  We study the communication complexity of combinatorial auctions via
interpolation mechanisms that interpolate between non-truthful and truthful
protocols. Specifically, an interpolation mechanism has two phases. In the
first phase, the bidders participate in some non-truthful protocol whose output
is itself a truthful protocol. In the second phase, the bidders participate in
the truthful protocol selected during phase one. Note that virtually all
existing auctions have either a non-existent first phase (and are therefore
truthful mechanisms), or a non-existent second phase (and are therefore just
traditional protocols, analyzed via the Price of Anarchy/Stability).
  The goal of this paper is to understand the benefits of interpolation
mechanisms versus truthful mechanisms or traditional protocols, and develop the
necessary tools to formally study them. Interestingly, we exhibit settings
where interpolation mechanisms greatly outperform the optimal traditional and
truthful protocols. Yet, we also exhibit settings where interpolation
mechanisms are provably no better than truthful ones. Finally, we apply our new
machinery to prove that the recent single-bid mechanism of Devanur et.
al.~\cite{DevanurMSW15} (the only pre-existing interpolation mechanism in the
literature) achieves the optimal price of anarchy among a wide class of
protocols, a claim that simply can't be addressed by appealing just to
machinery from communication complexity or the study of truthful mechanisms.
