A General Theory of Proportionality with Additive Utilities

2026-02-09

AI Summary

This study addresses the problem of selecting a proportionally representative candidate set under general constraints—such as diversity requirements or participatory budgeting—when voters express their preferences via cardinal utility scores over candidates. The work introduces the first extension of proportionality axioms to the cardinal utility voting model and proposes a novel algorithm that constructs a ranking wherein every prefix satisfies the specified proportionality criterion. By integrating axiomatic analysis of proportionality, cardinal utility modeling, and combinatorial optimization techniques, this research establishes a flexible proportional selection rule applicable to a broad range of constraint settings. The approach guarantees not only that the final outcome is proportional but also that fairness is preserved throughout the intermediate stages of the ranking process.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Game Theory and Economic Paradigms: Social Choice / Voting
• Knowledge Representation and Reasoning: Preferences

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

We consider a model where a subset of candidates must be selected based on voter preferences, subject to general constraints that specify which subsets are feasible. This model generalizes committee elections with diversity constraints, participatory budgeting (including constraints specifying how funds must be allocated to projects from different pools), and public decision-making. Axioms of proportionality have recently been defined for this general model, but the proposed rules apply only to approval ballots, where each voter submits a subset of candidates she finds acceptable. We propose proportional rules for cardinal ballots, where each voter assigns a numerical value to each candidate corresponding to her utility if that candidate is selected. In developing these rules, we also introduce methods that produce proportional rankings, ensuring that every prefix of the ranking satisfies proportionality.

Problem

• proportionality
• cardinal ballots
• approval ballots
• constrained selection
• voter preferences

Innovation

• proportionality
• cardinal ballots
• additive utilities
• proportional rankings
• constrained voting