Beyond Pairwise Comparisons in Social Choice: A Setwise Kemeny Aggregation Problem

TL;DR

This paper extends the Kemeny aggregation rule to setwise contests, proposing a generalized approach that minimizes k-wise disagreements, introduces new algorithms and graph structures, and demonstrates improved computation and approximation methods.

Contribution

It introduces a setwise Kemeny rule, develops algorithms and graph tools for efficient aggregation, and provides approximation guarantees and empirical validation.

Findings

Setwise Kemeny rule effectively generalizes pairwise aggregation.

Major graph-based decomposition speeds up exact computation.

A 2-approximation algorithm is developed with numerical validation.

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.