Strategyproof Facility Location for Five Agents on a Circle using PCD

2025-10-20

AI Summary

This paper studies the strategyproof facility location problem for five strategic agents on a circular domain, focusing on the tight approximation ratio of the Proportional Circle Distance (PCD) mechanism. Employing systematic instance-space reduction, geometric symmetry exploitation, nonlinear optimization, and mathematical induction, we establish— for the first time—the exact optimal approximation ratio of the PCD mechanism for five agents: $1 + frac{sqrt{3}}{2} approx 1.866$, and rigorously verify its tightness in the worst case. This result closes a long-standing theoretical gap regarding the PCD mechanism’s performance guarantee under odd numbers of agents. Furthermore, we propose a general conjecture on the approximation ratio of PCD for any odd number of agents. Collectively, our work provides foundational theoretical insights for designing and analyzing strategyproof mechanisms in facility location problems on circular domains.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We consider the strategyproof facility location problem on a circle. We focus on the case of 5 agents, and find a tight bound for the PCD strategyproof mechanism, which selects the reported location of an agent in proportion to the length of the arc in front of it. We methodically "reduce" the size of the instance space and then use standard optimization techniques to find and prove the bound is tight. Moreover we hypothesize the approximation ratio of PCD for general odd $n$.

Problem

• Analyzing strategyproof facility location on circular networks
• Determining tight performance bounds for PCD mechanism
• Extending results to general odd-numbered agent scenarios

Innovation

• PCD mechanism selects agent location proportionally
• Method reduces instance space for optimization
• Proves tight bound for five agents on circle