In this paper, we introduce and study the Facility Location Problem with Aleatory Agents (FLPAA), where the facility accommodates n agents larger than the number of agents reporting their preferences, namely n_r. The spare capacity is used by n_u=n-n_r aleatory agents sampled from a probability distribution \mu. The goal of FLPAA is to find a location that minimizes the ex-ante social cost, which is the expected cost of the n_u agents sampled from \mu plus the cost incurred by the agents reporting their position. We investigate the mechanism design aspects of the FLPAA under the assumption that the Mechanism Designer (MD) lacks knowledge of the distribution $\mu$ but can query k quantiles of \mu. We explore the trade-off between acquiring more insights into the probability distribution and designing a better-performing mechanism, which we describe through the strong approximation ratio (SAR). The SAR of a mechanism measures the highest ratio between the cost of the mechanisms and the cost of the optimal solution on the worst-case input x and worst-case distribution \mu, offering a metric for efficiency that does not depend on \mu. We divide our study into four different information settings: the zero information case, in which the MD has access to no quantiles; the median information case, in which the MD has access to the median of \mu; the n_u-quantile information case, in which the MD has access to n_u quantiles of its choice, and the k-quantile information case, in which the MD has access to k<n_u quantiles of its choice. For all frameworks, we propose a mechanism that is optimal or achieves a small constant SAR and pairs it with a lower bound on the SAR. In most cases, the lower bound matches the upper bound, thus no truthful mechanism can achieve a lower SAR. Lastly, we extend the FLPAA to include instances in which we must locate two facilities.

翻译：本文提出并研究了随机代理设施选址问题，其中设施可容纳的代理数量n大于实际报告偏好的代理数量n_r。剩余容量将由从概率分布μ中抽取的n_u=n-n_r个随机代理使用。FLPAA的目标是找到一个位置，以最小化事前社会成本，即从μ中抽取的n_u个代理的期望成本加上报告自身位置的代理所产生的成本。我们在机制设计者不知晓分布μ但可查询μ的k个分位数的假设下，研究FLPAA的机制设计问题。我们探讨了获取更多概率分布信息与设计性能更优机制之间的权衡，这一权衡通过强近似比进行刻画。机制的SAR衡量了在最坏输入x和最坏分布μ下，机制成本与最优解成本之间的最大比值，从而提供了一种不依赖于μ的效率度量标准。我们将研究分为四种不同的信息情境：零信息情境（MD无法获取任何分位数）、中位数信息情境（MD可获取μ的中位数）、n_u分位数信息情境（MD可自主选择n_u个分位数）以及k分位数信息情境（MD可自主选择k<n_u个分位数）。针对所有框架，我们提出了达到最优或较小常数SAR的机制，并给出了相应的SAR下界。在多数情况下，该下界与上界匹配，表明任何真实机制都无法获得更低的SAR。最后，我们将FLPAA扩展至需要部署两个设施的场景。