In voting systems, the Smith set is the smallest set of candidates in a particular election who, when paired off in pairwise elections, can beat all other candidates outside the set. Ideally, this set consists of only one candidate, the Condorcet winner. However, when the electorate is conflicted (as in Condorcet's paradox), the set has at least one cycle of candidates for whom A beats B, B beats C, and C beats A. See also Schwartz set, Directed Graph.
In voting systems, the Smithset is the smallest non-empty set of candidates in a particular election such that each member beats every other candidate outside the set in a pairwise election.
Voting systems that always elect a candidate from the Smithset pass the Smith criterion and are said to be "Smith-efficient".
Except for this case, the Smithset will almost always have either one member (a clear winner) or a cycle of three members (A beats B, B beats C, and C beats A, and all of them beat everyone else).
The Smithset is the smallest non-empty set of candidates such that every candidate in the set beats every candidate outside of the set in a pair-wise contest.
The Smithset is defined for any election in which every candidate is evaluated against every other candidate in pair-wise contests that each result either in a tie or with one of candidates beating the other.
The Schwartz set is equal to the Smithset except when there is a candidate in the Schwartz set that has a pair-wise tie with a candidate outside of the Schwartz set.
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