Smith 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 then 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, known as the Generalized Optimal-Choice Axiom or GOCHA, 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 whose 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 best known Condorcet method which does not satisfy 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;
}

The Smith set can be seen as defining a ranked voting method in which...

All members of the Smith set are winners. Usually combined with another method.[3]

A notable example of Smith's method in combination is Smith/IRV, which reduces the field of candidates to the Smith set and then uses instant-runoff voting to break the tie if this set has more than one element. A more complicated method (also invoking IRV) is Tideman's Alternative, whose article contains a table listing important properties of the two methods. Smith/minimax uses a significantly different procedure for breaking ties.[2]

• Condorcet criterion
• Condorcet method
• Landau set
• Preorder
• Partial order

• Ward, Benjamin (1961). "Majority Rule and Allocation". Journal of Conflict Resolution. 5 (4): 379–389. doi:10.1177/002200276100500405. S2CID 145231466. In an analysis of serial decision making based on majority rule, describes the Smith set and the Schwartz set.
• Smith, J.H. (1973). "Aggregation of Preferences with Variable Electorates". Econometrica. The Econometric Society. 41 (6): 1027–1041. doi:10.2307/1914033. JSTOR 1914033. Introduces a version of a generalized Condorcet Criterion that is satisfied when pairwise elections are based on simple majority choice, and for any dominating set, any candidate in the set is collectively preferred to any candidate not in the set. But Smith does not discuss the idea of a smallest dominating set.
• Fishburn, Peter C. (1977). "Condorcet Social Choice Functions". SIAM Journal on Applied Mathematics. 33 (3): 469–489. doi:10.1137/0133030. Narrows Smith's generalized Condorcet Criterion to the smallest dominating set and calls it Smith's Condorcet Principle.
• Schwartz, Thomas (1986). The Logic of Collective Choice. New York: Columbia University Press. Discusses the Smith set (named GETCHA) and the Schwartz set (named GOTCHA) as possible standards for optimal collective choice.
• Somdeb Lahiri (nd), "Group and multi-criteria decision making". Outlines some properties of choice sets.

• Example algorithms to calculate the Smith set