Achieving Maximin Share and EFX/EF1 Guarantees Simultaneously

2024-09-03

arXiv.org

AI Summary

This paper studies fair allocation of indivisible goods among multiple agents under additive valuations, aiming to simultaneously satisfy two fundamental fairness criteria: the maximin share (MMS) guarantee and approximate envy-freeness (EFX or EF1). Methodologically, we integrate greedy matching, value scaling, and hierarchical packing strategies to design both pseudopolynomial-time and polynomial-time algorithms under reasonable approximation relaxations. Our key contributions are constructive: we establish the existence of a *partial* allocation satisfying 2/3-MMS and EFX, and a *complete* allocation satisfying 2/3-MMS and EF1—breaking the prior best-known bound of 1/2-MMS + EFX. These results significantly improve the MMS approximation ratio while preserving strong envy-freeness guarantees. To date, this provides the strongest constructive guarantee for simultaneously achieving MMS and EFX/EF1 in the additive setting, advancing the theoretical foundations of fair division.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We study the problem of computing emph{fair} divisions of a set of indivisible goods among agents with emph{additive} valuations. For the past many decades, the literature has explored various notions of fairness, that can be primarily seen as either having emph{envy-based} or emph{share-based} lens. For the discrete setting of resource-allocation problems, emph{envy-free up to any good} (EFX) and emph{maximin share} (MMS) are widely considered as the flag-bearers of fairness notions in the above two categories, thereby capturing different aspects of fairness herein. Due to lack of existence results of these notions and the fact that a good approximation of EFX or MMS does not imply particularly strong guarantees of the other, it becomes important to understand the compatibility of EFX and MMS allocations with one another. In this work, we identify a novel way to simultaneously achieve MMS guarantees with EFX/EF1 notions of fairness, while beating the best known approximation factors [Chaudhury et al., 2021, Amanatidis et al., 2020]. Our main contribution is to constructively prove the existence of (i) a partial allocation that is both $2/3$-MMS and EFX, and (ii) a complete allocation that is both $2/3$-MMS and EF1. Our algorithms run in pseudo-polynomial time if the approximation factor for MMS is relaxed to $2/3-varepsilon$ for any constant $varepsilon>0$ and in polynomial time if, in addition, the EFX (or EF1) guarantee is relaxed to $(1-delta)$-EFX (or $(1-delta)$-EF1) for any constant $delta>0$. In particular, we improve from the best approximation factor known prior to our work, which computes partial allocations that are $1/2$-MMS and EFX in pseudo-polynomial time [Chaudhury et al., 2021].

Problem

• Fair Allocation
• Indivisible Goods
• Envy-Freeness

Innovation

• FairAllocation
• MMS-EFXCriteria
• EfficiencyImprovement