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In this paper, we study of the $m$-Capacitated Facility Location Problem($m$-CFLP) on the line from a Bayesian Mechanism Design perspective and proposea novel class of mechanisms: the \textit{Extended Ranking Mechanisms} (ERMs).We first show that an ERM is truthful if and only if it satisfies a system ofinequalities that depends on the capacities of the facilities we need to place.We then establish a connection between the $m$-CFLP and a norm minimizationproblem in the Wasserstein space, which enables us to show that if the numberof agent goes to infinity the limit of the ratio between the expected SocialCost of an ERM and the expected optimal Social Cost is finite and characterizeits value. Noticeably, our method generalizes to encompass other truthfulmechanisms and other metrics, such as the $l_p$ and Maximum costs. We concludeour theoretical analysis by characterizing the optimal ERM tailored to a$m$-CFLP and a distribution $\mu$, that is the ERM whose limit Bayesianapproximation ratio is the lowest compared to all other feasible ERMs. Weconsider mainly two frameworks: (i) in the first framework, the total facilitycapacity matches the number of agents, (ii) in the second framework, $m=2$.When we consider the Maximum Cost, we retrieve the optimal ERM for every $\mu$,while for the Social Cost, we characterize the solution when the measure $\mu$is symmetric. Lastly, we numerically compare the performance of the ERMsagainst other truthful mechanisms and evaluate how quickly the Bayesianapproximation ratio converges to its limit.

From Optimal Transport to Efficient Mechanisms for the $m$-Capacitated Facilities Location Problem with Bayesian Agent