arXiv 2411.033902024-11cs.GTcs.DMcs.DSmath.CO

六名候选人足以赢得选民多数

Six Candidates Suffice to Win a Voter Majority

Moses Charikar, Alexandra Lassota, Prasanna Ramakrishnan, Adrian Vetta, Kangning Wang

备注：: to appear at STOC 2025
DOI：: 10.1145/3717823.3718235

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AI总结：

本研究证明，无论选民和候选人数量如何，总能选出6名候选人组成委员会，使多数选民偏好它而非任何其他候选人，并推广到更一般的参数条件。

摘要：

社会选择理论的基石之一是孔多塞悖论，它指出在一次有$n$名选民对$m$名候选人进行排序的选举中，有可能无论哪位候选人被宣布为获胜者，大多数选民都会更倾向于另一位候选人。相反，我们能否总是选出一个由少数获胜候选人组成的小委员会，使得大多数选民相对于任何其他候选人都更偏好该委员会？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。他们证明了大小为$2$的获胜集可能不存在，但大小与候选人数的对数成正比的获胜集总是存在。在本工作中，我们证明了大小为$6$的孔多塞获胜集总是存在，无论候选人数或选民人数如何。更一般地，我们证明如果$\frac{\alpha}{1 - \ln \alpha} \geq \frac{2}{k + 1}$，那么总是存在一个大小为$k$的委员会，使得少于$\alpha$比例的选民更倾向于其他候选人。这些是适用于所有$k \geq 2$的首个非平凡正面结果。我们的证明使用了概率方法和极小极大定理，灵感来自于最近关于近似稳定委员会选择的工作。我们构造了一个在委员会上的分布，该分布表现足够好（当与任何小规模选民子集上的任何候选人比较时），因此该分布的支持中必定包含一个具有所需性质的委员会。

英文摘要：

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 $\fracα{1 - \ln α} \geq \frac{2}{k + 1}$, then there always exists a committee of size $k$ such that less than an $α$ 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.

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