Approval-Based Voting with Mixed Goods

TL;DR

This paper extends approval-based voting to scenarios involving both indivisible and divisible goods, proposing new fairness notions and analyzing how existing voting rules perform under these criteria.

Contribution

It introduces EJR-M and EJR-1 fairness notions for mixed goods, extends three voting rules, and analyzes their proportionality guarantees.

Findings

All three rules satisfy EJR-1.

Only GreedyEJR satisfies EJR-M.

Derived tight bounds on proportionality degree.

Abstract

We consider a voting scenario in which the resource to be voted upon may consist of both indivisible and divisible goods. This setting generalizes both the well-studied model of multiwinner voting and the recently introduced model of cake sharing. Under approval votes, we propose two variants of the extended justified representation (EJR) notion from multiwinner voting, a stronger one called EJR for mixed goods (EJR-M) and a weaker one called EJR up to 1 (EJR-1). We extend three multiwinner voting rules to our setting -- GreedyEJR, the method of equal shares (MES), and proportional approval voting (PAV) -- and show that while all three generalizations satisfy EJR-1, only the first one provides EJR-M. In addition, we derive tight bounds on the proportionality degree implied by EJR-M and EJR-1, and investigate the proportionality degree of our proposed rules.