Fair allocation of indivisible goods: Beyond additive valuations

Abstract

We conduct a study on the problem of fair allocation of indivisible goods when maximin share[1] is used as the measure of fairness. Most of the current studies on this notion are limited to the case that the valuations are additive. In this paper, we go beyond additive valuations and consider the cases that the valuations are submodular, fractionally subadditive, and subadditive. We give constant approximation guarantees for agents with submodular and XOS valuations, and a logarithmic bound for the case of agents with subadditive valuations. Furthermore, we complement our results by providing close upper bounds for each class of valuation functions. Finally, we present algorithms to find such allocations for submodular and XOS settings in polynomial time.