A Generalised Theory of Proportionality in Collective Decision Making

TL;DR

This paper introduces a unified framework for proportionality in diverse collective decision-making models, ensuring fair representation without predefined groups, and extends existing rules to this general setting.

Contribution

It formulates new proportionality axioms applicable to a broad class of voting models and adapts prominent election rules to satisfy these axioms under matroid constraints.

Findings

Axioms ensure proportionality for any cohesive voter subset.

Proportional Approval Voting and Phragmén rules satisfy axioms under matroid constraints.

Framework unifies diverse voting models with fairness guarantees.

Abstract

We consider a voting model, where a number of candidates need to be selected subject to certain feasibility constraints. The model generalises committee elections (where there is a single constraint on the number of candidates that need to be selected), various elections with diversity constraints, the model of public decisions (where decisions needs to be taken on a number of independent issues), and the model of collective scheduling. A critical property of voting is that it should be fair -- not only to individuals but also to groups of voters with similar opinions on the subject of the vote; in other words, the outcome of an election should proportionally reflect the voters' preferences. We formulate axioms of proportionality in this general model. Our axioms do not require predefining groups of voters; to the contrary, we ensure that the opinion of every subset of voters whose preferences are cohesive-enough are taken into account to the extent that is proportional to the size of the subset. Our axioms generalise the strongest known satisfiable axioms for the more specific models. We explain how to adapt two prominent committee election rules, Proportional Approval Voting (PAV) and Phragm\'{e}n Sequential Rule, as well as the concept of stable-priceability to our general model. The two rules satisfy our proportionality axioms if and only if the feasibility constraints are matroids.