Quantifying the Inefficiency of Multi-unit Auctions for Normal Goods

We study multi-unit auctions for homogenous goods in a private value setting where bidders have non-quasilinear preferences. Several recent impossibility results study this setting and find there is no mechanism that retains the Vickrey auction’s desired incentive and efficiency properties without quasilinearity. While a fully efficient mechanism is impossible, we show that any undominated outcome of the Vickrey auction has a negligible inefficiency when bidder wealth effects are sufficiently small. In order to show this, we first place bounds on undominated bid behavior in the Vickrey auction when bidders have non-quasilinear preferences. We use (Marshallian) deadweight loss as our inefficiency metric, and we derive a tight upper bound on the inefficiency associated with the Vickrey auction in terms of the degree of bidder wealth effects. As wealth effects diminish, the bound continuously approaches zero. Other common multi-unit auction formats do not have this property, and their worst-case inefficiencies are higher than that of the Vickrey auction.