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Article Excerpt
We analyze a dynamic second-price auction with an informed bidder and an uninformed bidder who, upon seeing a posted price, learns whether his valuation is above that price. In the essentially unique equilibrium, an informed bidder bids in the first period if her valuation is below some cutoff and bids only in the last period otherwise. An uninformed bidder bids in every period to optimally change the price unless the price is above his valuation or he is the high bidder. This model also provides a rationale behind the use of a secret reserve price in private-value settings.
1. Introduction
* In most economic models, an agent with complete preferences is assumed to be able to formulate her preferences perfectly. For example, a rational agent usually knows her valuation in a private-value setting. In this article, we introduce nonstandard agents who are not aware of exactly how much they like an object. In many real-life situations, people actually do not need to know their exact valuations for efficient trades to take place. When a person buys milk in the supermarket, he does not need to know exactly how much he values a gallon of milk. He only needs to know whether he values it more than the price. That is, knowing his preferences around the posted price is enough for decision making in this situation. Using this intuition, we introduce a stylized model where, when confronted with a price, an agent who does not know his exact valuation for a good easily learns whether his valuation is above or below the price. We apply this model of preference elicitation in a dynamic second-price auction such as those held on eBay. We show that this model explains the prevalence of late and multiple bidding in an independent private-value (IPV) setting. Moreover, the model provides a rationale behind the use of secret reserve prices in second-price auctions and is consistent with many other stylized facts from online auctions.
An eBay auction is essentially a second-price auction with a fixed closing time where a bidder can submit as many bids as she wants. A bidder's latest bid has to be higher than all of her previous bids and this bid is basically considered as her active bid. At any point during the auction, the current price equals the second-highest bid received so far, but the exact bids are kept undisclosed. At the pre-announced time when the auction ends, the highest bidder wins and pays the bid of the second-highest bidder plus a small bid increment. The winner, hence, does not pay her own bid. For a more detailed description of eBay auctions, see Roth and Ockenfels (2001). It is an empirical regularity that online auctions receive a significant number of bids in the last few minutes of an auction. In this article, we define sniping as bidding in the last three minutes of the auction and not placing any earlier bid. (1) Moreover, many bidders frequently update their bids by placing a new bid slightly above the current price if they are not the highest bidder. We refer to this multiple bidding as nibbling. (2) Figure 1 presents all the individual bids submitted in an eBay auction of a golf driver. The bidder "dsgn 101" submitted her only bid 17 seconds before the end of the auction, or sniped. On the other hand, all other bidders placed multiple bids. Each entry denotes submission of a new bid. When bidder "kalanisurf' entered, she saw that the opening price was $9.99 and the highest bidder was "fredburt," but did not know the current highest bid. She took two bids to become the current highest bidder with a bid of $120. Then bidder fredburt came back and placed three separate bids to overtake kalanisurf's bid. Bidders fredburt, kalanisurf, and later bidders "ogus" and "rbwaugh@aol.com" all nibbled. Bidder fredburt placed eight separate bids to bid $125. As eBay never raises the price above the bid of the second-highest bidder, placing a single bid of $125 would lead to the same response by other bidders with private values, but would eliminate the risk from a possible failure to place a future bid due to some unforeseen event. The fact that the price in eBay reflects only the second-highest bid makes nibbling, at best, superfluous in a private-value setting. Many of the late bidders also bid earlier in the auction, and this suggests that sniping does not only arise from buyers' desires to know the auction outcomes soon after they place a bid. Section 2 presents further evidence of prevalence of sniping and nibbling and some other anomalous empirical regularities in online auctions.
In Section 3, we introduce a second-price auction with a fixed number of periods where bidders can bid in every period. The current price at the beginning of a period equals the second-highest bid received in the previous periods. At the end of the auction, the bidder who placed the highest bid wins and pays the second-highest bid as the price. In an IPV setting where all bidders know their valuations, or are informed bidders, any strategy profile where bidders ultimately bid their private valuations is an outcome-invariant equilibrium. Thus, sniping and nibbling can be consistent with the standard theory. However, all bidders bid their valuations in the first period in the unique equilibrium in weakly undominated strategies if there is a small opportunity cost of waiting. Such an opportunity cost may arise if there is always a small probability that a bidder will not be able to place any bid in the remainder of the auction. In reality, this may correspond to a bidder being unable to go back to an auction because of unanticipated personal commitments, computer malfunctions, or forgetfulness.
The main innovation of this article is introducing a model of bidders who do not know their private valuations for the good exactly but learn more about those during the auction using a boundedly rational learning process. Specifically, when such a bidder is confronted with a minimum price that he has to pay if he wins, either as a posted price or as a minimum bid in an auction, he costlessly learns whether his valuation is above or below that price. As the minimum bid in our dynamic second-price auction can be changed many times, when a bidder does not know his valuation, he can experiment strategically by using his bids and learn more about his preferences. A point to note is that we do not assume that an uninformed bidder, a bidder who does not know his valuation, has to change the price with his own bids to get more information about his valuation. He gets more information any time the price changes whether the change results from his own bid or some other bidder's bid. New prices give him new information and this incentive leads him to place multiple bids. On the other hand, in some cases, it may be optimal for an informed bidder to restrict such learning by bidding by the uninformed bidder. This leads to late bidding or sniping by her.
Theorem 1 shows that, when both bidders bid in the first period of the auction with positive probability, there essentially is a unique equilibrium with the following properties:
(i) Bidder 1 bids above the opening price either only in the first period or only in the last period. There exists a cutoff value v v, then she bids [v.sub.1] in period T.
(ii) Bidder 2 bids in every period unless he is the high bidder or learns that [v.sub.2] < [p.sub.t].
This result illustrates that sniping and nibbling are equilibrium phenomena in a private-value setting. Some bidders will bid only early in an auction, some will bid both early and late, and some will bid only late. Moreover, both early and late bidding may occur in the same equilibrium. A significant fraction of bidders who bid toward the end of an auction will not bid earlier in that auction. A bidder who places multiple bids may raise her bid frequently until she becomes the high bidder. Occurrence of sniping and nibbling in equilibrium is robust to many modifications of the original learning model. This model explains many other stylized facts presented in Section 2 quite well. We then show, in Proposition 4, that when an uninformed bidder can also learn by comparing their valuations to hypothetical prices or prices in other auctions before the auction in which he participates, sniping and nibbling will still occur.
Sniping and nibbling with uninformed bidders is not an artifact of having exactly one informed bidder and one uninformed bidder. We concentrate on this case because of the simplicity of the model. In Section 4, we extend the model to auctions with arbitrary numbers of bidders. We show that sniping and nibbling occurs in any auction with at least one informed bidder and at least one uninformed bidder.
Learning by bidding may also explain the extensive use of a secret reserve price in second-price auctions. If all bidders are informed, secret or public reserve prices lead to the same outcome in an IPV setting. On the other hand, an uninformed bidder may win the object even when his valuation is below the reserve price if the reserve is secret but not if it is public. Proposition 7 shows that a secret reserve price auction may generate higher expected revenue than a public reserve price auction.
The rest of the article is organized as follows: the next section reports some stylized facts that led to the questions addressed in this article and relates this article to the existing literature. Section 3 introduces the theoretical model with two bidders where one is informed and the other is uninformed. Section 4 discusses auctions with more than two bidders. Section 5 analyzes secret reserve price auctions and Section 6 concludes the article. All proofs are in the Appendix.
2. Some stylized facts and relation to the literature
* We collected a data set of 2,026 completed auctions of "Titleist 975J" golf drivers conducted on eBay between February and April of 2003. In these auctions, 9,003 bidders placed 17,057 separate bids, on average placing 1.89 bids each. About 40% of all bidders placed more than one bid or nibbled. Of all bidders, 9.6% placed a bid in the last three minutes and 67% of these bidders did not place any bid earlier in the auction. Some bidders bid early in the auction, some bid late, and others bid both early and late.
Many studies found the same bid pattern in eBay auctions for a wide variety of goods. Roth and Ockenfels (2001) gathered a sample of over 1,000 auctions of various items. Of the auctions with two or more bidders, 18% received bids in the last minute and 74% had at least one bidder submitting multiple bids. In a study of coin auctions on eBay by Bajari and Hortacsu (2003), 32% of the bids were submitted after 97% of the duration of the auction had passed. Hossain and Morgan (2006) sold brand new popular music CDs and Xbox game cartridges on eBay. There was little uncertainty about the quality or popularity of these products; still, at least one bidder placed multiple bids in 76% of the auctions. Bids were placed in the last five minutes in 30% of the auctions. Ariely et al. (2005) found evidence of extensive sniping and nibbling in laboratory experiments of eBay-type auctions. We report some additional empirical regularities or stylized facts found in the existing literature or in our data set.
(i) Many late bidders on eBay bid only in the last few minutes of an auction. In our data set, more than two thirds of the bidders who placed a bid in the last three minutes of an auction did not bid earlier in that auction. A third of the late bidders placed bids both early and late.
(ii) Many bidders self-nibble; that is, they bid repeatedly below the highest outstanding bid (the exact outstanding bid is unknown to them) while the highest bidder stays unchanged. Moreover, many bidders place a new bid every time they are displaced from the highest bidder position. Almost 80% of all nibblers in our data set placed consecutive bids while they were not the high bidder. A nibbler placed more than 3.26 bids on average.
(iii) Roth and Ockenfels (2001) find that more experienced bidders are more likely to snipe than less experienced bidders. One may argue that experienced bidders are more proficient in thinking counterfactually, asking themselves repeatedly how much they would be willing to bid, and are thus more likely to be informed bidders. In our data set, a bidder with a feedback rating of at least 20 was one-third less likely to place multiple bids than a bidder with a lower feedback rating.
(iv) Now-defunct Amazon.com had a "soft" closing time: the length of the auction would be increased if there was a bid within the last ten minutes of the pre-announced closing time. After that, the auction would close when there had not been any activity for ten minutes. Roth and Ockenfels (2001) find that nibbling was common but sniping was relatively uncommon in Amazon auctions.
(v) Controlling for auction characteristics, an auction where all bidders placed only one bid each received almost 17% less revenue than an auction where all bidders placed multiple bids in our data set. According to our model, informed bidders are likely to bid only once and are also more likely to snipe. On the other hand, uninformed bidders are likely to bid multiple times. Thus, multiple bidding can be an indicator of being uninformed. However, whether a bidder places multiple bids depends on the opening price leading to an endogeneity problem in the regressions. Combined with the model's predictions, the third stylized fact suggests that an experienced bidder is more likely to be informed...
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