We develop a geometric approach to identify all possible profiles that support specified votes for separate initiatives or for a bundled bill. This disaggregation allows us to compute the likelihood of different scenarios describing how voters split over the alternatives and to offer new interpretations for pairwise voting. The source of the problems—an unanticipated loss of available information—also explains a variety of other phenomena, such as Simpson’s paradox (a statistical paradox in which the behavior of the “parts” disagrees with that of the “whole”) and Arrow’s theorem from social choice.