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Facility Location Games with Scaling Effects

Yu He, Alexander Lam, Minming Li

TL;DR

This paper introduces a multiplicative scaling model for facility location where each agent's cost is $c_i(q,y)=q(y)|y-x_i|$, with a continuous scaling function $q$ that depends on facility placement. It characterizes when agent preferences are single-peaked, identifies phantom-mechanism with scaling as the class of continuous, strategyproof, anonymous mechanisms, and derives tight worst-case approximation ratios for total and maximum cost under both continuous and piecewise-linear scaling. Key results show that, in general, the optimal mechanisms are not strategyproof, but under conditions guaranteeing single-peakedness, phantom mechanisms with scaling attain best possible performance within lower bounds that scale with $r_q$ and the number of linear pieces. The work provides a foundation for scalable, strategyproof facility-location mechanisms in settings where facility effectiveness varies with location, with implications for real-world deployment and extensions to multi-facility and multidimensional problems.

Abstract

We take the classic facility location problem and consider a variation, in which each agent's individual cost function is equal to their distance from the facility multiplied by a scaling factor which is determined by the facility placement. In addition to the general class of continuous scaling functions, we also provide results for piecewise linear scaling functions which can effectively approximate or model the scaling of many real world scenarios. We focus on the objectives of total and maximum cost, describing the computation of the optimal solution. We then move to the approximate mechanism design setting, observing that the agents' preferences may no longer be single-peaked. Consequently, we characterize the conditions on scaling functions which ensure that agents have single-peaked preferences. Under these conditions, we find a characterization of continuous, strategyproof, and anonymous mechanisms, and compute the total and maximum cost approximation ratios achievable by these mechanisms.

Facility Location Games with Scaling Effects

TL;DR

This paper introduces a multiplicative scaling model for facility location where each agent's cost is , with a continuous scaling function that depends on facility placement. It characterizes when agent preferences are single-peaked, identifies phantom-mechanism with scaling as the class of continuous, strategyproof, anonymous mechanisms, and derives tight worst-case approximation ratios for total and maximum cost under both continuous and piecewise-linear scaling. Key results show that, in general, the optimal mechanisms are not strategyproof, but under conditions guaranteeing single-peakedness, phantom mechanisms with scaling attain best possible performance within lower bounds that scale with and the number of linear pieces. The work provides a foundation for scalable, strategyproof facility-location mechanisms in settings where facility effectiveness varies with location, with implications for real-world deployment and extensions to multi-facility and multidimensional problems.

Abstract

We take the classic facility location problem and consider a variation, in which each agent's individual cost function is equal to their distance from the facility multiplied by a scaling factor which is determined by the facility placement. In addition to the general class of continuous scaling functions, we also provide results for piecewise linear scaling functions which can effectively approximate or model the scaling of many real world scenarios. We focus on the objectives of total and maximum cost, describing the computation of the optimal solution. We then move to the approximate mechanism design setting, observing that the agents' preferences may no longer be single-peaked. Consequently, we characterize the conditions on scaling functions which ensure that agents have single-peaked preferences. Under these conditions, we find a characterization of continuous, strategyproof, and anonymous mechanisms, and compute the total and maximum cost approximation ratios achievable by these mechanisms.
Paper Structure (13 sections, 28 theorems, 17 equations, 1 figure, 1 table)

This paper contains 13 sections, 28 theorems, 17 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Model
  5. Computing the Optimal Solution
  6. Total Cost
  7. Maximum Cost
  8. Achieving Single-Peaked Preferences
  9. Characterization of Strategyproof and Anonymous Mechanisms
  10. Approximation Ratio Results
  11. Total Cost
  12. Maximum Cost
  13. Discussion

Key Result

proposition 1

There exists a scaling function with a discontinuity and an agent location profile such that the optimal facility locations for total cost $y^*_{TC}$ and maximum cost $y^*_{MC}$ are not well-defined.

Figures (1)

  • Figure 1: Facility location example where the agents ($\bullet$) are located at $(0,0,0.6)$, and the scaling function (represented by the red lines) is $2-3y$ for $y\leq 0.5$, and $3y-1$ for $y>0.5$. The median and midpoint facility placements are at $y_{med}=0$ and $y_{mid}=0.3$ respectively, whilst the optimal facility placements in terms of total and maximum cost are at $y^*_{TC}=y^*_{MC}=0.5$.

Theorems & Definitions (54)

  • definition 1
  • proposition 1
  • proof
  • proposition 2
  • proof
  • theorem 1
  • proof
  • corollary 1
  • definition 2: Strategyproofness
  • proposition 3
  • ...and 44 more