  We study the problem of designing truthful and fair mechanisms when
allocating a mixture of divisible and indivisible goods. We first show that
there does not exist an EFM (envy-free for mixed goods) and truthful mechanism
in general. This impossibility result holds even if there is only one
indivisible good and one divisible good and there are only two agents. Thus, we
focus on some more restricted settings. Under the setting where agents have
binary valuations on indivisible goods and identical valuations on a single
divisible good (e.g., money), we design an EFM and truthful mechanism. When
agents have binary valuations over both divisible and indivisible goods, we
first show there exist EFM and truthful mechanisms when there are only two
agents or when there is a single divisible good. On the other hand, we show
that the mechanism maximizing Nash welfare cannot ensure EFM and truthfulness
simultaneously.
