Definitions
A sincere vote is one with no falsified preferences or preferences
left unspecified when the election method allows them to be specified
(in addition to the preferences already specified).
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.
The Smith set is the smallest set of candidates such that every
member of the set is preferred to every candidate not in the set. If the
Smith set consists of only one candidate, that candidate is the Ideal
Democratic Winner (IDW).
Statement of Criterion
If all votes are sincere, the winner should be a member of the Smith set.
Complying Methods
The Condorcet method complies with the Generalized Condorcet Criterion, but none of the other methods in the compliance table above comply.
Commentary
GCC generalizes the Condorcet Criterion (CC) to the case in which no Ideal Democratic Winner (IDW) exists, thereby covering all possible cases. If no IDW exists, then a cyclical ambiguity exists among the members of the Smith set, and that ambiguity must be resolved in such a way that the winner comes from that set. The commentary for CC above applies here also.