A cornerstone of social choice theory is Condorcet's paradox which says that in an election where $n$ voters rank $m$ candidates it is possible that, no matter which candidate is declared the winner, a majority of voters would have preferred an alternative candidate. Instead, can we always choose a small committee of winning candidates that is preferred to any alternative candidate by a majority of voters? Elkind, Lang, and Saffidine raised this question and called such a committee a Condorcet winning set. They showed that winning sets of size $2$ may not exist, but sets of size logarithmic in the number of candidates always do. In this work, we show that Condorcet winning sets of size $6$ always exist, regardless of the number of candidates or the number of voters. More generally, we show that if $\frac{\alpha}{1 - \ln \alpha} \geq \frac{2}{k + 1}$, then there always exists a committee of size $k$ such that less than an $\alpha$ fraction of the voters prefer an alternate candidate. These are the first nontrivial positive results that apply for all $k \geq 2$. Our proof uses the probabilistic method and the minimax theorem, inspired by recent work on approximately stable committee selection. We construct a distribution over committees that performs sufficiently well (when compared against any candidate on any small subset of the voters) so that this distribution must contain a committee with the desired property in its support.

翻译：社会选择理论的基石之一是孔多塞悖论，该悖论指出：在 $n$ 位选民对 $m$ 位候选人进行排序的选举中，无论宣布哪位候选人获胜，都可能存在多数选民更偏好另一位候选人的情况。反之，我们是否总能选出一个由获胜候选人组成的小型委员会，使得多数选民更偏好该委员会而非任何其他候选人？Elkind、Lang 和 Saffidine 提出了这个问题，并将这样的委员会称为孔多塞获胜集。他们证明，大小为 $2$ 的获胜集可能不存在，但大小为候选人数量对数的集合总是存在。在本研究中，我们证明无论候选人数量或选民数量如何，大小为 $6$ 的孔多塞获胜集总是存在。更一般地，我们证明若 $\frac{\alpha}{1 - \ln \alpha} \geq \frac{2}{k + 1}$，则总存在一个大小为 $k$ 的委员会，使得偏好其他候选人的选民比例小于 $\alpha$。这是首个适用于所有 $k \geq 2$ 的非平凡正向结果。我们的证明采用概率方法和极小极大定理，其灵感来源于近期关于近似稳定委员会选择的研究。我们构建了一个委员会分布，该分布在性能上（与任何候选人在任何选民小规模子集上相比）表现足够好，从而确保该分布的支撑集中必然存在满足所需性质的委员会。