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Approximate Vickrey Auctions

In a socially efficient or "competitive" equilibrium in a market for homogenous goods, firms produce until their marginal cost of production is equal to the prevailing market price; identically, firms on the demand side consume until their marginal utility, or profit, is equal to that price. In imperfectly competitive markets for homogenous goods, firms may attempt to sell (buy) less than their socially efficient quantity in order to increase (decrease) the price of both marginal and infra-marginal units and increase their total profits. This strategic behavior, or distortion, leads to deadweight loss. The extent to which firms are able to distort the market depends both on the elasticity of demand and on the elasticity of supply from other firms. In a series of pioneering papers, William Vickrey sought to overcome this inefficiency by severing the connection between each firm's bids and the price it faces for infra-marginal units. These Vickrey mechanisms have theoretically attractive properties but have been widely criticized as impractical and are, in fact, rare or nonexistent in real-world auctions, although many ascending auctions resemble the simplest of these mechanisms. This paper shows that the usual framework of uniform price, multi-unit homogenous goods auctions is a first-order approximation of the two-sided auction described in Vickrey (1961). The paper examines some properties of a second-order approximation involving quadratic payments.