The Art and Beauty of Voting Power

TL;DR

This paper explores the hidden beauty of weighted voting and voting power by analyzing how different voting rules influence players' pivotal influence, revealing structural differences through geometric representations.

Contribution

It introduces a generalized Penrose-Banzhaf index applied to social choice rules, providing a geometric visualization of voting power differences across various rules.

Findings

Voting rules exhibit distinct geometric structures in voting power triangles.

The geometry reflects differences in inclusiveness and transparency of voting rules.

Players' influence varies significantly depending on the voting rule used.

Abstract

We exhibit the hidden beauty of weighted voting and voting power by applying a generalization of the Penrose-Banzhaf index to social choice rules. Three players who have multiple votes in a committee decide between three options by plurality rule, Borda's rule, antiplurality rule, or one of the scoring rules in between. A priori influence on outcomes is quantified in terms of how players' probabilities to be pivotal for the committee decision compare to a dictator. The resulting numbers are represented in triangles that map out structurally equivalent voting weights. Their geometry and color variation reflect fundamental differences between voting rules, such as their inclusiveness and transparency.