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Facility Location Problem with Aleatory Agents

Gennaro Auricchio, Jie Zhang

Abstract

In this paper, we introduce and study the Facility Location Problem with Aleatory Agents (FLPAA), where the facility accommodates n agents larger than the number of agents reporting their preferences, namely n_r. The spare capacity is used by n_u=n-n_r aleatory agents sampled from a probability distribution μ. The goal of FLPAA is to find a location that minimizes the ex-ante social cost, which is the expected cost of the n_u agents sampled from μplus the cost incurred by the agents reporting their position. We investigate the mechanism design aspects of the FLPAA under the assumption that the Mechanism Designer (MD) lacks knowledge of the distribution $μ$ but can query k quantiles of μ. We explore the trade-off between acquiring more insights into the probability distribution and designing a better-performing mechanism, which we describe through the strong approximation ratio (SAR). The SAR of a mechanism measures the highest ratio between the cost of the mechanisms and the cost of the optimal solution on the worst-case input x and worst-case distribution μ, offering a metric for efficiency that does not depend on μ. We divide our study into four different information settings: the zero information case, in which the MD has access to no quantiles; the median information case, in which the MD has access to the median of μ; the n_u-quantile information case, in which the MD has access to n_u quantiles of its choice, and the k-quantile information case, in which the MD has access to k<n_u quantiles of its choice. For all frameworks, we propose a mechanism that is optimal or achieves a small constant SAR and pairs it with a lower bound on the SAR. In most cases, the lower bound matches the upper bound, thus no truthful mechanism can achieve a lower SAR. Lastly, we extend the FLPAA to include instances in which we must locate two facilities.

Facility Location Problem with Aleatory Agents

Abstract

In this paper, we introduce and study the Facility Location Problem with Aleatory Agents (FLPAA), where the facility accommodates n agents larger than the number of agents reporting their preferences, namely n_r. The spare capacity is used by n_u=n-n_r aleatory agents sampled from a probability distribution μ. The goal of FLPAA is to find a location that minimizes the ex-ante social cost, which is the expected cost of the n_u agents sampled from μplus the cost incurred by the agents reporting their position. We investigate the mechanism design aspects of the FLPAA under the assumption that the Mechanism Designer (MD) lacks knowledge of the distribution but can query k quantiles of μ. We explore the trade-off between acquiring more insights into the probability distribution and designing a better-performing mechanism, which we describe through the strong approximation ratio (SAR). The SAR of a mechanism measures the highest ratio between the cost of the mechanisms and the cost of the optimal solution on the worst-case input x and worst-case distribution μ, offering a metric for efficiency that does not depend on μ. We divide our study into four different information settings: the zero information case, in which the MD has access to no quantiles; the median information case, in which the MD has access to the median of μ; the n_u-quantile information case, in which the MD has access to n_u quantiles of its choice, and the k-quantile information case, in which the MD has access to k<n_u quantiles of its choice. For all frameworks, we propose a mechanism that is optimal or achieves a small constant SAR and pairs it with a lower bound on the SAR. In most cases, the lower bound matches the upper bound, thus no truthful mechanism can achieve a lower SAR. Lastly, we extend the FLPAA to include instances in which we must locate two facilities.
Paper Structure (18 sections, 20 theorems, 63 equations, 1 figure, 1 table)

This paper contains 18 sections, 20 theorems, 63 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Setting Statement
  3. Computing the Optimal Solution of the FLPAA
  4. The Mechanism Design Aspects for the FLPAA
  5. Tailor-making a Mechanism to different Levels of Information
  6. Zero Information case
  7. k-Quantile Information Case
  8. Median Information Case
  9. k-Quantile Information Case
  10. Conclusion and Future Works
  11. Appendix
  12. Missing Proof
  13. Extension to Two Facilities
  14. Setting Statement
  15. Mechanism Design for the 2-FLPAA
  16. ...and 3 more sections

Key Result

Theorem 1

Given $\vec{x}\in\mathbb{R}^{n_r}$ and $\mu\in\mathcal{P}(\mathbb{R})$, every median of $F_{\lambda,\mu,\vec{x}}$ is a solution, thus $y^*=\inf\{y\in\mathbb{R},\;\;\text{s.t.}\;\; F_{\lambda,\mu,\vec{x}}(t)\ge \frac{1}{2}\}$ is a solution. Moreover, given $\vec{x}\in\mathbb{R}^{n_r}$ and $\mu\in\mat

Figures (1)

  • Figure 1: Two sequences of probability measures that concentrate the probability at one and two points, respectively. The sequence of measures on the left concentrates all the probability at $x_1$, and in the limit, it converges to a probability measure whose random variable is equal to $x_1$ with a probability of $1$. On the right, the sequence concentrates the probability at $x_1$ and $x_2$. In the limit, it converges to a probability measure whose associated random variable is equal to $x_1$ with a probability of $0.8$ or equal to $x_2$ otherwise.

Theorems & Definitions (54)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Definition 1
  • Theorem 3
  • Theorem 4
  • proof : Sketch of the Proof
  • Example 1
  • Theorem 5
  • proof : Sketch of the Proof
  • ...and 44 more