Optimal Bundling for Truthful Auctions

We consider a multi-item VCG auction setting where a bidder’s valuation of a bundle of items is additive, i.e., the sum of the individual item valuations in the bundle. We propose a direct mechanism, called the Pairwise Bundler Auction (PBA), where bidders report their individual item valuations following which the auctioneer optimally partitions the set of items into bundles. This is a bilevel cubic binary optimization problem. We transform this cubic binary integer social welfare maximization problem into a binary integer linear program to compute the optimal bundling of items. We show that the dominant strategy for the bidders is to report their bids truthfully. We present numerical results and find that the benefit of bundling items decreases as the number of bidders per item increases. Our formulation allows for the construction of a Benders decomposition algorithm to compute the optimal bundling where the master problem is a binary integer program while the subproblem in the Benders decomposition algorithm is a linear program. We show how the dual variables of each linear program quantifies the marginal gain from having a given pair of items together in a bundle. As an important theoretical contribution, we identify a class of linearly-constrained binary quadratic optimization problems whose relaxations have integral solutions.