Weighted voting systems: A threshold- Boolean perspective

Abstract

Weighted voting systems play a crucial role in the investigation and modeling of manyengineering structures and political and socio-economic phenomena. There is an urgentneed to describe these systems in a simplified powerful mathematical way that can begeneralized to systems of any size. An elegant description of voting systems is presentedin terms of threshold Boolean functions. This description benefits considerably fromthe wealth of information about these functions, and of the potpourri of algebraic andmap techniques for handling them. The paper demonstrates that the prime implicantsof the system threshold function are its Minimal Winning Coalitions (MWC). Thepaper discusses the Boolean derivative (Boolean difference) of the system thresholdfunction with respect to each of its member components. The prime implicants of thisBoolean difference can be used to deduce the winning coalitions (WC) in which thepertinent member cannot be dispensed with. Each of the minterms of this Booleandifference is a winning coalition in which this member plays a pivotal role. However,the coalition ceases to be winning if the member defects from it. Hence, the numberof these minterms is identified as the Banzhaf index of voting power. The conceptsintroduced are illustrated with detailed demonstrative examples that also exhibit someof the known paradoxes of voting- system theory. Finally, the paper stresses the utilityof threshold Boolean functions in the understanding, study, analysis, and design ofweighted voting systems irrespective of size.