Towards a Switching-Algebraic Theory of Weighted Voting Systems: Exploring Restrictions on Coalition Formation

TL;DR

This paper develops a switching-algebraic framework for computing Banzhaf indices and related measures in weighted voting systems, including restricted coalition formations and non-monotone systems, with formulas and illustrative examples.

Contribution

It introduces new switching-algebraic formulas for Banzhaf indices and related measures applicable to both monotone and non-monotone voting systems, including restricted coalition scenarios.

Findings

Derived Boolean-quotient formulas for non-monotone systems.

Provided formulas for Banzhaf-related indices like PII and PPI.

Illustrated formulas with four voting system examples.

Abstract

We explore the switching-algebraic computation of the Banzhaf indices for general and monotone or unrestricted systems. This computation is achieved via (a) two Boolean-quotient formulas that are valid when the voting system is not necessarily monotone (e.g., when coalition formation is restricted), (b) four Boolean differencing formulas and six Boolean-quotient formulas that are applicable when the decision switching function is a positively polarized unate one. We also provide switching-algebraic formulas for certain Banzhaf-related indices, including the power-to-initiate index (PII), and the power-to-prevent index (PPI), as well as satisfaction indices. Moreover, we briefly address other Banzhaf-related indices, including the Strict Power Index (SPI) and the Public Good Index (PGI). We illustrate the various indices formulas by way of four examples of voting systems, each considered first as an unrestricted monotone system and then subjected to a restriction on the formation of a coalition between two particular voters.