Almost Envy-Free Allocations of Indivisible Goods or Chores with Entitlements

We consider the problem of fair allocation of indivisible goods or chores to $n$ agents with $\textit{weights}$ that describe their entitlements to a set of indivisible resources. Stemming from the well studied fairness notions envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX) for agents with $\textit{equal}$ entitlements, we here present the first impossibility results in addition to algorithmic guarantees on fairness for agents with $\textit{unequal}$ entitlements. In this paper, we extend the notion of envy-freeness up to any good or chore to the weighted context (WEFX and XWEF respectively), proving that these allocations are not guaranteed to exist for two or three agents. In spite of these negative results, we provide an approximate WEFX procedure for two agents -- a first result of its kind. We further present a polynomial time algorithm that guarantees a weighted envy-free up to one chore (1WEF) allocation for any number of agents with additive cost functions. Our work highlights the increased complexity of the weighted fair division problem as compared to its unweighted counterpart.