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Mechanism Design for Locating Facilities with Capacities with Insufficient Resources

Gennaro Auricchio, Harry J. Clough, Jie Zhang

TL;DR

This work studies mechanism design for the $m$-Capacitated Facility Location Problem under scarce resources, where the total facility capacity is less than the number of agents and a FCFS game follows facility placement. It introduces absolute truthfulness and Equilibrium Stable (ES) mechanisms, and shows percentile mechanisms are absolutely truthful; it fully characterizes when they are ES and provides tight approximation guarantees, with notable results such as a $4/3$-approximation in the two-facility balanced case and sublinear-ar scaling when many facilities are present. The analysis extends to higher dimensions, revealing stronger ES restrictions (often requiring coordinates to be All-In-One or a single SBS among otherwise AIO CPMs), and demonstrates that ES percentile mechanisms perform well empirically under Bayesian and average-case evaluations. The findings advance understanding of mechanism design in capacitated facility location with incomplete resources and offer practical mechanisms with provable performance bounds, applicable to networks and resource allocation scenarios.

Abstract

This paper explores the Mechanism Design aspects of the $m$-Capacitated Facility Location Problem where the total facility capacity is less than the number of agents. Following the framework outlined by Aziz et al., the Social Welfare of the facility location is determined through a First-Come-First-Served (FCFS) game, in which agents compete once the facility positions are established. When the number of facilities is $m > 1$, the Nash Equilibrium (NE) of the FCFS game is not unique, making the utility of the agents and the concept of truthfulness unclear. To tackle these issues, we consider absolutely truthful mechanisms, i.e. mechanisms that prevent agents from misreporting regardless of the strategies used during the FCFS game. We combine this stricter truthfulness requirement with the notion of Equilibrium Stable (ES) mechanisms, which are mechanisms whose Social Welfare does not depend on the NE of the FCFS game. We demonstrate that the class of percentile mechanisms is absolutely truthful and identify the conditions under which they are ES. We also show that the approximation ratio of each ES percentile mechanism is bounded and determine its value. Notably, when all the facilities have the same capacity and the number of agents is sufficiently large, it is possible to achieve an approximation ratio smaller than $1+\frac{1}{2m-1}$. Finally, we extend our study to encompass higher-dimensional problems. Within this framework, we demonstrate that the class of ES percentile mechanisms is even more restricted and characterize the mechanisms that are both ES and absolutely truthful. We further support our findings by empirically evaluating the performance of the mechanisms when the agents are the samples of a distribution.

Mechanism Design for Locating Facilities with Capacities with Insufficient Resources

TL;DR

This work studies mechanism design for the -Capacitated Facility Location Problem under scarce resources, where the total facility capacity is less than the number of agents and a FCFS game follows facility placement. It introduces absolute truthfulness and Equilibrium Stable (ES) mechanisms, and shows percentile mechanisms are absolutely truthful; it fully characterizes when they are ES and provides tight approximation guarantees, with notable results such as a -approximation in the two-facility balanced case and sublinear-ar scaling when many facilities are present. The analysis extends to higher dimensions, revealing stronger ES restrictions (often requiring coordinates to be All-In-One or a single SBS among otherwise AIO CPMs), and demonstrates that ES percentile mechanisms perform well empirically under Bayesian and average-case evaluations. The findings advance understanding of mechanism design in capacitated facility location with incomplete resources and offer practical mechanisms with provable performance bounds, applicable to networks and resource allocation scenarios.

Abstract

This paper explores the Mechanism Design aspects of the -Capacitated Facility Location Problem where the total facility capacity is less than the number of agents. Following the framework outlined by Aziz et al., the Social Welfare of the facility location is determined through a First-Come-First-Served (FCFS) game, in which agents compete once the facility positions are established. When the number of facilities is , the Nash Equilibrium (NE) of the FCFS game is not unique, making the utility of the agents and the concept of truthfulness unclear. To tackle these issues, we consider absolutely truthful mechanisms, i.e. mechanisms that prevent agents from misreporting regardless of the strategies used during the FCFS game. We combine this stricter truthfulness requirement with the notion of Equilibrium Stable (ES) mechanisms, which are mechanisms whose Social Welfare does not depend on the NE of the FCFS game. We demonstrate that the class of percentile mechanisms is absolutely truthful and identify the conditions under which they are ES. We also show that the approximation ratio of each ES percentile mechanism is bounded and determine its value. Notably, when all the facilities have the same capacity and the number of agents is sufficiently large, it is possible to achieve an approximation ratio smaller than . Finally, we extend our study to encompass higher-dimensional problems. Within this framework, we demonstrate that the class of ES percentile mechanisms is even more restricted and characterize the mechanisms that are both ES and absolutely truthful. We further support our findings by empirically evaluating the performance of the mechanisms when the agents are the samples of a distribution.
Paper Structure (21 sections, 14 theorems, 44 equations, 10 figures)

This paper contains 21 sections, 14 theorems, 44 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Setting Statement
  3. First-Come-First-Served Game
  4. Mechanism Design Aspects of the problem
  5. Absolutely Truthful and Equilibrium Stable Mechanisms to place more than one facility
  6. Mechanisms to Place Two Facilities
  7. Wide-Gap mechanisms
  8. The All-In-One and Side-By-Side mechanisms
  9. Beyond two facilities
  10. Case Lg.
  11. Case Lg.
  12. The All-aside mechanisms
  13. The AIO and SBS mechanisms
  14. The Higher Dimensional Problem
  15. The Setting
  16. ...and 6 more sections

Key Result

Theorem 1

For every $\vec{x}\in[0,1]^n$, every $\vec{y}\in[0,1]^m$, and every capacity vector $\vec{k}$, the FCFS game associated with $\vec{x}$, $\vec{y}$, and $\vec{k}$ admits at least one pure Nash Equilibrium.

Figures (10)

  • Figure 1: The two Nash Equilibria described in Example \ref{['ex:1']}. The circles represents the agents, the green squares the facilities and the red arrows the strategies played by the agents getting a non-null utility.
  • Figure 2: The two Nash Equilibria described in Example \ref{['example2high']}. The circles represents the agents (agents from $1$ to $4$ are represented by $x_l$ while agents from $8$ to $10$ are represented by $x_j$), the green squares the facilities and the red arrows the strategies played by the agents getting a non-null utility in the two different Nash Equilibria.
  • Figure 3: The two Nash Equilibria described in Example \ref{['ex_high2']}. The circles represents the agents (agents from $5$ to $10$ are represented by $x_j$), the green squares the facilities and the red arrows the strategies played by the agents getting a non-null utility in the two different Nash Equilibria.
  • Figure 4: The way $A_1$, $A_2$, $A_3$, and $A_4$ partition of $[0,1]^2$ in Theorem \ref{['thm:ar_highdim']}.
  • Figure 5: The Bayesian approximation ratio of $\mathcal{PM}_{best}$ and $\mathcal{PM}_{\vec{w}}$ in the balanced case, i.e. $k_1=k_2$ for $n=10,20,\dots,50$. Every column contains the results for different vector $\vec{k}$. In the first row, we consider a uniform distribution. In the second row a symmetric Beta, that is $\mathcal{B}(5,5)$. The third row contains the results for the triangular distribution $\mathcal{T}$.
  • ...and 5 more figures

Theorems & Definitions (36)

  • Theorem 1
  • proof
  • Example 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4: Percentile Mechanism, sui2013analysis
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 26 more