and C is ranked over A by a 60% majority. 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 it is the only candidate in 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 pairwise ties between members of the Smith set). Complying methods The Smith criterion is satisfied by Ranked Pairs, Schulze's method, Nanson's method, the Robert's Rules method for voting on motions & amendments, and several other methods. Methods failing the Condorcet criterion also fail the Smith criterion. Some Condorcet methods, such as Minimax, also fail the Smith criterion. Voting methods that fail the Smith criterion can be modified to satisfy it (typically at the expense of other criteria). One approach is to apply the voting method to the Smith set only. (In other words, begin by deleting the candidates not in the Smith set from the votes.) For example, the voting method Smith/Minimax is the application of Minimax to the candidates in the Smith set. Another approach is to elect the member of the Smith set that is highest in the voting method's order of finish. Examples Minimax Mutual majority criterion#Minimax The Smith criterion implies the Mutual majority criterion, therefore Minimax's failure to satisfy the Mutual majority criterion is also a failure to satisfy the Smith criterion. Observe that the set S = {A, B, C} in the example is the Smith set and D is the Minimax winner. See also Independence of Smith-dominated alternatives 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. Electoral system criteria