On the Edge of Core (Non-)Emptiness: An Automated Reasoning Approach to Approval-Based Multi-Winner Voting

2025-12-18

AI Summary

This paper addresses the long-standing open problem of core stability existence in approval-based multiwinner voting: Does a nonempty core necessarily exist for any number of voters and a fixed number of candidates? To resolve this, the authors propose the first voter-count-independent, formally verifiable decision procedure—based on mixed-integer linear programming (MILP) modeling and dual-reconstruction analysis—that enables automated generation of formal existence proofs. Theoretically, the work establishes an equivalence between core stability and priceability, along with implication relationships, yielding novel existence theorems for several special cases. Practically, it introduces the first computational framework capable of automatically generating rigorous, machine-verifiable proofs of both existence and nonexistence of core-stable committees. These contributions provide a rigorous theoretical foundation and a practical, certifiable tool for ensuring group fairness in multiwinner elections.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity
• Security and Privacy: Large-scale security measurements

Abstract

Core stability is a natural and well-studied notion for group fairness in multi-winner voting, where the task is to select a committee from a pool of candidates. We study the setting where voters either approve or disapprove of each candidate; here, it remains a major open problem whether a core-stable committee always exists. In this work, we develop an approach based on mixed-integer linear programming for deciding whether and when core-stable committees are guaranteed to exist. In contrast to SAT-based approaches popular in computational social choice, our method can produce proofs for a specific number of candidates independent of the number of voters. In addition to these computational gains, our program lends itself to a novel duality-based reformulation of the core stability problem, from which we obtain new existence results in special cases. Further, we use our framework to reveal previously unknown relationships between core stability and other desirable properties, such as notions of priceability.

Problem

• Determines existence of core-stable committees in approval-based multi-winner voting.
• Develops automated reasoning via mixed-integer linear programming for core stability.
• Analyzes relationships between core stability and other fairness properties.

Innovation

• Mixed-integer linear programming for core stability analysis
• Proof generation independent of voter count
• Duality-based reformulation revealing new existence results