九席位认可选举的确定性野兔核心非空
The Deterministic Hare Core Is Nonempty for Nine-Seat Approval Elections
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中文总结 AI 辅助
该研究证明九席位认可选举存在确定性野兔配额核心，通过结合PAV最大化者性质、封闭响应系统分析与计算机辅助验证完成，解决了九席位情况但未推广至任意规模。
中文摘要 AI 辅助
核心稳定性为基于认可的委员会选举提供了一种强形式的联盟比例代表制，但对于任意席位数量，其确定性存在性尚未明确。我们证明，每个具有九个席位的有限选举都存在一个确定性的野兔配额核心委员会。从一个既是全局比例认可投票（PAV）最大化者又满足核心稳定性的八席位委员会出发，假设所有单候选人扩展均被阻止，并提取一个包含意义最小的封闭响应系统。通过严格等式刚性消除单元素响应。对于剩余响应，有理Farkas证书将每个非正的PAV附加边际漂移下界限定为-n/189，并将每个后继漂移上界限定为n/126。均匀响应链是不可约的，因此平稳性使得这些边界相互矛盾。计算机辅助部分包含对7356种选民类型、36个单阻止者单元以及19个对称完全双阻止者基元族的逐点精确检查。精确证书存档和标准库验证器支持这些检查。该定理解决了九席位情况，但未确定任意委员会规模的确定性核心存在性。
英文摘要
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.
发表机构
- Boston University（波士顿大学）
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