To EFX OR to MMS, That is the Question

2026-08-10

AI Summary

This study introduces and systematically analyzes the novel notion of agent-level disjunctive fairness—denoted EFX∨MMS—where each agent is required to satisfy either envy-freeness up to any good (EFX) or maximin share (MMS) fairness. Focusing on the allocation of indivisible goods, the work investigates the feasibility boundaries of this flexible fairness criterion under additive and submodular valuations through combinatorial constructions, counterexample design, and case-based existence proofs. The main contributions include the first counterexample demonstrating the non-existence of an EFX∨MMS allocation for three agents, a proof of guaranteed existence in specific additive mixed-item settings accompanied by an approximation algorithm, and the revelation that this model simultaneously expands the space of feasible solutions while, surprisingly, failing to admit allocations in certain instances.

Abstract

We study the agent-wise disjunction of two central fairness notions for indivisible items, where every agent must be either envy-free up to any item (EFX) or maximin-share (MMS) satisfied. One might expect this flexibility to restore existence, especially because the existence of EFX itself resisted resolution for nearly a decade. Surprisingly, it does not. We construct counterexamples with three agents and eight submodular goods, and with three agents and seven submodular chores, significantly strengthening recent EFX impossibility results. On the positive side, we prove existence for additive mixed items with at most three valuation types when one type is a singleton. Our proof extends beyond additivity for goods and yields approximation schemes for goods and three-agent chores instances. We also identify a clean separation between the disjunction and its constituents: For additive chores with two valuation types, EFX and MMS are both known to fail, whereas an EFX$\vee$MMS allocation always exists. Finally, we show that identical additive valuations even admit the conjunction EFX$\wedge$MMS for mixed items. Overall, our results show that allowing flexibility in choosing agent-specific fairness certificates expands the frontier of fair solutions while also uncovering surprising impossibilities.

Problem

• EFX
• MMS
• fair allocation
• indivisible items
• submodular valuations

Innovation

• EFX
• MMS
• fair division
• submodular valuations
• mixed items