Fair Representation and a Linear Shapley Rule

TL;DR

This paper proposes replacing the Penrose square root rule with a linear Shapley rule for fair representation in voting systems, especially when voters have correlated preferences, extending the concept beyond binary decisions.

Contribution

It introduces a linear Shapley rule as a new fairness criterion for weighted voting, applicable to correlated preferences and alternative intervals, improving upon the Penrose rule.

Findings

Linear Shapley rule aligns delegate influence with constituency size.

Applicable to correlated preferences and interval-based decisions.

Offers a more general fairness criterion than Penrose rule.

Abstract

When delegations to an assembly or council represent differently sized constituencies, they are often allocated voting weights which increase in population numbers (EU Council, US Electoral College, etc.). The Penrose square root rule (PSRR) is the main benchmark for fair representation of all bottom-tier voters in the top-tier decision making body, but rests on the restrictive assumption of independent binary decisions. We consider intervals of alternatives with single-peaked preferences instead, and presume positive correlation of local voters. This calls for a replacement of the PSRR by a linear Shapley rule: representation is fair if the Shapley value of the delegates is proportional to their constituency sizes.