Analyses of Cardinal Auctions

Abstract

We study cardinal auctions for selling multiple copies of a good, in which bidders specify not only their bid or how much they are ready to pay for the good, but also a cardinality constraint on the number of copies that will be sold via the auction. We perform first known Price of Anarchy type analyses with detailed comparison of the classical Vickrey-Clarke-Groves (VCG) auction and one based on minimum pay property (MPP) which is similar to Generalized Second Price auction commonly used in sponsored search. Without cardinality constraints, MPP has the same efficiency (total value to bidders) and at least as much revenue (total income to the auctioneer) as VCG; this also holds for certain other generalizations of MPP (e.g., prefix constrained auctions, as we show here). In contrast, our main results are that, with cardinality constraints, (a) equilibrium efficiency of MPP is 1/2 of that of VCG and this factor is tight, and (b) in equilibrium MPP may collect as little as 1/2 the revenue of VCG. These aspects arise because in presence of cardinality constraints, more strategies are available to bidders in MPP, including bidding above their value, and this makes analyses nontrivial.