Towards a Switching-Algebraic Theory of Weighted Monotone Voting Systems: The Case of Banzhaf Voting Indices

TL;DR

This paper develops a switching-algebraic framework for analyzing weighted monotone voting systems, focusing on computing the Banzhaf voting index using Boolean techniques and establishing connections with game theory and system reliability.

Contribution

It introduces novel Boolean-based symmetry-aware methods for calculating the Banzhaf index in weighted voting systems, advancing the theoretical foundation of switching-algebraic voting analysis.

Findings

New Boolean symmetry-aware techniques for Banzhaf index computation

Application of switching algebra to weighted monotone voting systems

Establishment of links between voting theory, switching algebra, and reliability analysis

Abstract

This paper provides a serious attempt towards constructing a switching-algebraic theory for weighted monotone voting systems, whether they are scalar-weighted or vector-weighted. The paper concentrates on the computation of a prominent index of voting powers, viz., the Banzhaf voting index. This computation involves two distinct operations: (a) either Boolean differencing (Boolean differentiation) or Boolean quotient construction (Boolean restriction), and (b) computation of the weight (the number of true vectors or minterms) of a switching function. We introduce novel Boolean-based symmetry-aware techniques for computing the Banzhaf index by way of four voting systems. The paper finally outlines further steps needed towards the establishment of a full-fledged switching-algebraic theory of weighted monotone voting systems. Througout the paper, a tutorial flavour is retained, multiple solutions of consistent results are given, and a liasion is established among game-theoretic voting theory, switching algebra, and sytem reliability analysis.