Switching-Algebraic Calculation of Banzhaf Voting Indices

TL;DR

This paper introduces switching-algebraic methods for calculating the Banzhaf voting power index, simplifying the process through symmetry-aware techniques and Boolean operations, especially for systems with partial symmetry.

Contribution

It presents novel Boolean-based techniques for efficient Banzhaf index computation in voting systems, extending switching-algebraic approaches to voting theory.

Findings

Simplified calculation of Banzhaf index using switching algebra.

Effective handling of symmetric and partially symmetric voting functions.

Application to scalar voting systems with six and nine variables.

Abstract

This paper employs switching-algebraic techniques for the calculation of a fundamental index of voting powers, namely, the total Banzhaf power. This calculation involves two distinct operations: (a) Boolean differencing or differentiation, and (b) computation of the weight (the number of true vectors or minterms) of a switching function. Both operations can be considerably simplified and facilitated if the pertinent switching function is symmetric or it is expressed in a disjoint sum-of-products form. We provide a tutorial exposition on how to implement these two operations, with a stress on situations in which partial symmetry is observed among certain subsets of a set of arguments. We introduce novel Boolean-based symmetry-aware techniques for computing the Banzhaf index by way of two prominent voting systems. These are scalar systems involving six variables and nine variables, respectively. The paper is a part of our ongoing effort for transforming the methodologies and concepts of voting systems to the switching-algebraic domain, and subsequently utilizing switching-algebraic tools in the calculation of pertinent quantities in voting theory.