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.

翻译：在序数偏好的社会选择理论背景下，我们定义可辩护集为这样的备选方案集合：对于集合中的任何备选方案$x$，若存在另一备选方案$y$在双头多数比较中击败$x$，则必存在第三个备选方案$z$在双头多数比较中以至少等于$y$击败$x$的差额击败$y$。我们证明，任何满足投票理论中两个已知公理——积极参与性和孔多塞胜者准则——的序数投票方法都对可辩护集构成精炼。基于这一引理，我们证明了一个不可能性定理：不存在同时满足孔多塞败者准则、可解性以及孔多塞方法通用不变性（即胜者选择仅取决于多数差额的相对大小）的此类投票方法。