发表机构
School of Mathematical Science, Zhejiang University（浙江大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究不可分割商品完全在线分配的随机算法，针对非自适应对手提出改进的PROP1近似，并证明EFX和EF1的固有局限性。
AI 中文摘要
我们研究在$n\ge2$个具有非负可加估值的智能体之间完全在线分配不可分割商品的随机算法。商品逐一到达，必须立即且不可撤销地分配。除智能体数量外，事先一无所知。由于精确的事前无嫉妒性和比例性易于实现，而此处考虑的公平概念不可能有正的事后近似，我们研究高概率公平性和期望实现公平性这两个中间概念。针对非自适应对手，我们给出一个随机算法，对每个$\delta\in(0,1)$，以至少$1-\delta$的概率实现对至多一件商品的比例性（PROP1）的$\Omega(\sqrt{\log n}/\log(n/\delta))$近似，并达到$\Omega(1/\sqrt{\log n})$的期望实现PROP1因子。与独立均匀分配（Rand）相比，这些保证在高概率和期望实现标准下均带来$\Omega(\sqrt{\log n})$的改进。对于至多任意一件商品的无嫉妒性（EFX），任何正因子的成功概率可以任意小，即使估值相同。因此，每个随机完全在线算法的期望实现因子为零。对于至多一件商品的无嫉妒性（EF1），没有随机完全在线算法能以超过$\frac{n+1}{2n}$的置信度保证正因子。期望实现的EF1保证也至多为$\frac{n+1}{2n}$。
英文摘要
We study randomized algorithms for the fully online allocation of indivisible goods among $n\ge2$ agents with nonnegative additive valuations. Goods arrive sequentially and must be allocated immediately and irrevocably, with only $n$ known in advance. Since exact ex-ante envy freeness and proportionality are readily achievable, while no positive ex-post approximation is possible for the fairness notions considered here, we study the intermediate notions of high-probability fairness and expected realized fairness. Against a non-adaptive adversary, we give a randomized algorithm for proportionality up to one good (PROP1) whose parameter depends only on $n$ and that preserves exact ex-ante envy-freeness and proportionality. At confidence $1-δ$, its PROP1 guarantee improves on independent uniform allocation (Rand) by a factor of $Ω(\log n)$, uniformly over $δ\in(0,1/2]$. As $n\to\infty$, its expected realized PROP1 factor is at least $\frac{3-\sqrt5}{2}-o(1)$. We also show that the expected realized PROP1 factor of Rand is $(1+o(1))/\log n$, yielding an improvement of at least $\bigl(\frac{3-\sqrt5}{2}-o(1)\bigr)\log n$ for our algorithm. For every randomized online algorithm and every positive approximation factor, the success probability can be made arbitrarily small for envy freeness up to any good (EFX) and at most $\frac{n+1}{2n}$ for envy freeness up to one good (EF1). Consequently, every randomized fully online algorithm has an expected realized EFX guarantee of zero and an expected realized EF1 guarantee of at most $\frac{n+1}{2n}$.
Comments22 pages. Improved high-probability and expected realized PROP1 guarantees; revised proofs and exposition
