Designing optimal–that is, revenue‐maximizing–combinatorial auctions (CAs) is an important elusive problem. It is unsolved even for two bidders and two items for sale. Rather than pursuing the manual approach of attempting to characterize the optimal CA, we introduce a family of CAs and then seek a high‐revenue auction within that family. The family is based on bidder weighting and allocation boosting; we coin such CAs virtual valuations combinatorial auctions (VVCAs). VVCAs are the Vickrey‐Clarke‐Groves (VCG) mechanism executed on virtual valuations that are affine transformations of the bidders’ valuations. The auction family is parameterized by the coefficients in the transformations. The problem of designing a CA is thereby reduced to search in the parameter space of VVCA–or the more general space of affine maximizer auctions. We first construct VVCAs with logarithmic approximation guarantees in canonical special settings: (1) limited supply with additive valuations and (2) unlimited supply. In the main part of the paper, we develop algorithms that design high‐revenue CAs for general valuations using samples from the prior distribution over bidders’ valuations. (Priors turn out to be necessary for achieving high revenue.) We prove properties of the problem that guide our design of algorithms. We then introduce a series of algorithms that use economic insights to guide the search and thus reduce the computational complexity. Experiments show that our algorithms create mechanisms that yield significantly higher revenue than the VCG and scale dramatically better than prior automated mechanism design algorithms. The algorithms yielded deterministic mechanisms with the highest known revenues for the settings tested, including the canonical setting with two bidders, two items, and uniform additive valuations.1
