# 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 the smallest non-empty subset of the candidates such that every candidate in the subset is majority-preferred over every candidate not in the subset. (A candidate X is said to be majority-preferred over another candidate Y if, in a one-on-one competition between X & Y, the number of voters who prefer X over Y exceeds the number of voters who prefer Y over X.) [1] The Smith set is named for mathematician John H Smith, whose version of the Condorcet criterion is actually stronger than that defined above for social welfare functions. Benjamin Ward was probably the first to write about this set, which he called the "majority set".

The Smith set is also called the top cycle.[2] The term top cycle may be somewhat misleading, however, since the Smith set can contain candidates that do not cycle. For examples, when there is a Condorcet winner it doesn't cycle with any alternatives,[3] and when the Smith set consists only of two alternatives that tie pairwise, the two do not cycle with any alternatives.