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.

翻译：在将物品分配给具有平等权利的主体时，人们常通过比较个体所获效用与基准份额（仅取决于其自身估值和主体数量的函数）来评估分配规则的公平性。若对任意估值分布均存在一种分配使得每个主体都能达到该份额，则称此份额为保证。当物品为不可分割商品时，Budish [J. Political Econ., 2011] 提出了最大最小份额（MaxMinShare），即物品最优划分中某个子集的最小效用，可惜该份额并非保证。相比之下，Hill [Ann. Probab., 1987] 的开创性工作更早提出了一种份额——在所有具有相同最大单项价值的估值中取最差情形下的最大最小份额。尽管Hill份额比最大最小份额更为保守，但它是一种实际可行的保证，且其计算是初等的，而最大最小份额的计算涉及NP难问题。我们将Hill方法应用于不可分割厌恶品（具有负效用或成本的物品）的分配，并针对给定最差厌恶品价值，刻画了最差情形下最小最大份额（MinMaxShare）的紧致封闭形式。我们论证了Hill份额在分配厌恶品时是有效的，因为其值接近原始最小最大份额值，并且从个体最差单一厌恶品的负效用中可以获知该主体可被提供的保证。此外，我们证明Hill份额的单调覆盖是在Hill模型下对所有分配实例可实现的最佳保证。