The Facility Location Problem with Aleatory Agents

In this paper, we introduce and study the Facility Location Problem with Aleatory Agents (FLPAA), where the facility can accommodate a number of agents, denoted as n, which is larger than the number of agents reporting their preferences, denoted as n_r. The spare capacity is used by nu := n - nr aleatory agents distributed according to μ. The goal of FLPAA is to find a location y that minimises the ex-ante social cost, defined as the expected cost of the nu agents sampled from μ plus the classic social cost incurred by the agents reporting their position. We investigate the mechanism design aspects of the FLPAA under the assumption that the mechanism designer lacks knowledge of the distribution μ but can query k quantiles of μ. 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 mechanism and the cost of the optimal solution on the worst-case input ⃗x and worst-case distribution μ, offering a stringent metric that does not depend on μ. In most cases, the lower bound matches the upper bound, proving that our mechanisms are tight, thus no truthful mechanism can achieve a lower SAR. Lastly, we extend our study to the case in which we must locate two facilities.