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

Wesley H. Holliday

TL;DR

The paper proves an impossibility: no preferential voting method can satisfy the Condorcet winner/loser criteria, positive involvement, and resolvability simultaneously on five-candidate elections. It builds on the margin-graph framework and the defensible-set concept, leveraging a Moulin-Perez style lemma to constrain outcomes and then constructs a chain of profiles to force a contradiction under the axioms. The result sharpens our understanding of the incompatibilities among desirable voting properties and shows that removing ordinal margin invariance is sufficient to obtain the impossibility in this domain. The findings guide future work toward relaxing one axiom or restricting the domain to avoid the conflict, impacting theoretical and practical design of voting rules.

Abstract

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 resolvability (if $x$ initially ties for winning, then $x$ can be made the unique winner by adding a single voter). In a previous note, we proved an analogous result assuming an additional axiom of ordinal margin invariance, which we now show is unnecessary for an impossibility theorem, at least if the desired voting method is defined for five-candidate elections.

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

TL;DR

The paper proves an impossibility: no preferential voting method can satisfy the Condorcet winner/loser criteria, positive involvement, and resolvability simultaneously on five-candidate elections. It builds on the margin-graph framework and the defensible-set concept, leveraging a Moulin-Perez style lemma to constrain outcomes and then constructs a chain of profiles to force a contradiction under the axioms. The result sharpens our understanding of the incompatibilities among desirable voting properties and shows that removing ordinal margin invariance is sufficient to obtain the impossibility in this domain. The findings guide future work toward relaxing one axiom or restricting the domain to avoid the conflict, impacting theoretical and practical design of voting rules.

Abstract

We prove that there is no preferential voting method satisfying the Condorcet winner and loser criteria, positive involvement (if a candidate wins in an initial preference profile, then adding a voter who ranks uniquely first cannot cause to lose), and resolvability (if initially ties for winning, then can be made the unique winner by adding a single voter). In a previous note, we proved an analogous result assuming an additional axiom of ordinal margin invariance, which we now show is unnecessary for an impossibility theorem, at least if the desired voting method is defined for five-candidate elections.
Paper Structure (3 sections, 2 theorems, 2 figures)

This paper contains 3 sections, 2 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Setup
  3. Result

Key Result

Lemma 3

Let $F$ be a voting method satisfying positive involvement and the Condorcet winner criterion. Let $\mathbf{P}$ be a profile in which for all $x,x',y,y'\in X(\mathbf{P})$ with $x\neq x'$ and $y\neq y'$, if $\text{Margin}_\mathbf{P}(x,x')>\text{Margin}_\mathbf{P}(y,y')$, then $\text{Margin}_\mathbf{P

Figures (2)

  • Figure 1: the profile $\mathbf{P}_1$ used in the proof of Theorem \ref{['MainThm']}
  • Figure 2: margin graphs used in the proof of Theorem \ref{['MainThm']} with defensible candidates shaded gray

Theorems & Definitions (6)

  • Definition 1
  • Definition 2
  • Lemma 3: Moulin1988, Perez1995
  • proof
  • Theorem 4
  • proof