Proportional Representation in Metric Spaces and Low-Distortion Committee Selection

TL;DR

This paper introduces a new notion of gamma-proportional representation for a small committee R of size k representing a larger set V embedded in a metric space, and strengthens it via resource augmentation to the (b alpha,b gamma) framework. In the ordinal information model, where only rankings are available, the Expanding Approvals Rule (EAR) yields a constant-factor representativeness: for every b alpha>1, the output is (b alpha,b gamma(b alpha)) with b gamma(b alpha)=1+((7+b sqrt{41})/2) * b alpha/(b alpha-1) Approximately, EAR attains a b 5.71-proportional fairness bound in the ordinal model, and achieves approximate core fairness under the same model. When full distances are known, a variant of Greedy Capture improves constants, yielding b gamma(b alpha)=1+(2+b sqrt{2}) * b alpha/(b alpha-1)  and a simple single-winner rule attains distortion at most 44. The results illuminate a fundamental trade-off between augmentation and representational accuracy, provide the first constant-factor ordinal-proportional fairness results, and establish groundwork for practical, fair multi-winner selection in elections and clustering contexts.

Abstract

We introduce a novel definition for a small set R of k points being "representative" of a larger set in a metric space. Given a set V (e.g., documents or voters) to represent, and a set C of possible representatives, our criterion requires that for any subset S comprising a theta fraction of V, the average distance of S to their best theta*k points in R should not be more than a factor gamma compared to their average distance to the best theta*k points among all of C. This definition is a strengthening of proportional fairness and core fairness, but - different from those notions - requires that large cohesive clusters be represented proportionally to their size. Since there are instances for which - unless gamma is polynomially large - no solutions exist, we study this notion in a resource augmentation framework, implicitly stating the constraints for a set R of size k as though its size were only k/alpha, for alpha > 1. Furthermore, motivated by the application to elections, we mostly focus on the "ordinal" model, where the algorithm does not learn the actual distances; instead, it learns only for each point v in V and each candidate pairs c, c' which of c, c' is closer to v. Our main result is that the Expanding Approvals Rule (EAR) of Aziz and Lee is (alpha, gamma) representative with gamma <= 1 + 6.71 * (alpha)/(alpha-1). Our results lead to three notable byproducts. First, we show that the EAR achieves constant proportional fairness in the ordinal model, giving the first positive result on metric proportional fairness with ordinal information. Second, we show that for the core fairness objective, the EAR achieves the same asymptotic tradeoff between resource augmentation and approximation as the recent results of Li et al., which used full knowledge of the metric. Finally, our results imply a very simple single-winner voting rule with metric distortion at most 44.

Theorems & Definitions (27)

• Definition 2.1: $\gamma$-proportionally representative committee
• Definition 2.2: $(\alpha, \gamma)$-proportionally representative committee
• Definition 2.3: Approximate Core
• Proposition 2.4
• Proposition 2.5
• Definition 2.6: $\gamma$-proportional fairness
• Theorem 4.1
• Theorem 4.2
• Corollary 4.3
• Corollary 4.4