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

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. It reveals useful to divide the aggregation problem into several sub-problems. We conclude with numerical tests.

Preference Aggregation in the Generalised Unavailable Candidate Model

“…Lemma 3 allows us to apply Corollary 4 to subsets of C. We can then show Theorem 3 which proves that MyopicTop algorithm is a PTAS for the ex post dissatisfaction rule when the normalised DF are bounded 11 . Note that here parameter p is assumed to be fixed and independent on n and m. This differs from the assumptions made to establish the NP-hardness result in Theorem 2.…”

“…We now provide some theoretical directions of research. First of all, we are interested in studying other probability distributions (including non impartial ones) than Bernoulli and thereby emphasising the fact that ex post model generalises voting rules studied in [11,2,15]. Other aggregators than the sum could enable us to include fairness considerations [16].…”

“…We will see in the following that both ex ante and ex post models are interestingly linked to several well known voting rules. Note that the ex post approach is also considered in [2,11] and, in particular, by Lu and Boutilier in [15] as the following proposition shows. Proposition 3.…”

“…The optimal rankings are computed by minimisation of the expected number of disagreements over all the possible subsets of available candidates. In [15], the probability distribution on these subsets is supposed to follow a Bernoulli law whereas several other authors [12,2,11] used other distributions. Lu and Boutilier provide a clear decision-theoretic justification for producing a ranking instead of a single winner.…”

“…Indeed, a ranking serves as a very compact decision policy: the winner is the output ranking's best candidate among the available ones. Yet, the binary disagreement used in [15,12,2,11] relies on strong hypotheses that can be discussed, as acknowledged by the authors: a voter is satisfied only if its favourite available candidate is elected (as in "plurality" rule) and fully unsatisfied otherwise.…”