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.
If one candidate is preferred over each of the other candidates, that candidate is the Ideal Democratic Winner (IDW).
Statement of Criterion
If all votes are sincere, the Ideal Democratic Winner should win if one exists.
Complying Methods
The Condorcet method complies with the Condorcet Criterion, but none of the other methods in the compliance table above comply.
Commentary
The Condorcet criterion is one of the most basic criteria for election methods. When an Ideal Democratic Winner exists, common sense tells us that ideally he or she should win. However, the only method listed in Table 1 that complies with the Condorcet criterion is the Condorcet method itself, which is designed specifically to comply with the criterion named after it.
Non-ranking methods such as Plurality and Approval could not possibly comply with the Condorcet Criterion because they do not allow each voter to fully specify their preferences. But IRV allows each voter to rank the candidates, yet it still does not comply. A simple example will prove that IRV fails to comply with the Condorcet Criterion.
Consider, for example, the following vote count with three candidates {A,B,C}:
In this case, B is preferred to A by 12 votes to 8, and B is preferred to C by 13 to 7, hence B is preferred to both A and C. So according to common sense and the Condorcet criteria, B should win. But under IRV, B does not win. According to the rules of IRV, B is ranked first by the fewest voters and is eliminated. Again, an election method that allows such nonsensical anomalies should be rejected. (See The Problem with IRV.)