SIAM Journal on Discrete Mathematics, Volume 35, Issue 2, Page 788-813, January 2021.
We investigate the query complexity of the fair allocation of indivisible goods. For two agents with arbitrary monotonic utilities, we design an algorithm that computes an allocation satisfying envy-freeness up to one good (EF1), a relaxation of envy-freeness, using a logarithmic number of queries. We show that the logarithmic query complexity bound also holds for three agents with additive utilities and that a polylogarithmic bound holds for three agents with monotonic utilities. These results suggest that it is possible to fairly allocate goods in practice even when the number of goods is extremely large. By contrast, we prove that computing an allocation satisfying envy-freeness and another of its relaxations, envy-freeness up to any good (EFX), requires a linear number of queries even when there are only two agents with identical additive utilities.

中文翻译：

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以较少的查询公平分配许多商品

SIAM 离散数学杂志，第 35 卷，第 2 期，第 788-813 页，2021 年 1 月。
我们调查了不可分割商品公平分配的查询复杂性。对于具有任意单调效用的两个代理，我们设计了一种算法，该算法使用查询的对数数来计算满足无嫉妒（EF1）的分配，这是无嫉妒的松弛。我们表明，对数查询复杂度界限也适用于三个具有加性效用的代理，而多对数界限适用于三个具有单调效用的代理。这些结果表明，即使在商品数量非常大的情况下，在实践中也可以公平地分配商品。相比之下，我们证明了计算一个满足嫉妒自由度和另一个放松的分配，嫉妒自由度达到任何好处（EFX），