Is the preference of the majority representative?

Given a profile of preferences on a set of alternatives, a majoritarian relation is a complete binary relation that agrees with the strict preference of a strict majority of these preferences whenever such strict strict majority is observed. We show that a majoritarian relation is, among all conceivable binary relations, the most representative of the profile of preferences from which it emanates. We define "the most representative" to mean "the closest in the aggregate". This requires a definition of what it means for a pair of preferences to be closer to each other then another. We assume that this definition takes the form of a distance function defined over the set of all conceivable preferences. We identify a necessary and sufficient condition for such a distance to be minimized by a majoritarian relation. This condition requires the distance to be additive with respect to a plausible notion of compromise between preferences. The well-known Kemeny distance between preferences does satisfy this property. We also provide a characterization of the class of distances satisfying this property as numerical representations of a primitive qualitative proximity relation between preferences.