To EFX OR to MMS, That is the Question

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 that having a flexible fairness requirement for individual agents will 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. Additionally, we obtain polynomial-time approximation schemes for additive goods-only and chores-only 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.