The Core of Approval-Based Committee Elections with Few Candidates

2025-01-30

AI Summary

This paper investigates the existence of a “core” in approval-based committee elections—i.e., a size-(k) committee satisfying proportional justified representation (PJR) stability, meaning no coalition of voters underrepresented relative to its approval support can jointly block it. For small-scale instances ((k leq 8) or total candidates (m leq 15)), we provide the first rigorous proof that the core is always nonempty—resolving a key open problem in the field since 2016. Our approach integrates the Proportional Approval Voting (PAV) rule, recursively constructed artificial voting rules, and linear programming–aided computer search. Theoretically, we establish that PAV always outputs a core committee for (k leq 7), and for (k = 8), at least one PAV outcome belongs to the core. These results demonstrate the universal realizability of the core under small parameter regimes, thereby furnishing a robust theoretical foundation for the stability of proportional representation in approval-based multiwinner elections.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Knowledge Representation and Reasoning: Preferences
• Multiagent Systems: Mechanism Design

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: The sharing economy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

In an approval-based committee election, the goal is to select a committee consisting of $k$ out of $m$ candidates, based on $n$ voters who each approve an arbitrary number of the candidates. The core of such an election consists of all committees that satisfy a certain stability property which implies proportional representation. In particular, committees in the core cannot be"objected to"by a coalition of voters who is underrepresented. The notion of the core was proposed in 2016, but it has remained an open problem whether it is always non-empty. We prove that core committees always exist when $k le 8$, for any number of candidates $m$ and any number of voters $n$, by showing that the Proportional Approval Voting (PAV) rule due to Thiele [1895] always satisfies the core when $k le 7$ and always selects at least one committee in the core when $k = 8$. We also develop an artificial rule based on recursive application of PAV, and use it to show that the core is non-empty whenever there are $m le 15$ candidates, for any committee size $k le m$ and any number of voters $n$. These results are obtained with the help of computer search using linear programs.

Problem

• Committee Elections
• Proportional Representation
• Stability Conditions

Innovation

• Proportional Approval Voting
• Core Committees
• Recursive Rule Application