发表机构
University of Oxford（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
针对不知情对手的在线公平分配，提出改进算法，将PROP1近似因子提升至Ω(1/log log(n/δ))，并揭示EF1、EFX和MMS的随机算法概率下界。
AI 中文摘要
我们研究在$n$个智能体之间不可分割物品的在线分配问题，其中每个物品到达时必须立即且不可撤销地分配。针对自适应对手，Neoh和Teh [2026]证明了没有任何算法能够保证与物品数量无关的、对至多一个物品的比例性（PROP1）的正近似，并且对于任意固定的$k$，对至多$k$个物品的比例性（PROP$k$）同样如此。我们转而考虑不知情对手，其预先固定输入。Choo等人 [2026]表明，均匀随机分配以至少$1-δ$的概率返回一个$Θ(1/\log(n/δ))$-PROP1分配。我们将其改进为$Ω(1/\log\log(n/δ))$；我们的算法不将$δ$作为输入，因此同一算法对每个$δ\in(0,1)$都实现该界。此外，以相同概率，我们算法的一个变体给予每个智能体几乎其全部束，并且即使在没有单个物品相对于该份额过于有价值时，也不添加任何物品。相比之下，对于至多一个物品的无嫉妒性（EF1），我们证明，对于每个$α\in(0,1]$，每个随机算法都存在一个输入，其返回$α$-EF1分配的概率至多为$e^{-Ω(n)}$。对于至多任意物品的无嫉妒性（EFX），该概率在仅有$n+1$个物品时至多为$1/n!$，这一界在因子$(n+1)/2$内是最优的。对于最大最小份额（MMS），该概率至多为$5/6$，无论$α$多小。允许更多移除则给出正的无嫉妒保证：将每个物品分配给具有正价值的智能体中的均匀随机一个，以高概率实现对至多对数多个物品的无嫉妒性的任意接近一的近似因子，并且对数多个物品对于该规则是必要的。
英文摘要
We study the online allocation of indivisible goods among $n$ agents, where each good must be allocated immediately and irrevocably upon arrival. Against an adaptive adversary, Neoh and Teh [2026] proved that no algorithm can guarantee a positive approximation to proportionality up to one good (PROP1) that is independent of the number of goods, and the same holds for proportionality up to $k$ goods (PROP$k$) for any fixed $k$. We instead consider an oblivious adversary, which fixes the input in advance. Choo et al. [2026] showed that the uniformly random allocation returns a $Θ(1/\log(n/δ))$-PROP1 allocation with probability at least $1-δ$. We improve this to $Ω(1/\log\log(n/δ))$; our algorithm does not take $δ$ as input, so the same algorithm achieves this bound for every $δ\in(0,1)$. Moreover, with the same probability, a variant of our algorithm gives every agent almost her bundle, and even without adding any good when no single good is too valuable relative to this share. In contrast, for envy-freeness up to one good (EF1), we show that, for every $α\in(0,1]$, every randomized algorithm has an input on which its probability of returning an $α$-EF1 allocation is at most $e^{-Ω(n)}$. For envy-freeness up to any good (EFX), this probability is at most $1/n!$ with only $n+1$ goods, a bound that is optimal within a factor of $(n+1)/2$. For the maximin share (MMS), this probability is at most $5/6$, however small $α$ is. Allowing more removals gives a positive envy-freeness guarantee: allocating each good to a uniformly random agent among those with positive values achieves, with high probability, an approximation factor arbitrarily close to one for envy-freeness up to logarithmically many goods, and logarithmically many goods are necessary for this rule.
