Designing Optimal Mechanisms to Locate Facilities with Insufficient Capacity for Bayesian Agents

In this paper, we study the Facility Location Problem with Scarce Resources (FLPSR) under the assumption that agents' type follow a probability distribution. In the FLPSR, the objective is to identify the optimal locations for one or more capacitated facilities to maximize Social Welfare (SW), defined as the sum of the utilities of all agents. The total capacity of the facilities, however, is not enough to accommodate all the agents, who thus compete in a First-Come-First-Served game to determine whether they get accommodated and what their utility is. The main contribution of this paper ties Optimal Transport theory to the problem of determining the best truthful mechanism for the FLPSR tailored to the agents' type distributions. Owing to this connection, we identify the mechanism that maximizes the expected SW as the number of agents goes to infinity. For the case of a single facility, we show that an optimal mechanism always exists. We examine three classes of probability distributions and characterize the optimal mechanism either analytically represent the optimal mechanism or provide a routine to numerically compute it. We then extend our results to the case in which we have two capacitated facilities to place. While we initially assume that agents are independent and identically distributed, we show that our techniques are applicable to scenarios where agents are not identically distributed. Finally, we validate our findings through several numerical experiments, including: (i) deriving optimal mechanisms for the class of beta distributions, (ii) assessing the Bayesian approximation ratio of these mechanisms for small numbers of agents, and (iii) assessing how quickly the expected SW attained by the mechanism converges to its limit.