  We study the fair division of divisible bad resources with strategic agents
who can manipulate their private information to get a better allocation. Within
certain constraints, we are particularly interested in whether truthful
envy-free mechanisms exist over piecewise-constant valuations. We demonstrate
that no deterministic truthful envy-free mechanism can exist in the
connected-piece scenario, and the same impossibility result occurs for hungry
agents. We also show that no deterministic, truthful dictatorship mechanism can
satisfy the envy-free criterion, and the same result remains true for
non-wasteful constraints rather than dictatorship. We further address several
related problems and directions.
