Smith set

In voting systems, the Smith set, named after John H. Smith, but also known as the top cycle, or as Generalized Top-Choice Assumption (GETCHA), is the smallest non-empty set of candidates in a particular election such that each member defeats every candidate outside the set in a pairwise election. The Smith set provides one standard of optimal choice for an election outcome. Voting systems that always elect a candidate from the Smith set pass the Smith criterion and are said to be 'Smith-efficient' or to satisfy the Smith criterion. A set of candidates each of whose members pairwise defeats every candidate outside the set is known as a dominating set. The Smith set can be seen as defining a voting method (Smith's method) which is most often encountered when extended by an IRV tie-break as Smith/IRV or as Tideman's Alternative, or by minimax as Smith/minimax. The Smith set always exists and is well defined (see next section). The Smith set can have more than one candidate, either because of pairwise ties or because of cycles, such as in Condorcet's paradox. The Condorcet winner, if one exists, is the sole member of the Smith set. If weak Condorcet winners exist then they are in the Smith set. The Smith set is always a subset of the mutual majority-preferred set of candidates, if one exists. Theorem: Dominating sets are nested; that is, of any two dominating sets in an election, one is a subset of the other. Proof: Suppose on the contrary that there exist two dominating sets, D and E, neither of which is a subset of the other. Then there must exist candidates d ∈ D, e ∈ E such that d ∉ E and e ∉ D. But by hypothesis d defeats every candidate not in D (including e) while e defeats every candidate not in E (including d), which is a contradiction. ∎ Corollary: It follows that the Smith set is the smallest non-empty dominating set, and that it is well defined. Theorem: If D is a dominating set, then there is some threshold θD such that the elements of D are precisely the candidates whose Copeland scores are at least θD.