欧几里得空间中的策略证明聚合：刚性与中位数最优性
Strategyproof Aggregation in Euclidean Spaces: Rigidity and Median Optimality
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中文总结 AI 辅助
本文证明在有限维欧几里得空间中，对于连续、匿名、确定性策略证明机制，坐标中位数在总欧几里得距离的最坏情况近似比上达到最优，适用于任意奇数或特定偶数个智能体。
中文摘要 AI 辅助
我们研究了有限维欧几里得空间中的确定性策略证明聚合问题。对于任意奇数个智能体 $n\ge3$ 和任意有限维度，我们证明了在所有连续、匿名、确定性的策略证明机制中，坐标中位数最小化了总欧几里得距离的最坏情况近似比。当每个坐标使用固定的下或上中位秩选择时，同样的最优性结果也适用于每个偶数 $n\ge4$。该证明结合了一个刚性定理与一个不增加近似比的归一化：任何假设优于中位数的机制都有一个归一化代表，该代表在单一正交框架中是固定的坐标顺序统计规则。然后，一个反射论证表明，没有这样的规则能优于中位数。
英文摘要
We study deterministic strategyproof aggregation in finite-dimensional Euclidean spaces. For every odd number $n\ge3$ of agents and every finite dimension, we prove that the coordinate-wise median minimizes the worst-case approximation ratio for total Euclidean distance among all continuous, anonymous, deterministic strategyproof mechanisms. The same optimality result holds for every even $n\ge4$ when each coordinate uses a fixed choice of the lower or upper middle rank. The proof combines a rigidity theorem with a normalization that does not increase the approximation ratio: any hypothetical mechanism outperforming the median has a normalized representative that is a fixed coordinate-wise order-statistic rule in a single orthonormal frame. A reflection argument then shows that no such rule improves on the median.
