On Banzhaf and Shapley-Shubik Fixed Points and Divisor Voting Systems

TL;DR

This paper characterizes fixed points of Banzhaf and Shapley-Shubik power indices in weighted voting systems, providing algebraic conditions, classifying specific fixed points, and analyzing differences in divisor voting systems of abundant numbers.

Contribution

It offers a complete characterization of fixed points for the Shapley-Shubik index and algebraic criteria for Banzhaf fixed points, advancing understanding of power indices in voting systems.

Findings

Characterized all fixed points of Shapley-Shubik index of the form (a,b,...,b)

Derived algebraic equations to verify Banzhaf fixed points

Proved differences between Banzhaf and Shapley-Shubik indices in divisor voting systems of abundant numbers

Abstract

The Banzhaf and Shapley-Shubik power indices were first introduced to measure the power of voters in a weighted voting system. Given a weighted voting system, the fixed point of such a system is found by continually reassigning each voter's weight with its power index until the system can no longer be changed by the operation. We characterize all fixed points under the Shapley-Shubik power index of the form $(a,b,\ldots,b)$ and give an algebraic equation which can verify in principle whether a point of this form is fixed for Banzhaf; we also generate Shapley-Shubik fixed classes of the form $(a,a,b,\ldots,b)$. We also investigate the indices of divisor voting systems of abundant numbers and prove that the Banzhaf and Shapley-Shubik indices differ for some cases.