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Voting Methods that always elect a candidate from the Smith set pass the "Smith criterion," and are said to be "Smith-efficient." Some argue that Smith-efficient methods have the best claim to providing majority rule in multi-candidate elections.

It is clear that a Smith set exists by observing that we can construct a Directed Graph where the vertices are the candidates and there is an edge from A to B if A is pairwise preferred to B. Such a graph equals its own Transitive Closure , since the preference relation is transitive, and its Strongly Connected Component s are Clique s of members which all beat one another. If we contract each of these cliques to a single vertex representing a set of candidates, we have a Directed Acyclic Graph , which necessarily has a vertex with zero in-degree, and the set of candidates this vertex corresponds to is the Smith set.