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An impossibility theorem concerning positive involvement in voting

Wesley H. Holliday

TL;DR

This note establishes an impossibility: no ordinal voting method can satisfy positive involvement together with the Condorcet winner and loser criteria, resolvability, and ordinal margin invariance. The authors formalize profiles with margins, introduce the defensible set $D(\mathbf{P})$ and ordinal margin graphs $\mathbb{M}(\mathbf{P})$, and show that any method meeting positive involvement and the Condorcet criteria refines $D(\mathbf{P})$. Under ordinal margin invariance, such methods would yield a unique winner on linearly edge-ordered tournaments, but combining this with resolvability leads to contradictions, implying the incompatibility of all four axioms with ordinal margin invariance. Consequently, the paper argues that one must drop ordinal margin invariance to potentially construct a method satisfying the remaining properties, marking an open direction for bit-axiomatic design in social choice. The result clarifies why several prominent ordinal-margin methods cannot satisfy all desired axioms and motivates exploring alternatives outside ordinal margin invariance.

Abstract

In social choice theory with ordinal preferences, a voting method satisfies the axiom of positive involvement if adding to a preference profile a voter who ranks an alternative uniquely first cannot cause that alternative to go from winning to losing. In this note, we prove a new impossibility theorem concerning this axiom: there is no ordinal voting method satisfying positive involvement that also satisfies the Condorcet winner and loser criteria, resolvability, and a common invariance property for Condorcet methods, namely that the choice of winners depends only on the ordering of majority margins by size.

An impossibility theorem concerning positive involvement in voting

TL;DR

This note establishes an impossibility: no ordinal voting method can satisfy positive involvement together with the Condorcet winner and loser criteria, resolvability, and ordinal margin invariance. The authors formalize profiles with margins, introduce the defensible set and ordinal margin graphs , and show that any method meeting positive involvement and the Condorcet criteria refines . Under ordinal margin invariance, such methods would yield a unique winner on linearly edge-ordered tournaments, but combining this with resolvability leads to contradictions, implying the incompatibility of all four axioms with ordinal margin invariance. Consequently, the paper argues that one must drop ordinal margin invariance to potentially construct a method satisfying the remaining properties, marking an open direction for bit-axiomatic design in social choice. The result clarifies why several prominent ordinal-margin methods cannot satisfy all desired axioms and motivates exploring alternatives outside ordinal margin invariance.

Abstract

In social choice theory with ordinal preferences, a voting method satisfies the axiom of positive involvement if adding to a preference profile a voter who ranks an alternative uniquely first cannot cause that alternative to go from winning to losing. In this note, we prove a new impossibility theorem concerning this axiom: there is no ordinal voting method satisfying positive involvement that also satisfies the Condorcet winner and loser criteria, resolvability, and a common invariance property for Condorcet methods, namely that the choice of winners depends only on the ordering of majority margins by size.
Paper Structure (3 sections, 3 theorems, 2 equations, 5 figures, 2 tables)

This paper contains 3 sections, 3 theorems, 2 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Impossibility

Key Result

Lemma 2

$\,$

Figures (5)

  • Figure 1: Ordinal margin graphs for the proof of Theorem \ref{['MainThm']}.
  • Figure 2: Profiles realizing $\mathbb{M}_1$ and $\mathbb{M}_2$ and the transition between them in Figure \ref{['RelMarginsFig']}.
  • Figure 3: Profiles realizing $\mathbb{M}_2$ and $\mathbb{M}_3$ and the transition between them in Figure \ref{['RelMarginsFig']}.
  • Figure 4: Profiles realizing $\mathbb{M}_1$ and $\mathbb{M}_4$ and the transition between them in Figure \ref{['RelMarginsFig']}.
  • Figure 5: Profiles realizing $\mathbb{M}_4$ and $\mathbb{M}_5$ and the transition between them in Figure \ref{['RelMarginsFig']}.

Theorems & Definitions (7)

  • Definition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof