Before revision:  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;
After revision:  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 , and weak, 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;