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Asymptotic prices in uniform-price multi-unit auctions.

Article Excerpt
Summary. This paper considers a uniform-price auction in which each of n symmetric bidders can place, say, M bids. Each bidder has privately known, decreasing marginal values from an arbitrary M-dimensional distribution. We provide a quantile-type description of the asymptotic price that appropriately generalizes the characterization of the unit-demand asymptotic price. Specifically, the limiting price equals the (1 - [alpha])-th quantile of the "average" of the marginal distributions if a fraction [alpha] of the demand is met asymptotically. The result also implies that the expected price in the limit as n becomes large depends only on the aggregate of the marginal distributions of each bidder's marginal values (and not on the correlation between the marginal values).

Keywords and Phrases: Multi-unit auctions, Uniform price.

JEL Classification Numbers: D44.

1 Introduction

Prices play a central role in the analysis of auctions. Precise expressions for expected prices make comparisons of revenues from different auction formats and econometric estimation of auction models possible. Moreover, certain characterizations of prices provide interesting insights into various aspects of auctions.

On the other hand, auctions play a central role in the analysis of competitive pricing. Agents in large auctions have been often shown to display a price-taking behavior. A price established in the limit when the demand and supply can both grow indefinitely in the auction is then a competitive market-clearing price, as well. Characterization of this limiting price, therefore, serves as a description of the competitive market outcome. Moreover, strategic agent behavior in auctions provide interesting insights into various aspects of price formation.

One interesting description of expected prices can be obtained by considering, for example, the standard Vickrey or second-price auction (a single-unit, sealed-bid auction with the price set equal to the second-highest...

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