# Full Proportional Justified Representation

arXiv: https://arxiv.org/abs/2501.12015 · HTML: https://mlsanity.com/paper/2501.12015/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 35 references that are in the map.
- First posted on arXiv 21 Jan 2025.

## Builds on (most cited first, 19 of 19)

- [Justified Representation in Approval-Based Committee Voting](https://mlsanity.com/paper/1407.8269/) (2014)
- [Proportional Participatory Budgeting with Additive Utilities](https://mlsanity.com/paper/2008.13276/) (2020)
- [Proportional Justified Representation](https://mlsanity.com/paper/1611.09928/) (2016)
- [Participatory Budgeting: Models and Approaches](https://mlsanity.com/paper/2003.00606/) (2020)
- [Robust and Verifiable Proportionality Axioms for Multiwinner Voting](https://mlsanity.com/paper/2302.01989/) (2023)
- [Proportionality and the Limits of Welfarism](https://mlsanity.com/paper/1911.11747/) (2019)
- [Phragm'en's Voting Methods and Justified Representation](https://mlsanity.com/paper/2102.12305/) (2021)
- [Computational Aspects of Multi-Winner Approval Voting](https://mlsanity.com/paper/1407.3247/) (2014)
- [Knapsack Voting for Participatory Budgeting](https://mlsanity.com/paper/2009.06856/) (2020)
- [The (Computational) Social Choice Take on Indivisible Participatory Budgeting](https://mlsanity.com/paper/2303.00621/) (2023)
- [Proportional Rankings](https://mlsanity.com/paper/1612.01434/) (2016)
- [Proportionally Representative Participatory Budgeting: Axioms and Algorithms](https://mlsanity.com/paper/1711.08226/) (2017)
- [Proportionally Representative Participatory Budgeting with Ordinal Preferences](https://mlsanity.com/paper/1911.00864/) (2019)
- [The Metric Distortion of Multiwinner Voting](https://mlsanity.com/paper/2201.13332/) (2022)
- [Phragm'en's and Thiele's election methods](https://mlsanity.com/paper/1611.08826/) (2016)
- [Proportionality in Approval-Based Participatory Budgeting](https://mlsanity.com/paper/2302.03672/) (2023)
- [Boosting Sortition via Proportional Representation](https://mlsanity.com/paper/2406.00913/) (2024)
- [Proportional Representation in Metric Spaces and Low-Distortion Committee Selection](https://mlsanity.com/paper/2312.10369/) (2023)
- [Approval-Based Elections and Distortion of Voting Rules](https://mlsanity.com/paper/1901.06709/) (2019)

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

- [Metritocracy: Representative Metrics for Lite Benchmarks](https://mlsanity.com/paper/2506.09813/) (2025)
- [Strengthening Full Justified Representation: Efficient Verification and Computation](https://mlsanity.com/paper/2608.11500/) (2026)
- [A General Theory of Proportionality with Additive Utilities](https://mlsanity.com/paper/2602.08504/) (2026)

## Abstract (arXiv)

In multiwinner approval voting, forming a committee that proportionally represents voters' approval ballots is an essential task. The notion of justified representation (JR) demands that any large "cohesive" group of voters should be proportionally "represented". The "cohesiveness" is defined in different ways; two common ways are the following: (C1) demands that the group unanimously approves a set of candidates proportional to its size, while (C2) requires each member to approve at least a fixed fraction of such a set. Similarly, "representation" have been considered in different ways: (R1) the coalition's collective utility from the winning set exceeds that of any proportionally sized alternative, and (R2) for any proportionally sized alternative, at least one member of the coalition derives less utility from it than from the winning set. Three of the four possible combinations have been extensively studied: (C1)-(R1) defines Proportional Justified Representation (PJR), (C1)-(R2) defines Extended Justified Representation (EJR), (C2)-(R2) defines Full Justified Representation (FJR). All three have merits, but also drawbacks. PJR is the weakest notion, and perhaps not sufficiently demanding; EJR may not be compatible with perfect representation; and it is open whether a committee satisfying FJR can be found efficiently. We study the combination (C2)-(R1), which we call Full Proportional Justified Representation (FPJR). We investigate FPJR's properties and find that it shares PJR's advantages over EJR: several proportionality axioms (e.g. priceability, perfect representation) imply FPJR and PJR but not EJR. We also find that efficient rules like the greedy Monroe rule and the method of equal shares satisfy FPJR, matching a key advantage of EJR over FJR. However, the Proportional Approval Voting (PAV) rule may violate FPJR, so neither of EJR and FPJR implies the other.