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...auction, uniform-price auction which the price equals the lowest winning bid, and the uniform-price auction in which the price equals the highest losing bid. We also consider the Vickrey pricing rule. In the case we examine, the auctions are all efficient and thus are revenue equivalent. The equilibria illustrate several phenomona that cannot arise in single-unit auctions. Even though the auctions are expected-revenue equivalent, different bidders may end up paying very different amounts. Also, in contrast to single-unit auctions, changing the seller's reservation price affects revenues, even if it remains below the lowest possible value to bidders.
1. Introduction
Recent work suggests that multiunit auctions are much more complicated than their single-unit 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 1 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.
2. The Model
Consider a multiunit auction in which K(N + 1) - 1 identical units will be auctioned to N + 1 symmetric, risk-neutral bidders, where K [greater than or equal to] 2 and N + 1 [greater than or equal to] 2.
Consider the problem from the perspective of any one of the bidders, namely "me." Let v = ([v.sub.1], [v.sub.2], ..., [v.sub.k]) denote my privately known, nonnegative marginal valuations for any K units. The components of an individual bidder's v need not be independently distributed, but we do assume that the v's are independent across bidders. (We do not need to assume the marginal values to be decreasing.) As we will see, the bidding strategies will turn out to be independent of all valuations except [v.sub.k]. Let [v.sub.*] denote the bottom of the support of VK and let B* denote the top of the support.
Each bidder makes K bids, and may win up to K units. (2) Let b = ([b.sub.1], [b.sub.2], ..., [b.sub.k]) represent my bids sorted in descending order. In particular, [b.sub.K] is the smallest of my bids. The total number of bids will be K(N + 1).
The K(N + 1) - 1 highest bids win. Thus each bidder wins at least K - 1 units and all but the bidder with the lowest of the K(N + 1) bids win K units. Exactly one bid will lose.
Unlike in single-unit auctions, our multiunit auction will not have an equilibrium unless there is some minimum allowable bid. Let r denote the "reserve price"--that is, the minimum price at which the auctioneer will sell a unit. (3) The reserve price is known in advance by all bidders. Clearly there could be reserve prices high enough that bidders are unwilling to win K units; we will only consider reserve prices sufficiently low that bidders are willing to submit bids of at least the reserve price on K units. Sufficient for this to be the case is that the reserve price be below the lower end of the support of the marginal valuation of all bidders for each of the K units. (Much weaker conditions, of course, exist: For example, if marginal valuations are increasing, then it suffices that with probability one the average marginal value over all his units exceeds the reserve price.) (4)
In short, we assume each bidder must submit K bids greater than or equal to r, where we restrict attention to situations in which r [less than or equal to] [v.sup.*].
For convenience, we establish some additional notation. Let F and f denote the distribution and density, respectively, of the lowest of the other bidders' values. We assume that the densities exist mainly for ease of exposition. Similarly, for any fixed bidding strategies followed by the other bidders, let H denote the distribution of the lowest of the other bidders' bids. Define V = [v.sub.1] + [v.sub.2] + ... + [v.sub.k-1].
3. The Pay-Your-Bid Auction
In the pay-your-bid auction (sometimes referred to...
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