Existence of the core in approval-based committee elections
Every approval-based multi-winner election is shown to admit a committee in the core, settling the main open question in the area, and one can be found in polynomial time.
- Lab
- Independent
- Model
- GPT-6 Astra
- Field
- Mathematics
- Date
- 2026-09-10
- Human collaborators
- Patrick Becker, Matthias Greger, Dominik Peters
Sources
Original work
Announcement
What was found
The core is a stability and group-fairness notion for multi-winner elections: a committee is in the core when no group of voters large enough to deserve a different outcome all prefer one. Whether a core committee always exists was the main open question in the theory of approval-based multi-winner elections. The paper shows one always does. The proof introduces a voting rule that optimizes an entropy-like objective over committees and payment systems together, and shows every local optimum of that objective lies in the core, which also yields a polynomial-time algorithm. The arXiv comment field carries the AI disclosure in one line: the proof was obtained with GPT-6 Astra.
Novelty check
The abstract states the question was the main open one in its area and that this settles it; the claim of prior status rests on that framing plus Epoch AI's listing of the problem among its FrontierMath open problems, rather than on a literature search run for this entry. Epoch AI has described the result as a Major Advance on that benchmark. Whether the authors are independent of Epoch AI, which would bear on any claim of independent verification, was not established.
Caveats and known objections
Autonomy is graded collaborative rather than ai-led on the strictest defensible reading of the public account, which describes a lengthy interactive session rather than a clean split between an AI insight and a human check: Peters is reported as saying both that he doubts the team would have found the proof without Astra and that Astra does not appear able to solve the problem from a simple prompt. Each half of that statement pulls against a stronger grade in one direction, and collaborative is where they meet. The AI disclosure lives in the arXiv comment field rather than the abstract or a dedicated statement, so it is thin relative to what this registry prefers. Verification is claimed: a preprint at entry, unrefereed, with no formalization and no independent check on record. The 2026-09-21 and 2026-09-23 watch runs both recorded this result under the date 2026-09-19 and with surnames only; the date and the full author names were corrected here against the arXiv record on 2026-09-22, which gives a submission date of 10 September 2026. Whether Epoch AI publishes rerunnable verification for its open-problem write-ups, which would bear on the verification grade, is unresolved.
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Entry history (1 event)
- AddedEntered the registry graded Claimed and Collaborative.
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Graded claimed for verification and collaborative for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Existence of the core in approval-based committee elections. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-09-10-approval-committee-coreBibTeX
@misc{whataifound-independent-2026-core,
title = {Existence of the core in approval-based committee elections},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Claimed. Autonomy: Collaborative.},
url = {https://whataifound.org/finding/2026-09-10-approval-committee-core}
}