Envy Freeness Up to One Item: Shall We Duplicate or Remove Resources?

We consider a fair division model in which agents have general valuations for bundles of indivisible items. We propose two new approximate properties for envy freeness of allocations in this model: DEFX and DEF1. We compare these with two existing axiomatic properties: EFX and EF1. For example, we give the first result confirming that EFX allocations may not exist with general but identical valuations. However, even when they do exist in such problems, we prove that DEFX (and, therefore DEF1) and PO allocations exist whereas EFX and PO allocations may not exist. Our results assert eloquently that DEFX and DEF1 approximate fairness better than EFX and EF1.