  We study the trade-off between the Price of Anarchy (PoA) and the Price of
Stability (PoS) in mechanism design, in the prototypical problem of unrelated
machine scheduling. We give bounds on the space of feasible mechanisms with
respect to the above metrics, and observe that two fundamental mechanisms,
namely the First-Price (FP) and the Second-Price (SP), lie on the two opposite
extrema of this boundary. Furthermore, for the natural class of anonymous
task-independent mechanisms, we completely characterize the PoA/PoS Pareto
frontier; we design a class of optimal mechanisms $\mathcal{SP}_\alpha$ that
lie exactly on this frontier. In particular, these mechanisms range smoothly,
with respect to parameter $\alpha\geq 1$ across the frontier, between the
First-Price ($\mathcal{SP}_1$) and Second-Price ($\mathcal{SP}_\infty$)
mechanisms.
  En route to these results, we also provide a definitive answer to an
important question related to the scheduling problem, namely whether
non-truthful mechanisms can provide better makespan guarantees in the
equilibrium, compared to truthful ones. We answer this question in the
negative, by proving that the Price of Anarchy of all scheduling mechanisms is
at least $n$, where $n$ is the number of machines.
