AI 中文总结
该研究针对社会选择理论中的孔多塞悖论，通过GPT-5.6 Sol Ultra构造证明，选举选出的k人委员会相关结果在放宽多数阈值后可达到常数因子范围内的紧致性。
AI 中文摘要
孔多塞悖论是社会选择理论的基础结论，表明无论选举中哪位候选人获胜，都可能有多数选民偏好某位落败的候选人。更糟糕的是，即便选举选出k名获胜者组成委员会，也可能存在某位落败者在多数投票中击败所有获胜者。近期研究显示，通过放宽多数阈值可规避这一障碍：对所有ε>0，任何选举都能选出由O(1/ε²)名获胜者组成的委员会，使得没有落败者能被超过1/2+ε比例的选民偏好于所有获胜者。我们提出一种由GPT-5.6 Sol Ultra找到的简单构造，证明该结果在常数因子范围内是紧的。
英文摘要
Condorcet's paradox is a foundational result in social choice theory, showing that no matter which candidate wins an election, a majority of voters may prefer some losing candidate. Worse still, even if the election can choose a committee of $k$ winners, some loser may beat every winner in a majority vote. Recent work showed that this obstruction can be sidestepped by relaxing the majority threshold. For all $\varepsilon > 0$, any election can select a committee of $O(1/\varepsilon^2)$ winners such that no loser is preferred to every winner by $\frac12 + \varepsilon$ fraction of voters. We present a simple construction, found by GPT-5.6 Sol Ultra, which proves that this result is tight up to a constant factor.
