Proportional Fairness in Clustering: A Social Choice Perspective
Leon Kellerhals, Jannik Peters
TL;DR
This work reframes proportional clustering as a social-choice problem by linking PF, IF, and the transferable core via a distance-based metric, and introduces metric JR axioms (mJR/mPJR) from multiwinner voting to capture proportional representation. It proves tight connections between PF and IF, and between PF and TC, showing that PF approximations yield strong guarantees for IF and core concepts, with corresponding lower bounds. The paper also analyzes mJR and mPJR outcomes, giving near-optimal PF/IF/TC bounds and providing efficient algorithms for finite candidate sets, while addressing unbounded spaces through reduction techniques. In the context of sortition, stronger results via fair greedy capture yield improved core bounds, and the work highlights open questions on the best attainable approximation factors and further generalizations.
Abstract
We study the proportional clustering problem of Chen et al. [ICML'19] and relate it to the area of multiwinner voting in computational social choice. We show that any clustering satisfying a weak proportionality notion of Brill and Peters [EC'23] simultaneously obtains the best known approximations to the proportional fairness notion of Chen et al. [ICML'19], but also to individual fairness [Jung et al., FORC'20] and the "core" [Li et al. ICML'21]. In fact, we show that any approximation to proportional fairness is also an approximation to individual fairness and vice versa. Finally, we also study stronger notions of proportional representation, in which deviations do not only happen to single, but multiple candidate centers, and show that stronger proportionality notions of Brill and Peters [EC'23] imply approximations to these stronger guarantees.