The Voter Basis and the Admissibility of Tree Characters

When making simultaneous decisions, our preference for the outcomes on one subset can depend on the outcomes on a disjoint subset. In referendum elections, this gives rise to the separability problem, where a voter must predict the outcome of one proposal when casting their vote on another. A set $S \subset [n]$ is separable for preference order $\succeq$ when our ranking of outcomes on $S$ is independent of outcomes on its complement $[n]-S$. The admissibility problem asks which characters $\mathcal{C} \subset \mathcal{P}([n])$ can arise as the collection of separable subsets for some preference order. We introduce a linear algebraic technique to construct preference orders with desired characters. Each vector in our $2^n$-dimensional voter basis induces a simple preference ordering with nice separability properties. Given any collection $\mathcal{C} \subset \mathcal{P}([n])$ whose subset lattice has a tree structure, we use the voter basis to construct a preference order with character $\mathcal{C}$.