In fair division applications, agents may have unequal entitlements reflecting their different contributions. Moreover, the contributions of agents may depend on the allocation itself. Previous fairness notions designed for agents with equal or pre-determined entitlement fail to characterize fairness in these collaborative allocation scenarios. We propose a novel fairness notion of average envy-freeness (AEF), where the envy of agents is defined on the average value of items in the bundles. Average envy-freeness provides a reasonable comparison between agents based on the items they receive and reflects their entitlements. We study the complexity of finding AEF and its relaxation, average envy-freeness up to one item (AEF-1). While deciding if an AEF allocation exists is NP-complete, an AEF-1 allocation is guaranteed to exist and can be computed in polynomial time. We also study allocation with quotas, i.e. restrictions on the sizes of bundles. We prove that finding AEF-1 allocation satisfying a quota is NP-hard. Nevertheless, in the instances with a fixed number of agents, we propose polynomial-time algorithms to find AEF-1 allocation with a quota for binary valuation and approximated AEF-1 allocation with a quota for general valuation.

翻译：在公平分配应用中，代理人可能具有不同的权益，以反映其不同贡献。此外，代理人的贡献可能取决于分配本身。以往为具有平等或预设权益的代理人设计的公平性概念，难以刻画这些合作分配场景中的公平性。我们提出了一种新颖的公平性概念——平均无嫉妒性（AEF），其中代理人的嫉妒基于其所得束中物品的平均价值定义。平均无嫉妒性提供了基于所得物品的合理比较，并反映了代理人的权益。我们研究了寻找AEF分配及其松弛形式——至多容忍一件物品的平均无嫉妒性（AEF-1）的复杂性。虽然判断是否存在AEF分配是NP完全的，但AEF-1分配的存在性有保证且可在多项式时间内计算。我们还研究了带配额分配，即对束大小的限制。我们证明，寻找满足配额的AEF-1分配是NP难的。然而，在代理人数量固定的实例中，我们提出了多项式时间算法，用于寻找二值估值下带配额的AEF-1分配，以及一般估值下带配额的近似AEF-1分配。