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Article Excerpt
Online auction sites often enable sellers to add a buy-out price. In one-shot auctions, this has been motivated by appeal to impatience or risk aversion. We offer additional justification in a dynamic model, by showing that an early seller has an incentive to use a buy-out price, if a similar product is offered later by another seller, and bidders desire multiple objects. Revenue in the first auction increases, but revenue in the second auction decreases, as does the sum of revenues. The buy-out price causes the auction sequence to become inefficient, because the first item may be awarded to a bidder who should have received none.
1. Introduction
* The proliferation of the Internet and online deal engines has brought a renewed focus on the details of auction design and their implications for bidder behavior and seller revenues. To name a few features which have drawn attention, we can list the relationship between closing rules and bidder sniping, open versus secret reserve prices, jump bidding, signalling, and default. In this article, we study the role of buy-outs in auctions. Roughly, the introduction of a buy-out option in an auction implies that the seller stipulates ex ante a price at which he is willing to end the auction immediately, if someone shows a willingness to pay this price.
In the economics literature, the presence of buy-out prices in online auctions (1) has thus far been explained by focusing on a single auction and assuming that potential buyers and/or sellers exhibit either risk aversion or impatience (see Budish and Takeyama, 2001; Mathews, 2003, 2004; Mathews and Katzman, 2006; Reynolds and Wooders, forthcoming; Hidvegi, Wang, and Whinston, 2006). In this article, we take a somewhat broader view of auction markets, realizing, in particular, that buyers and sellers alike are aware of the fact that new products will be offered on the market in the future. This will tend to depress revenue in today's auctions, as buyers know that close substitutes will be offered tomorrow. In this dynamic environment, we will show that there is good reason for an early seller to introduce a buy-out price, even if agents are patient and risk neutral. (2)
Buy-out prices or maximum prices in online auctions were noted by Lucking-Reiley (2000) in his empirical overview of auction activities on the Internet. Because (sell) auctions are ostensibly staged to elicit high prices in situations where markets are thin and sellers are short on information about the willingness to pay of potential buyers, such buy-out prices may appear surprising. In fact, Lucking-Reiley explicitly posed this as a challenge to theorists. In addition, he quoted evidence to suggest that the exercise of posted buy-out options is not uncommon in online auctions. Hence, buy-out prices do more than just attract attention.
Reynolds and Wooders (forthcoming) provide some additional information on the frequency of buy-out prices in Yahoo! and eBay auctions, though, not on the frequency with which the option was exercised by some bidder. The categories sampled on March 27, 2002, were automobiles, clothing, DVD players, VCRs, digital cameras, and TV sets. A total of 1,248 auctioned items were sampled from Yahoo!, of which 842 had a buy-out price posted by the seller (roughly, 66%). In similar fashion, 31,142 auctioned items were sampled from eBay, of which 12,480 had a buy-out price posted by the seller (roughly, 40%). There is some variation across the categories of goods sampled, but the frequency of buy-out prices never drops below 25% in the sample. Hence, in these categories, at least, the appearance of buy-out prices is very frequent.
For eBay, Mathews (2004) presents some numbers on the frequency with which buy-out options are exercised when offered. For two categories of games (racing and sports) for Sony PS2, Mathews reports that on January 29-30, 2001, 210 items were on offer. A buy-out option was available on 124 items (59%), and it was exercised 34 times (27% of the times it was offered). So, at least in these categories, the exercise frequency is high.
Formally, we analyze eBay's version of a buy-out price, termed the Buy It Now price (introduced in January 2001). Here is how the Buy It Now price roughly works from the seller's viewpoint (3): "If a buyer is willing to meet your Buy It Now price before the first bid comes in, your item sells instantly and your auction ends. Or, if a bid comes in first, the Buy It Now option disappears. Then your auction proceeds normally." Hence, in eBay auctions, the buy-out price is temporary.
Throughout this article, we assume that potential buyers or bidders have multi-unit demands, with diminishing marginal utility. With two objects for sale, at least two bidders and absent any buy-out option, it has been shown by Black and de Meza (1992) that expected auction revenue will increase over time and that the auction outcome is efficient under these assumptions. In particular, in a sequence of standard second-price or English auctions, the seller offering his good today will not earn as much as a competing seller offering a similar good tomorrow, that is, prices are increasing in expectation. In fact, Black and de Meza (1992) were mainly interested in what has been referred to as the declining price anomaly. Therefore, they went on to consider an option of the following kind: the winner of the first item is given the option of buying the second item at the same price. This type of option is observed in certain multi-unit auctions, and it is enough to lead to a declining price path.
However, for the case with two competing sellers, we show that the first (i.e., the early) seller can always increase his revenue by introducing a buy-out price. The revenue to the second seller is adversely affected, as is overall revenue. An optimally chosen buy-out price in the first auction also introduces inefficiency, in the sense that a bidder who should have won no object wins one. Our analysis is partial in the following sense. We consider a sequence of two second-price (or English) auctions, allowing the first seller the possibility of introducing a buy-out price without giving the second seller the opportunity to respond in kind. (4) Thus, we essentially show that an auction market without buy-out prices is unstable, in the sense that current sellers will try to persuade the auction site to (at least temporarily) allow buy-out prices. (5)
At eBay and other auction sites, a seller can also introduce a reserve price, which we ignore in this article. As is well known, reserve prices are generally useful because they allow sellers to ration or withhold items by effectively excluding potential buyers with low valuations. Reserve prices, thus, affect the probability with which objects are sold. As suggested in the previous paragraph, the effect of buy-out prices is fundamentally different. In particular, buy-out prices do not influence the probability that objects are sold, but they may change the identity of the winners. It follows that a buy-out price is not a substitute for a reserve price, and that it may have a role to play, even when a reserve price is present.
The rest of the article is organized as follows. In Section 2, we set up a simple model and present the results for the benchmark case where a sequence of two, standard second-price auctions is staged. Then, Section 3 shows that the first of two sellers can improve his lot by offering a buy-out price and presents results on the path of revenues, the properties of an optimal buy-out price, and the overall efficiency of the string of auctions. In Section 4, we comment further on the relationship between the buy-out price, total revenue, and efficiency. In addition, this section remarks on the robustness of our main result to changes in the auction format (second-price versus English auction) and the nature of the buy-out option (temporary versus permanent). To further illustrate the results, Section 5 briefly outlines an example with uniformly distributed valuations. Section 6 contains a few concluding remarks, and a selection of proofs is in the Appendix. Full details of the proofs as well as the example outlined in Section 5 are in the Web Appendix (see www.tobeadded.com).
2. Model and benchmark
* In this section, we first set up the model and then derive results for the benchmark case where a sequence of two second-price auctions is staged.
We assume that two identical objects are offered for sale sequentially, and that there are n potential buyers on the market. Each buyer i, i = 1, 2, ..., n, is characterized by a type, [v.sub.i], drawn from a continuously differentiable distribution function, F([v.sub.i]), without mass points. The i) associated density is referred to as f([v.sub.i]) = dF([v.sub.i])/d[v.sub.i], whereas further regularity assumptions on F will be imposed below (as the need arises). We assume that [v.sub.i] [member of] [0, [bar.[v]]. The value to bidder i of the first unit purchased is [v.sub.i], whereas the value of the second unit is k[v.sub.i], < k < 1. Hence, each bidder desires both units, but individual demands are downward sloping.
Below, we shall occasionally take the perspective of a particular bidder, i, and label his rivals j, j = 1, 2, ..., n - 1. Now, i's competitors have random valuations of the first item denoted [y.sub.j], which we order as [y.sub.1] [greater than or equal to] [y.sub.2] [greater than or equal to] ... [greater than or equal to] [y.sub.n-1]. This allows us to refer to bidder j as bidders i's jth-strongest rival. When appealing to the order statistics, we shall generally refer to [F.sub.m,n](x) as the distribution function of the mth-highest of n draws, with associated density [f.sub.m,n](x) = d[F.sub.m,n](x)/dx.
Throughout, we assume that two different sellers each own one object initially. The two objects are offered sequentially, and we allow the first seller to stipulate a buy-out price of the eBay variety (Buy It Now). Thus, in the general case, we consider the following augmented game:
(i) Seller 1 announces a buy-out price, B. At this stage, bidders can submit a bid of B or refrain from bidding. The object is sold at the price B if at least one bidder bids B. If several bidders bid B, one bidder is picked at random to win. If no one bids B, a normal second-price auction is staged. The price can exceed B in this event.
(ii) Seller 2 auctions off the second item, using a second-price auction.
In stage 1 of this game, the bidders first have to decide whether to take the buy-out price B or leave it. If one or more bidders take the buy-out price, the first auction ends, and the winner pays B. If no one takes the buy-out price, then the first stage continues to a standard second-price auction. Stage 2 simply consists of a standard second-price auction.
First, though, we summarize the results of the benchmark case, where no buy-out price can be stipulated by the first seller (or, that it is set so high as to be irrelevant for...
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