Smith criterion

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.

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

Example
When there is a Condorcet winner—a candidate that is majority-preferred over all other candidates—the Smith set consists of only that candidate. Here is an example in which there is no Condorcet winner: There are four candidates: A, B, C and D. 40% of the voters rank D>A>B>C. 35% of the voters rank B>C>A>D. 25% of the voters rank C>A>B>D. The Smith set is {A,B,C}. All three candidates in the Smith set are majority-preferred over D (since 60% rank each of them over D). The Smith set is not {A,B,C,D} because the definition calls for the smallest subset that meets the other conditions. The Smith set is not {B,C} because B is not majority-preferred over A; 65% rank A over B. (Etc.)

In this example, under minimax, A and D tie; under Smith//Minimax, A wins.

In the example above, the three candidates in the Smith set are in a "rock/paper/scissors" majority cycle: A is ranked over B by a 65% majority, B is ranked over C by a 75% majority, and C is ranked over A by a 60% majority.

Other criteria
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).

Complying methods
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.

Minimax

 * Mutual majority criterion

The Smith criterion implies the mutual majority criterion, so Minimax's failure to satisfy the Mutual majority criterion is also a failure to satisfy the Smith criterion. Observe that the set S = {A, B, C} in the example is the Smith set and D is the Minimax winner.