On Hill’s Worst-Case Guarantee for Indivisible Bads

We evaluate the fairness of a rule allocating among agents with equal rights by computing their utility for a hypothetical worst-case share, which only depends on their own valuation and the number of agents. For indivisible goods, Budish proposed the maximin share : the least utility of a bundle in the best partition of the objects; unfortunately maximin share is not always satisfiable. Earlier, Hill proposed the worst maximin share over all utilities with the same largest possible single-object value. More conservative than maximin share, it is guaranteed to be satisfiable for any possible profile of utilities and its computation is elementary, whereas it is NP-hard to compute maximin share. We apply Hill’s approach to the allocation of indivisible bads (objects with disutilities) and compute in closed form the worst-case minimax share for a given value of the worst single bad. We show that the worst-case minimax share is close to the original minimax share and that its monotonic closure is the best guaranteed share for all allocation instances with a common upper bound on the value of the worst single bad.