Strategy Proof Mechanisms for Facility Location in Euclidean and Manhattan Space
This work addresses theoretical limitations in mechanism design for facility location, highlighting challenges in multi-dimensional spaces that impact fairness and efficiency for agents in spatial allocation problems.
The paper investigates the difficulty of achieving anonymity, Pareto optimality, and strategy proofness in facility location mechanisms when moving from one-dimensional to two-dimensional Euclidean or Manhattan spaces, showing that no mechanism satisfies all three properties in these higher-dimensional settings, unlike in one dimension.
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.