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

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.