We study the problem of fairly allocating $m$ indivisible items among $n$ agents. Envy-free allocations, in which each agent prefers her bundle to the bundle of every other agent, need not exist in the worst case. However, when agents have additive preferences and the value $v_{i,j}$ of agent $i$ for item $j$ is drawn independently from a distribution $D_i$, envy-free allocations exist with high probability when $m \in \Omega( n \log n / \log \log n )$. In this paper, we study the existence of envy-free allocations under stochastic valuations far beyond the additive setting. We introduce a new stochastic model in which each agent's valuation is sampled by first fixing a worst-case function, and then drawing a uniformly random renaming of the items, independently for each agent. This strictly generalizes known settings; for example, $v_{i,j} \sim D_i$ may be seen as picking a random (instead of a worst-case) additive function before renaming. We prove that random renaming is sufficient to ensure that envy-free allocations exist with high probability in very general settings. When valuations are non-negative and ``order-consistent,'' a valuation class that generalizes additive, budget-additive, unit-demand, and single-minded agents, SD-envy-free allocations (a stronger notion of fairness than envy-freeness) exist for $m \in \omega(n^2)$ when $n$ divides $m$, and SD-EFX allocations exist for all $m \in \omega(n^2)$. The dependence on $n$ is tight, that is, for $m \in O(n^2)$ envy-free allocations don't exist with constant probability. For the case of arbitrary valuations (allowing non-monotone, negative, or mixed-manna valuations) and $n=2$ agents, we prove envy-free allocations exist with probability $1 - \Theta(1/m)$ (and this is tight).

翻译：我们研究在 $m$ 个不可分割物品中公平分配给 $n$ 个智能体的问题。无嫉妒分配，即每个智能体偏好自己的束超过其他所有智能体的束，在最坏情况下可能不存在。然而，当智能体具有可加偏好且智能体 $i$ 对物品 $j$ 的价值 $v_{i,j}$ 独立地从分布 $D_i$ 中抽取时，当 $m \in \Omega( n \log n / \log \log n )$ 时，无嫉妒分配以高概率存在。在本文中，我们研究在远超可加设定的随机估值下无嫉妒分配的存在性。我们引入一种新的随机模型，其中每个智能体的估值首先通过固定一个最坏情况函数，然后独立地为每个智能体随机重命名物品来采样。这严格推广了已知设定；例如，$v_{i,j} \sim D_i$ 可视为在重命名前选取一个随机（而非最坏情况）的可加函数。我们证明，在非常一般的设定下，随机重命名足以确保无嫉妒分配以高概率存在。当估值为非负且“序一致”（一种推广可加、预算可加、单位需求及单一意图体的估值类）时，若 $n$ 整除 $m$，则对于 $m \in \omega(n^2)$ 存在 SD-无嫉妒分配（一种比无嫉妒更强的公平性概念），且对于所有 $m \in \omega(n^2)$ 存在 SD-EFX 分配。对 $n$ 的依赖是紧的，即当 $m \in O(n^2)$ 时，无嫉妒分配以常数概率不存在。对于任意估值（允许非单调、负值或混合曼纳估值）且 $n=2$ 个智能体的情形，我们证明无嫉妒分配以概率 $1 - \Theta(1/m)$ 存在（且这是紧的）。