates in subsets of size ≤k.
• Develop a dynamic programming approach to compute the k-wise Kemeny score exactly, leveraging the k-wise majority graph for preprocessing.
• Propose a 2-approximation algorithm based on minimizing a k-wise variant of the Spearman footrule distance.
• Use the Diaconis-Graham inequality to derive the theoretical worst-case approximation ratio of 2.
• Implement and evaluate the algorithm on synthetic data generated via the Mallows model, using SciPy's linear sum assignment for efficiency.

Experimental results

Research questions

• RQ1 Can a k-wise generalization of the Kemeny rule improve consensus ranking by capturing collective preferences beyond pairwise comparisons?
• RQ2 What is the computational complexity of computing a consensus ranking under the k-wise Kemeny rule for k ≥3?
• RQ3 How can a k-wise majority graph be constructed and used to speed up exact computation of the k-wise Kemeny consensus?
• RQ4 What is the practical approximation ratio of the proposed 2-approximation algorithm, and how does it vary with k and preference correlation?
• RQ5 Can the k-wise Kemeny rule resolve tensions between Condorcet consistency and plurality support, as illustrated in real-world preference profiles?

Key findings

• The k-wise Kemeny aggregation problem is NP-hard for any k ≥3, extending the known hardness of the standard Kemeny rule.
• The proposed 2-approximation algorithm guarantees a worst-case approximation ratio of 2, derived from a k-wise variant of the Spearman distance and the Diaconis-Graham inequality.
• In practice, the approximation ratio is consistently much better than 2, with the worst observed ratio being 1.04 across 50 test instances.
• The practical performance improves with higher preference correlation (lower φ in the Mallows model) and larger k, as shown in box plots of the approximation ratio.
• The algorithm runs in under 12 milliseconds for 50 voters and 12 candidates using standard libraries, demonstrating strong practical efficiency.
• The k-wise majority graph enables effective preprocessing and reduces the search space, making exact computation feasible for up to 18 candidates.