Before revision:  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  .
After revision:  We propose two variants of envy-freeness up to one item for the weighted setting: in the strong variant, the (where 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 .