[Paper Review] Strategyproof Mechanism for Two Heterogeneous Facilities with Constant Approximation Ratio

Minming Li, Pinyan Lu+2 more|arXiv (Cornell University)|Jul 21, 2019

Game Theory and Voting Systems41 references4 citations

TL;DR

This paper presents a deterministic, strategyproof mechanism for the two-heterogeneous-facility location problem on a line with private preferences, achieving a constant approximation ratio of 2.75. The mechanism first computes optimal facility locations based on public agent locations, then selects the best configuration from four candidate placements using reported preferences, ensuring strategyproofness through a non-trivial structural argument.

ABSTRACT

In this paper, we study the two-facility location game on a line with optional preference where the acceptable set of facilities for each agent could be different and an agent's cost is his distance to the closest facility within his acceptable set. The objective is to minimize the total cost of all agents while achieving strategyproofness. We design a deterministic strategyproof mechanism for the problem with approximation ratio of 2.75, improving upon the earlier best ratio of n/2+1.

Motivation & Objective

• To design a deterministic, strategyproof mechanism for two heterogeneous facilities with constant approximation ratio.
• To close the large gap between the best-known approximation ratio and the theoretical lower bound in the two-heterogeneous-facility model.
• To explore whether non-trivial deterministic mechanisms exist beyond dictatorship or majority voting in strategyproof mechanism design.
• To investigate the feasibility of generalizing the mechanism to three or more facilities.

Proposed method

• The mechanism first computes the optimal facility locations $s_\ell$ and $s_r$ assuming all agents accept both facilities, based on public agent locations.
• It then evaluates four candidate configurations: $(s_\ell, s_\ell)$, $(s_\ell, s_r)$, $(s_r, s_\ell)$, and $(s_r, s_r)$, selecting the one minimizing social cost using agents' reported preferences.
• The mechanism ensures strategyproofness via a non-trivial argument that prevents agents from benefiting by misreporting preferences.
• It leverages the structure of single-peaked preferences and the linearity of the problem to bound the approximation ratio.
• The mechanism is proven strategyproof by analyzing cost changes under preference misreporting and showing no agent can benefit from lying.
• For generalization, the mechanism extends to $k$ facilities by evaluating all $k^k$ configurations, but this fails to preserve strategyproofness for $k \geq 3$.

Experimental results

Research questions

• RQ1 Can a deterministic strategyproof mechanism achieve a constant approximation ratio in the two-heterogeneous-facility location problem?
• RQ2 What is the tight approximation ratio of the proposed mechanism, and does it achieve the theoretical lower bound of $1 + \sqrt{2}$?
• RQ3 Why does the mechanism fail to be strategyproof when generalized to three or more facilities?
• RQ4 Is there a non-trivial deterministic strategyproof mechanism for $k \geq 3$ facilities with constant approximation ratio?
• RQ5 What preference structure enables the mechanism to be strategyproof despite not being a dictatorship or majority vote?

Key findings

• The proposed mechanism achieves an approximation ratio of at most 2.75, significantly improving upon the previous best of $n/2 + 1$.
• The mechanism is strategyproof, as no agent can reduce their cost by misreporting their acceptable facility set.
• A lower bound of $1 + \sqrt{2} \approx 2.414$ is established for the approximation ratio of the mechanism, suggesting the true ratio may be $1 + \sqrt{2}$.
• The mechanism's strategyproofness relies on a non-trivial structural argument that does not fall into the two exception classes of Gibbard-Satterthwaite's impossibility theorem.
• Generalizing the mechanism to three or more facilities fails to preserve strategyproofness, as demonstrated by a counterexample where an agent benefits from misreporting.
• The counterexample shows that when $k \geq 3$, the mechanism can be manipulated by agents who report a smaller acceptable set than their true one, reducing their cost.