Approximation guarantees of Median Mechanism in $\mathbb{R}^d$

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.