  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.
