voting system An electoral system or voting system is a set of rules that determine how elections and referendums are conducted and how their results are determined. Electoral systems are used in politics to elect governments, while non-political elections m ...

s, 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 The Smith criterion (sometimes generalized Condorcet criterion, but this can have other meanings) is a voting systems criterion defined such that it's satisfied when a voting system always elects a candidate that is in the Smith set, which is th ...

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 Minimax (sometimes MinMax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for ''mini''mizing the possible loss for a worst case (''max''imum loss) scenario. Whe ...

as Smith/minimax.

Properties of Smith sets

*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 paradox (also known as the voting paradox or the paradox of voting) in social choice theory is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic, even if the prefer ...

. *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.

Properties of dominating sets

''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 such that and But by hypothesis ''d'' defea