  We consider a multi-agent model for fair division of mixed manna (i.e. items
for which agents can have positive, zero or negative utilities), in which
agents have additive utilities for bundles of items. For this model, we give
several general impossibility results and special possibility results for three
common fairness concepts (i.e. EF1, EFX, EFX3) and one popular efficiency
concept (i.e. PO). We also study how these interact with common welfare
objectives such as the Nash, disutility Nash and egalitarian welfares. For
example, we show that maximizing the Nash welfare with mixed manna (or
minimizing the disutility Nash welfare) does not ensure an EF1 allocation
whereas with goods and the Nash welfare it does. We also prove that an EFX3
allocation may not exist even with identical utilities. By comparison, with
tertiary utilities, EFX and PO allocations, or EFX3 and PO allocations always
exist. Also, with identical utilities, EFX and PO allocations always exist. For
these cases, we give polynomial-time algorithms, returning such allocations and
approximating further the Nash, disutility Nash and egalitarian welfares in
special cases.
