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,YS and any Z ⊂ [n] − S, the ordering of X and Y is the same as the ordering of XZ and YZ. 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 \(\mathcal {C}\).

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Acknowledgements

We thank Tom Halverson for many insightful conversations and his help in developing the voter basis; Jeremy Martin for suggesting Lemma 3; and Trung Nguyen and Tuyet-Anh Tran for their careful readings of earlier drafts. We also thank the anonymous referees for their feedback, which has significantly improved the exposition.

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Correspondence to Andrew Beveridge.

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Beveridge, A., Calaway, I. The Voter Basis and the Admissibility of Tree Characters. Order (2021). https://doi.org/10.1007/s11083-021-09553-8

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Keywords

  • Preference order
  • Separability
  • Admissibility
  • Poset
  • Referendum elections