Beyond Pairwise Comparisons in Social Choice: A Setwise Kemeny Aggregation Problem
This addresses a social choice problem for decision-making systems, offering an incremental improvement by extending pairwise methods to setwise comparisons.
The paper tackles the problem of aggregating multiple rankings by generalizing the Kemeny rule to minimize k-wise disagreements instead of pairwise ones, resulting in algorithms that speed up exact computation and provide a 2-approximation solution.
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.