事前与事后：平等主义设施选址机制设计
Ex-ante versus Ex-post: Egalitarian Facility Location Mechanism Design
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中文总结 AI 辅助
该研究针对欧氏空间的设施选址问题，对比事前与事后评估，在低维中设计出打破确定性壁垒的机制，高维中则证明随机旋转中位数机制为最优平等主义机制。
中文摘要 AI 辅助
我们研究设施选址机制设计问题，其中n个策略智能体在欧氏空间中报告位置，机制输出单个设施位置。每个智能体的成本是其与设施的距离，我们的目标是以防策略方式最小化平等主义成本，即最大智能体成本。最优确定性近似比为2，由任何独裁者机制实现。我们研究随机期望防策略机制的能力。先前工作集中于事后评估，定义为期望最大智能体成本。我们转而研究事前评估，定义为最大期望智能体成本，其自然与期望防策略性对齐。我们得出以下结果：(1)低维：事前与事后的严格分离。在R中，我们给出一种简单的防策略机制，实现最优事前近似比1。在R²中，我们设计“随机旋转角落”机制，事前近似比最多为1.598，打破确定性壁垒。对于事后目标，我们证明下界为1.605，在R²中产生严格分离。(2)高维：不可能性。在Rᵈ（d≫1）中，我们表明没有期望防策略机制能在o_d(1)之外改进确定性独裁者机制。因此，事后和事前评估均未在高维中产生改进的公平性保证。一个推论是，当前功利主义目标的最佳已知机制“逐坐标随机旋转中位数”（RRCWM），在高维中也是平等主义目标的最佳机制：我们证明其对每个d≥1的Rᵈ中，事后和事前目标的近似比均为2。
英文摘要
We study the facility location mechanism design problem where $n$ strategic agents report locations in Euclidean space and the mechanism outputs a single facility location. Each agent's cost is its distance from the facility, and our objective is to minimize the egalitarian cost, i.e., the maximum agent cost, in a strategyproof way. The optimal deterministic approximation ratio is $2$, achieved by any dictator mechanism. We study the power of randomized strategyproof-in-expectation mechanisms. Prior work has focused on ex-post evaluation, defined as the expected maximum agent cost. We instead study ex-ante evaluation, defined as the maximum expected agent cost, which is naturally aligned with strategyproofness in expectation. We establish the following results: (1) Low dimensions: Strict ex-ante vs. ex-post separation. In $\mathbb{R}$, we give a simple strategyproof mechanism achieving the optimal ex-ante approximation ratio of $1$. In $\mathbb{R}^2$, we design the "Random Rotated Corner" mechanism, with ex-ante approximation ratio at most $1.598$, breaking the deterministic barrier. For the ex-post objective, we prove a lower bound of $1.605$, yielding a strict separation in $\mathbb{R}^2$. (2) High dimensions: Impossibility. In $\mathbb{R}^d$ for $d \gg 1$, we show that no strategyproof-in-expectation mechanism improves on the deterministic dictator mechanism beyond $o_d(1)$. Thus neither ex-post nor ex-ante evaluation yields improved fairness guarantees in high dimensions. An implication is that the "Random Rotation Coordinate-Wise Median" (RRCWM), currently the best known mechanism for the utilitarian objective, is also best possible for the egalitarian objective in high dimension: we show it achieves an approximation ratio of $2$ for both ex-post and ex-ante objectives in $\mathbb{R}^d$ for every $d \ge 1$.
