基于最优传输的k-设施选址问题：百分位机制的贝叶斯研究
The k-Facility Location Problem Via Optimal Transport: A Bayesian Study of the Percentile Mechanisms
AI总结：
该研究在贝叶斯机制设计框架下，通过关联k-设施选址问题与Wasserstein空间投影问题，刻画百分位机制的渐近行为、贝叶斯近似比上界及最优百分位机制，并估算分布近似带来的最优性损失。
摘要：
本文在贝叶斯机制设计框架下研究k-设施选址问题（k-FLP），其中智能体的偏好是直线上某概率分布的样本。我们的核心贡献是刻画百分位机制的渐近行为，其行为随支配智能体类型的分布不同而变化。为此，我们将k-FLP与Wasserstein空间中的投影问题建立关联。借助这一关联，我们证明百分位机制的期望成本与期望最优成本的比值具有渐近有界性，进一步刻画了该比值的极限并分析其收敛速度。我们还推导了贝叶斯近似比的一个上界，适用于智能体数量n超过设施数量k的情形，以此补充渐近研究的结论。此外，我们通过一个含k个方程的方程组，刻画了针对给定智能体分布的最优百分位机制。最后，我们估算了当使用智能体分布的近似分布而非真实分布推导最优百分位机制时产生的最优性损失。
英文摘要：
In this paper, we investigate the $k$-Facility Location Problem ($k$-FLP) within the Bayesian Mechanism Design framework, in which agents' preferences are samples of a probability distributed on a line. Our primary contribution is characterising the asymptotic behavior of percentile mechanisms, which varies according to the distribution governing the agents' types. To achieve this, we connect the $k$-FLP and projection problems in the Wasserstein space. Owing to this relation, we show that the ratio between the expected cost of a percentile mechanism and the expected optimal cost is asymptotically bounded. Furthermore, we characterize the limit of this ratio and analyze its convergence speed. Our asymptotic study is complemented by deriving an upper bound on the Bayesian approximation ratio, applicable when the number of agents $n$ exceeds the number of facilities $k$. We also characterize the optimal percentile mechanism for a given agent's distribution through a system of $k$ equations. Finally, we estimate the optimality loss incurred when the optimal percentile mechanism is derived using an approximation of the agents' distribution rather than the actual distribution.
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