Plurality rule and Condorcet criterion over restricted domains

Given a nonempty set of voters and a nonempty set of candidates, we provide a characterization of preference domains over which the plurality rule is Condorcet consistent; these are preference domains over which it is possible to design a social choice function that always chooses a plurality winner and also respects the Condorcet criterion. We take into account the possibility that some voters abstain and that some candidates withdraw freely from the competition, just as in real elections. In the general case with at least five potential voters and at least three potential candidates, we show that being defined on a quasi-cyclic permutation domain is a necessary and sufficient condition for the plurality rule to be Condorcet consistent. As points of discussion, we also investigate on the probability that a Condorcet winner exists in an admissible profile over a cyclic permutation domain as well as the existence of an ideal preference domain on which all scoring rules are Condorcet consistent. On this latter issue, it turns out that only essentially top-trivial domains are left.