Single-peaked Compatible Preference Profiles: Some Combinatorial Results

This paper examines combinatorial problems in connection with single-peaked preference orderings on a unidimensional scale. A binary relation, dominance, is defined on the set of connected orderings. Relevant properties of the corresponding poset and cover graph are discussed. A formula for the number of pairs of connected orderings consistent with spatial single-peakedness is derived. The total number of such pairs is shown to be expressible in a simple form involving binomial probabilities. A possible application is a coalition formation process of the kind examined by Brams et al. (J Theor Polit 14:359–383, 2002), where actors have single-peaked preferences on a common scale.