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.

翻译：重要的Kemeny问题（即基于Kemeny投票规则计算选举中位共识排序）在生物学与计算社会选择领域具有重要应用，且近期由Gilbert等人通过一种有趣的集合式方法进行了推广。我们的首要结果建立了经典Kemeny中位数问题中一致同意性质与著名的Betzler等人3/4多数规则的最优量化扩展。此外，通过系统阐述3-元Kemeny规则的量化公理性质（包括孔多塞准则、Smith准则、5/6多数规则等）——该规则不仅考虑成对比较，还纳入了三个候选子集胜者间的不一致性——我们得出结论：由3-元Kendall-tau距离导出的3-元Kemeny投票方案相较于经典Kemeny规则具有显著优势。例如，该方案满足若干改进的防操纵性质。由于3-元Kemeny问题属于NP困难问题，我们的结果还通过确定备选方案对的相对排序，首次提供了若干实用的空间约简技术。本研究揭示了高阶集合式Kemeny投票方案的类似有趣性质，这些性质证明并弥补了其相较于经典Kemeny方案更高的计算成本。