发表机构
University of Texas, Austin; University of Chicago; University of Illinois Urbana-Champaign; University of California, Irvine（德克萨斯大学奥斯汀分校; 芝加哥大学; 伊利诺伊大学厄巴纳-香槟分校; 加州大学尔湾分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对在线到达的组合型智能体，提出Single-or-Sample随机算法，在无需先验估值信息下以常数概率实现MMS常数因子近似，并揭示近似度与成功概率的紧权衡。
AI 中文摘要
我们研究在最大化份额（MMS）公平性概念下，将 $m$ 个不可分割商品公平分配给 $n$ 个在线到达的智能体的问题。具有在线到达的公平分配以极具挑战性而闻名：先前的工作仅在智能体的偏好属于事先已知的一组估值函数时才能实现常数因子 MMS 保证，而在没有此类先验信息的情况下则没有任何保证。我们为加性估值和次模估值开发了一种新的随机化在线算法，称为 Single-or-Sample，该算法以常数概率同时为所有智能体实现 MMS 的常数因子近似。该算法不需要关于智能体估值的先验知识，并且能够对抗对抗性（不知情的）输入。我们进一步建立了近似度与成功概率之间的基本权衡。具体来说，对于任何 $c \ge 1$，即使对于二元加性估值，也没有在线算法能够以超过 $1 - 1/c^2$ 的概率保证 MMS 的 $1/c$ 近似。这排除了以渐近接近 1 的常数概率实现常数 MMS 的可能性。对于 XOS，下界要强得多：没有任何算法能够以常数概率为所有智能体实现甚至 $1/\log\log n$-MMS。我们用一种适用于 $c\in \Omega(\log n)$ 范围的算法来补充这一下界，即以 $(1-O(1/c))$ 的概率为所有智能体实现 $1/c$-MMS，并表明这种权衡对于 XOS 是紧的。我们的常数因子算法引入了几个新思想，将贪心次模最大化与随机化分配和单项归约相结合。一个关键的技术要素是一种分析无放回迭代抽样的新方法。我们开发了适用于一类广泛的、在元素和轮次之间具有复杂依赖关系的自适应过程的集中界，这可能具有独立的意义。
英文摘要
We study the problem of fairly allocating $m$ indivisible goods among $n$ agents who arrive online, under the notion of maximin share (MMS) fairness. Fair allocation with online arrivals is notoriously challenging: prior work achieves constant-factor MMS guarantees only when agents' preferences belong to a set of valuation functions known in advance, while no guarantees were known without such prior information. We develop a new randomized online algorithm for additive and submodular valuations, that we call Single-or-Sample, and that achieves a constant-factor approximation to MMS simultaneously for all agents, with constant probability. The algorithm requires no prior knowledge about the agents' valuations, and works against adversarial (oblivious) inputs. We further establish a fundamental tradeoff between approximation and success probability. Specifically, for any $c \ge 1$, no online algorithm can guarantee a $1/c$-approximation to MMS with probability exceeding $1 - 1/c^2$, even for binary additive valuations. This rules out constant MMS with probability asymptotically closer to $1$ than a constant. For XOS, the lower bound is much stronger: no algorithm can achieve even $1/\log\log n$-MMS to all agents with a constant probability. We complement this lower bound with an algorithm for the regime $c\in Ω(\log n)$, namely $1/c$-MMS to all agents with probability $(1-O(1/c))$, and show that this tradeoff is tight for XOS. Our constant-factor algorithm introduces several new ideas, combining greedy submodular maximization with randomized allocation and single-item reduction. A key technical ingredient is a new approach for analyzing iterative sampling without replacement. We develop concentration bounds that apply to a broad class of adaptive processes with complex dependencies across elements and rounds, which may be of independent interest.
