The Smith set, sometimes called the top-cycle generalizes the idea of a Condorcet winner to cases where no such winner exists. It does so by allowing cycles of candidates to be treated jointly, as if they were a single Condorcet winner. Voting systems that always elect a candidate from the Smith set pass the Smith criterion. The Smith set and Smith criterion are both named for mathematician John H. Smith.

The Smith set provides one standard of optimal choice for an election outcome. An alternative, stricter criterion is given by the Landau set.

The Smith set is formally defined as the smallest set such that every candidate inside the set S pairwise defeats every candidate outside S.

Alternatively, it can be defined as the set of all candidates with a (non-strict) beatpath to any candidate who defeats them.

A set of candidates each of whose members pairwise defeats every candidate outside the set is known as a dominating set. Thus the Smith set is also called the smallest dominating set.

The Schwartz set is equivalent to the Smith set, except it ignores tied votes. Formally, the Schwartz set is the set such that any candidate inside the set has a strict beatpath to any candidate who defeats them.

The Smith set can be constructed from the Schwartz set by repeatedly adding two types of candidates until no more such candidates exist outside the set:

Note that candidates of the second type can only exist after candidates of the first type have been added.