# Generalized Condorcet criterion

The **Generalized Condorcet criterion** or **Smith criterion** for a voting system is that it picks the winner from the Smith set, the smallest set of candidates such that every member of the set is preferred to every candidate not in the set. One candidate is preferred over another candidate if, in a one-on-one competition, more voters prefer the first candidate than prefer the other candidate.

Any election method that complies with the Generalized Condorcet 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. Voting methods that pass the Smith criterion are called "Smith-efficient" (or often "Smith-efficient Condorcet methods"). They also pass the mutual majority criterion, as the Smith Set will always be a subset of the mutual majority-preferred set of candidates (each of the mutual majority-preferred candidates pairwise beats all non-mutual majority-preferred candidates by a majority, but some of them may pairwise beat each other, so the Smith Set will at most be all of them.)

## Complying Methods[edit | edit source]

Among methods that comply with the Condorcet criterion, Schulze and Ranked Pairs comply with the Generalized Condorcet Criterion.

Methods that do not comply with the Condorcet criterion, such as Approval voting, Cardinal Ratings, Borda count, Plurality voting, and Instant-Runoff Voting, do not with the Generalized Condorcet Criterion.

Any voting method can be made Smith-efficient by first eliminating everyone outside the Smith set and then running the voting method. An even better approach from the perspective of the Smith set may be to eliminate everyone outside the Smith set, then among the remaining candidates, eliminate the candidate the voting method considers worst, and repeat. See Tideman's Alternative methods#Notes for more information.

## Multi-winner generalizations[edit | edit source]

Schulze has proposed a multi-winner generalization of the Smith criterion which can be roughly described as: "if for a group of candidates equal to or smaller than the number of winners, a certain number of them would always win when up to that certain number of any of them and enough of the candidates not in the group face off such that there are always at most (one more than the number of winners) candidates running, and the same holds for any number smaller than the certain number, then at least that certain number of candidates from the group must be in every winner set in the Smith set, and the voting method must pick one of the Smith winner sets as the final winner set." ^{[1]}

Note that this should be considered as only one of several conditions a voting methods should pass to be considered a Smith-efficient Condorcet PR method, since Bloc Score Voting passes it yet is not proportional or a Condorcet method.

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