Kemeny's rule is one of the most studied and well-known voting schemes with various important applications in computational social choice and biology. Recently, Kemeny's rule was generalized via a set-wise approach by Gilbert et. al. This paradigm presents interesting advantages in comparison with Kemeny's rule since not only pairwise comparisons but also the discordance between the winners of subsets of three alternatives are also taken into account in the definition of the $3$-wise Kendall-tau distance between two rankings. In spite of the NP-hardness of the 3-wise Kemeny problem which consists of computing the set of $3$-wise consensus rankings, namely rankings whose total $3$-wise Kendall-tau distance to a given voting profile is minimized, we establish in this paper several generalizations of the Major Order Theorems, as obtained by Milosz and Hamel for Kemeny's rule, for the $3$-wise Kemeny voting schemes to achieve a substantial search space reduction by efficiently determining in polynomial time the relative orders of pairs of alternatives. Essentially, our theorems quantify precisely the nontrivial property that if the preference for an alternative over another one in an election is strong enough, not only in the head-to-head competition but even when taking into account one or two more alternatives, then the relative order of these two alternatives in all $3$-wise consensus rankings must be as expected. As an application, we also obtain an improvement of the Major Order Theorems for Kememy's rule. Moreover, we show that the well-known $3/4$-majority rule of Betzler et al. for Kemeny's rule is only valid in general for elections with no more than $5$ alternatives with respect to the $3$-wise Kemeny scheme. Several simulations and tests of our algorithms on real-world and uniform data are provided.

The important Kemeny problem, which consists of computing median consensus rankings of an election with respect to the Kemeny voting rule, admits important applications in biology and computational social choice and was generalized recently via an interesting setwise approach by Gilbert et. al. Our first results establish optimal quantitative extensions of the Unanimity property and the well-known $3/4$-majority rule of Betzler et al. for the classical Kemeny median problem. Moreover, by elaborating an exhaustive list of quantified axiomatic properties (such as the Condorcet and Smith criteria, the $5/6$-majority rule, etc.) of the $3$-wise Kemeny rule where not only pairwise comparisons but also the discordance between the winners of subsets of three candidates are also taken into account, we come to the conclusion that the $3$-wise Kemeny voting scheme induced by the $3$-wise Kendall-tau distance presents interesting advantages in comparison with the classical Kemeny rule. For example, it satisfies several improved manipulation-proof properties. Since the $3$-wise Kemeny problem is NP-hard, our results also provide some of the first useful space reduction techniques by determining the relative orders of pairs of alternatives. Our works suggest similar interesting properties of higher setwise Kemeny voting schemes which justify and compensate for the more expensive computational cost than the classical Kemeny scheme.

Rank aggregation aims at producing a single ranking from a co llection of rankings of a fixed set of alternatives. In social choice theory (e.g., Moulin 1991), where the alternatives are candidates to an election and each ranking represents the preferences o f a voter, aggregation rules are called Social Welfare Functions (SWFs). Apart from social choice, rank aggregation has prov ed useful in many applications, including preference learning (Cheng a nd H ullermeier, 2009; Cl emen con et al., 2018), collaborative filtering (Wang et al., 2014), genetic map creation (Jackson et al., 2008), similarity search in databases systems (Fagin et al., 2003) and design of web search engines (Altman and Tennenholtz, 2008; Dwork et al., 2001). In the fo llowing, we use interchangeably the terms "input rankings" and "preferences", "output rank ing" and "consensus ranking", as well as "alternatives" and "'candidates". The well-known Arrow's impossibility theorem states that t here exists no aggregation rule satisfying a small set of desirable properties (Arrow, 1950). In the absense of an "ideal" rule, various aggregation rules have been proposed and studied. F ollowing Fishburn's classification (1977), we can distinguish between the SWFs for which the out put ranking can be computed from the majority graph alone, those for which the output ranking can be computed fro m the 1 Table 1: Results of setwise contests in Example 1. set c

We study rank aggregation algorithms that take as input the opinions of players over their peers, represented as rankings, and output a social ordering of the players (which reflects, e.g., relative contribution to a project or fit for a job). To prevent strategic behavior, these algorithms must be impartial, i.e., players should not be able to influence their own position in the output ranking. We design several randomized algorithms that are impartial and closely emulate given (non-impartial) rank aggregation rules in a rigorous sense. Experimental results further support the efficacy and practicability of our algorithms.

We explore the relationship between two approaches to rationalizing voting rules: the maximum likelihood estimation (MLE) framework originally suggested by Condorcet and recently studied by Conitzer, Rognlie, and Xia, and the distance rationalizability (DR) framework of Elkind, Faliszewski, and Slinko. The former views voting as an attempt to reconstruct the correct ordering of the candidates given noisy estimates (i.e., votes), while the latter explains voting as search for the nearest consensus outcome. We provide conditions under which an MLE interpretation of a voting rule coincides with its DR interpretation, and classify a number of classic voting rules, such as Kemeny, Plurality, Borda and Single Transferable Vote (STV), according to how well they fit each of these frameworks. The classification we obtain is more precise than the ones that result from using MLE or DR alone: indeed, we show that the MLE approach can be used to guide our search for a more refined notion of distance rationalizability and vice versa.