Quantifying the inefficiency of multi-unit auctions for normal goods

We study multi-unit auctions for homogeneous goods in a private-value setting where bidders have multi-unit demand and non-negative wealth effects. When bidders have quasilinear preferences, the Vickrey auction implements an efficient outcome in dominant strategies. When bidders have positive wealth effects, recent impossibility results find that no auction implements an efficient outcome.

We quantify the worst-case inefficiency of the Vickrey auction and other multi-unit auctions when bidders have positive wealth effects. We measure an auction's worst-case inefficiency as the largest compensating variation associated with any Pareto improvement over an undominated auction outcome. We show that the Vickrey auction is ‘nearly’ efficient when the strength of bidder wealth effects is sufficiently small. This result follows because the set of undominated bids in the Vickrey auction collapses to truthfully reporting demand as bidder wealth effects become small. We also compare the worst-case inefficiency of the Vickrey auction with that of the uniform-price and discriminatory auctions.