The incompatibility of the Condorcet winner and loser criteria with positive involvement and resolvability

We prove that there is no preferential voting method satisfying the Condorcet winner and loser criteria, positive involvement (if a candidate $x$ wins in an initial preference profile, then adding a voter who ranks $x$ uniquely first cannot cause $x$ to lose), and $n$-voter resolvability (if $x$ initially ties for winning, then $x$ can be made the unique winner by adding some set of up to $n$ voters). This impossibility theorem holds for any positive integer $n$. It also holds if either the Condorcet loser criterion is replaced by independence of clones or positive involvement is replaced by negative involvement.

💡 Research Summary

The paper establishes a new impossibility theorem in the theory of preferential voting. It shows that no voting rule can simultaneously satisfy four natural criteria: the Condorcet winner criterion (CWC), the Condorcet loser criterion (CLC), positive involvement (PI), and n‑voter resolvability (Rₙ) for any positive integer n. The authors also demonstrate that the same incompatibility holds if the CLC is replaced by independence of clones, or if PI is replaced by negative involvement.

The formal framework defines a voting method F as a function that maps every finite profile of strict weak orders over a finite set of candidates to a non‑empty set of winners. The four axioms are precisely stated: CWC requires that a candidate beating every other candidate in pairwise margins be the unique winner; CLC excludes a candidate who loses to every other candidate; PI demands that adding a single voter who ranks a current winner uniquely first cannot make that candidate lose; Rₙ requires that whenever there is a tie among winners, each tied candidate can be made the unique winner by adding at most n new voters.

The core of the proof is a concrete construction involving five candidates a, b, c, d, e and a series of profiles P₁,…,P₅. Profile P₁ is designed so that candidate a is the only “defensible” candidate (a candidate that cannot be beaten in a specific margin‑dominance sense). Using Lemma 3 (originally due to Moulin and Pérez), which links defensibility to the winner set under CWC and PI, the authors infer that F(P₁)={a}. They then modify the profile step‑by‑step: adding voters who rank a, d, b, and finally d uniquely first, while also using resolvability to insert “empty” or single‑voter profiles S₂ and S₄ that preserve uniqueness of the current winner. At each stage the margin graph is carefully kept such that the differences between any two edges are at least 4, satisfying the separation condition required by Lemma 3.

Through repeated application of PI, the winner a survives the first modification, and resolvability guarantees a profile S₂ that keeps a as the unique winner. After adding d‑first voters, PI forces d out of the winner set, but the margin structure now makes b the only defensible candidate, so Lemma 3 forces F(P₃+S₂)={b}. A similar argument with S₄ makes b survive the next modification, and finally the addition of d‑first voters forces d to become the sole defensible candidate, leading Lemma 3 to conclude F(P₅+S₂+S₄)={d}. This chain yields a contradiction because the same method would have to output both a and d as the unique winner under the same axioms.

The authors generalize the construction for any n by replicating each voter n times, which multiplies all margins by n and preserves the required separation, thereby proving that no rule can satisfy CWC, CLC, PI, and Rₙ for any n. They also show that replacing CLC with independence of clones, or PI with negative involvement, yields the same impossibility. Moreover, weaker variants such as singleton positive involvement, quasi‑resoluteness, or “r‑resolvability” (where the number of added voters is bounded proportionally to the original electorate) are examined; the proof adapts to show that even these relaxed notions cannot coexist with the other three axioms.

In summary, the paper demonstrates a fundamental incompatibility between candidate‑centric fairness (Condorcet criteria), voter‑centric monotonicity (positive/negative involvement), and a strong tie‑breaking capability (n‑voter resolvability). This result adds a new dimension to classic impossibility theorems (Arrow, Gibbard‑Satterthwaite) by highlighting that attempts to design voting systems that are simultaneously Condorcet‑compatible, monotone with respect to new voters, and able to resolve ties with a bounded number of additional votes are mathematically impossible.