[Paper Review] Strategy Proof Mechanisms for Facility Location in Euclidean and Manhattan Space

Toby Walsh|arXiv (Cornell University)|Sep 17, 2020

Game Theory and Voting Systems15 references4 citations

TL;DR

This paper investigates strategyproof mechanisms for facility location in two or more dimensions using Euclidean or Manhattan distances. It shows that moving from one-dimensional to multi-dimensional spaces severely limits the feasibility of achieving anonymity, Pareto optimality, and strategyproofness simultaneously—impossibility results emerge even for single facilities in Manhattan space or two facilities in Euclidean space, contrasting with the existence of such mechanisms in one dimension.

ABSTRACT

We study the impact on mechanisms for facility location of moving from one dimension to two (or more) dimensions and Euclidean or Manhattan distances. We consider three fundamental axiomatic properties: anonymity which is a basic fairness property, Pareto optimality which is one of the most important efficiency properties, and strategy proofness which ensures agents do not have an incentive to mis-report. We also consider how well such mechanisms can approximate the optimal welfare. Our results are somewhat negative. Moving from one dimension to two (or more) dimensions often makes these axiomatic properties more difficult to achieve. For example, with two facilities in Euclidean space or with just a single facility in Manhattan space, no mechanism is anonymous, Pareto optimal and strategy proof. By contrast, mechanisms on the line exist with all three properties.We also show that approximation ratios may increase when moving to two (or more) dimensions. All our impossibility results are minimal. If we drop one of the three axioms (anonymity, Pareto optimality or strategy proofness) multiple mechanisms satisfy the other two axioms.

Motivation & Objective

• To analyze how moving from one-dimensional to two or more dimensions affects the design of strategyproof, fair, and efficient facility location mechanisms.
• To evaluate the impact of using Euclidean or Manhattan distances on axiomatic properties like anonymity, Pareto optimality, and strategyproofness.
• To investigate whether approximation ratios for optimal welfare (total or maximum distance) remain bounded in higher dimensions compared to one dimension.
• To determine whether mechanisms that work in one dimension can be extended to multi-dimensional spaces while preserving key axiomatic and approximation properties.
• To identify minimal impossibility results—i.e., whether dropping any single axiom allows mechanisms to satisfy the other two.

Proposed method

• Formalizing the facility location problem in d-dimensional Euclidean and Manhattan spaces with n agents and m facilities.
• Defining three core axiomatic properties: anonymity (fairness), Pareto optimality (efficiency), and strategyproofness (incentive compatibility).
• Analyzing the performance of standard mechanisms such as the Percentile mechanism, Median mechanism, and serial dictatorship (SD) in multi-dimensional settings.
• Extending mechanisms to capacitated settings where facilities have limited capacity and agents must be assigned without necessarily going to the nearest facility.
• Using mathematical proofs to demonstrate impossibility results when all three axioms are required simultaneously in 2D or higher dimensions.
• Comparing approximation ratios for total and maximum distance objectives between 1D and multi-dimensional spaces, particularly for multi-dimensional Percentile mechanisms.

Experimental results

Research questions

• RQ1 Can mechanisms that are anonymous, Pareto optimal, and strategyproof in one dimension be extended to two or more dimensions without losing these properties?
• RQ2 What is the impact of switching from Euclidean to Manhattan distance metrics on the feasibility of achieving strategyproofness and efficiency in multi-dimensional facility location?
• RQ3 Do bounded approximation ratios for total or maximum distance remain achievable in multi-dimensional spaces when they are achievable in one dimension?
• RQ4 Are there minimal impossibility results—i.e., can any two of the three axioms (anonymity, Pareto optimality, strategyproofness) be satisfied simultaneously in 2D Euclidean or Manhattan space if the third is dropped?
• RQ5 How do capacitated facility constraints affect the design and performance of strategyproof mechanisms in higher-dimensional spaces?

Key findings

• In two-dimensional Euclidean space with two facilities, no mechanism can be simultaneously anonymous, Pareto optimal, and strategyproof.
• In Manhattan space with a single facility, no mechanism satisfies all three axioms—anonymous, Pareto optimal, and strategyproof—whereas such mechanisms exist in one dimension.
• For two facilities in 2D Manhattan space, no multi-dimensional Percentile mechanism has a bounded approximation ratio for total or maximum distance, unlike in 1D where bounded ratios exist.
• The 2D Median mechanism achieves optimal total Manhattan distance but only provides a √2-approximation for optimal Euclidean distance.
• Approximation ratios for total and maximum distance increase when moving from 1D to 2D Manhattan space, becoming unbounded in higher dimensions.
• All impossibility results are minimal: dropping any one of the three axioms (anonymity, Pareto optimality, strategyproofness) allows mechanisms to satisfy the other two in 2D settings.