Exact Maximin Share Fairness via Adjusted Supply

Abstract

This work addresses fair allocation of indivisible items in settings wherein it is feasible to create copies of resources or dispose of tasks. We establish that exact maximin share (MMS) fairness can be achieved via limited duplication of goods even under monotone valuations. We also show that, when allocating chores under monotone costs, MMS fairness is always feasible with limited disposal of chores. Since monotone valuations do not admit any nontrivial approximation guarantees for MMS, our results highlight that such barriers can be circumvented by post facto adjustments in the supply of the items. We prove that, for division of $m$ goods among $n$ agents with monotone valuations, there always exists an assignment of subsets of goods to the agents such that they receive at least their maximin shares and no single good is allocated to more than $3 \log m$ agents. In addition, the sum of the sizes of the assigned subsets does not exceed $m$. For identically ordered valuations, we obtain an upper bound of $O(\sqrt{\log m})$ on the maximum assignment multiplicity across goods and an $m + \widetilde{O}\left(\frac{m}{\sqrt{n}} \right)$ bound for the total number of goods assigned. Further, for additive valuations, we prove that there always exists an MMS assignment in which no single good is allocated to more than $2$ agents and the total number of goods assigned is at most $2m$. For chores, we upper bound the number of chores that need to be discarded for ensuring MMS fairness. We prove that, under monotone costs, there exists an MMS assignment in which at most $\frac{m}{e}$ remain unassigned. For identically ordered costs, we establish that MMS fairness can be achieved while keeping at most $\widetilde{O} \left(\frac{m}{n^{1/4}} \right)$ chores unassigned. We also prove that the obtained bounds for monotone valuations and monotone costs are essentially tight.