SIAM Journal on Discrete Mathematics, Volume 34, Issue 3, Page 1505-1521, January 2020.
We consider a fair division setting in which $m$ indivisible items are to be allocated among $n$ agents, where the agents have additive utilities and the agents' utilities for individual items are independently sampled from a distribution. Previous work has shown that an envy-free allocation is likely to exist when $m=\Omega(n\log n)$ but not when $m=n+o(n)$, and left open the question of determining where the phase transition from non-existence to existence occurs. We show that, surprisingly, there is in fact no universal point of transition---instead, the transition is governed by the divisibility relation between $m$ and $n$. On the one hand, if $m$ is divisible by $n$, an envy-free allocation exists with high probability as long as $m\geq 2n$. On the other hand, if $m$ is not “almost” divisible by $n$, an envy-free allocation is unlikely to exist even when $m=\Theta(n\log n/\log\log n)$.

中文翻译：

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无羡慕分配何时存在？

SIAM离散数学杂志，第34卷，第3期，第1505-1521页，2020年1月。
我们考虑一个公平的划分设置，其中将$ m $个不可分割的项目分配给$ n $个代理商，其中代理商具有累加效用，代理商对单个项目的效用是从分配中独立采样的。先前的工作表明，当$ m = \ Omega（n \ log n）$时，很可能存在无羡慕的分配，而当$ m = n + o（n）$时，可能没有羡慕的分配。从不存在到存在的相变发生了。我们惊奇地表明，实际上并没有普遍的过渡点，而是由$ m $和$ n $之间的可除性关系决定的。一方面，如果$ m $可被$ n $整除，则只要$ m \ geq 2n $，就有很高的可能性实现无羡慕的分配。另一方面，如果$ m $不能被$ n $整除，