Generalized roll-call model for the Shapley-Shubik index

TL;DR

This paper extends the roll-call model for the Shapley-Shubik index by providing a combinatorial proof, relaxing assumptions, and generalizing to multiple alternatives, enhancing understanding of voting power in decision-making.

Contribution

It offers a simplified combinatorial proof, weakens model assumptions, and generalizes the model to more than two alternatives, broadening the applicability of the Shapley-Shubik index.

Findings

Probabilistic pivotal voter coincides with the Shapley-Shubik index

Model assumptions are weakened for broader applicability

Generalization to multiple alternatives is achieved

Abstract

In 1996 Dan Felsenthal and Mosh\'e Machover considered the following model. An assembly consisting of $n$ voters exercises roll-call. All $n!$ possible orders in which the voters may be called are assumed to be equiprobable. The votes of each voter are independent with expectation $0<p<1$ for an individual vote {\lq\lq}yea{\rq\rq}. For a given decision rule $v$ the \emph{pivotal} voter in a roll-call is the one whose vote finally decides the aggregated outcome. It turned out that the probability to be pivotal is equivalent to the Shapley-Shubik index. Here we give an easy combinatorial proof of this coincidence, further weaken the assumptions of the underlying model, and study generalizations to the case of more than two alternatives.