Mechanism Design for Locating Facilities with Capacities with Insufficient Resources

This paper explores the Mechanism Design aspects of the $m$-Capacitated Facility Location Problem where the total facility capacity is less than the number of agents. Following the framework outlined by Aziz et al., the Social Welfare of the facility location is determined through a First-Come-First-Served (FCFS) game, in which agents compete once the facility positions are established. When the number of facilities is $m > 1$, the Nash Equilibrium (NE) of the FCFS game is not unique, making the utility of the agents and the concept of truthfulness unclear. To tackle these issues, we consider absolutely truthful mechanisms, i.e. mechanisms that prevent agents from misreporting regardless of the strategies used during the FCFS game. We combine this stricter truthfulness requirement with the notion of Equilibrium Stable (ES) mechanisms, which are mechanisms whose Social Welfare does not depend on the NE of the FCFS game. We demonstrate that the class of percentile mechanisms is absolutely truthful and identify the conditions under which they are ES. We also show that the approximation ratio of each ES percentile mechanism is bounded and determine its value. Notably, when all the facilities have the same capacity and the number of agents is sufficiently large, it is possible to achieve an approximation ratio smaller than $1+\frac{1}{2m-1}$. Finally, we extend our study to encompass higher-dimensional problems. Within this framework, we demonstrate that the class of ES percentile mechanisms is even more restricted and characterize the mechanisms that are both ES and absolutely truthful. We further support our findings by empirically evaluating the performance of the mechanisms when the agents are the samples of a distribution.