  The classic fair division problems assume the resources to be allocated are
either divisible or indivisible, or contain a mixture of both, but the agents
always have a predetermined and uncontroversial agreement on the
(in)divisibility of the resources. In this paper, we propose and study a new
model for fair division in which agents have their own subjective divisibility
over the goods to be allocated. That is, some agents may find a good to be
indivisible and get utilities only if they receive the whole good, while others
may consider the same good to be divisible and thus can extract utilities
according to the fraction of the good they receive. We investigate fairness
properties that can be achieved when agents have subjective divisibility.
First, we consider the maximin share (MMS) guarantee and show that the
worst-case MMS approximation guarantee is at most $2/3$ for $n \geq 2$ agents
and this ratio is tight in the two- and three-agent cases. This is in contrast
to the classic fair division settings involving two or three agents. We also
give an algorithm that produces a $1/2$-MMS allocation for an arbitrary number
of agents. Second, we adapt the notion of envy-freeness for mixed goods (EFM)
to our model and show that EFM is incompatible with non-wastefulness, a rather
weak economic efficiency notion. On the positive side, we prove that an EFM and
non-wasteful allocation always exists for two agents if at most one good is
discarded.
