Quantifying the Inefficiency of Multi-unit Auctions for Normal Goods

Abstract

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 there is no auction that 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 from an undominated auction outcome. We show that Vickrey auction is ‘nearly’ efficient when the strength of bidder wealth effects is sufficiently small. The 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 uniform-price and discriminatory auctions. 

Keywords: Undominated actions, wealth effects, multi-unit auctions, non-quasilinear preferences, Pareto efficiency.

JEL Classification: D44, D47, D61, D82.