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

If a measure of voting power assigns greater voting power to a player because it no longer effectively cooperates with another, then the measure displays the quarrelling paradox and violates the quarrel postulate. We provide formal criteria by which to judge whether a given conception of quarrelling is (a) reasonable and (b) fit to serve as the basis for a reasonable quarrel postulate. To achieve this, we formalize a general framework distinguishing between three degrees of quarrelling (weak, strong, cataclysmic), symmetric vs. asymmetrical quarrels, and reciprocal vs. non-reciprocal quarrels, and which thereby yields twelve conceptions of quarrelling, which encompasses the two conceptions proposed by Felsenthal and Machover and by Laruelle and Valenciano, respectively. We argue that the two existing formulations of the quarrel postulate based on these conceptions are unreasonable. In contrast, we prove that the symmetric, weak conception of quarrelling identified by our framework -- whether reciprocal or not -- is fit to serve as the basis for a reasonable quarrel postulate. Furthermore, the classic Shapley-Shubik index and Penrose-Banzhaf measure both satisfy the quarrel postulate based on a symmetric weak quarrel.