We introduce and analyze new envy-based fairness concepts for agents with weights that quantify their entitlements in the allocation of indivisible items. We propose two variants of weighted envy-freeness up to one item (WEF1): strong, where envy can be eliminated by removing an item from the envied agents bundle, and weak, where envy can be eliminated either by removing an item (as in the strong version) or by replicating an item from the envied agents bundle in the envying agents bundle. We show that for additive valuations, an allocation that is both Pareto optimal and strongly WEF1 always exists and can be computed in pseudo-polynomial time; moreover, an allocation that maximizes the weighted Nash social welfare may not be strongly WEF1, but always satisfies the weak version of the property. Moreover, we establish that a generalization of the round-robin picking sequence algorithm produces in polynomial time a strongly WEF1 allocation for an arbitrary number of agents; for two agents, we can efficiently achieve both strong WEF1 and Pareto optimality by adapting the adjusted winner procedure. Our work highlights several aspects in which weighted fair division is richer and more challenging than its unweighted counterpart.
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