# Approval-Based Voting with Mixed Goods

arXiv: https://arxiv.org/abs/2211.12647 · HTML: https://mlsanity.com/paper/2211.12647/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Nov 2022.

## Built on by (most cited first, 3 of 3)

- [Method of Equal Shares with Bounded Overspending](https://mlsanity.com/paper/2409.15005/) (2024)
- [Proportional Committee Elections with Positive and Negative Votes](https://mlsanity.com/paper/2503.01985/) (2025)
- [Fairness in the Multi-Secretary Problem](https://mlsanity.com/paper/2511.23097/) (2025)

## Abstract (arXiv)

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.