A note on limit results for the Penrose-Banzhaf index

TL;DR

This paper investigates conditions under which the Penrose-Banzhaf index converges to the weight distribution in weighted games, providing parametric examples and necessary conditions for such limit results.

Contribution

It introduces necessary conditions and parametric examples that clarify when the Penrose-Banzhaf index approaches weights in weighted games.

Findings

Identifies conditions for convergence of the Penrose-Banzhaf index to weights

Provides parametric examples illustrating these conditions

Clarifies the contrast between index and weight disparities

Abstract

It is well known that the Penrose-Banzhaf index of a weighted game can differ starkly from corresponding weights. Limit results are quite the opposite, i.e., under certain conditions the power distribution approaches the weight distribution. Here we provide parametric examples that give necessary conditions for the existence of limit results for the Penrose-Banzhaf index.