Classifying interdependence in multidimensional binary preferences

When individual preferences over multiple dimensions are interdependent, the resulting collective decisions can be unsatisfactory and even paradoxical. The notion of separability formalizes this idea of interdependence, and preferences that are completely free from interdependence are said to be separable. In this paper, we develop a mechanism for classifying preferences according to the extent to which they achieve or fail to achieve the desirable property of separability. We show that binary preferences over multiple dimensions are surprisingly complex, in that their interdependence structures defy the most natural attempts at characterization. We also extend previous results pertaining to the rarity of separable preferences by showing that the probability of complete nonseparability approaches 1 as the number of dimensions increases without bound.