锚定求真：多设施选址的随机锚点体积机制
Anchoring for Truthfulness: The Random-Anchor Volume Mechanism for Multi-Facility Location
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中文总结 AI 辅助
该研究针对多设施选址问题，提出随机锚点体积机制，证明其在k=1、2、3时为期望防策略机制，k≥4时可被操纵，k=3时期望社会成本最多为8OPT₃。
中文摘要 AI 辅助
我们研究在实数轴上放置k个设施的防策略（strategyproof）问题，针对n个私密报告自身位置的智能体，且不涉及货币转移。对于两个设施，Lu、Sun、Wang和Zhu（2010）提出的比例机制（Proportional Mechanism）在期望意义上是防策略的，且能实现最优社会成本的常数因子近似。在标准模型中，每个智能体由最近的开放设施提供服务，三个设施是否能达到此类保证一直是未解决的问题。我们通过引入随机锚点体积（Random-Anchor Volume）机制肯定地解决了这一问题。该机制首先在均匀随机选中的智能体报告处开放一个设施，称为锚点（anchor），然后联合选择另外两个报告，为每对分配概率，其大小与该对和锚点形成的两个连续间隙的乘积成正比。我们证明该机制在期望意义上是防策略的，且期望社会成本最多为8OPT₃，其中OPTₖ表示使用至多k个设施可实现的最小社会成本。该机制可自然扩展到所有k≥2的情况，即选择k-1个额外报告，概率与这些报告和锚点之间的连续间隙的乘积成正比。在真实报告下，该扩展的期望社会成本最多为4(k-1)OPTₖ。然而，其激励保证存在明确边界：当k∈{1,2,3}时，该机制在期望意义上是防策略的，但对于所有k≥4的情况，该机制是可被操纵的。
英文摘要
We study the strategyproof placement of \(k\) facilities on the real line for \(n\) agents who privately report their locations, without monetary transfers. For two facilities, the Proportional Mechanism of Lu, Sun, Wang, and Zhu (2010) is strategyproof in expectation and achieves a constant-factor approximation to the optimal social cost. Whether such a guarantee is possible for three facilities in the standard model, where each agent is served by her nearest open facility, has remained open. We resolve this question affirmatively by introducing the \emph{Random-Anchor Volume} mechanism. The mechanism first opens a facility at the report of a uniformly random agent, called the \emph{anchor}, and then jointly selects two additional reports, assigning each pair probability proportional to the product of the two consecutive gaps formed by the pair and the anchor. We prove that the mechanism is strategyproof in expectation and has expected social cost at most \(8 OPT_3\), where \(OPT_k\) denotes the minimum social cost achievable using at most \(k\) facilities. The mechanism naturally extends to every \(k\geq 2\) by selecting \(k-1\) additional reports with probability proportional to the product of the consecutive gaps among them and the anchor. Under truthful reporting, this generalization has expected social cost at most \(4(k-1)OPT_k\). Its incentive guarantee, however, has a sharp boundary: the mechanism is strategyproof in expectation for \(k\in\{1,2,3\}\), but is manipulable for every \(k\geq 4\).
