Justified Representation: From Hare to Droop (2508.00811v1)

Abstract: The study of proportionality in multiwinner voting with approval ballots has received much attention in recent years. Typically, proportionality is captured by variants of the Justified Representation axiom, which say that cohesive groups of at least $\ell\cdot\frac{n}{k}$ voters (where $n$ is the total number of voters and $k$ is the desired number of winners) deserve $\ell$ representatives. The quantity $\frac{n}{k}$ is known as the Hare quota in the social choice literature. Another -- more demanding -- choice of quota is the Droop quota, defined as $\lfloor\frac{n}{k+1}\rfloor+1$. This quota is often used in multiwinner voting with ranked ballots: in algorithms such as Single Transferable Voting, and in proportionality axioms, such as Droop's Proportionality Criterion. A few authors have considered it in the context of approval ballots, but the existing analysis is far from comprehensive. The contribution of our work is a systematic study of JR-style axioms (and voting rules that satisfy them) defined using the Droop quota instead of the Hare quota. For each of the standard JR axioms (namely, JR, PJR, EJR, FPJR, FJR, PJR+ and EJR+), we identify a voting rule that satisfies the Droop version of this axiom. In some cases, it suffices to consider known rules (modifying the corresponding Hare proof, sometimes quite substantially), and in other cases it is necessary to modify the rules from prior work. Each axiom is more difficult to satisfy when defined using the Droop quota, so our results expand the frontier of satisfiable proportionality axioms. We complement our theoretical results with an experimental study, showing that for many probabilistic models of voter approvals, Droop JR/EJR+ are considerably more demanding than standard (Hare) JR/EJR+.

• The paper's main contribution is demonstrating that Droop-based JR axioms enforce stronger proportionality than their Hare-based versions.
• It details modifications to established voting rules like MES and Greedy Justified Candidate Rule to satisfy Droop proportionality axioms.
• Experimental evaluations show that Droop-EJR+ is significantly more demanding, influencing the design of fairer multiwinner voting systems.

Justified Representation: From Hare to Droop

This paper investigates the implications of using the Droop quota instead of the Hare quota in the definitions of Justified Representation (JR) axioms for multiwinner voting with approval ballots. The paper systematically examines several JR-style axioms and identifies voting rules that satisfy the Droop version of these axioms, expanding the frontier of satisfiable proportionality axioms. The paper provides both theoretical results and experimental evaluations that highlight the increased stringency of Droop-based JR axioms compared to their Hare-based counterparts.

Core Concepts and Definitions

The paper begins by defining fundamental concepts related to multiwinner elections with approval ballots, including candidates, voters, approval ballots, and committees. It introduces key multiwinner voting rules such as Approval Voting (AV) and Proportional Approval Voting (PAV), which are used later to illustrate the properties of the JR axioms.

Central to the paper is the notion of cohesive groups of voters. The authors define both Hare and Droop $\ell$-cohesive groups, where $\ell$ represents the minimum number of candidates a group must jointly approve to warrant representation. The Droop quota, defined as $\lfloor\frac{n}{k+1}\rfloor+1$, is shown to be more demanding than the Hare quota, $\frac{n}{k}$, particularly for small values of $k$. The paper formally defines seven representation axioms: JR, PJR, EJR, FPJR, FJR, PJR+, and EJR+, for both Hare and Droop quotas. The Droop versions are shown to be at least as demanding as their Hare counterparts.

Main Results

The paper presents a comprehensive analysis of voting rules satisfying Droop versions of JR axioms. The main results are summarized as follows:

• Droop-EJR/EJR+: It is shown that these axioms are satisfied by $\varepsilon$-lsPAV (with appropriately chosen $\varepsilon$), a modified Greedy Justified Candidate Rule, and two variants of the Method of Equal Shares (MES) executed with artificially inflated budgets.
• Droop-FPJR: Modifications of the Monroe rule and its greedy variant satisfy Droop-FPJR when $k+1$ divides $n$, as do all priceable rules that select committees of size $k$. The paper provides a separate proof that MES and Exact Equal Shares (EES) satisfy Droop-FPJR.
• Droop-FJR: A modification of the Greedy Cohesive Rule satisfies Droop-FJR.

The paper includes a table summarizing these results, indicating which rules satisfy each axiom. The paper also includes several negative results. E.g., for several rules the paper shows that, even though they satisfy the Hare version of some proportionality axiom, they fail the Droop version of the same axiom; this justifies our proposed modifications of these rules.

Implications and Significance

The research demonstrates that existing voting rules can be adapted to satisfy stronger proportionality axioms based on the Droop quota. This has practical implications for the design and implementation of multiwinner voting systems, potentially leading to more representative and equitable outcomes. The finding that Droop-EJR+ is significantly more difficult to satisfy than Hare-EJR+ suggests that Droop-EJR+ serves as a more stringent test of proportionality.

The paper extends the theoretical understanding of proportionality in multiwinner voting by providing a complete picture of the Droop proportionality landscape. It reproduces many key results from the literature for the Droop quota versions of the axioms. The work clarifies the relationships between different JR axioms and identifies rules that can satisfy them.

Experimental Evaluation

The paper presents an experimental paper that compares the satisfiability of Hare and Droop versions of JR and EJR+ under various probabilistic models of voter approvals. The key observation is that, for the models considered, the Droop versions of the axioms are substantially more demanding than their Hare versions. In particular, the results show that Droop-EJR+ is significantly harder to satisfy than Hare-EJR+, especially for certain parameter ranges. The experimental results also indicate that the standard MES rule may not provide Droop-EJR+ in practice, motivating the use of MES with inflated budgets.

Conclusion

The paper offers a comprehensive paper of justified representation in multiwinner voting with approval ballots, focusing on the Droop quota. The theoretical results establish the satisfiability of Droop versions of key JR axioms, while the experimental findings demonstrate the increased stringency of these axioms in practice. The research contributes to a deeper understanding of proportionality in voting and provides insights for designing more equitable voting systems.

Future research directions include filling in the remaining unknown entries in the results table, recovering other known results from the Hare setting (such as for average satisfaction and hardness of verification), exploring the use of the Droop quota in proportional clustering, and conducting more extensive experiments using real-world election data.