发表机构
Harvard University; University of Oxford（哈佛大学; 牛津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对在线不可分割物品公平分配的三个开放问题，证明了PROPk的不可能性，提出了确定性PROP1算法，并确定了Like规则的紧高概率PROP1保证。
AI 中文摘要
我们研究不可分割物品的在线公平分配，物品逐个到达且必须立即不可撤销地分配。我们解决了文献中三个核心开放问题：第一，我们证明，针对适应性对手，任何在线算法都无法保证对任意k件物品的比例性（PROPk）有任何正的乘性近似，即便提前知晓物品总数、所有价值在[0,1]区间内、每件物品最多仅两名代理赋予正价值，该结论依然成立，且这种不可能性扩展到文献中当前研究的广泛标准基于嫉妒、比例性和份额的概念，我们也为家务品（chores）建立了类似的不可能性；第二，在有预测的场景中，此前已知对物品最大价值的轻量级预测可给出1/n-PROP1，留下了对n的依赖是否必要的问题，我们针对适应性对手给出了确定性1/2-PROP1算法，更一般地，若算法获得任意物品赋予正价值的代理数量的上界κ∈[2,n]，则保证提升为n/(n+κ)，且在单侧预测误差下保持恒定，当提前知晓物品总数m且m≥n log n时，对每个固定的β∈(0,1/2)，我们还给出一种确定性算法，在将每个代理的价值按其最大物品价值归一化后，同时保证β-PROP1和O(√(m log n/n))的最大加性嫉妒；第三，针对非适应性对手，我们确定了经典Like规则的紧高概率PROP1保证，该规则将每件物品在对其赋予正价值的代理中均匀分配，与均匀随机分配不同，当更少代理对同一件物品赋予价值时，其保证会提升。
英文摘要
We study the online fair division of indivisible items, where items arrive one at a time and must be allocated immediately and irrevocably. We address three central open questions in the literature. First, we show that for every $n\ge 2$ agents, every fixed $k\ge 1$, and every $α\in(0,1]$, no online algorithm can guarantee $α$-PROP$k$ against an adaptive adversary. This remains true even when the total number of goods is known in advance, all values lie in $[0,1]$, and every good is positively valued by at most two agents. The impossibility extends to a broad range of standard envy-based, proportionality-based, and share-based fairness notions. We also establish an analogous impossibility for chores. Second, in the setting with predictions, a lightweight form of future information, the maximum item value, was previously known to guarantee only $1/n$-PROP1, leaving open whether the dependence on $n$ is necessary. We give a deterministic $19/30$-PROP1 algorithm against adaptive adversaries that does not require knowing the total number of goods. Given an additional upper bound $κ\in[2,n]$ on the number of agents who value any good positively, the guarantee improves to $\max \{ 19/30, n/(n+κ) \}$. The guarantee remains a positive constant under any fixed one-sided prediction error below one. When predictions are exact and the total number of goods $m\ge n\log n$ is known in advance, a deterministic algorithm achieves the same $19/30$-PROP1 factor together with $O(\sqrt{m\log n/n})$ maximum additive envy after normalizing each agent's values by their maximum item value. Third, against a non-adaptive adversary, we determine the tight high-probability PROP1 guarantee of the classical Like rule, which assigns each good uniformly among the agents who value it positively. Its guarantee improves when fewer agents value the same good, unlike uniform random allocation.
