希尔对不可分割的坏的最坏情况保证
On Hill's Worst-Case Guarantee for Indivisible Bads
摘要：
当在具有平等权利的代理人之间分配对象时，人们通常通过将他们收到的效用与基准份额进行比较来评估分配规则的公平性-基准份额只是她自己的估值和代理人数量的函数。这个份额被称为保证，如果对于任何估值概况，存在确保每个代理人的份额的分配。当对象是不可分割的商品时，Budish [J. Political Econ.，2011]提出了MaxMinShare，即，在对象的最佳划分中，bundle的效用最小，不幸的是，这并不能保证。相反，早期的开创性工作希尔[安。1987年]提出了一个份额的最坏情况最大最小份额在所有估值相同的最大可能的单一对象的价值。虽然希尔的份额比MaxMinShare更保守，但它是一个实际的保证，并且它的计算是基本的，不像MaxMinShare涉及解决NP-难问题。我们应用希尔的方法来分配不可分割的坏（与disutilities或成本的对象），并将最坏情况下的MinMaxShare的紧闭形式的最坏坏的给定值。我们认为，希尔的份额分配坏是有效的意义上接近原来的MinMaxShare值，有很多要了解的保证代理可以提供从她最坏的单坏的负效用。此外，我们证明了单调覆盖希尔的份额是最好的保证，可以实现在希尔模型的所有分配实例。
英文摘要：
When allocating objects among agents with equal rights, people often evaluate the fairness of an allocation rule by comparing their received utilities to a benchmark share - a function only of her own valuation and the number of agents. This share is called a guarantee if for any profile of valuations there is an allocation ensuring the share of every agent. When the objects are indivisible goods, Budish [J. Political Econ., 2011] proposed MaxMinShare, i.e., the least utility of a bundle in the best partition of the objects, which is unfortunately not a guarantee. Instead, an earlier pioneering work by Hill [Ann. Probab., 1987] proposed for a share the worst-case MaxMinShare over all valuations with the same largest possible single-object value. Although Hill's share is more conservative than the MaxMinShare, it is an actual guarantee and its computation is elementary, unlike that of the MaxMinShare which involves solving an NP-hard problem. We apply Hill's approach to the allocation of indivisible bads (objects with disutilities or costs), and characterise the tight closed form of the worst-case MinMaxShare for a given value of the worst bad. We argue that Hill's share for allocating bads is effective in the sense of being close to the original MinMaxShare value, and there is much to learn about the guarantee an agent can be offered from the disutility of her worst single bad. Furthermore, we prove that the monotonic cover of Hill's share is the best guarantee that can be achieved in Hill's model for all allocation instances.
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