Source-linked AI summary
Mechanism Design for Facility Location Games Under a Prelocated Facility
Genjie Qin, Qizhi Fang, Wenjing Liu
TL;DR
The paper asks how to truthfully locate a new homogeneous facility when a prelocated facility already exists, under maximum-cost and social-cost objectives. It studies line and circle models in general and one-sided settings, deriving approximation bounds for deterministic and randomized mechanisms. Its main results include tight or explicit bounds such as 2, 5/3, n, n−1, 1.5, 1.5−ε, and 1.0425, while leaving gaps between some deterministic upper and lower bounds.
Problem
The paper studies strategy-proof facility location with a prelocated facility, where privately located agents may lie and mechanisms should approximately optimize maximum or social cost.
Method
It models a publicly known existing facility, a privately reported agent profile, and deterministic or randomized mechanisms for line and circle settings with general or one-sided locations.
Results
The paper establishes approximation bounds including 2 for deterministic maximum cost on the general line setting, 5/3 for randomized maximum cost in the special setting, and 2 for deterministic maximum cost on the circle.
Takeaways & Limitations
The results provide strategy-proof approximation guarantees and introduce improvement ratio to assess how effectively a mechanism uses the added facility.
Takeaways & Limitations
For deterministic social cost, the paper leaves a large gap between the n upper bound and 1.5 lower bound and conjectures an Ω(n) lower bound without proving it.
Abstract
from arXiv · showhide
We study the problem of locating a new homogeneous facility under a prelocated facility. Here, a set of $n$ agents is located on a real line or a circle, each of whom has her location as private information, and her cost is the (expected) distance from her location to the nearest facility. Our goal is to design mechanisms which can approximately minimize the maximum cost or the social cost while eliciting agents' private information truthfully (i.e., strategy-proof). Based on real-life scenarios, we consider the problem in two settings: the general setting where each agent can be located at both sides of the prelocated facility, and the special setting where all the agents are located at the same side of the prelocated facility. For agents on a line, in the general setting, we design the best possible deterministic strategy-proof mechanism with $2$-approximation and provide a lower bound of $1.5-ε\textbf{ }(ε>0)$ for any randomized strategy-proof mechanism under the maximum cost objective. For the social cost, we obtain an upper bound of $n$ for deterministic strategy-proof mechanisms and lower bounds of $1.5$ and $1.0425$ for any deterministic strategy-proof mechanism and any randomized strategy-proof mechanism, respectively. In the special setting, we further provide a randomized strategy-proof $5/3$-approximation mechanism for the maximum cost and a deterministic strategy-proof $(n-1)$-approximation mechanism for the social cost. For agents on a circle, we provide a deterministic strategy-proof 2-approximation mechanism under the maximum cost objective.
1 Introduction
The paper studies strategy-proof mechanisms for locating a new homogeneous facility when a facility already exists, addressing both general and one-sided agent locations. It develops approximation bounds for line and circle settings and introduces improvement ratio as an additional performance measure.
- Motivation: Prelocated facilities matter because new facilities should account for existing resources in applications such as schools, seafood markets, and building services.
- Problem settings: The model covers a general setting with agents on either side of the prelocated facility and a special setting with all agents on one side.
- Additional measure: The improvement ratio measures the worst-case fraction of optimal improvement obtained by a mechanism when adding a facility.
- Contributions: The paper derives upper and lower approximation bounds for strategy-proof mechanisms on a line or circle under maximum-cost and social-cost objectives.
2 Model
The model assumes a publicly known facility on the real line and privately located agents who receive service from the nearest of the existing and new facilities. Mechanisms are evaluated for strategy-proofness and approximation under maximum and social cost.
- Model: A publicly known facility is fixed at y0 = 0, while each of n agents privately reports a location xi ∈ R.
- Model: The new facility is homogeneous, and each agent is served by whichever of the two facilities is nearest.
- Mechanisms: A deterministic mechanism maps a location profile to one facility location, whereas a randomized mechanism maps it to a probability distribution over facility locations.
- Strategy-proofness: Strategy-proofness requires that no agent benefits from misreporting her location regardless of the other agents’ reports.
- Objectives: Maximum cost is the largest agent cost and social cost is the sum of agent costs, with randomized mechanisms evaluated by expected cost.
- Approximation: Approximation ratios compare a strategy-proof mechanism’s objective value with the corresponding optimal solution.
3 Line
On the real line, the paper analyzes strategy-proof facility placement with agents on either side of a prelocated facility and with all agents on one side. It establishes approximation guarantees and lower bounds for maximum and social cost objectives, while identifying a remaining deterministic social-cost gap.
- 3.1.1 Maximum Cost: 2-approximation is achieved by a deterministic strategy-proof mechanism for maximum cost in the general line setting.The mechanism is also shown to be best possible among deterministic strategy-proof mechanisms.
- 3.1.1 Maximum Cost: 1.5 − ε is the lower bound for any randomized strategy-proof mechanism under the maximum cost objective.Here ε > 0.
- 3.1.2 Social cost: n-approximation is obtained for deterministic strategy-proof mechanisms under the social cost objective.A tight example reaches this factor, with n agents yielding social cost n times the optimum.
- 3.1.2 Social cost: 1.5 and 1.0425 are lower bounds for deterministic and randomized strategy-proof mechanisms, respectively, under social cost.The randomized lower bound uses strategy-proofness together with partial group strategy-proofness.
- 3.1.3 Discussion: A remaining limitation is the gap between the deterministic social-cost upper bound n and lower bound 1.5.The authors conjecture an Ω(n) deterministic lower bound but do not verify it.
- 3.2 Special Setting: In the special one-sided setting, a randomized strategy-proof mechanism achieves 5/3-approximation for maximum cost.The mechanism is presented as Mechanism 3 and is strategy-proof.
- 3.2 Special Setting: (n − 1)-approximation is achieved by a deterministic strategy-proof mechanism for social cost in the special setting.The same section states that the general-setting impossibility results remain except for deterministic social cost.
4 Circle
For agents on a circle with a prelocated facility, the paper designs a strategy-proof mechanism achieving 2-approximation for maximum cost and specifies its placement rule.
- 2-approximation is achieved by Mechanism 4 under the maximum cost objective.The mechanism is deterministic and strategy-proof.
- The circle has length 2, the prelocated facility is at 0, and each agent’s position is represented by clockwise distance xi ∈[0,2).The setting is illustrated in Figure 4.
- Mechanism 4 identifies xa and xb as the farthest agent locations on the two semicircles relative to the prelocated facility.xa is selected from [0,1), while xb is selected from [1,2).
- When xa ≥ 2 − xb, the new facility is placed according to a piecewise rule based on xa, xb, and min{4 − 2xb, 1}.When xa < 2 − xb, the placement is defined symmetrically.
- Mechanism 4 is strategy-proof, and its social-cost guarantee is n-approximation.The n-approximation result concerns the social cost objective rather than maximum cost.
5 Improvement Ratio
The paper introduces improvement ratio to evaluate how effectively a strategy-proof mechanism uses a new facility, then establishes upper and lower bounds across objectives and settings.
- The improvement ratio is the worst-case ratio between a mechanism’s improvement and the optimal improvement from adding a new facility.A lower ratio indicates near-optimal use of the new facility.
- 1.5-improvement is achieved by Mechanism 1 under the maximum cost objective.The result is stated for the relevant real-line setting analyzed in this section.
- 1.01 is a lower bound on the improvement ratio of any randomized strategy-proof mechanism under maximum cost.This lower bound applies to randomized strategy-proof mechanisms.
- 1.5 is a lower bound on the improvement ratio of any deterministic strategy-proof mechanism under social cost.The result rules out deterministic mechanisms with a smaller improvement ratio.
- 15/13-improvement is achieved by Mechanism 3, a randomized strategy-proof mechanism under maximum cost.The accompanying proof bounds the ratio by 15/13.
6 Conclusions and Open Problems
The paper studies strategy-proof mechanisms for locating a new homogeneous facility alongside a prelocated facility on a line or circle, across general and same-side settings. It establishes approximation bounds for maximum and social cost, proposes the improvement ratio, and identifies gaps and extensions for future work.
- The model covers line and circle settings, with agents either on both sides of the prelocated facility or entirely on one side.
- A deterministic mechanism achieves a tight 2-approximation for maximum cost on the line in the general setting, while randomized mechanisms face a 1.5−ϵ lower bound.
- For social cost on the line, deterministic mechanisms have an upper bound of n, with lower bounds of 1.5 for deterministic and 1.0425 for randomized mechanisms.
- In the special setting, the paper gives a randomized 5/3 approximation for maximum cost and a deterministic n −1 approximation for social cost.
- On the circle, a deterministic mechanism provides a 2-approximation for maximum cost, while the improvement ratio receives additional bounds across settings and objectives.
- Future work includes narrowing upper–lower-bound gaps, extending the model to other facility preferences, and imposing facility-location constraints.