  We consider a fair division model in which agents have general valuations for
bundles of indivisible items. We propose two new axiomatic properties for
allocations in this model: EF1+- and EFX+-. We compare these with the existing
EF1 and EFX. Although EF1 and EF1+- allocations often exist, our results assert
eloquently that EFX+- and PO allocations exist in each case where EFX and PO
allocations do not exist. Additionally, we prove several new impossibility and
incompatibility results.
