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. Following this paradigm, we have shown in \cite{Phung-Hamel-2023} 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. While the $3$-wise Kemeny problem, which consists of computing the set of $3$-wise consensus rankings of a voting profile, is NP-hard, we establish in this paper several generalizations of the Major Order Theorems, as obtained in \cite{Milosz-Hamel-2020} for the classical Kemeny rule, for the $3$-wise Kemeny voting scheme 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 non-trivial 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 consideration one or two more alternatives, then the relative order of these two alternatives in every $3$-wise consensus ranking must be as expected. Moreover, we show that the well-known $3/4$-majority rule of Betzler et al. for the classical Kemeny rule is only valid for elections with no more than $5$ alternatives with respect to the $3$-wise Kemeny scheme. Examples are also provided to show that the $3$-wise Kemeny rule is more resistant to manipulation than the classical one.

翻译：Kemeny规则是计算社会选择与生物学中应用最广泛、研究最深入的投票方案之一，具有多种重要应用。近年来，Gilbert等人通过集合论方法对Kemeny规则进行了推广。遵循这一范式，我们在文献\cite{Phung-Hamel-2023}中证明，基于三向Kendall-tau距离的三向Kemeny投票方案相较于经典Kemeny规则展现出显著优势。尽管计算投票档案的三向共识排序集（即三向Kemeny问题）属于NP难问题，本文仍建立了若干Major Order定理的推广形式——这些定理最初由文献\cite{Milosz-Hamel-2020}针对经典Kemeny规则提出——从而通过多项式时间内高效确定候选对间的相对顺序，实现搜索空间的大幅缩减。本质上，我们的定理精确量化了如下非平凡性质：当选举中某一候选相对于另一候选的偏好强度足够显著时（不仅体现在两两对决中，更需考虑引入一至两个额外候选的情形），则这两个候选在任何三向共识排序中的相对顺序必然符合预期。此外，我们证明Betzler等人针对经典Kemeny规则提出的著名3/4多数规则，对于三向Kemeny方案仅适用于不超过5个候选的选举。文中还提供示例表明，三向Kemeny规则比经典规则具有更强的抗操纵性。