AI 中文总结
该研究提出乘积间隙机制用于多设施选址，证明其对社会成本的近似比，分析其策略防卫性，还结合比例机制得到更优近似比的混合机制。
AI 中文摘要
我们研究在实数轴上定位多个设施的随机策略防卫机制。我们提出乘积间隙（Product-Gap）机制，该机制以所选k个报告位置之间连续间隙的乘积为概率比例选择k个报告位置，并在所选位置开设设施。我们证明，对于每个k≥2，该机制对社会成本实现了紧近似比2k。随后我们研究其激励性质，表明对于k=2和k=3，该机制是期望策略防卫的，但对于k≥4则不是策略防卫的。特别地，该机制分别为两个和三个设施提供策略防卫的4倍和6倍近似。最后，对于两个设施，我们将乘积间隙机制与Lu等人的比例（Proportional）机制相结合，证明一种优化后的与报告无关的混合机制是策略防卫的，且在实数轴上具有紧近似比(74+4√3)/23≈3.519，优于之前的因子4。
英文摘要
We study randomized strategyproof mechanisms for locating multiple facilities on the real line. We introduce the \emph{Product-Gap mechanism}, which selects $k$ reported locations with probability proportional to the product of the consecutive gaps between them and opens facilities at the selected locations. We prove that, for every $k\geq 2$, the mechanism achieves a tight approximation ratio of $2k$ for social cost. We then study its incentive properties and show that it is strategyproof in expectation for $k=2$ and $k=3$, but is not strategyproof for $k\geq 4$. In particular, the mechanism gives strategyproof $4$- and $6$-approximations for two and three facilities, respectively. Finally, for two facilities, we combine Product-Gap with the Proportional mechanism of Lu et al. We show that an optimized report-independent mixture is strategyproof and has a tight approximation ratio of $(74+4\sqrt{3})/23\approx 3.519$ on the line, improving upon the previous factor of $4$.
