The Voter Basis and the Admissibility of Tree Characters

Abstract

Let ≽ be a total order on the power set of a finite set [n]. A subset S ⊂ [n] is separable when for any X,Y ⊂ S and any Z ⊂ [n] − S, the ordering of X and Y is the same as the ordering of X ∪ Z and Y ∪ Z. The character of a preference order is the collection of all separable subsets. Motivated by questions in the theories of voting, marketing and social choice, the admissibility problem asks which collections $\mathcal {C} \subset \mathcal {P}({[n]})$ can arise as characters of preference orders. We introduce a linear algebraic technique to construct preference orders. Each vector in our 2ⁿ-dimensional voter basis induces a simple preference preorder (where ties are allowed) with nice separability properties. Given any collection $\mathcal {C} \subset \mathcal {P}({[n]})$ that contains both ∅ and [n], and such that all pairs of subsets are either nested or disjoint, we use the voter basis to construct a preference order with character $\mathcal {C}$.