Unified Fair Allocation of Goods and Chores via Copies

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