几乎所有代理的公平份额分配
Fair Share Allocations for Almost All Agents
- Tel Aviv University（特拉维夫大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对可加估值下的不可分割商品公平分配问题，提出了保证几乎所有代理获得精确任意价格份额（APS）的分配方案，并扩展到最大最小份额（MMS）及两全其美的比例性保证，同时给出了权利上限的量化结果与不可能性界限。
AI中文摘要：
我们研究了在可加估值下不可分割商品在代理间的公平分配问题。我们的主要结果保证了当总权利至多为 $1-\varepsilon$ 且每个个体权利足够小时，每个代理都能获得其精确的任意价格份额（APS）。对于平等权利，这产生了一个 $1$-out-of-$(n+O(\sqrt{n\log n}))$ 的最大最小份额（MMS）分配。我们还建立了两个兼顾两全其美的保证，这些保证在期望中保持比例性：每个代理以至少 $1-O((\log n/n)^{1/3})$ 的概率获得其完整的 MMS，或者每个代理在每种结果中都获得其 $1$-out-of-$(n+O(n^{2/3}(\log n)^{1/3}))$ 的 MMS。这两个保证都扩展到足够小的不平等权利，用 APS 替代 MMS。在数量上，我们的确定性结果允许权利上限为 $\varepsilon^2/\log(1/\varepsilon)$ 的量级，而一个不可能性构造表明任何普遍充分的上限必须是 $O(\varepsilon)$。
英文摘要：
We study fair allocation of indivisible goods among agents with additive valuations. Our main result guarantees every agent her exact AnyPrice Share (APS) whenever the total entitlement is at most $1-\varepsilon$ and each individual entitlement is sufficiently small. For equal entitlements, this yields a $1$-out-of-$(n+O(\sqrt{n\log n}))$ maximin share (MMS) allocation. We also establish two best-of-both-worlds guarantees that preserve proportionality in expectation: each agent receives her full MMS with probability at least $1-O((\log n/n)^{1/3})$, or every agent receives her $1$-out-of-$(n+O(n^{2/3}(\log n)^{1/3}))$ MMS in every outcome. Both guarantees extend to sufficiently small unequal entitlements, with APS replacing MMS. Quantitatively, our deterministic result allows an entitlement cap of order $\varepsilon^2/\log(1/\varepsilon)$, while an impossibility construction shows that any universally sufficient cap must be $O(\varepsilon)$.
