[Paper Review] Strategyproof Mechanisms For Group-Fair Facility Location Problems

Houyu Zhou, Minming Li+1 more|arXiv (Cornell University)|Jul 12, 2021

Game Theory and Voting Systems4 citations

TL;DR

This paper proposes strategyproof mechanisms for group-fair facility location problems, where agents are partitioned into groups (e.g., by race or age), and the goal is to locate a facility to fairly minimize group costs. It introduces novel group-based mechanisms and a new fairness metric—interruption and intragroup fairness (IIF)—achieving tight constant approximation ratios, with the Majority Group Deterministic Mechanism (MGDM) achieving a 3-approximation for minimizing maximum total group cost.

ABSTRACT

We study the facility location problems where agents are located on a real line and divided into groups based on criteria such as ethnicity or age. Our aim is to design mechanisms to locate a facility to approximately minimize the costs of groups of agents to the facility fairly while eliciting the agents' locations truthfully. We first explore various well-motivated group fairness cost objectives for the problems and show that many natural objectives have an unbounded approximation ratio. We then consider minimizing the maximum total group cost and minimizing the average group cost objectives. For these objectives, we show that existing classical mechanisms (e.g., median) and new group-based mechanisms provide bounded approximation ratios, where the group-based mechanisms can achieve better ratios. We also provide lower bounds for both objectives. To measure fairness between groups and within each group, we study a new notion of intergroup and intragroup fairness (IIF) . We consider two IIF objectives and provide mechanisms with tight approximation ratios.

Motivation & Objective

• To design strategyproof mechanisms that fairly serve agents grouped by criteria such as ethnicity or age in facility location problems.
• To identify group-fair cost objectives that allow bounded approximation ratios despite the impossibility of bounded ratios for many natural objectives.
• To introduce and formalize a new fairness notion—interruption and intragroup fairness (IIF)—capturing fairness both between and within groups.
• To provide mechanisms with tight approximation ratios for the new IIF objectives, ensuring truthfulness and fairness.
• To close the gap between upper and lower bounds for key objectives by establishing tight bounds and improved mechanisms.

Proposed method

• Propose the Majority Group Deterministic Mechanism (MGDM), which locates the facility at the median of the largest group to minimize maximum total group cost.
• Introduce the Narrow Randomized Mechanism (NRM), which refines the randomization range of the RM mechanism using group information to improve approximation for average group cost.
• Define two new IIF objectives: $IIF_1$ treats intergroup and intragroup fairness as separate indicators, while $IIF_2$ combines them into a single metric per group.
• Use the k-LDM mechanism to achieve a 4-approximation for both IIF objectives by placing the facility at the k-th location based on group structure.
• Apply rigorous game-theoretic analysis, including partial group strategyproofness and lower bound proofs via contradiction, to validate mechanism robustness.
• Leverage existing classical mechanisms (MDM, LDM, RM) and extend them with group-aware logic to improve approximation ratios under fairness constraints.

Experimental results

Research questions

• RQ1 Can strategyproof mechanisms achieve bounded approximation ratios for group-fair facility location objectives, despite the unbounded ratios of many natural objectives?
• RQ2 What mechanisms can effectively minimize the maximum total group cost while ensuring truthfulness and leveraging group structure?
• RQ3 How can intergroup and intragroup fairness be formally modeled and optimized in facility location problems?
• RQ4 What are the tight approximation ratios achievable for the new IIF objectives, and can mechanisms be designed to match these bounds?
• RQ5 Are there lower bounds that establish the limits of approximation for group-fair objectives, particularly under strategyproofness constraints?

Key findings

• The Majority Group Deterministic Mechanism (MGDM) achieves a 3-approximation ratio for minimizing the maximum total group cost, improving upon classical mechanisms.
• For minimizing the maximum average group cost, the Narrow Randomized Mechanism (NRM) achieves a 2-approximation ratio, outperforming the classical RM.
• The k-LDM mechanism achieves a tight 4-approximation ratio for both $IIF_1$ and $IIF_2$ objectives, matching the established lower bound of 4.
• A lower bound of 2 is proven for the maximum total group cost objective, showing that no deterministic strategyproof mechanism can achieve better than 2-approximation.
• The lower bound of 4 for $IIF_1$ and $IIF_2$ is tight, as demonstrated by contradiction using partial group strategyproofness and strategic agent misreporting.
• Group-based mechanisms significantly outperform classical mechanisms like MDM and LDM in terms of approximation ratio, especially when group size and structure are leveraged.