Core-Stable Committees under Restricted Domains

TL;DR

This paper investigates the existence and computation of core-stable committees in various restricted preference domains, providing polynomial algorithms for many cases and insights into the structure of preferences.

Contribution

It proves non-emptiness and polynomial-time computability of the core in several restricted domains and introduces relaxations for classes where the core may be empty.

Findings

Core is non-empty and computable in polynomial time for several restricted domains.

Classic proportional rules do not always produce core committees.

Structural insights into top-monotonic preferences and other domains.

Abstract

We study the setting of committee elections, where a group of individuals needs to collectively select a given size subset of available objects. This model is relevant for a number of real-life scenarios including political elections, participatory budgeting, and facility-location. We focus on the core -- the classic notion of proportionality, stability and fairness. We show that for a number of restricted domains including voter-interval, candidate-interval, single-peaked, and single-crossing preferences the core is non-empty and can be found in polynomial time. We show that the core might be empty for strict top-monotonic preferences, yet we introduce a relaxation of this class, which guarantees non-emptiness of the core. Our algorithms work both in the randomized and discrete models. We also show that the classic known proportional rules do not return committees from the core even for the most restrictive domains among those we consider (in particular for 1D-Euclidean preferences). We additionally prove a number of structural results that give better insights into the nature of some of the restricted domains, and which in particular give a better intuitive understanding of the class of top-monotonic preferences.