  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.
