We analyze the structure of the disagreement among a population of voters over a set of alternatives. Surveys typically ask either for pairwise comparisons, simple and intuitive for participants, or full rankings over alternatives, eliciting the entire voters' preferences. Building on the observation that pairwise comparisons cannot distinguish structural disagreement from noise, we propose a stratified framework to identify the minimal aggregated preference information needed to compute a number of disagreement measures from the literature. Specifically, we introduce the plurality matrix, a generalization of pairwise comparisons that records, for every subset $S$ of alternatives, the probability that each $a \in S$ ranks first in $S$. We define the level of a disagreement measure as the smallest subset size needed to express it, showing that many existing notions, including rank-variance and divisiveness, sit at level $3$, proving that pairwise comparisons are not enough. In addition, we demonstrate the interest of going beyond level $3$ both theoretically and experimentally. To make these results actionable, we design two elicitation protocols to estimate the plurality matrix, exploring the trade-off between the number of required participants and the cognitive load requested to each of them.

翻译：我们分析了一组选民在多个备选方案中分歧的结构。调查通常要求参与者进行两两比较（简单直观）或提供备选方案的完整排名（以揭示选民的完整偏好）。基于两两比较无法区分结构性分歧与噪声这一观察，我们提出一个分层框架，用于识别计算文献中多种分歧度量所需的最小聚合偏好信息。具体而言，我们引入了“多数矩阵”（plurality matrix），这是一种两两比较的推广形式，记录每个备选方案子集$S$中各个$a \in S$在$S$中排名第一的概率。我们将分歧度量的层次定义为表达该度量所需的最小子集规模，证明包括排名方差和分裂性在内的许多现有概念均处于第三层次，说明两两比较不足以捕捉这些分歧。此外，我们从理论和实验两方面展示了超越第三层次的价值。为使这些结果具有可操作性，我们设计了两种收集协议来估计多数矩阵，并探讨了所需参与者人数与每个参与者认知负荷之间的权衡关系。