The coordinate-wise median is a classic and most well-studied strategy-proof mechanism in social choice and facility location scenarios. Surprisingly, there is no systematic study of its approximation ratio in $d$-dimensional spaces. The best known approximation guarantee in $d$-dimensional Euclidean space $\mathbb{L}_2(\mathbb{R}^d)$ is $\sqrt{d}$ via embedding $\mathbb{L}_1(\mathbb{R}^d)$ into $\mathbb{L}_2(\mathbb{R}^d)$ metric space, that only appeared in appendix of [Meir 2019].This upper bound is known to be tight in dimension $d=2$, but there are no known super constant lower bounds. Still, it seems that the community's belief about coordinate-wise median is on the side of $\Theta(\sqrt{d})$. E.g., a few recent papers on mechanism design with predictions [Agrawal, Balkanski, Gkatzelis, Ou, Tan 2022], [Christodoulou, Sgouritsa, Vlachos 2024], and [Barak, Gupta, Talgam-Cohen 2024] directly rely on the $\sqrt{d}$-approximation result.In this paper, we systematically study approximate efficiency of the coordinate-median in $\mathbb{L}_{q}(\mathbb{R}^d)$ spaces for any $\mathbb{L}_q$ norm with $q\in[1,\infty]$ and any dimension $d$. We derive a series of constant upper bounds $UB(q)$ independent of the dimension $d$. This series $UB(q)$ is growing with parameter $q$, but never exceeds the constant $UB(\infty)= 3$. Our bound $UB(2)=\sqrt{6\sqrt{3}-8}<1.55$ for $\mathbb{L}_2$ norm is only slightly worse than the tight approximation guarantee of $\sqrt{2}>1.41$ in dimension $d=2$. Furthermore, we show that our upper bounds are essentially tight by giving almost matching lower bounds $LB(q,d)=UB(q)\cdot(1-O(1/d))$ for any dimension $d$ with $LB(q,d)=UB(q)$ when $d\to\infty$. We also extend our analysis to the generalized median mechanism in [Agrawal, Balkanski, Gkatzelis, Ou, Tan 2022] for $\mathbb{L}_2(\mathbb{R}^2)$ space to arbitrary dimensions $d$ with similar results.

翻译：坐标中位数是社会选择和设施选址场景中一个经典且被研究得最为透彻的策略证明机制。令人惊讶的是，对于其在$d$维空间中的近似比，目前尚缺乏系统性的研究。在$d$维欧几里得空间$\mathbb{L}_2(\mathbb{R}^d)$中，已知的最佳近似保证是$\sqrt{d}$，这是通过将$\mathbb{L}_1(\mathbb{R}^d)$嵌入到$\mathbb{L}_2(\mathbb{R}^d)$度量空间得到的，该结果仅出现在[Meir 2019]的附录中。已知该上界在维度$d=2$时是紧的，但目前尚未知存在超常数的下界。尽管如此，学界对于坐标中位数的普遍看法似乎倾向于$\Theta(\sqrt{d})$。例如，最近几篇关于带预测的机制设计的论文[Agrawal, Balkanski, Gkatzelis, Ou, Tan 2022]、[Christodoulou, Sgouritsa, Vlachos 2024]和[Barak, Gupta, Talgam-Cohen 2024]都直接依赖于$\sqrt{d}$-近似的结果。在本文中，我们系统地研究了坐标中位数在任意$\mathbb{L}_q$范数（$q\in[1,\infty]$）和任意维度$d$的$\mathbb{L}_{q}(\mathbb{R}^d)$空间中的近似效率。我们推导出了一系列与维度$d$无关的常数上界$UB(q)$。该上界序列$UB(q)$随参数$q$增长，但始终不超过常数$UB(\infty)= 3$。我们对于$\mathbb{L}_2$范数得到的界$UB(2)=\sqrt{6\sqrt{3}-8}<1.55$，仅略逊于在维度$d=2$时的紧近似保证$\sqrt{2}>1.41$。此外，我们证明了我们的上界本质上是紧的，因为我们给出了对于任意维度$d$几乎匹配的下界$LB(q,d)=UB(q)\cdot(1-O(1/d))$，且当$d\to\infty$时，$LB(q,d)=UB(q)$。我们还将我们的分析扩展到[Agrawal, Balkanski, Gkatzelis, Ou, Tan 2022]中针对$\mathbb{L}_2(\mathbb{R}^2)$空间的广义中位数机制，并将其推广到任意维度$d$，得到了类似的结果。