  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.
