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

TL;DR

The paper addresses facility location under scarce capacity where agents report positions probabilistically. It builds a bridge between Bayesian mechanism design and Optimal Transport to characterize the asymptotic Social Welfare and identify optimal percentile mechanisms for one facility, extending to two facilities with Equilibrium Stability considerations. For monotone, single-peaked, and single-dipped distributions, it provides analytic or computable prescriptions for the best percentile mechanisms, and shows robust performance through numerical experiments, including Beta-distributed and non-identically distributed populations. The results offer a principled, distribution-aware approach to facility placement that yields near-optimal welfare and rapid convergence, with practical implications for planning under capacity constraints in public services and related domains.

Abstract

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.

Figures (8)

• Figure 1: In this Figure we plot the agents distribution $f_\mu$ along with the facility identified by the Median Mechanism (in red) and the facility identified by the Decile Mechanism (in green). The red area describes the set of agents accommodated by the red facility, while the green area represents the set of agents accommodated by the green facility.
• Figure 2: Bayesian approximation ratio attained by the Mechanism found in Theorem \ref{['thm:optmechanism']} or algorithmically. In the leftmost and central plot, we plot the absolute error incurred when $\mu\sim\mathcal{B}(2,2)$ and $\mu\sim\mathcal{B}(6,2)$, respectively. Finally, the rightmost plot reports the error incurred when the agents are not identically distributed.
• Figure 3: Logaritmic plot of the absolute error between the expected Social Welfare attained by the Mechanism characterized in Theorem \ref{['thm:optmechanism']} or algorithmically. In the leftmost and central plot, we report the absolute error incurred when $\mu\sim\mathcal{B}(2,2)$ and $\mu\sim\mathcal{B}(6,2)$, respectively. Finally, the rightmost plot reports the error incurred when the agents are not identically distributed. In all three figures, we plot the function $\frac{1}{\sqrt{n}}$ (in pink) along the errors for comparison.
• Figure 4: Results for two facilities with different capacity vectors $\vec{q}=(q_1,q_2)$. The agents are sampled from a Beta distribution with parameter $\alpha=6$ and $\beta=2$. On the left-hand side, we plot the Bayesian approximation ratio. On the right-hand side, we plot the absolute error in logarithmic scale along with the function $\frac{1}{\sqrt{n}}$ (in pink) for comparison.
• Figure 5: Bayesian approximation ratio attained by the Mechanism characterized in Theorem \ref{['thm:optmechanism']} or algorithmically to locate one facility. Each plot showcases the result for a different eta distribution and for different capacity vectors..

Theorems & Definitions (37)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Example 1
• Theorem 5
• Lemma 1
• Theorem 6
• Theorem 7
• Theorem 8