Maximin Share Guarantees for Few Agents with Subadditive Valuations

2025-02-07

AI Summary

This paper studies fair allocation of indivisible goods among a small number of agents under subadditive valuations, focusing on approximation guarantees for the maximin share (MMS). For at most four subadditive agents, it establishes— for the first time—that 1/2-MMS is a tight lower bound and thus the optimal guarantee; develops a complete analytical framework for three agents; introduces a scalable new fairness model; derives tight bounds for multi-agent settings with two valuation types; and strengthens impossibility results for three submodular agents. Methodologically, the work integrates combinatorial game theory, subadditive and submodular function analysis, constructive lower-bound proofs, and tailored counterexample design. The core contribution lies in breaking theoretical barriers for MMS approximation under subadditivity, delivering tight characterizations and structured algorithmic tools for fair division in small-scale settings.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

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.

Problem

• Fair allocation of indivisible items
• Maximin share guarantees
• Subadditive valuations analysis

Innovation

• Improved lower bound for MMS
• New fair division model
• Impossibility result for submodular agents