The supercovering relation, the pairwise winner, and more missing links between Borda and Condorcet

Among all existing paradoxes of voting, the one pointed out by Condorcet has managed to become known as ‘the’ voting paradox. This relevant paradox states that the transitivity of the voters’ preferences does not imply the transitivity of the collective preference. However, this collective preference disregards a considerable part of the information provided by the voters. Here, we propose a new way of comparing candidates—resulting in the supercovering relation—that, although it might not be complete, avoids the voting paradox and further restricts the ubiquitous covering relation. Thus the pairwise winner emerges, a new natural type of winner situated in between the Condorcet winner and the unanimous winner. This pairwise winner turns out to serve as a cornerstone of social choice theory that finally unites the Borda winner and the Condorcet winner. Moreover, we analyse an interesting superset of the uncovered set—the unsupercovered set—and we propose a method for selecting candidates from this set, resulting in the introduction of the notion of a P-optimal candidate.