Strategyproof Facility Location for Five Agents on a Circle using PCD

TL;DR

The paper resolves the strategyproof facility location problem on a circle for five agents by exactly characterizing the worst-case behavior of the Proportional Circle Distance (PCD) mechanism. It achieves a tight upper bound on the approximation ratio, $\gamma=7-4\sqrt{2}$, by a structured reduction of the instance space (two-pair and large-arc analyses) and bounding the social cost to $SC\le 1.2$, then solving a boundary-focused optimization over reduced configurations. The result improves the previous $\frac{8}{5}$ bound from Random Dictator and positions PCD as the best-known mechanism for five agents on a circle, with a conjectured closed form for general odd $n$ and insights into clustering as $n$ grows. The work combines rigorous combinatorial reasoning with convex-analytic optimization on a compact, rotated representation of circle instances, and provides an interactive visualization tool to illustrate mechanism behavior.

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 $n$.

Figures (4)

• Figure 1: The hypothesized approximation ratio of PCD as number of agents grows, compared to a numerical computation, and other known mechanisms.
• Figure 2:
• Figure 3: An example of an instance where $d(x_1,x_4)=d_A(x_1,x_4)=P_1+P_5+P_4$ and $d(x_2,x_5)=d_D(x_2,x_5)=P_3+P_4$ and thus an instance for $\gamma_{AD}$. Moreover, all distances from $x_1$ are shown, which demonstrates the ability to calculate $C_1$.
• Figure 4: The "clustering" phenomena - two clusters with $k$ agents, one lone agent, and an arc of length 0.5

Theorems & Definitions (36)

• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 4
• Theorem 5
• proof : Proof sketch
• Lemma 6
• proof
• Theorem 7
• proof