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

TL;DR

This paper develops a comprehensive framework for analyzing quarrels in voting power, showing that certain conceptions align with normative criteria while others do not, thereby clarifying the quarrel paradox.

Contribution

It formalizes a general framework for quarrels in voting power, classifies twelve conceptions, and evaluates their suitability for normative postulates, addressing limitations of existing models.

Findings

Symmetric, weak conception satisfies the quarrel postulate.

Existing conceptions by Felsenthal-Machover and Laruelle-Valenciano do not.

Framework helps distinguish suitable from unsuitable quarrel conceptions.

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.