The problem of allocating indivisible resources to agents arises in a wide range of domains, including treatment distribution and social support programs. An important goal in algorithm design for this problem is fairness, where the focus in previous work has been on ensuring that the computed allocation provides equal treatment to everyone. However, this perspective disregards that agents may start from unequal initial positions, which is crucial to consider in settings where fairness is understood as equality of outcome. In such settings, the goal is to create an equal final outcome for everyone by leveling initial inequalities through the allocated resources. To close this gap, focusing on agents with additive utilities, we extend the classic model by assigning each agent an initial utility and study the existence and computational complexity of several new fairness notions following the principle of equality of outcome. Among others, we show that complete allocations satisfying a direct analog of envy-freeness up to one resource (EF1) may fail to exist and are computationally hard to find, forming a contrast to the classic setting without initial utilities. We propose a new, always satisfiable fairness notion, called minimum-EF1-init and design a polynomial-time algorithm based on an extended round-robin procedure to compute complete allocations satisfying this notion.

翻译：不可分割资源在多个主体间的分配问题广泛存在于治疗分配与社会支持计划等诸多领域。该问题算法设计的一个重要目标是公平性，先前研究的焦点在于确保计算所得的分配方案能为所有人提供平等对待。然而，这种视角忽视了主体可能从不平等的初始状态出发的事实，这在将公平性理解为结果平等的场景中至关重要。在此类场景中，目标是通过分配资源来消除初始不平等，从而为所有人创造平等的最终结果。为填补这一空白，我们聚焦于具有加性效用的主体，通过为每个主体赋予初始效用来扩展经典模型，并依据结果平等原则研究了若干新公平性概念的存在性与计算复杂性。我们发现，满足嫉妒至多一项资源（EF1）直接类比形式的完全分配方案可能不存在且计算上难以寻找，这与不存在初始效用的经典设定形成鲜明对比。我们提出了一种新的、恒可满足的公平性概念——最小初始EF1，并基于扩展的轮询流程设计了一种多项式时间算法，用于计算满足该概念的完全分配方案。