Multiunit Auctions in Which Almost Every Bid Wins

Abstract

1. Introduction Recent work suggests that multiunit auctions are much more complicated than their singleunit counterparts (Back and Zender 1993; Katzman 1995; Noussair 1995; Ausubel and Cramton 1996; Chakraborty 1997; Engelbrecht-Wiggans and Kahn 1998a). Because multiunit auctions are of such importance in economics and finance applications-treasury auctions, auctions for electromagnetic spectrum, electric power, privatizations, procurement-it becomes important for economic theorists to be able to make legitimate predictions about the performance of various commonly used auction forms. This is an enormous task. To date, only certain natural but extreme cases have yielded to analysis, for example restriction to auctions in which purchasers only want one or two units (Engelbrecht-Wiggans and Kahn 1998a, b; Katzman 1995). Other results have been demonstrated asymptotically,1 a problematic case since expanding the number of bidders eliminates the very feature that generates a need for organized auctions: a limited number of purchasers. Thus, to have a comprehensive understanding of the behavior of multiunit auctions, it will likely be necessary to devote attention to a large variety of cases. As part of that process, this paper examines a different sector of the parameter space, one unexamined thus far, but one that is readily amenable to solution by simple analytic techniques without resort to asymptotic approximations. We consider auctions in which the number of units available is so large that every bid but one wins a unit. We examine, in the case of independent private values, three different auctions: the pay-your-bid auction (section 3) and two different forms of uniform-price auction (sections 4 and 5). We find that when all but one bid wins, the three auctions are efficient. In section 6, we use a multiunit revenue equivalence result of Engelbrecht-Wiggans (1988) to calculate their expected revenue by comparison to a Vickrey auction (Vickrey 1961, 1962). In most previously studied cases, specifically those in which each bidder wins at most one object, Vickrey auctions turn out to be uniform-price auctions. In the case where all but one bid wins, the Vickrey auction differs significantly from uniform-price auctions. For example, in the case of two bidders and a reservation price of zero, efficiency implies that each bidder in the Vickrey auction wins approximately half of the units. Nonetheless, we show that all the revenue from the Vickrey auction comes from one bidder; the other pays nothing. The analysis of the Vickrey auction shows the importance of the reserve price in a multiunit setting. We also find that the bids from the two forms of uniform-price auction are identical, providing some justification for the common practice of using one as a proxy for the other in theoretical work. Section 8 shows that this equivalence continues to hold for some cases beyond the pure private values setting of the rest of the paper. Of course, the case where every bid but one wins is an extreme example. Nonetheless, as we briefly argue in the final section, the equivalence of the two uniform-price auctions in this case is still significant, since the extreme example we consider is in important ways biased against similarity of the two auction forms. The final section also provides some conjectures about when the two forms of uniform-price auction are likely to yield similar bids. 9. Summary In this paper we have used the case of auctions in which almost every bid wins as a natural laboratory for comparison of different auction rules. We have seen three useful points. The first is extremely simple: If a player is guaranteed to win a unit, the reserve price plays a central role in determining revenue. In single-unit auctions with more than one bidder, reserve prices play no role, as long as they are below the support of the bidders' valuations. In contrast, when some bidder is assured of winning at least one unit no matter what that bidder bids, we need a lower limit on the bids to keep revenue from drifting to minus infinity. …