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Voting criterion requiring a kind of majority rule
The Smith criterion (sometimes the generalized Condorcet criterion) is a voting system criterion that formalizes the concept of a majority rule. A voting system satisfies the Smith criterion if it always elects a candidate from the Smith set, which generalizes the idea of a "Condorcet winner" to cases where there may be cycles or ties, by allowing for several who together can be thought of as being "Condorcet winners." A Smith method will always elect a candidate from the Smith set. The Smith criterion is also called the top (Condorcet) cycle criterion, but this can be slightly misleading: the Smith set sometimes consists of a degenerate cycle with only one candidate who "cycles" with themselves (a Condorcet winner), or a pair of tied candidates who "cycle" with each other. An alternative, stricter criterion is given by the Landau set.

The Smith set can be calculated with the Floyd–Warshall algorithm in time Θ(n3) or Kosaraju's algorithm in time Θ(n2).

Any election method that complies with the Smith criterion also complies with the Condorcet winner criterion, since if there is a Condorcet winner, then it is the only candidate in the Smith set. Smith methods also comply with the Condorcet loser criterion, because a Condorcet loser will never fall in the Smith set. It also implies the mutual majority criterion, since the Smith set is a subset of the MMC set. The Smith set and Schwartz set are sometimes confused in the literature. Miller (1977, p. 775) lists as an alternate name for the Smith set, but it actually refers to the Schwartz set. The Schwartz set is actually a subset of the Smith set (and equal to it if there are no pairwise ties between members of the Smith set).

The Smith criterion is satisfied by Ranked Pairs, Schulze's method, Nanson's method, and several other methods. Moreover, any voting method can be modified to satisfy the Smith criterion, by finding the Smith set and eliminating any candidates outside of it. For example, the voting method Smith//Minimax applies Minimax to the candidates in the Smith set. Another approach is to elect the member of the Smith set that is highest in the voting method's order of finish. Methods failing the Condorcet criterion also fail the Smith criterion. However, some Condorcet methods (such as Minimax) can fail the Smith criterion.