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Design and Characterization of Strategy-Proof Mechanisms for Two-Facility Game on a Line

Pinyan Lu, Zihan Luo, Jialin Zhang

TL;DR

This work investigates the two-facility location game on a line under deterministic, money-free, strategy-proof mechanisms with bounded social-cost approximation. It provides a refined characterization of mechanisms with a unique dictator, showing the second facility must satisfy tight end- or dictator-aligned constraints, and expands the landscape with three new classes of switching-point mechanisms that remain strategy-proof and have provable approximation ratios. The paper also studies mechanism design with prediction, proving that achieving sublinear consistency is impossible under bounded robustness, even with perfect predictions, establishing a fundamental consistency-robustness trade-off. Collectively, these results deepen the understanding of the structural limits of strategy-proof mechanisms in facility location and point toward broader implications for general facility-allocation problems on a line.

Abstract

We focus on the problem of placing two facilities along a linear space to serve a group of agents. Each agent is committed to minimizing the distance between her location and the closest facility. A mechanism is an algorithm that maps the reported agent locations to the facility locations. We are interested in mechanisms without money that are deterministic, strategy-proof, and provide a bounded approximation ratio for social cost. It is a fundamental problem to characterize the family of strategy-proof mechanisms with a bounded approximation ratio. Fotakis and Tzamos already demonstrated that the deterministic strategy-proof mechanisms for the 2-facility game problem are mechanisms with a unique dictator and the leftmost-rightmost mechanism. In this paper, we first present a more refined characterization of the first family. We then reveal three new classes of strategy-proof mechanisms that show the intricacy of structure within this family. This helps us get a more complete picture of the characterization of the 2-facility game problem, and may also have value in understanding and solving more general facility allocation game problems. Besides, based on our refined characterization, we surprisingly find that prediction cannot effectively improve the performance of the mechanism in the two-facility game problem, while this methodology to overcome bad approximation ratio works in many other mechanism design problems. We show that if we require that the mechanism admits a bounded approximation ratio when the prediction is arbitrarily bad, then at the same time, the mechanism can never achieve sublinear approximation ratios even with perfect prediction.

Design and Characterization of Strategy-Proof Mechanisms for Two-Facility Game on a Line

TL;DR

This work investigates the two-facility location game on a line under deterministic, money-free, strategy-proof mechanisms with bounded social-cost approximation. It provides a refined characterization of mechanisms with a unique dictator, showing the second facility must satisfy tight end- or dictator-aligned constraints, and expands the landscape with three new classes of switching-point mechanisms that remain strategy-proof and have provable approximation ratios. The paper also studies mechanism design with prediction, proving that achieving sublinear consistency is impossible under bounded robustness, even with perfect predictions, establishing a fundamental consistency-robustness trade-off. Collectively, these results deepen the understanding of the structural limits of strategy-proof mechanisms in facility location and point toward broader implications for general facility-allocation problems on a line.

Abstract

We focus on the problem of placing two facilities along a linear space to serve a group of agents. Each agent is committed to minimizing the distance between her location and the closest facility. A mechanism is an algorithm that maps the reported agent locations to the facility locations. We are interested in mechanisms without money that are deterministic, strategy-proof, and provide a bounded approximation ratio for social cost. It is a fundamental problem to characterize the family of strategy-proof mechanisms with a bounded approximation ratio. Fotakis and Tzamos already demonstrated that the deterministic strategy-proof mechanisms for the 2-facility game problem are mechanisms with a unique dictator and the leftmost-rightmost mechanism. In this paper, we first present a more refined characterization of the first family. We then reveal three new classes of strategy-proof mechanisms that show the intricacy of structure within this family. This helps us get a more complete picture of the characterization of the 2-facility game problem, and may also have value in understanding and solving more general facility allocation game problems. Besides, based on our refined characterization, we surprisingly find that prediction cannot effectively improve the performance of the mechanism in the two-facility game problem, while this methodology to overcome bad approximation ratio works in many other mechanism design problems. We show that if we require that the mechanism admits a bounded approximation ratio when the prediction is arbitrarily bad, then at the same time, the mechanism can never achieve sublinear approximation ratios even with perfect prediction.
Paper Structure (13 sections, 12 theorems, 9 equations, 3 figures)

This paper contains 13 sections, 12 theorems, 9 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Characterization of strategy-proof mechanisms
  3. Mechanism design with prediction
  4. Related works
  5. Preliminaries
  6. Property of Nice Mechanisms for Two-facility Game on a Line
  7. Two-Facility Game with Prediction
  8. Strategy-Proof Mechanisms
  9. Extension 1: Switching Points with Different Position
  10. Extension 2: Multiple Switching Points
  11. Extension 3: Switching Points Related to All Agents
  12. Conclusion
  13. Appendix A: Proving the robustness of Mechanism 1

Key Result

theorem 1

If $f$ is a nice mechanism for the two-facility game on a straight line applied to $n \geq 5$ agents, for arbitrary instances $\mathbf{x}$, it must satisfy the following property: assuming that $f(\mathbf{x}) = \{l_1, l_2\}$, either there exists $l \in f(\mathbf{x})$ such that $l \geq \max \mathbf{x

Figures (3)

  • Figure 1: The correspondence between $\mathbf{x}$ and $\mathbf{x}^\prime$. The positions labeled above the line segment are the locations of the points, and below are the agents or the sets of agents located at those positions. There is a possibility that min $\mathbf{x}$ and $x_j$ overlap, in which case $j \in S_l$.
  • Figure 2: The correspondence between $\mathbf{x}^\prime$ and $\mathbf{x}^{\prime\prime}$. The positions labeled above the line segment are the locations of the points, and below are the agents or the sets of agents located at those positions.
  • Figure 3: location profile $\mathbf{x}$. The labels below the line segment indicate the number of agents located at this position, while the labels above the line segment indicate the positions of the points.

Theorems & Definitions (22)

  • definition 1: strategy-proofness
  • theorem 1
  • lemma 1: Theorem 4.1 fotakis2014power
  • lemma 2: Corollary 3.2 lu2010asymptotically
  • proof
  • lemma 3: Corollary 6.1 fotakis2014power
  • proof
  • theorem 2
  • proof
  • theorem 3
  • ...and 12 more