Proportional Representation in Rank Aggregation

TL;DR

This paper introduces new social welfare functions for rank aggregation that ensure proportional representation of input rankings, addressing fairness issues in classical methods.

Contribution

The paper proposes the Proportional Sequential Borda rule and two variants, extending rank aggregation methods with fairness guarantees inspired by voting theory.

Findings

Proportional Sequential Borda satisfies the fairness condition.

Ranked Method of Equal Shares offers a utilitarian approach with proportionality.

Flow-adjusting Borda satisfies a stronger fairness condition.

Abstract

In rank aggregation, the task is to aggregate multiple weighted input rankings into a single output ranking. While numerous methods, so-called social welfare functions (SWFs), have been suggested for this problem, all of the classical SWFs tend to be majoritarian and are thus not acceptable when a proportional ranking is required. Motivated by this observation, we will design SWFs that guarantee that every input ranking is proportionally represented by the output ranking. Specifically, our central fairness condition requires that the number of pairwise comparisons between candidates on which an input ranking and the output ranking agree is proportional to the weight of the input ranking. As our main contribution, we present a simple SWF called the Proportional Sequential Borda rule, which satisfies this condition. Moreover, we introduce two variants of this rule: the Ranked Method of Equal Shares, which has a more utilitarian flavor while still satisfying our fairness condition, and the Flow-adjusting Borda rule, which satisfies an even stronger fairness condition. Many of our axioms and techniques are inspired by results on approval-based committee voting and participatory budgeting, where the concept of proportional representation has been studied in depth.