The Sealed-Bid Abstraction in Online Auctions

ion holds, the following five things should be true, the first four are as follows: (T1) the top bid should be equally likely to arrive before or after the second-highest bid, (T2) the top two bids should not be exactly one bid increment apart too often, (T3) the difference between the top two bids should not be a function of which was placed first, and (T4) the difference between the top two bids should not be a function of time remaining in the auction. Our last test (T5) is a nontrivial contribution to the empirical auction literature. Instead of relying on timing or increment data, it considers only the joint distribution of the top two bids and asks what should be true about the conditional distribution of the top bid Zeithammer and Adams: The Sealed-Bid Abstraction in Online Auctions 966 Marketing Science 29(6), pp. 964–987, © 2010 INFORMS given the second-highest bid. The sealed-bid abstraction implies that the conditional distribution of the top bid given the second-highest bid should have the same right tail for any particular value of the second-highest bid. When bids depend on auction-level observables as well as private signals, the T5 test can be applied to the residuals of the appropriate truncated regression of bids on observables. In this paper, we specify the different assumptions under which each of these five tests works, and we propose a way to operationalize the novel test T5. Our operationalization of T5 generalizes the nonparametric Wilcoxon test to deal with draws from tails of distributions. Theoretically, there is nothing special about the top two bids in an auction—most of the above tests would be valid with any other pair of order statistics of the bidding distribution. Practically speaking, however, the top two order statistics are the only reliably observable ones because of entry truncation in online auctions: on eBay, one can submit a bid only if it exceeds the highest bid at the moment; thus eBay data contain relatively more high bidders and relatively fewer low bidders than the underlying population. Although many latent bidders may thus be truncated, the highest and the second-highest bidder in each auction are always recorded. Conversely, the lower order statistics of the observed bids do not necessarily correspond to the order statistics of the latent bidders. Another reason to focus on the top two bids is the obvious alternative “incremental” model of bidding in which each bidder manually raises his proxy bid by the minimum increment up to his maximum (i.e., up to his valuation in an IPV context). The top two bids are special in that incremental bidding by all participants predicts sharply different outcomes of T1–T5 from sealed bidding. Specifically, when bidders bid in an incremental fashion, the top bid always arrives after the second-highest bid, the top two bids are always exactly one bid increment apart, and the conditional distribution of the top bid given the second bid degenerates to a mass point one increment above the second-highest bid. We apply the tests to three different data sets from eBay (MP3 players, movies on DVD, and used cars). Our data are provided directly by eBay, so we observe the proxy bid of the winner not available from the eBay website. Therefore we report on empirical regularities of uniquely complete data sets.2 Taken together, our application of all five tests rejects the sealed-bid abstraction as a general property of bidding in eBay auctions. Three tests reject consistently, 2 These data are not completely unique anymore because eBay now sells its data through third-party providers such as Advanced Ecommerce Research Systems. Obtaining data from eBay is not the only way to observe the proxy bid of the winner. For example, Bapna et al. (2008) observe it by running their own sniping agent. with surprising empirical regularity of the test statistics across the three diverse data sets. First (T1), the top bid is placed after second-highest bid in about two thirds of the auctions. Second (T2), about 15% of the auctions end with the two bids exactly one increment apart (compared to only 4% that end in an exact tie). Finally (T3), the one-increment-apart outcome is about three times more likely when the top bid comes after the second-highest bid compared with the reverse order. The easiest way to explain these findings is that a significant proportion (more than 15%) of the auctions within each data set is better described by incremental bidding. Although sealed bidding is thus not a general property of all auctions, it may still describe a subset of auctions large enough for demand estimation based on pairs of order statistics. We explore this possibility with two “plausibly sealed” subsets: the “OverInc” auctions in which the top bid exceeds the second-highest bid by more than an increment, and the “HighFirst” auctions in which the top bid is placed before the secondhighest bid. Interestingly, tests T1–T4 still consistently reject the sealed-bid abstraction in these plausibly sealed subsets. We also applied the novel T5 test to the OverInc data after conditioning on auction-level observables, and it rejected the abstraction in two of our three data sets (MP3 players and cars). To assess the direction and magnitude of bias one would encounter if one were to rely on the abstraction in demand estimation following Song (2004), we develop an alternative empirical model. When we examine the root causes of the T1–T5 rejections, we find the bidding style of the auction winner is an important correlate. Specifically, auctions won by multibid bidders (bidders observed submitting multiple bids per auction) tend to conform to the abstraction less than auctions won by singlebid bidders (bidders observed bidding only once per auction). Although multiple bidding is not clear evidence against the sealed-bid abstraction (as discussed above), our data indicate that multibid bidders bid systematically differently in that they tend to submit smaller bids than single-bid bidders. Therefore we propose that bidders have different personal bidding styles, and only the “sealed style” conforms to the abstraction, whereas the “reactive-style” bidders initially bid only a fraction of their valuation and subsequently raise their bid gradually toward their valuation whenever they are outbid. Reactive bidders can be at least partially detected by bidding multiple times in a single auction, and we can rely on bidder characteristics, such as experience, to estimate the probability that any given bidder is reactive. The link between experience and multiple bidding replicates previous findings by Wilcox (2000) and Borle et al. (2006). Consistent with the results presented in Zeithammer and Adams: The Sealed-Bid Abstraction in Online Auctions Marketing Science 29(6), pp. 964–987, © 2010 INFORMS 967 List (2003) and Simonsohn and Ariely (2008), we find that experienced traders are more likely to behave “rationally” in the sense of conforming to the sealedbid abstraction. Given the bidding-style probability for each bidder and a standard IPV assumption, we can estimate the underlying distribution of valuations (a.k.a. “demand”) by interpreting the observed bid distribution as a weighted combination of bids coming from sealed bidders and reactive bidders. The estimation uses the conditional order statistic approach, and so it does not need to infer the number of latent bidders. We estimate the model on the DVD data (in which the rejection of the sealed-bid abstraction was the weakest), and we compare the estimates to an alternative model that assumes all bidders are using the sealedbidding style. The proposed model fits the data better than the all-sealed model and corrects a downward bias caused by the reactive bidders not bidding their true valuations. We find that this potential bias is large: specifically, the estimation results suggest that valuations of DVDs have a population mean and variance that are both more than double the all-sealed estimates. In addition, the two models imply substantially different public reserve prices. For example, a seller with a realistic marginal cost of $3 per DVD should use a starting price of $4.20 under the allsealed model estimates versus $6.00 under the proposed model. Therefore our rejection of the sealed-bid abstraction has substantial managerial implications. The rest of this paper is organized as follows. The next section introduces the tests and the assumptions on which they are based. Section 3 then describes our data and applies the tests. Section 4 considers robustness of our results to relaxation of the assumptions. Section 5 discusses the new model with reactive bidders. Section 6 concludes by summarizing our results and outlining how the empirical regularities we document constrain theories of online auction behavior and econometric methods for estimating demand from eBay data. 2. Tests of the Sealed-Bid Abstraction in Online Auctions 2.1. Definitions Our tests are geared toward understanding whether the economically important final bids in eBay auctions behave analogously to bids in a second-price sealedbid auction. However, the tests’ applicability extends to other auctions and other sealed-bid scenarios. To facilitate the widest possible scope of application, we define our primitives in maximum generality. Our concept of an online auction encompasses any auction that receives bids over time, with each bid associated with a unique time stamp. For the purposes of this paper, every current Internet auction is thus an online auction, but so is every other auction that receives mail-in or call-in bids. The feedback to bidders during the auction can range from none (as in a government auction with mail-in bids) to a full record of successful bids to date (as in an eBay auction). A model of an online auction involves a sealed-bid abstraction whenever the bidders bid as if they were in a standard sealed-bid auction. Mathematically speaking, each bidder in a sealed-bid auction receives a private scalar signal x and bids according to a strictly increasing function x x →bid that depends only on x. For example, Bajari and Hortaçsu (2003) assume x is a private signal about the common value of the auctioned good (a collectible coin). In such a common value environment, a symmetric equilibrium exists in which all bidders bid at the last moment as if they were in a second-price sealed-bid auction. The equilibrium bid function is increasing and mitigates winner’s curse by x < x. Similarly, bidders in Song’s (2004) model receive independent signals about their private valuations, and each bidder has an exogenous last opportunity to bid. In such a private value model, each bidder has a dominant strategy to bid x at his last opportunity to bid. The observable bidding behavior may not look sealed in the sense that bidders might be submitting multiple bids and following arbitrary dynamic bidding strategies before their last opportunity. However, the final bid each bidder submits should be the bid he would submit in a second-price sealed-bid auction. Note that in both examples, the second-price nature of the sealed-bid auction arises from eBay’s proxy-bidding agent. Our tests do not depend on the equilibrium assumption— x could be any ad hoc behavioral regularity. For example, bidders may not actually be strategic but may follow eBay’s instructions that direct them to submit their maximum willingness to pay as their proxy bids on arrival to the auction. 2.2. Model Properties Having defined the two key primitives of our theory, we now turn to the properties of auction models from which our tests arise. (Please refer to Table A.5 in the appendix for notation used throughout this paper.) We will present a series of tests, and each test will rely on different properties of the auction model. Therefore, even if one or two of the following properties do not hold, some of the proposed tests will still work. We will be using the following properties. • A1 (timing independent of signals): time ti is independent from signal xi for every bidder i. • A2 (continuity): signals xij of bidder i in auction j are drawn from a continuous distribution. • A3 (conditional iid): conditional on auctionspecific observables Zj , signals xij are independent Zeithammer and Adams: The Sealed-Bid Abstraction in Online Auctions 968 Marketing Science 29(6), pp. 964–987, © 2010 INFORMS and identically distributed (iid) across auctions j and bidders i. The function log x is additively separable in auction-specific observables Zj and residual private valuation shocks. • A4 (increment): the auction is actually an ascending auction with a minimum bid increment inc, which the econometrician observes. The first property (A1) says no link exists between the magnitude of private signals and the timing of the bids associated with those private signals. As discussed in §1, this property does not hold in the models of Peters and Severinov (2006), Bradlow and Park (2007), and Hossain (2008). The second property (A2) says the signals are drawn from a continuous distribution that may vary from auction to auction and may involve arbitrary correlations across bidders, auctions, or both. A popular property that allows pooling of data across auctions in empirical research is that the signals are iid across auctions and bidders. This iid property is the essence of property A3, with A3 also conditioning on auction-level observables. The additional additive separability assumption is not needed if one can obtain a large number of observations with the same Zj ; then the implied test T5 can be validly carried out on the bids themselves. To pool across observations with different Zj , we will approximate conditioning on Zj with a linear regression of log bids on observables and focus on the residuals. One setting that satisfies the additive separability assumption is an IPV setting with x = x and valuations that are multiplicatively separable into Zj and private shocks. Finally, property A4 is an assumption about the rules of the auction that generates the data. 2.3. Tests Suppose we have the data on the top two bids in an auction b1 ≥ b2, their timing ti = time bi , and the increment inc. In the eBay setting, b1 b2 are the top two proxy bids submitted in the auction and inc is eBay’s minimum increment.3 Given data on t1 t2 b1 b2 inc , the simplest test is based on timing alone: as long as timing of bids is independent of private signals (A1), the sealed-bid abstraction predicts neither of the bids should be more likely to appear first: A1 ⇒ T1 Pr t1 > t2 = 1/2 (1) In the alternative incremental bidding model, Pr t1 > t2 = 1; thus the highest bid usually coming in after the second-highest suggests a departure toward incremental bidding. To operationalize this test, we compute the empirical probabilities with their associated standard errors. 3 eBay’s increment is a function of b2 and varies from 5 cents to $100 as b2 increases from $1 to $5,000. In online ascending auctions with a minimum increment (A4), another simple test uses only data on bids and the continuity property (A2): A2,A4 ⇒ T2 Pr b= inc = 0 (2) where b= b1−b2. In contrast, the incremental model predicts that Pr b = inc = 1, so T2 failing should be a strong indication of incremental behavior. To operationalize T2, we again compute the empirical probability with its standard error. A2 is an unrealistic assumption about the auction environment when some bidders value goods in whole dollar amounts. Section 4 will generalize A2 to allow mass points in the distribution and discuss how such a generalization would impact testing. With mass points, Pr b= inc > 0, and test T2 becomes weaker. Combining the bid data with the timing data allows for more detailed testing based on the independence assumption (A1). By definition of independence between timing and bidding, A1 implies b should be completely invariant to the timing of the bids. Two fruitful tests arise, again motivated by the incremental behavior alternative. First, the sealed-bid abstraction predicts that the difference between the top two bids should not depend on which bid came first: A1 ⇒ T3 cdf b t1 > t2 = cdf b t1 < t2 (3) where cdf stands for cumulative distribution function. In contrast, if some auctions behave more like incremental auctions and others more like sealed-bid auctions, b t1 > t2 will be smaller and related to the minimum bid increment, whereas b t1 < t2 will be “more continuous” with a tail. To operationalize this test idea, we use the nonparametric Wilcoxon– Mann–Whitney (hereafter WMW) rank test and compare the subsample b t1 > t2 to the subsample b t1 < t2 .4 Analogous to conditioning on the relative timing of the top two bids, conditioning on time remaining in the auction should also leave b unchanged. Define bidding as late versus early by doing a median split on the time left t̄ −min t1 t2 , where t̄ is the ending time of the auction. The sealedbid abstraction implies A1 ⇒ T4 cdf b late = cdf b early (4) To operationalize this test, we again use theWMW test. Bringing in the increment information allows us to zoom in on the value of b = inc to conduct special cases of T3 and T4 with a sharp alternative hypothesis 4 In parallel, we also computed the standard t-test of the E log b t1 > t2 = E log b t1 < t2 hypothesis, implicitly assuming lognormality of b. This parametric test almost perfectly agrees