Switching-Algebraic Calculation of Banzhaf Voting Indices

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 on- going 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.

Received: 13 February 2023 | Revised: 5 April 2023 | Accepted: 19 May 2023

Conflicts of Interest

The authors declare that they have no conflicts of interest to this work.

Data Availability Statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study.