Smith criterion

The Smith criterion (sometimes generalized Condorcet criterion, but this can have other meanings) is a voting systems criterion defined such that its satisfaction by a voting system occurs when the system always picks the winner from the Smith set, the smallest set of candidates such that every member of the set is pairwise preferred to every candidate not in the set. One candidate is pairwise preferred over another candidate if, in a one-on-one competition, more voters prefer the first candidate than prefer the other candidate. The Smith set is named for mathematician John H Smith, whose version of the Condorcet criterionref|Smith is actually stronger than that defined above for social welfare functions. Benjamin Wardref|Ward was probably the first to write about this set, which he called the "majority set".

The Smith set can be calculated with the Floyd-Warshall algorithm in time Θ("n"³).

Other criteria

Any election method that complies with the Smith criterion also complies with the Condorcet criterion, since if there is a Condorcet winner, then that winner is the only member of the Smith set. Obviously, this means that failing the Condorcet criterion automatically implies the non-compliance with the Smith criterion as well. Additionally, such sets comply with the Condorcet loser criterion. This is notable, because even some Condorcet methods do not (Minimax). It also implies the mutual majority criterion, since the Smith set is a subset of the MMC set.

The Smith set and Schwartz set are sometimes confused in the literature. Miller (1977, p. 775) lists GOCHA as an alternate name for the Smith set, but it actually refers to the Schwartz set. The Schwartz set is actually a subset of the Smith set (and equal to it if there are no ties between members of the Smith set).

Complying methods

Schulze, Nanson's method, and Ranked Pairs comply with the Smith Criterion. Compliance with the Smith criterion can be "forced" by explicitly applying a voting system to the Smith set. For example, Smith/Minimax is the application of Minimax to only the candidates inside the Smith set.

Methods failing the Condorcet criterion never comply with the Smith criterion, and many Condorcet methods also fail the Smith criterion.

References

#J. H. Smith, "Aggregation of preferences with variable electorate", "Econometrica", vol. 41, pp. 1027–1041, 1973.
#Benjamin Ward, "Majority Rule and Allocation", "The Journal of Conflict Resolution", Vol. 5, No. 4. (1961), pp. 379–389.