Maximin Share Guarantees for Few Agents with Subadditive Valuations

TL;DR

This paper advances the understanding of fair division by establishing tight bounds for maximin share guarantees among few agents with subadditive valuations, introducing a new model, and analyzing special cases.

Contribution

It provides tight bounds for MMS guarantees for up to four agents with subadditive valuations and introduces a novel model for fair division analysis.

Findings

Established a tight 1/2 lower bound for up to four agents with subadditive valuations.

Derived a tight lower bound for multiple agents with specific valuation types.

Proposed a new model extending previous fair division frameworks and analyzed it for three agents.

Abstract

We study the problem of fairly allocating a set of indivisible items among a set of agents. We consider the notion of (approximate) maximin share (MMS) and we provide an improved lower bound of $1/2$ (which is tight) for the case of subadditive valuations when the number of agents is at most four. We also provide a tight lower bound for the case of multiple agents, when they are equipped with one of two possible types of valuations. Moreover, we propose a new model that extends previously studied models in the area of fair division, which will hopefully give rise to further research. We demonstrate the usefulness of this model by employing it as a technical tool to derive our main result, and we provide a thorough analysis for this model for the case of three agents. Finally, we provide an improved impossibility result for the case of three submodular agents.