On the Edge of Core (Non-)Emptiness: An Automated Reasoning Approach to Approval-Based Multi-Winner Voting
Ratip Emin Berker, Emanuel Tewolde, Vincent Conitzer, Mingyu Guo, Marijn Heule, Lirong Xia
TL;DR
This work tackles the long-standing question of core stability in approval-based multi-winner elections by reframing the problem through mixed-integer linear programming (MILP). The authors introduce a nested optimization (M3) that, via a vote-distribution perspective, yields a practical MILP formulation whose optimum reflects the sizewise-least core value, enabling non-emptiness proofs and new duality-based upper bounds. They derive key results, including lower bounds that are tight in certain regimes (e.g., singleton deviations) and non-emptiness in the m=k+1 regime under Droop-like quotas, while also linking core stability to priceability notions and disproving conjectures about Lindahl priceability. The framework provides computational proofs and counterexamples, clarifying the landscape of core stability and its relationship to other fairness criteria, with open-source code to replicate and extend the analyses. Overall, the paper demonstrates how MILP and duality can uncover structural properties of core stability and its connections to fair allocation concepts in social choice and related AI settings.
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.