On Minimal Achievable Quotas in Multiwinner Voting

TL;DR

This paper explores minimizing quota requirements in approval-based multiwinner voting by introducing instance-dependent proportionality notions, analyzing their computational complexity, and proposing ILP solutions for specific domains.

Contribution

It introduces a new framework for instance-dependent quotas in multiwinner voting, analyzes their optimal bounds, and provides ILP methods for practical computation.

Findings

All studied voting rules have an additive gap of .25 to the optimal quota.

Determining the optimal  for -JR is NP-complete.

ILP formulations are effective in voter and candidate interval domains.

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 in mind, we take a step beyond the fixed-quota paradigm by studying instance-dependent proportionality notions. More specifically, we minimize the quota requirements for JR and EJR using the parameter $\alpha$. We demonstrate that all commonly studied voting rules can have an additive gap to the optimum of $\frac{k^2}{(k+1)^2}$. Moreover, we examine the computational aspects of our instance-dependent quota and prove that determining the optimal value of $\alpha$ for a given approval profile that allows some committee to satisfy $\alpha$-JR is NP-complete. To address this, we introduce an integer linear programming (ILP) formulation for computing committees that satisfy $\alpha$-JR, and we provide positive computational results in the voter interval (VI) and candidate interval (CI) domains.