Compact bid languages and core-pricing in large multi-object auctions

Combinatorial auctions address the fundamental problem of allocating multiple items in the presence of complex bidder preferences including complements or substitutes. They have found application in public and private sector auctions. Many real-world markets involve the sale of a large number of items, limiting the direct application of combinatorial auctions due to both computational intractability for the auctioneer and communication difficulty for the bidders. More specifically for the latter, an enumerative XOR bidding language (widely discussed in the literature and used in recent government spectrum auctions) grows too quickly to be practical. Market designs for large markets with many items and similar incentive properties have previously received little attention in the literature. We introduce an auction design framework for large markets with hundreds of items and complex bidder preferences. The framework comprises compact bid languages in a sealed-bid auction and methods to compute second-price rules such as the Vickrey-ClarkeGroves or bidder-optimal, core-selecting payment rules. The latter have been introduced in spectrum auctions worldwide as a means to encourage incentives for truthful bidding, but at the same time avoid some problems of the Vickrey-Clarke-Groves mechanism. We discuss compact bidding languages for TV ads markets and volume-discount procurement auctions, and investigate the resulting winner-determination problem and the computation of core payments. For realistic instances of the respective winner determination problems, very good solutions with a small integrality gap can be found quickly, though closing the integrality gap to find marginally better solutions or prove optimality can take a prohibitively large amount of time. Our subsequent adaptation of a constraint-generation technique for the computation of bidder-optimal core paymants to this environment is a new, practically viable paradigm by which core-selecting auction designs can be applied to large markets with potentially hundreds of items.