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个智能体报告其在欧几里得空间中的位置，输出为一个单一设施的位置。每个智能体的成本函数为与返回设施之间的距离，目标是在策略规避方式下最小化社会成本函数（智能体成本之和）。我们的贡献如下：1. 突破确定性障碍。对于ℝ²，我们给出一个随机策略规避机制（RR-CWM），其期望近似比为4/π ≈ 1.27，严格优于最佳确定性策略规避机制（其近似比为√2 ≈ 1.41）。这解决了ℝ²中功利主义设施选址机制设计问题上确定性机制与随机机制分离的开放问题。对于ℝ^d，我们证明该机制的期望近似比位于[1.41 - O(1/√d), 1.547]区间内。2. 通过随机化改进学习增强机制。我们证明在ℝ²的学习增强设置中，我们的思路可实现更优性能，该设置下机制除输入外还接收预测。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。我们的结果还意味着Barak等人2024年输入MAC预测模型性能的改进。3. 随机独裁者的局限性。我们给出通用机制类GRD（广义随机独裁者）机制的下界，该类机制仅能返回智能体报告的位置。我们证明任何GRD机制的期望近似比均大于我们的RR-CWM机制，因为ℝ²的下界为4/π（与RR-CWM的上界匹配，而RR-CWM并非GRD机制）。对于ℝ^d，我们证明下界为√2 - O(1/d)。