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A General Framework for a Class of Quarrels: The Quarrelling Paradox Revisited

Arash Abizadeh, Adrian Vetta

TL;DR

The paper develops a general, symmetry- and reciprocity-aware framework for quarrelling in binary voting games, introducing Cooperative-Success-Reduction (CSR) as the core criterion and distinguishing transformation monotonicity from non-monotonicity. It shows that only a symmetric weak quarrel can serve as a valid basis for a normative quarrel postulate, with Shapley-Shubik and Penrose-Banzhaf satisfying this postulate, while stronger or cataclysmic quarrels can increase some players’ absolute power and enable strategic sabotage. By classifying twelve quarrel conceptions (reciprocal/non-reciprocal, weak/strong/cataclysmic, symmetric), the framework clarifies when quarrelling is a meaningful normative test versus a descriptive vehicle for strategic interaction. The results highlight both normative guidance for voting-power measures and the potential political significance of quarrelling as a tool for internal sabotage or strategic coalition formation.

Abstract

If a measure of voting power assigns players greater voting power because they no longer effectively cooperate, then it displays the quarrelling paradox and violates the quarrel postulate. However, we prove that certain types of quarrel increase some quarrellers' voting power on any proposed measure. On the one hand, such quarrels are politically significant because they incentivize players to strategically join coalitions in order to sabotage them from within; on the other, a postulate based on them cannot provide a reasonable normative criterion for evaluating measures of voting power. We therefore formalize a general framework of quarrels -- comprising twelve conceptions distinguished according to symmetry, reciprocality, and strength -- and provide criteria for whether a conception provides a suitable basis for a reasonable quarrel postulate. Although the two existing conceptions, proposed by Felsenthal and Machover and by Laruelle and Valenciano, do not, our framework's symmetric, weak conception does.

A General Framework for a Class of Quarrels: The Quarrelling Paradox Revisited

TL;DR

The paper develops a general, symmetry- and reciprocity-aware framework for quarrelling in binary voting games, introducing Cooperative-Success-Reduction (CSR) as the core criterion and distinguishing transformation monotonicity from non-monotonicity. It shows that only a symmetric weak quarrel can serve as a valid basis for a normative quarrel postulate, with Shapley-Shubik and Penrose-Banzhaf satisfying this postulate, while stronger or cataclysmic quarrels can increase some players’ absolute power and enable strategic sabotage. By classifying twelve quarrel conceptions (reciprocal/non-reciprocal, weak/strong/cataclysmic, symmetric), the framework clarifies when quarrelling is a meaningful normative test versus a descriptive vehicle for strategic interaction. The results highlight both normative guidance for voting-power measures and the potential political significance of quarrelling as a tool for internal sabotage or strategic coalition formation.

Abstract

If a measure of voting power assigns players greater voting power because they no longer effectively cooperate, then it displays the quarrelling paradox and violates the quarrel postulate. However, we prove that certain types of quarrel increase some quarrellers' voting power on any proposed measure. On the one hand, such quarrels are politically significant because they incentivize players to strategically join coalitions in order to sabotage them from within; on the other, a postulate based on them cannot provide a reasonable normative criterion for evaluating measures of voting power. We therefore formalize a general framework of quarrels -- comprising twelve conceptions distinguished according to symmetry, reciprocality, and strength -- and provide criteria for whether a conception provides a suitable basis for a reasonable quarrel postulate. Although the two existing conceptions, proposed by Felsenthal and Machover and by Laruelle and Valenciano, do not, our framework's symmetric, weak conception does.
Paper Structure (28 sections, 33 theorems, 15 equations, 1 table)

This paper contains 28 sections, 33 theorems, 15 equations, 1 table.

Table of Contents

  1. Introduction
  2. Voting Games
  3. What is a Reasonable Conception of a Quarrel?
  4. Reducing Effective Cooperation
  5. Three Quarrel Properties
  6. Cooperative-Success-Reduction (CSR)
  7. Symmetry
  8. Reciprocality
  9. Reasonable Conceptions of Quarrelling
  10. What is a Reasonable Quarrel Postulate?
  11. Transformation Monotonicity
  12. Symmetry
  13. Felsenthal and Machover's Conception
  14. Laruelle and Valenciano's Conception
  15. A General Quarrel Framework
  16. ...and 13 more sections

Key Result

Theorem 4.1

If a reasonable conception of quarrelling $\mathcal{Q}$ (which satisfies CSR) is disposed to induce non-monotonicity over quarrellers, then the standard quarrel postulate based on $\mathcal{Q}$ will be violated by any measure of voting power $\Psi$.

Theorems & Definitions (49)

  • Theorem 4.1
  • Theorem 4.3
  • Theorem 5.1
  • Theorem 6.1
  • Theorem 6.2
  • Theorem 8.1
  • Theorem 8.2
  • Corollary 8.3
  • Theorem 8.4
  • Theorem 8.5
  • ...and 39 more