Unanimity Overruled: Majority Voting and the Burden of History

Sequential majority voting over interconnected binary propositions can lead to the overruling of unanimous consensus. We characterize, within the general framework of judgement aggregation, under what circumstances this happens for some sequence of the voting process. It turns out that the class of aggregation spaces for which this difficulty arises is very large, including 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. Remarkably, while this is not possible in general, it can be accomplished in interesting special cases. Adapting and generalizing a classic result by Shepsle and Weingast, we show that respect for unanimity can indeed be thus guaranteed in case of the aggregation of weak orderings, strict orderings and equivalence relations. A well-known problem from the theory of preference aggregation is that sequential pairwise majority voting can yield Pareto-dominated alternatives, provided that there are at least four alternatives (Moulin, 1988, p.242). Within the general framework of judgement aggregation this is an instance of the fact that sequential majority voting over interconnected propositions can lead to the situation that a previous majority decision forces the rejection of a proposition that is unanimously accepted (see Section 1.5 below for examples). In this paper, we characterize the aggregation spaces on which sequential majority voting always respects unanimous consent, no matter in which sequence the majority decisions are taken and no matter what the individual judgements are. We shall call such spaces robustly unanimity consistent. It turns out that the robustly unanimity consistent spaces represent a simple and natural generalization of the median spaces which play a distinguished role in the theory of aggregation (see Barthélémy and Monjardet (1981), Nehring ∗Department of Economics, University of California at Davis, U.S.A. [email protected]. †Department of Mathematics, Trent University, Canada. [email protected] ‡Department of Economics and Business Engineering, Karlsruhe Institute of Technology, Germany. [email protected]