Efficient Elicitation of Collective Disagreements

This paper focuses on how to measure disagreements in a population that faces alternative proposals, be them political candidates, policy issues, or projects to be funded. Disagreement measures require increasing levels of information to be elicited from individuals, from one single pairwise comparison to complete rankings over the alternatives. In this paper, we propose a stratified framework that classifies disagreement measures according to this information, allowing us in turn to design efficient elicitation protocols for their computation. Specifically, we introduce the plurality matrix, an aggregated preference representation 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 compute it. We prove that rank variance and divisiveness require exactly level $3$ and that the hierarchy is strict, with skewness and excess kurtosis at levels $4$ and $5$, respectively. Electoral data illustrate the contrasting rank distributions that these higher moments describe. To make our results practically actionable, we design two elicitation protocols to estimate the various levels of the plurality matrix, exploring the trade-off between the number of required participants and the cognitive load requested to each of them.