The Deterministic Hare Core Is Nonempty for Nine-Seat Approval Elections
Abstract
Core stability gives approval-based committee elections a strong form of coalitional proportionality, but deterministic existence is not known for an arbitrary number of seats. We prove that every finite election with nine seats has a deterministic Hare-quota core committee. Starting from an eight-seat committee that is both a global Proportional Approval Voting (PAV) maximizer and core stable, we suppose that all one-candidate extensions are blocked and extract an inclusion-minimal closed response system. Singleton responses are eliminated by exact equality rigidity. For the remaining responses, rational Farkas certificates bound every nonpositive PAV add-marginal drift below by −n/189 and force every successor drift above n/126. The uniform response chain is irreducible, so stationarity makes these bounds incompatible. The computer-assisted component comprises an exact pointwise check over 7,356 voter types, 36 one-blocker cells, and a symmetry-complete family of 19 two-blocker motifs. Exact certificate archives and standard-library verifiers support these checks. The theorem settles the nine-seat case but does not decide deterministic core existence for arbitrary committee size.
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BibTeX
@misc{fang2026deterministic,
title = {{The Deterministic Hare Core Is Nonempty for Nine-Seat Approval Elections}},
author = {Jiarui Fang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04497},
eprint = {2609.04497},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04497},
}
RIS
TY - UNPB
TI - The Deterministic Hare Core Is Nonempty for Nine-Seat Approval Elections
AU - Jiarui Fang
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.04497
UR - https://arxiv.org/abs/2609.04497
AB - Core stability gives approval-based committee elections a strong form of coalitional proportionality, but deterministic existence is not known for an arbitrary number of seats. We prove that every finite election with nine seats has a deterministic Hare-quota core committee. Starting from an eight-seat committee that is both a global Proportional Approval Voting (PAV) maximizer and core stable, we suppose that all one-candidate extensions are blocked and extract an inclusion-minimal closed response system. Singleton responses are eliminated by exact equality rigidity. For the remaining responses, rational Farkas certificates bound every nonpositive PAV add-marginal drift below by −n/189 and force every successor drift above n/126. The uniform response chain is irreducible, so stationarity makes these bounds incompatible. The computer-assisted component comprises an exact pointwise check over 7,356 voter types, 36 one-blocker cells, and a symmetry-complete family of 19 two-blocker motifs. Exact certificate archives and standard-library verifiers support these checks. The theorem settles the nine-seat case but does not decide deterministic core existence for arbitrary committee size.
ER -