Existence of a voting method with positive involvement, Condorcet winner/loser criteria, and resolvability without ordinal margin invariance

Determine whether there exists an ordinal voting method that simultaneously satisfies positive involvement, the Condorcet winner criterion, the Condorcet loser criterion, and resolvability, without imposing ordinal margin invariance; alternatively, establish an impossibility theorem proving that no such method exists.

Background

The paper proves that no ordinal voting method can satisfy the combination of positive involvement, the Condorcet winner and loser criteria, resolvability, and ordinal margin invariance. This rules out a broad class of Condorcet-style methods that depend only on the ordering of majority margins.

Given the impossibility with ordinal margin invariance, the authors suggest dropping this invariance requirement while retaining the other axioms (positive involvement, Condorcet winner/loser criteria, resolvability). The central open question is whether such a method exists or whether a new impossibility result also precludes this combination of axioms.

References

Thus, in the search for the "holy grail" of a voting method satisfying positive involvement, the Condorcet winner and loser criteria, and resolvability, we must drop the restriction of ordinal margin invariance. Whether such a method exists or another impossibility theorem awaits us is an important open question.

— An impossibility theorem concerning positive involvement in voting (2401.05657 - Holliday, 11 Jan 2024) in Section 3 (Impossibility), concluding paragraph after Theorem 4