Unified Fair Allocation for Indivisible Goods and Chores via Copies

We consider fair allocation of indivisible items in a model with goods, chores, and copies, as a unified framework for studying: (1)~the existence of EFX and other solution concepts for goods with copies; (2)~the existence of EFX and other solution concepts for chores. We establish a tight relation between these issues via two conceptual contributions: First, a refinement of envy-based fairness notions that we term envy \emph{without commons} (denoted $\efxwc$ when applied to EFX). Second, a formal \emph{duality theorem} relating the existence of a host of (refined) fair allocation concepts for copies to their existence for chores. We demonstrate the usefulness of our duality result by using it to characterize the existence of EFX for chores through the dual environment, as well as to prove EFX existence in the special case of leveled preferences over the chores. We further study the hierarchy among envy-freeness notions without commons and their $\alpha$-MMS guarantees, showing for example that any $\efxwc$ allocation guarantees at least $\frac{4}{11}$-MMS for goods with copies.