  We study the problem of scheduling $m$ tasks to $n$ selfish, unrelated
machines in order to minimize the makespan, where the execution times are
independent random variables, identical across machines. We show that the VCG
mechanism, which myopically allocates each task to its best machine, achieves
an approximation ratio of $O\left(\frac{\ln n}{\ln \ln n}\right)$. This
improves significantly on the previously best known bound of
$O\left(\frac{m}{n}\right)$ for prior-independent mechanisms, given by Chawla
et al. [STOC'13] under the additional assumption of Monotone Hazard Rate (MHR)
distributions. Although we demonstrate that this is in general tight, if we do
maintain the MHR assumption, then we get improved, (small) constant bounds for
$m\geq n\ln n$ i.i.d. tasks, while we also identify a sufficient condition on
the distribution that yields a constant approximation ratio regardless of the
number of tasks.
