AI 中文总结
该研究针对平面度量选举，将相关问题建模为Stackelberg博弈，证明不同范数下孔多塞获胜集的规模上界，并给出单轮Voronoi博弈的新范数无关界。
AI 中文摘要
在排序复选制中，孔多塞获胜集是一组候选人，不存在外部候选人能被多数选民优先于该组中的每一位成员。我们研究平面度量选举中的孔多塞获胜集，其中选民根据给定范数下的距离对候选人进行排序。我们将一般度量选举、同行选择以及单轮Voronoi博弈建模为两玩家Stackelberg博弈的实例，并将这些问题置于一个共同的层级结构中。我们还引入了一个新变体，称之为强同行选择。我们的主要结果涉及ℓ₁范数和ℓ_∞范数下的同行选择，证明每一个平面实例都存在大小至多为3的孔多塞获胜集，即使在强同行选择下亦是如此，这一结论源于强矩形ε网的新结果：对于平面中的任意点集，可选择至多3个输入点，使其与所有包含超过一半点的轴对齐矩形相交，改进了Ashok等人给出的9/16的先前阈值。在ℓ₂范数下，我们证明每一个平面度量选举都存在大小至多为4的孔多塞获胜集，改进了Song等人给出的5的一般界。最后，我们给出单轮Voronoi博弈的新的与范数无关的界。
英文摘要
In ranked-choice voting, a Condorcet-winning set is a group of candidates for which no outside candidate is preferred to every member of the group by a majority of voters. We study Condorcet-winning sets in planar metric elections, where voters rank candidates according to their distance under a given norm. We formulate general metric elections, peer selection, and the one-round Voronoi game as instances of a two-player Stackelberg game and place these problems in a common hierarchy. We also introduce a new variant, which we call strong peer selection. Our main result concerns peer selection under the $\ell_1$ and $\ell_\infty$ norms. We prove that every planar instance admits a Condorcet-winning set of size at most three, even under strong peer selection. This follows from a new result for strong rectangular $\varepsilon$-nets. We show that, for every set of points in the plane, one can choose at most three input points that intersect every axis-parallel rectangle containing more than half of the points, improving the previous threshold of $9 / 16$ due to Ashok et al. Under the $\ell_2$ norm, we prove that every planar metric election admits a Condorcet-winning set of size at most four, improving on the general bound of five due to Song et al. Finally, we give new norm-independent bounds for the one-round Voronoi game.
