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The k-Facility Location Problem Via Optimal Transport: A Bayesian Study of the Percentile Mechanisms

Gennaro Auricchio, Jie Zhang

TL;DR

The paper studies the $k$-Facility Location Problem on the line under a Bayesian framework, focusing on percentile mechanisms. It establishes a tight connection to Wasserstein projection problems from Optimal Transport, enabling an exact asymptotic characterization of the Bayesian approximation ratio for percentile mechanisms in terms of transport distances $W_1$. It proves the existence of an optimal percentile vector $\\mathbf{v}_{\\mu}$ for any distribution $\\mu$, characterized by a system of $k$ equations, and shows scale-invariance and robustness to distributional mis-specification via $W_\infty$-based bounds. The results yield an $O(n^{-1/2})$ convergence rate under mild moment assumptions and provide concrete computations for common distributions (Uniform, Gaussian, Exponential). Overall, the work guides mechanism design for location problems under uncertainty by linking percentile rules to optimal transport geometry.

Abstract

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.

The k-Facility Location Problem Via Optimal Transport: A Bayesian Study of the Percentile Mechanisms

TL;DR

The paper studies the -Facility Location Problem on the line under a Bayesian framework, focusing on percentile mechanisms. It establishes a tight connection to Wasserstein projection problems from Optimal Transport, enabling an exact asymptotic characterization of the Bayesian approximation ratio for percentile mechanisms in terms of transport distances . It proves the existence of an optimal percentile vector for any distribution , characterized by a system of equations, and shows scale-invariance and robustness to distributional mis-specification via -based bounds. The results yield an convergence rate under mild moment assumptions and provide concrete computations for common distributions (Uniform, Gaussian, Exponential). Overall, the work guides mechanism design for location problems under uncertainty by linking percentile rules to optimal transport geometry.

Abstract

In this paper, we investigate the -Facility Location Problem (-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 -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 exceeds the number of facilities . We also characterize the optimal percentile mechanism for a given agent's distribution through a system of 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.
Paper Structure (13 sections, 5 theorems, 41 equations)

This paper contains 13 sections, 5 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. Preliminaries
  5. The k-Facility Location Problem.
  6. Mechanism Design and the Worst-Case analysis.
  7. Bayesian Analysis.
  8. The Percentile Mechanisms.
  9. Basic Notions on Optimal Transport.
  10. Basic Assumptions
  11. The Bayesian Analysis of the Percentile Mechanism
  12. The k-FLP as a Wasserstein Projection problem
  13. The Bayesian Analysis of the Percentile Mechanisms

Key Result

theorem thmcountertheorem

Let $\hbox{\boldmath$x$}$ be the reports of $n$ agents. Let $\hbox{\boldmath$y$}$ be the solution to the $k$-FLP, i.e. the facility locations that minimize the Social Cost. Then the set $\{y_j\}_{j\in [k]}$ is the support of a measure $\nu^{(k,n)}$ that solves problem eq:projection_problem_gen. More Vice-versa, if $\nu\in\mathcal{P}_k(\mathbb{R})$ is a solution to problem eq:projection_problem_gen

Theorems & Definitions (10)

  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • theorem thmcountertheorem
  • proof