发表机构
Ramakrishna Mission Vivekananda Educational and Research Institute（拉马克里希纳使命维韦卡南达教育与研究机构）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究一维单轮离散 Voronoi 博弈，计算单响应设施下的博弈值与最优策略，并证明设施优势的上界与下界，提出猜想 $k^*(\ell)=\ell+1$。
AI 中文摘要
在单轮离散 Voronoi 博弈中，给定一条直线上由 $n$ 个投票者组成的多重集 $V$；玩家 P 放置 $k$ 个设施，随后玩家 Q 放置 $\ell$ 个设施，每个投票者被较近的设施赢得，平局时归 P。若 P 至少保住 $n/2$ 个投票者则获胜。在竞争选址的术语中，这等价于路径上具有单位需求的绝对 $(\ell|k)$-质心问题，而响应者的问题为 $(\ell|X_k)$-中位数问题，其在路径上的闭式解——至多 $2k$ 个显式边际量中最大的 $\ell$ 个之和——归功于 Spoerhase 和 Wirth。我们记录这一结构，给出完整证明，并得出两个我们认为新颖的推论。第一，我们针对单个响应设施计算博弈值 $\Gamma_{k,1}(V)$ 以及 P 的最优策略，对任意正实数需求和每个 $k$ 均在 $O(n\log n)$ 时间内完成。这改进了 Lazar 和 Tamir 针对路径上绝对 $(1|k)$-质心问题的 $O(kn\log^2 n)$ 界。第二，我们研究设施优势 $k^*(\ell)$，即 P 针对 $\ell$ 个设施赢得所有实例所需的最小 $k$。我们证明 $k^*(\ell)\le 2\ell-1$，展示实例证明对 $2\le\ell\le6$ 有 $k^*(\ell)\ge\ell+1$（一个基于半整数离散化的精确计算机辅助证明），确定 $k^*(1)=1$ 和 $k^*(2)=3$，并表明在均匀实例上 $k=\ell$ 已经足够，因此极端实例是加权的，且 Q 以单个投票者之差获胜。我们猜想对所有 $\ell\ge2$ 有 $k^*(\ell)=\ell+1$。
英文摘要
In the one-round discrete Voronoi game a multiset $V$ of $n$ voters on a line is given; player P places $k$ facilities, player Q then places $\ell$, and each voter is won by the nearer facility, ties going to P. P wins if it keeps at least $n/2$ voters. In the vocabulary of competitive location this is the absolute $(\ell|k)$-centroid problem on a path with unit demands, and the responder's problem is the $(\ell|X_k)$-medianoid, whose closed form on a path -- the sum of the $\ell$ largest of at most $2k$ explicit marginals -- is due to Spoerhase and Wirth. We record this structure, with complete proofs, and draw two consequences that we believe are new. First, we compute the value of the game against a single responding facility, $Γ_{k,1}(V)$, together with an optimal strategy for P, in $O(n\log n)$ time for arbitrary positive real demands and every $k$. This improves the $O(kn\log^2 n)$ bound of Lazar and Tamir for the absolute $(1|k)$-centroid on a path. Second, we study the facility advantage $k^*(\ell)$, the least $k$ for which P wins every instance against $\ell$ facilities. We prove $k^*(\ell)\le 2\ell-1$, exhibit instances proving $k^*(\ell)\ge\ell+1$ for $2\le\ell\le6$ (an exact, computer-assisted proof resting on a half-integer discretisation), determine $k^*(1)=1$ and $k^*(2)=3$, and show that on uniform instances $k=\ell$ already suffices, so the extremal instances are weighted and Q wins them by a single voter. We conjecture $k^*(\ell)=\ell+1$ for all $\ell\ge2$.
