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.

翻译：本文在贝叶斯机制设计框架下研究k设施选址问题，其中智能体偏好服从直线上概率分布的抽样样本。我们的核心贡献在于刻画了百分位数机制的渐近行为特征，该行为随智能体类型分布规律的变化而改变。为实现这一目标，我们将k设施选址问题与Wasserstein空间中的投影问题建立联系。基于该关联关系，我们证明百分位数机制的期望成本与期望最优成本之比具有渐近有界性。进一步地，我们刻画了该比值的极限特征并分析了其收敛速度。通过推导适用于智能体数量n大于设施数量k情形的贝叶斯近似比上界，我们对渐近研究进行了补充。同时，我们通过k元方程组刻画了给定智能体分布下的最优百分位数机制。最后，我们评估了当使用智能体分布的近似而非真实分布推导最优百分位数机制时所产生的优化损失。