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：我们研究将一组不可分割物品或一组混合可分割与不可分割物品（即混合物品）分配给具有可加效用函数的智能体的问题，采取最佳两全视角，旨在同时保证事前与事后的公平性。本文考虑的事后公平性概念是嫉妒自由度的松弛形式，具体而言，对于不可分割物品分配是EFX，对于混合物品分配是EFM。对于两个智能体，我们证明存在一种多项式时间随机算法，能够同时实现事前嫉妒自由与事后EFX/EFM。对于具有二值效用的$n$个智能体，我们证明存在满足以下条件的随机分配：（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。