Sequential Phragm\'en Guarantees 2-Approximate Core Stability

2026-09-11

AI Summary

This study addresses the long-standing open problem concerning the existence of core stability and the determination of the optimal stability coefficient λ in approval-based committee elections. By integrating theoretical analysis of proportional representation, this work investigates the stability properties of weighted sequential Phragmén voting rules. It rigorously proves that this rule invariably returns a 2-stable committee, thereby establishing that the 2-core is non-empty. Notably, this result significantly reduces the best-known optimal stability guarantee for classical rules from 3.651 to 2 for the first time. By setting a new state-of-the-art record in the field and substantially approximating ideal stability, this research provides a critical theoretical breakthrough for ensuring fairness in computational social choice.

Abstract

In an approval-based committee election, a committee of size $k$ is selected from a set of candidates to represent voters of total weight $n$, each of whom has positive weight and approves a subset of the candidates. A size-$k$ committee is $\lambda$-stable if, for every nonempty subset $T$ of candidates, the total weight of voters who strictly prefer $T$ is less than $\lambda$ times the proportional share $n|T|/k$. Whether there exists a committee that is exactly stable, corresponding to $\lambda=1$, remains a major open problem in approval-based committee voting. Thus, a natural objective is to identify small values of $\lambda>1$ for which $\lambda$-stability can always be guaranteed. We prove that weighted sequential Phragm\'en, a classical and natural rule, always returns a $2$-stable committee of size $k$. Since the $\lambda$-core is the set of all $\lambda$-stable committees, our result implies that the $2$-core is always nonempty. This improves upon the previously best-known guarantee, due to Gao, Sun, and Vondr\'ak~[EC~2026], that a $3.651$-stable committee always exists.

Problem

• approval-based committee election
• core stability
• lambda-stability
• proportional representation

Innovation

• Sequential Phragmén
• Core Stability
• Approval-based Committee Election
• Approximation Guarantee