Proportional Representation in Rank Aggregation

2025-08-22

AI Summary

This paper addresses the lack of proportional representation in weighted rank aggregation, proposing the first social welfare function framework satisfying proportional fairness. Existing approaches—largely majority-based—fail to ensure that each input ranking receives recognition commensurate with its weight. To resolve this, the authors introduce Proportional Sequential Borda and two variants: Ranked Method of Equal Shares and Flow-adjusting Borda. These methods systematically adapt proportional representation theory—originally developed for approval-based committee elections and participatory budgeting—to rank aggregation. Proportional fairness is formally defined via pairwise consistency constraints, and the framework integrates sequential Borda scoring, equal-share allocation, and flow-adjustment mechanisms. Experiments demonstrate that the proposed methods significantly outperform classical aggregation rules across diverse scenarios, while remaining computationally feasible and scalable. Thus, the work unifies theoretical fairness guarantees with practical aggregation performance.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Social Choice / Voting

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Designing proportional social welfare functions for rank aggregation
• Ensuring output rankings fairly represent all weighted input rankings
• Introducing rules satisfying proportional pairwise agreement conditions

Innovation

• Proportional Sequential Borda rule
• Ranked Method of Equal Shares
• Flow-adjusting Borda rule