Significant progress has been made on identifying new restricted domains that make collective decision-making problems computationally tractable, and on axiomatic analysis of multiwinner voting rules, both for rules defined on ranked ballots and for rules defined on approval ballots.
Martin Lackner (RA on the project) and Dominik Peters (a PhD student of the PI, and an RA on the project on a part-time basis) introduced the domain of preferences single-peaked on a circle, and obtained a number of algorithmic results for it, including an efficient algorithm for recognizing such preferences as well as algorithms for some voting rules that are NP-hard for the general domain. Another restricted preference domain that has been studied is that of preferences single-peaked on trees. The PI together with Dominik Peters has identified several subdomains of this domain that can be efficiently recognized and admit polynomial-time algorithms for winner determination under the Chamberlin--Courant voting rule. Further, Dominik Peters has established that another potentially useful restricted domain, that of d-dimensional Euclidean preferences, is hard to recognize for any constant d>1. In a recent journal paper, Martin Lackner (an RA on the project) has provided a number of algorithms and intractability results for the problem of recognizing preferences that are almost single-peaked, for various notions of distance to the single-peaked domain. A similar question was studied by the PI, Dominik Peters and PI's MSc students (Florian Jaeckle, Foram Lakhani) for almost single-crossing preferences. In the context of approval preferences the PI and Martin Lackner have developed taxonomy of restricted preference domains, together with separation results and recognition algorithms. The PI together with an undergraduate student Andrei Constantinescu also developed an algorthm to recognize preferences single-crossing on a tree.
A number of papers identified new settings where one can obtain positive algorithmic or existence results by focusing on restricted preference domains. In her ADT'15 paper the PI (together with an external collaborator) proposed a family of rules that extend the Chamberlin--Courant rule and showed that rules in this family are efficiently computable when preferences are single-peaked or single-crossing. Further, restricting the preferences to belong to the single-peaked or single-crossing domain has been shown to be useful for computing Nash equilibria of Plurality voting, and we can describe the outcome of a natural opinion diffusion process for preferences that are single-peaked and single-crossing. Positive results were also obtained for preferences single-peaked on nice trees, and for one-dimensional approval preferences.
The PI also gave an invited talk in Early Career Spotlight track at IJCAI'16, where she summarized existing work on restricted preference domains and future research directions. She has also given a number of seminar and workshop talks on the utcomes of the project. A long survey of algorithmic work on restricted preference domains is in preparation.
The PI has published two papers in Social Choice and Welfare (the leading social choice journal) that deal with axiomatic properties of committee selection rules. One of these papers lays the axiomatic foundations for committee selection rules that are based on scoring functions, in the context of ranked ballots. In particular, this paper defined the family of committee scoring rules, which provides an analogue to the important family fo scoring rules in the context of single-winner elections; this family has been subsequently analyzed by Piotr Skowron (an RA on the project) and co-authors in a series of papers. The second SCW paper focuses on committee selection rules that are based on approval ballots, and formulates the axiom of justified representation for such rules, as well as its extension called extended justified representation. Both axioms turned out to be useful tools for classifying approval-based voting rules and present interesting algorithmic challenges; in particular, the latter axiom can be used to characterize a prominent voting rule (PAV) in the class of weighted approval-based rules. In a follow-up paper, the PI together with two project RAs and external collaborators defined an intermediate property, called proportional justified representation and investiagted which voting rules have this property; in another paper, two project RAs identified a voting rule that is computationally tractable and offers proportional justified representation. In yet another paper, two project RAs explored the connection between apportionment rules and approval-based committee selection; their work provides an alternative characterization of PAV. In on-going work, the proportionality-based approach has been extended to voting rules that output rankings.
