  In this paper, we investigate the $k$-Facility Location Problem ($k$-FLP)
within the Bayesian Mechanism Design framework, in which agents' preferences
are samples of a probability distributed on a line. Our primary contribution is
characterising the asymptotic behavior of percentile mechanisms, which varies
according to the distribution governing the agents' types. To achieve this, we
connect the $k$-FLP and projection problems in the Wasserstein space. Owing to
this relation, we show that the ratio between the expected cost of a percentile
mechanism and the expected optimal cost is asymptotically bounded. Furthermore,
we characterize the limit of this ratio and analyze its convergence speed. Our
asymptotic study is complemented by deriving an upper bound on the Bayesian
approximation ratio, applicable when the number of agents $n$ exceeds the
number of facilities $k$. We also characterize the optimal percentile mechanism
for a given agent's distribution through a system of $k$ equations. Finally, we
estimate the optimality loss incurred when the optimal percentile mechanism is
derived using an approximation of the agents' distribution rather than the
actual distribution.
