On Minimal Achievable Quotas in Multiwinner Voting

TL;DR

This paper introduces instance-dependent quotas for proportionality in approval-based multiwinner voting, defining $\alpha$-JR, $\alpha$-EJR, and $\alpha$-EJR+ to measure representation relative to adaptive cohesiveness. It shows that commonly used rules can be far from optimal under these dynamic notions, with an additive gap that can reach $\frac{k^2}{(k+1)^2}$, and proves that computing the optimal $\alpha$-value is NP-hard, providing an ILP formulation to decide $\alpha$-JR feasibility. The authors also identify positive results in restricted domains: party-list profiles, VI, and CI domains, where efficient computation of $\alpha^*$ is possible, and they present an empirical study indicating that many instances admit small optimal $\alpha$-values, suggesting practical benefits of adaptive quotas. Overall, the work lays a theoretical and empirical foundation for adaptive fairness in multiwinner approval voting, with clear directions for future research and potential real-world impact.

Abstract

Justified representation (JR) and extended justified representation (EJR) are well-established proportionality axioms in approval-based multiwinner voting. Both axioms are always satisfiable, but they rely on a fixed quota (typically Hare or Droop), with the Droop quota being the smallest one that guarantees existence across all instances. With this observation in mind, we take a first step beyond the fixed-quota paradigm and introduce proportionality notions where the quota is instance-dependent. We demonstrate that all commonly studied voting rules can have an additive distance to the optimum of $\frac{k^2}{(k+1)^2}$. Moreover, we look into the computational aspects of our instance-dependent quota and prove that determining the optimal value of $α$ for a given approval profile satisfying $α$-JR is NP-complete. To address this, we introduce an integer linear programming (ILP) formulation for computing committees that satisfy $α$-JR, and we provide positive results in the voter interval (VI) and candidate interval (CI) domains.

Figures (8)

• Figure 1: A voting instance with $N= [10]$, $C = \{c_1, c_2, c_3,c_4\}$ and $k =2$ in which any committee containing $c_3$ satisfies JR. Voters are denoted by an integer and precisely approve the candidates stacked above them.
• Figure 2: A voting instance with $N= [12]$, $C = \{c_1, \dots, c_7\}$ and $k =3$ in which the committee $W = \{c_1, c_2, c_3\}$ lies within the core even though a large group of voters remains unrepresented (example taken from peters2021market).
• Figure 3: A voting instance $N = [x+y]$, $C =\{c_1, \dots c_{k+1}\}$ in which the optimal $\alpha$-value for JR is $0$, whereas the optimal $\alpha$-value for EJR is $\min(\frac{x}{x+y},\frac{k\cdot y}{x + y})$.
• Figure 4: Visualization of Example \ref{['example1']} for $k=5$. Voters are indicated by dots, and hyperedges correspond to the candidates. A committee $W$ returned by most voting rules is indicated in violet, and the largest violation for $W$ is shown in orange. The committee $D = \{d_1, \dots, d_5\}$ minimizes the $\alpha$-value in this instance.
• Figure 5: Distribution of optimal $\alpha$-values, $\alpha_{\text{EJR}}^*$, under the Impartial Culture and the Euclidean Threshold model, based on 6400 generated instances each.

Theorems & Definitions (64)

• Definition 4.1: $\ell$-Cohesiveness ABC+16a
• Definition 4.2: $(\alpha, \ell)$-Cohesiveness
• Remark 1
• Definition 4.3: $\alpha$-JR
• Definition 4.4: $\alpha$-EJR
• Definition 4.5: $\alpha$-EJR+
• Theorem 4.6: casey2025justified
• Proposition 4.6
• proof : Proof Sketch
• Proposition 4.6