  Black-box transformations have been extensively studied in algorithmic
mechanism design as a generic tool for converting algorithms into truthful
mechanisms without degrading the approximation guarantees. While such
transformations have been designed for a variety of settings, Chawla et al.
showed that no fully general black-box transformation exists for
single-parameter environments. In this paper, we investigate the potentials and
limits of black-box transformations in the prior-free (i.e., non-Bayesian)
setting in \emph{downward-closed} single-parameter environments, a large and
important class of environments in mechanism design. On the positive side, we
show that such a transformation can preserve a constant fraction of the welfare
at every input if the private valuations of the agents take on a constant
number of values that are far apart, while on the negative side, we show that
this task is not possible for general private valuations.
