AI 中文总结
研究非线性估值下可分物品的公平分配问题，针对最大最小份额保证设计算法，给出\(\frac{1}{2n - 1}\)-MMS分配，证明对MMS的近似比例几乎是紧的；还研究无嫉妒性，证明其检查存在性对\(n\)个主体和至少三种物品是NP难的，对单个物品给出多项式时间算法。
AI 中文摘要
我们研究在具有非线性估值的主体间分配同质可分物品的问题。具体而言，主体从给定物品获得的价值仅取决于其获得的物品数量，且不一定与数量呈线性关系。例如，在单断点分段常数估值下，每个主体为每种物品指定一个阈值，当获得的数量低于（或至少）该阈值时，该主体获得的效用为零（或该物品的全部效用）。给定物品间可加的非线性估值，我们专注于设计公平分配算法，并考虑两个著名的公平属性：最大最小份额（MMS）保证和无嫉妒性（EF）。对于MMS，我们设计了一种算法，对于具有任意非递减估值的\(n\)个主体，总能产生\(\frac{1}{2n - 1}\)-MMS分配。值得注意的是，该算法结果几乎是紧的，因为即使主体具有单断点分段常数估值时，我们给出了不可能保证对MMS有超过\(1/n\)的近似。对于\(n\leq3\)个主体，我们表明\(1/n\)的比例是紧的。关于无嫉妒性，我们表明即使主体具有单断点分段常数估值，检查\(n\)个主体和至少三种物品的EF和帕累托最优（PO）分配的存在性是NP难的。我们通过考虑单个可分物品的情况来补充硬度结果，并设计了一种多项式时间算法来检查具有分段线性估值的主体是否存在EF和PO分配。
英文摘要
We study the problem of dividing homogeneous divisible goods among agents with non-linear valuations. Specifically, the value that an agent gains from a given good depends only on the amount of the good they receive, and is not necessarily linear with respect to the amount. For instance, under one-breakpoint piecewise-constant valuations, each agent specifies a threshold for each good such that this agent receives utility zero (resp., full utility of the good) when getting an amount below (resp., at least) the threshold. Given non-linear valuations that are additive across the goods, we focus on designing fair allocation algorithms and consider two well-known fairness properties: the maximin share (MMS) guarantee and envy-freeness (EF). For MMS, we devise an algorithm which always produces a $\frac{1}{2n-1}$-MMS allocation for $n$ agents with arbitrary non-decreasing valuations. It is worth noting that this algorithmic result is almost tight as we give an impossibility of guaranteeing more than $1/n$ approximation to MMS, even when agents have one-breakpoint piecewise-constant valuations. For $n \leq 3$ agents, we show the ratio $1/n$ is tight. Regarding envy-freeness, we show it is NP-hard to check the existence of an EF and Pareto optimal (PO) allocation for $n$ agents and at least three goods, even when agents have one-breakpoint piecewise-constant valuations. We complement the hardness result by considering the case with a single divisible good, and devising a polynomial-time algorithm to check whether an EF and PO allocation exists or not for agents with piecewise-linear valuations.
CommentsAppears in the 24th International Conference on Autonomous Agents and Multiagent Systems (AAMAS), 2025
