In this paper, we present new results on the fair and efficient allocation of indivisible goods to agents whose preferences correspond to matroid rank functions. This is a versatile valuation class, with several desirable properties (monotonicity, submodularity) which naturally models several real-world domains. We use these properties to our advantage: first, we show that when agent valuations are matroid rank functions, a socially optimal (i.e. utilitarian social welfare-maximizing) allocation that achieves envy-freeness up to one item (EF1) exists and is computationally tractable. We also prove that the Nash welfare-maximizing and the leximin allocations both exhibit this fair- ness/efficiency combination, by showing that they can be achieved by minimizing any symmetric strictly convex function of agents’ valuations over utilitarian optimal outcomes. Moreover, for a subclass of these valuation functions based on maximum (unweighted) bipartite matching, we show that a leximin allocation can be computed in polynomial time.
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