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.

@article{holliday2025_2401.05657,
  title={ An impossibility theorem concerning positive involvement in voting },
  author={ Wesley H. Holliday },
  journal={arXiv preprint arXiv:2401.05657},
  year={ 2025 }
}