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Closing Gaps in Asymptotic Fair Division
SIAM Journal on Discrete Mathematics ( IF 0.8 ) Pub Date : 2021-04-08 , DOI: 10.1137/20m1353381
Pasin Manurangsi , Warut Suksompong

SIAM Journal on Discrete Mathematics, Volume 35, Issue 2, Page 668-706, January 2021.
We study a resource allocation setting where $m$ discrete items are to be divided among $n$ agents with additive utilities, and the agents' utilities for individual items are drawn at random from a probability distribution. Since common fairness notions like envy-freeness and proportionality cannot always be satisfied in this setting, an important question is when allocations satisfying these notions exist. In this paper, we close several gaps in the line of work on asymptotic fair division. First, we prove that the classical round-robin algorithm is likely to produce an envy-free allocation provided that $m=\Omega(n\log n/\log\log n)$, matching the lower bound from prior work. We then show that a proportional allocation exists with high probability as long as $m\geq n$, while an allocation satisfying envy-freeness up to any item (EFX) is likely to be present for any relation between $m$ and $n$. Finally, we consider a related setting where each agent is assigned exactly one item and the remaining items are left unassigned, and show that the transition from nonexistence to existence with respect to envy-free assignments occurs at $m=en$.


中文翻译:

缩小渐近公平分部的差距

SIAM 离散数学杂志,第 35 卷,第 2 期,第 668-706 页,2021 年 1 月。
我们研究了一种资源分配设置,其中 $m$ 离散项目将在具有附加效用的 $n$ 代理之间分配,并且代理对单个项目的效用是从概率分布中随机抽取的。由于在这种情况下不能总是满足常见的公平概念,如无嫉妒和比例性,一个重要的问题是何时存在满足这些概念的分配。在本文中,我们弥补了渐近公平除法工作中的几个差距。首先,我们证明经典循环算法很可能会产生无嫉妒分配,前提是 $m=\Omega(n\log n/\log\log n)$,匹配先前工作的下界。然后我们证明,只要 $m\geq n$,比例分配就以高概率存在,而对于 $m$ 和 $n$ 之间的任何关系,很可能存在满足任何项目(EFX)的无嫉妒的分配。最后,我们考虑一个相关的设置,其中每个智能体只分配了一个项目,其余项目未分配,并表明从不存在到存在的过渡就无嫉妒分配发生在 $m=en$ 处。
更新日期:2021-04-08
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