The Voter Basis and the Admissibility of Tree Characters

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 𝒞⊂𝒫([n]) can arise as characters of preference orders. We introduce a linear algebraic technique to construct preference orders. Each vector in our 2 n -dimensional voter basis induces a simple preference preorder (where ties are allowed) with nice separability properties. Given any collection 𝒞⊂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 𝒞 .