The Smith set, also known as the top cycle, is a concept from the theory of electoral systems that generalizes the Condorcet winner to cases where no such winner exists, by allowing cycles of candidates to be treated jointly as if they were a single Condorcet winner. Named after John H. Smith, the Smith set is the smallest non-empty set of candidates in a particular election, such that each member defeats every candidate outside the set in a pairwise election. The Smith set provides one standard of optimal choice for an election outcome. Voting systems that always elect a candidate from the Smith set pass the Smith criterion.

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.

An alternative generalization of the Condorcet criterion (which is stricter than the Smith criterion) is given by the Landau set.

• The Smith set always exists and is well defined (see next section).
• The Smith set can have more than one candidate, either because of pairwise ties or because of cycles, such as in Condorcet's paradox.
• The Condorcet winner, if one exists, is the sole member of the Smith set. If weak Condorcet winners exist they are in the Smith set.
• The Smith set is always a subset of the mutual majority-preferred set of candidates, if one exists.^[1]

Theorem: Dominating sets are nested; that is, of any two dominating sets in an election, one is a subset of the other.

Proof: Suppose on the contrary that there exist two dominating sets, D and E, neither of which is a subset of the other. Then there must exist candidates d ∈ D, e ∈ E such that d ∉ E and e ∉ D. But by hypothesis d defeats every candidate not in D (including e) while e defeats every candidate not in E (including d), which is a contradiction. ∎

Corollary: It follows that the Smith set is the smallest non-empty dominating set, and that it is well defined.

Theorem: If D is a dominating set, then there is some threshold θ_D such that the elements of D are precisely the candidates whose Copeland scores are at least θ_D. (A candidate's Copeland score is the number of other candidates whom he or she defeats plus half the number of other candidates with whom he or she is tied.)

Proof: Choose d as an element of D with minimum Copeland score, and identify this score with θ_D. Now suppose that some candidate e ∉ D has a Copeland score not less than θ_D. Then since d belongs to D and e doesn't, it follows that d defeats e; and in order for e's Copeland score to be at least equal to d's, there must be some third candidate f against whom e gets a better score than does d. If f ∈ D, then we have an element of D who does not defeat e, and if f ∉ D then we have a candidate outside of D whom d does not defeat, leading to a contradiction either way. ∎

The Schwartz set is closely related to and is always a subset of the Smith set. The Smith set is larger if and only if a candidate in the Schwartz set has a pair-wise tie with a candidate that is not in the Schwartz set.

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:

• candidates that have pairwise ties with candidates in the set,
• candidates that defeat a candidate in the set.

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

The Smith criterion is satisfied by any voting method where the winner always belongs to the Smith set. Any method which satisfies the Smith criterion must also satisfy the Condorcet criterion; hence any method (such as IRV) which is not Condorcet-consistent must also fail the Smith criterion. Minimax is the most well-known Condorcet method that fails the Smith criterion.

The logical properties of dominating sets stated above can be used to construct an efficient algorithm for computing the Smith set. We have seen that the dominating sets are nested by Copeland score. It follows that by adjusting the Copeland threshold it is possible to work through the nested sets in increasing order of size until a dominating set is reached; and this set is necessarily the Smith set. Darlington sketches this method.^[2]

Testing whether a set is a dominating set at each stage might repeat some calculations, but this can fairly easily be avoided leading to an algorithm whose work factor is quadratic in the number of candidates.

int smithset(int ** r, int * s, int n) {
  int row, col, lhs, rhs;
  for (rhs = 1, lhs = 0; lhs < rhs; lhs = rhs, rhs = row + 1) {
    for (; rhs < n && s[rhs] == s[rhs - 1]; rhs++); /* this line optional */
    for (col = rhs, row = n; col == rhs && row >= rhs; row--)
      for (col = lhs; col < rhs && r[row - 1][col] == 0; col++);
  }
  return lhs;
}

• Condorcet criterion
• Condorcet method
• Landau set
• Preorder
• Partial order
• Maximal and minimal elements - the Smith set can be defined as the maximal elements of a particular partial order.