We study the facility location mechanism design problem where $n$ agents report their locations in Euclidean space, and the output is a single facility location. The cost function of each agent is the distance from the returned facility, and the objective is to minimize the social cost function (the sum of agent costs) in a strategyproof way. Our contributions: 1. Breaking the deterministic barrier. For $\mathbb{R}^2$, we give a random strategyproof mechanism (RR-CWM) achieving an expected approximation ratio of $\frac{4}π \approx 1.27$, which strictly improves upon the best deterministic strategyproof mechanism (which has a $\sqrt{2} \approx 1.41$ ratio). This closes the open problem of separating deterministic and random mechanisms for utilitarian facility location mechanism design in $\mathbb{R}^2$. For $\mathbb{R}^d$, we show that the expected approximation ratio of our mechanism is in $[1.41 - O(1/\sqrt{d}), 1.547]$. 2. Improved learning augmented mechanisms through randomization. We show our ideas can achieve better performance in the learning augmented setting in $\mathbb{R}^2$, where in addition to the input the mechanism also receives predictions. For the output prediction model of Agrawal et al. 2022 we show an improved expected consistency-robustness trade-off. Our results also imply improved performance for the input MAC predictions model of Barak et al. 2024. 3. The limitations of Random Dictators. We show a lower bound for the common mechanism class of GRD (Generalized Random Dictator) mechanisms, where only locations reported by the agents may be returned. We show that any GRD mechanism has a larger expected approximation ratio than our RR-CWM mechanism, as our lower bound for $\mathbb{R}^2$ is $\frac{4}π$ (matching the upper bound of RR-CWM, which is not a GRD mechanism). For $\mathbb{R}^d$, we show a lower bound of $\sqrt{2} - O(1/d)$.

翻译：我们研究设施选址机制设计问题，其中 $n$ 个代理报告其在欧几里得空间中的位置，输出为单一设施选址。每个代理的成本函数是到返回设施的距离，目标是以策略证明的方式最小化社会成本函数（代理成本之和）。我们的贡献：1. 突破确定性障碍。对于 $\mathbb{R}^2$，我们给出一个随机策略证明机制（RR-CWM），其期望近似比为 $\frac{4}{\pi} \approx 1.27$，严格优于最佳确定性策略证明机制（其比率为 $\sqrt{2} \approx 1.41$）。这解决了 $\mathbb{R}^2$ 中功利主义设施选址机制设计的确定性与随机机制分离的开放问题。对于 $\mathbb{R}^d$，我们证明该机制的期望近似比在 $[1.41 - O(1/\sqrt{d}), 1.547]$ 范围内。2. 通过随机化改进学习增强机制。我们证明，在 $\mathbb{R}^2$ 的学习增强设置中（除输入外，机制还接收预测），我们的思想可实现更优性能。针对 Agrawal 等人 2022 年的输出预测模型，我们展示了改进的期望一致性-鲁棒性权衡。我们的结果也暗示了对 Barak 等人 2024 年输入 MAC 预测模型的性能提升。3. 随机独裁的局限性。我们对 GRD（广义随机独裁）机制这一常见机制类给出下界，该机制仅能返回代理报告的位置。我们证明任何 GRD 机制都有比我们 RR-CWM 机制更大的期望近似比，因为对于 $\mathbb{R}^2$，下界为 $\frac{4}{\pi}$（与 RR-CWM 的上界匹配，而 RR-CWM 并非 GRD 机制）。对于 $\mathbb{R}^d$，我们证明下界为 $\sqrt{2} - O(1/d)$。