The defensible set and a new impossibility theorem in voting

In the context of social choice theory with ordinal preferences, we say that the defensible set is the set of alternatives x such that for any alternative y, if y beats x in a head-to-head majority comparison, then there is an alternative z that beats y in a head-to-head majority comparison by a margin at least as large as the margin by which y beat x. We show that any ordinal voting method satisfying two well-known axioms from voting theory–positive involvement and the Condorcet winner criterion–refines the defensible set. Using this lemma, we prove an impossibility theorem: there is no such voting method that also satisfies the Condorcet loser criterion, resolvability, and a common invariance property for Condorcet methods, namely that the choice of winners depends only on the relative sizes of majority margins.