We study strategyproof mechanisms for eliciting agents' location preferences truthfully in the Euclidean plane $\mathbb R^2$ and locating a facility so as to minimize the $L_p$-norm social cost, defined as the $L_p$-norm of the vector of distances from the facility to the agents' preferred locations, for any $p \ge 1$. While the cases $p=1$ and $p=\infty$ have been well-studied, open questions remain about the optimal approximation ratios achievable by strategyproof mechanisms for general $p$. Our first result resolves an open question of Goel and Hann-Caruthers [Soc. Choice Welf. 2023]. They showed that the coordinate-wise median (CM) mechanism achieves an approximation ratio lying between \(2^{1-\frac{1}{p}}\) and \(2^{\frac{3}{2}-\frac{2}{p}}\) for $p\ge 2$, and they conjectured that it is exactly \(2^{1-\frac{1}{p}}\). We confirm this conjecture, and we further show that CM has a tight $\sqrt 2$-approximation for $1\le p\le 2$. Our second and third results demonstrate that two randomized mechanisms can yield better approximation ratios. In particular, we first consider the uniformly rotated coordinate-wise median (URCM) mechanism, and prove that, for \(1\le p<2\), its approximation ratio strictly improves over the deterministic bound \(\sqrt{2}\), while no such improvement is possible for $p\ge 2$. We then study the centroid random dictatorship mechanism that returns the average location (i.e., centroid) and the random dictatorship each with half probability, and show that its approximation ratio strictly improves over CM and URCM for every finite \(p\gtrsim 1.6\). Moreover, our analysis independently recovers the classical deterministic and randomized results for $p=1$ [Meir, SAGT 2019] [Barak, EC 2026] and $p=\infty$ [Goel and Hann-Caruthers, SCW 2023] [Tang et al., EC 2020] using significantly different techniques.

翻译：我们研究在欧几里得平面 $\mathbb R^2$ 上，为如实获取代理人位置偏好并选址以最小化 $L_p$ 范数社会成本（定义为设施到代理人偏好位置的距离向量的 $L_p$ 范数，其中 $p \ge 1$）的防策略机制。尽管 $p=1$ 和 $p=\infty$ 的情形已得到充分研究，但关于一般 $p$ 下防策略机制所能达到的最优近似比仍存在开放问题。我们的第一个结果解决了 Goel 和 Hann-Caruthers [Soc. Choice Welf. 2023] 提出的一个开放问题。他们指出，坐标中位数机制在 $p\ge 2$ 时取得的近似比介于 \(2^{1-\frac{1}{p}}\) 和 \(2^{\frac{3}{2}-\frac{2}{p}}\) 之间，并猜想精确值为 \(2^{1-\frac{1}{p}}\)。我们证实了这一猜想，并进一步证明当 $1\le p\le 2$ 时，坐标中位数机制具有紧的 $\sqrt 2$ 近似比。我们的第二和第三个结果表明，两种随机化机制能实现更优的近似比。具体而言，我们首先考虑均匀旋转坐标中位数机制，证明当 \(1\le p<2\) 时，其近似比严格优于确定性下界 \(\sqrt{2}\)，而对于 $p\ge 2$ 则无法实现此类改进。随后我们研究以各 1/2 概率返回平均位置（即质心）和随机独裁的质心随机独裁机制，并证明其近似比对于每个有限 \(p\gtrsim 1.6\) 均严格优于坐标中位数机制和均匀旋转坐标中位数机制。此外，我们的分析采用显著不同的技术，独立复现了 $p=1$ [Meir, SAGT 2019] [Barak, EC 2026] 和 $p=\infty$ [Goel and Hann-Caruthers, SCW 2023] [Tang et al., EC 2020] 的经典确定性和随机化结果。