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

TL;DR

The paper addresses the problem of understanding how well the coordinate-wise median performs as a truthful mechanism for facility location in ${\mathbb{R}}^d$ under ${\ell}_q$ costs. It introduces a tractable optimization framework to bound the worst-case approximation ratio, proving constant, dimension-free upper bounds $UB(q)$ with $UB(2)=\sqrt{6\sqrt{3}-8}$ and $UB(\infty)=3$, and showing $LB(q,d)=UB(q)(1- O(1/d))$—i.e., near-tightness as $d$ grows. It also extends these results to the generalized median CMP$(c)$ in the ${\ell}_2$ setting across arbitrary dimensions, providing explicit consistency and robustness guarantees and illustrating only modest degradation relative to the two-dimensional case. Together, these results significantly strengthen the understanding of median-based mechanisms in high-dimensional strategic facility location and inform learning-augmented mechanism design.

Abstract

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 $Θ(\sqrt{d})$. 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.

Figures (2)

• Figure 1: UB(q)
• Figure 2: plot of $r_a$ and $r_b$

Theorems & Definitions (28)

• Claim 1
• proof
• Claim 2
• proof
• Lemma 1: Optimal $\mathbf{p}$ with given signature $\sigma^o$
• proof
• Lemma 2: Expression of $g(\mathbf{p})$ as a function of $\Delta_{_S}$
• proof : Proof sketch
• Theorem 1
• proof