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. [DMSW15] (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.
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