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.

翻译：我们研究了在考虑代理权重（描述其对不可分割资源集的应得权益）情况下，将不可分割商品或杂务公平分配给$n$个代理的问题。基于针对具有平等权益代理的成熟公平概念——至多一件商品的无嫉妒性（EF1）和任意商品的无嫉妒性（EFX），本文首次展示了针对具有不平等权益代理的不可行性结果，并给出了算法公平性保证。我们扩展了至多任意商品或杂务的无嫉妒性概念到加权情境（分别为WEFX和XWEF），证明对于两或三个代理，此类分配的存在性无法保证。尽管存在这些否定结果，我们为两个代理提出了一个近似的WEFX算法——这是此类结果的首次尝试。此外，我们提出一个多项式时间算法，对于任意数量具有加法成本函数的代理，保证产生加权至多一件杂务的无嫉妒（1WEF）分配。我们的工作凸显了加权公平分配问题相较于无权重情形具有更高的复杂度。