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 $alphageq 1$ 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.
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