We consider the facility location problem in two dimensions. In particular, we consider a setting where agents have Euclidean preferences, defined by their ideal points, for a facility to be located in $\mathbb{R}^2$. For the utilitarian objective and an odd number of agents, we show that the coordinate-wise median mechanism (CM) has a worst-case approximation ratio (WAR) of $\sqrt{2}\frac{\sqrt{n^2+1}}{n+1}$. Further, we show that CM has the lowest WAR for this objective in the class of strategyproof, anonymous, continuous mechanism. For the $p-norm$ social welfare objective, we find that the WAR for CM is bounded above by $2^{\frac{3}{2}-\frac{2}{p}}$ for $p\geq 2$. Since it follows from previous results in one-dimension that any deterministic strategyproof mechanism must have WAR at least $2^{1-\frac{1}{p}}$, our upper bound guarantees that the CM mechanism is very close to being the best deterministic strategyproof mechanism for $p\geq 2$.

中文翻译：

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坐标中位数：不错，不错，不错

我们从两个维度考虑设施选址问题。特别地，我们考虑了一个设置，其中代理具有欧几里德偏好，由他们的理想点定义，设施位于 $\mathbb{R}^2$。对于功利目标和奇数个代理，我们表明坐标中值机制 (CM) 的最坏情况近似比 (WAR) 为 $\sqrt{2}\frac{\sqrt{n^2+ 1}}{n+1}$。此外，我们表明 CM 在策略证明、匿名、连续机制类中为此目标具有最低的 WAR。对于 $p-norm$ 社会福利目标，我们发现 CM 的 WAR 在 $2^{\frac{3}{2}-\frac{2}{p}}$ 上为 $p\geq 2 $. 由于从先前的一维结果得出，任何确定性策略证明机制必须具有至少 $2^{1-\frac{1}{p}}$ 的 WAR，