The Smith criterion 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. 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 can be calculated with the Floyd–Warshall algorithm in time Θ or Kosaraju's algorithm in time Θ. 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. All three candidates in the Smith set are majority-preferred over D. The Smith set is not because the definition calls for the smallest subset that meets the other conditions. The Smith set is not because B is not majority-preferred over A; 65% rank A over B.
pro\con
A
B
C
D
A
—
65
40
60
B
35
—
75
60
C
60
25
—
60
D
40
40
40
—
max opp
60
65
75
60
minimax
60
60
In this example, under minimax, A and D tie; under Smith/Minimax, A wins. The Smith set is also called the top cycle. 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. 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, and when the Smith set consists only of two alternatives that tie pairwise, the two do not cycle with any alternatives.
Other criteria
Any election method that complies with the Smith criterion also complies with the Condorcet criterion, since if there is a Condorcet winner, then it is the only candidate in the Smith set. Obviously, this means that failing the Condorcet criterion automatically implies the non-compliance with the Smith criterion as well. Additionally, such sets comply with the Condorcet loser criterion. This is notable, because even some Condorcet methods do not. 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 lists GOCHA 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.
Complying methods
The Smith criterion is satisfied by Ranked Pairs, Schulze's method, Nanson's method, the Robert's Rules method for voting on motions & amendments, and several other methods. Methods failing the Condorcet criterion also fail the Smith criterion. Some Condorcet methods, such as Minimax, also fail the Smith criterion. Voting methods that fail the Smith criterion can be modified to satisfy it. One approach is to apply the voting method to the Smith set only. For example, the voting method Smith/Minimax is the application of 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.
Examples
Minimax
The Smith criterion implies the Mutual majority criterion, therefore Minimax's failure to satisfy the Mutual majority criterion is also a failure to satisfy the Smith criterion. Observe that the set S = in the example is the Smith set and D is the Minimax winner.
