We study the problem of fairly allocating either a set of indivisible goods or a set of mixed divisible and indivisible goods (i.e., mixed goods) to agents with additive utilities, taking the best-of-both-worlds perspective of guaranteeing fairness properties both ex ante and ex post. The ex-post fairness notions considered in this paper are relaxations of envy-freeness, specifically, EFX for indivisible-goods allocation, and EFM for mixed-goods allocation. For two agents, we show that there is a polynomial-time randomized algorithm that achieves ex-ante envy-freeness and ex-post EFX / EFM simultaneously. For $n$ agents with bi-valued utilities, we show there exist randomized allocations that are (i) ex-ante proportional and ex-post EFM, and (ii) ex-ante envy-free, ex-post EFX, and ex-post fractionally Pareto optimal.

翻译：本研究探讨了在加性效用假设下，如何公平分配不可分割物品集合或可分割与不可分割混合物品集合（即混合物品）的问题，采用"最佳两界"视角，旨在同时保证事前与事后的公平性。本文考虑的事后公平概念是嫉妒自由性的松弛形式：针对不可分割物品分配采用EFX准则，针对混合物品分配采用EFM准则。对于双智能体场景，我们提出了一种多项式时间随机算法，能够同时实现事前嫉妒自由性与事后EFX/EFM。针对具有二值效用的n个智能体，我们证明了存在满足以下条件的随机分配方案：（i）事前比例公平且事后EFM；（ii）事前嫉妒自由、事后EFX且事后分数帕累托最优。