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The paper offers a reassessment of canonical attempts to address a fundamental question about majority rule: what is the relationship between the preferences held by the participants and the outcomes that emerge from their interactions? Previous work, based on the analysis of abstract spatial models or relying on data from real-world spatial experiments, has yielded a mass of contradictory findings. Our work applies a new technique for estimating the uncovered set, a concept that describes a fundamental constraint on majority rule: given the preferences of decision makers, which outcomes can emerge from majority-rule decision making? By applying the uncovered set to a series of previous experiments on majority rule, we show that their seemingly bizarre and incompatible findings are in fact consistent with a clearly specified theory of how sophisticated individuals make decisions in majority-rule settings.