A Note on Revenue Maximization and E¢ciency in Multi-Object Auctions

We consider an auction with risk neutral agents having independent private valuations for several heterogenous objects. Most of the literature on revenuemaximizing auctions has focused on the sale of one good or on the sale of several identical units (thus yielding one-dimensional informational models). Two reasons for ine¢ciency in revenue-maximizing auctions have been identi...ed 1) The (monopolist) seller can increase revenue by restricting supply. 2) A revenue maximizing seller will sell to bidders with the highest ”virtual” valuations (see Myerson, 1981). Virtual valuations are adjusted valuations that take into account bidders’ informational rents, and depend on the distribution of private information. Asymmetries among bidders (and possibly other properties of these distributions) drive a wedge between virtual and true valuations, leading to ine¢ciencies (see Ausubel and Cramton, 1998 for a recent discussion of these issues). Our purpose here is to illustrate in the simplest possible way that a revenuemaximizing seller of several heterogenous objects has incentives to ”misallocate” the sold objects even in symmetric settings, and no matter what the (symmetric) function governing the distribution of private information is. This ine¢ciency result should be contrasted with the e¢ciency result in Armstrong (1998) that applies only to some cases with discrete distributions of valuations.