Unanimity overruled: Majority voting and the burden of history

Abstract

Sequential majority voting over interconnected binary decisions can lead to the overruling of unanimous consensus. We characterize, within the general framework of judgement aggregation, when this happens for some sequence of decisions. The large class of aggregation spaces for which this vulnerability is present includes the aggregation of preference orderings over at least four alternatives, the aggregation of equivalence relations over at least four objects, resource allocation problems, and most committee selection problems. We also ask whether it is possible to design respect for unanimity by choosing appropriate decision sequences (independently from the ballot). Remarkably, while this is not possible in general, it can be accomplished in some interesting special cases. Generalizing and sharpening a classic result by Shepsle and Weingast, we show that respect for unanimity can indeed be thus guaranteed in the cases of the aggregation of weak preference orderings, linear preference orderings, and equivalence relations. By contrast, impossibility results can be obtained for the aggregation of acyclic relations and separable preference orderings. As a key technical tool, we introduce the notion of a covering fragment that serves as a counterpart and generalization of the notions of covering relation/uncovered set in voting theory.