arXiv 1907.045962019-07cs.GT

一点慈善保证几乎没有嫉妒

A Little Charity Guarantees Almost Envy-Freeness

Bhaskar Ray Chaudhury, Tellikepalli Kavitha, Kurt Mehlhorn, Alkmini Sgouritsa

备注：: Preliminary version appeared in SODA 2020

摘要：

不可分割货物的公平分割是一个研究得很好的问题。这个问题的目标是以“公平”的方式将$m$商品分配给$n$代理，其中每个代理对每个商品子集都有一个估价。我们假设一般估值。无嫉妒是研究最广泛的公平概念。然而，当商品不可分割时，不嫉妒的分配并不总是存在。我们在这里考虑的公平性概念是“任何好的嫉妒自由”（EFX），其中没有代理人嫉妒另一个代理人后，从其他代理人的包中删除任何单一的好。不知道这样的分配是否总是存在，即使当$n=3$。我们表明，商品集合总是被划分为$n+1$子集$(X_1,\ldots,X_n,P)$，其中对于$i \in [n]$，$X_i$是分配给代理$i$的捆绑包，而集合$P$是未分配的（或捐赠给慈善机构），使得我们有$\colon$ 1）任何好的无嫉妒度，2）没有代理将$P$的价值高于她自己的捆绑包，以及3）少于$n$的商品捐给慈善机构，即，$|P| < n$（通常为$m \gg n$）。我们的证据是有建设性的。当代理具有加性赋值并且$\lvert P \rvert$很大时（即，当$|P|$接近$n$时），我们的分配也具有良好的最小共享（MMS）保证。此外，我们的算法的一个小的变化也表明存在的分配，这是$4/7$ groupwise最大最小份额（GMMS）：这是一个概念的公平性强于MMS。这改进了已知用于近似GMMS分配的$1/2$的当前最佳界限。

英文摘要：

Fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a "fair" manner, where every agent has a valuation for each subset of goods. We assume general valuations. Envy-freeness is the most extensively studied notion of fairness. However, envy-free allocations do not always exist when goods are indivisible. The notion of fairness we consider here is "envy-freeness up to any good" (EFX) where no agent envies another agent after the removal of any single good from the other agent's bundle. It is not known if such an allocation always exists even when $n=3$. We show there is always a partition of the set of goods into $n+1$ subsets $(X_1,\ldots,X_n,P)$ where for $i \in [n]$, $X_i$ is the bundle allocated to agent $i$ and the set $P$ is unallocated (or donated to charity) such that we have$\colon$ 1) envy-freeness up to any good, 2) no agent values $P$ higher than her own bundle, and 3) fewer than $n$ goods go to charity, i.e., $|P| < n$ (typically $m \gg n$). Our proof is constructive. When agents have additive valuations and $\lvert P \rvert$ is large (i.e., when $|P|$ is close to $n$), our allocation also has a good maximin share (MMS) guarantee. Moreover, a minor variant of our algorithm also shows the existence of an allocation which is $4/7$ groupwise maximin share (GMMS): this is a notion of fairness stronger than MMS. This improves upon the current best bound of $1/2$ known for an approximate GMMS allocation.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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