欧几里得平面中均匀旋转坐标中位数的精确近似比

中文摘要 AI 辅助

均匀旋转坐标中位数方法选择一个随机的标准正交坐标系，在每个坐标上取中位数，并将所得点映射回欧几里得平面。当社会成本为智能体欧几里得距离的$L_p$范数且$1<p<2$时，我们确定了其最坏情况下的期望近似比。该比值为$\frac{2^{2-1/p}}{\pi}\int_0^{\pi/2}(\cos^p\theta+\sin^p\theta)^{1/p}\dd\theta$。此表达式此前由Chan、Lin和Wang作为下界建立；我们的贡献是匹配的上界。证明建立了一个相对于任意参考点的加强的坐标中位数不等式。其右侧是方向相关范数之和的线性函数，这使得可以直接对旋转进行平均，而无需通过$p$阶矩界所带来的损失。我们给出了所有辅助不等式，并使用已建立的两簇加离群点构造提供了紧性的自包含证明。上界对每个有限配置以及坐标中位数区间内的每个可测选择都成立，而奇数规模的配置足以匹配下界。该结果刻画了这一固定机制，而非所有随机策略性防操纵机制中的最优近似比。

英文摘要

Uniformly rotated coordinate-wise median chooses a random orthonormal coordinate system, takes a median in each coordinate, and maps the resulting point back to the Euclidean plane. We determine its exact worst-case expected approximation ratio when social cost is the $L_p$ norm of the agents' Euclidean distances and $1<p<2$. The ratio is \(\frac{2^{2-1/p}}π\int_0^{π/2}(\cos^pθ+\sin^pθ)^{1/p}\ddθ\). This expression was previously established as a lower bound by Chan, Lin, and Wang; our contribution is the matching upper bound. The proof establishes a strengthened coordinate-wise median inequality relative to an arbitrary reference point. Its right-hand side is linear in a sum of direction-dependent norms, which permits direct averaging over rotations without the loss incurred by passing through a $p$th-moment bound. We give all auxiliary inequalities and a self-contained proof of tightness using the established two-cluster-and-outlier construction. The upper bound holds for every finite profile and every measurable choice within the coordinate median intervals, while odd-size profiles suffice for the matching lower bound. The result characterizes this fixed mechanism, rather than the optimal approximation ratio among all randomized strategyproof mechanisms.

发表机构

• City University of Hong Kong（香港城市大学）

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