In this paper, we introduce and analyze new envy-based fairness concepts for agents with weights: these weights regulate their mutual envy in a situation where indivisible goods are allocated to the agents  . We propose two variants of  envy-freeness up to one item for the weighted setting: in the strong variant, the  envy can be eliminated by removing an item from the envied agent's bundle , whereas in the weak variant,  envy can be eliminated by either removing an item from the envied agent's bundle or by replicating an item from the envied agent's bundle in the envying agent's bundle  . We prove that for additive valuations, a strongly weighted envy-free allocation up to one item always existsand can be efficiently computed by means of a weight-based picking sequence . For two agents, we can also efficiently achieve strong weighted envy-freeness up to one item in conjunction with Pareto optimality using a weighted version of the classic adjusted winner algorithm. In addition, we show that an allocation that maximizes the weighted Nash social welfare always satisfies weak weighted envy-freeness up to one item, but may fail to satisfy the strong version of this property .
In this paper, 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 the envy can be eliminated by removing an item from the envied agent's bundle ) andweak (where the envy can be eliminated either by removing an item as in the strong version or by replicating an item from the envied agent's bundle in the envious agent's bundle ) . We prove that for additive valuations, an allocation that is both Pareto optimal and strongly WEF1 always exists; however, 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 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 classic adjusted winner algorithm. We also explore the connections of WEF1 with approximations to the weighted versions of two other fairness concepts: proportionality and the maximin share guarantee .
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weights: these weights regulate their mutual envy in a situation where indivisible goods are allocated to the agents
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for the weighted setting: in the strong variant, the
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envying
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a strongly weighted envy-free allocation up to one item always existsand can be efficiently computed by means of a weight-based picking sequence . For
an allocation that is both Pareto optimal and strongly WEF1 always exists; however, 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 produces in polynomial time a strongly WEF1 allocation for an arbitrary number of agents; for
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also efficiently achieve strong weighted envy-freeness up to one item in conjunction with Pareto optimality using a weighted version of
efficiently achieve both strong WEF1 and Pareto optimality by adapting
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In addition, we show that an allocation that maximizes the weighted Nash social welfare always satisfies weak weighted envy-freeness up to one item, but may fail to satisfy the strong version of this property
We also explore the connections of WEF1 with approximations to the weighted versions of two other fairness concepts: proportionality and the maximin share guarantee