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MIT LIBRARIES 3 9080 019 7 6095 ilMPiiiiSSIiiia iiiljiiij;j!l}lr:^: immlMlk iB^Biiiiii I' ^ftfeii ifli:;:ii;H.;i;i i|i^ Digitized by the Internet Archive in 2011 with funding from Boston Library Consortium IVIember Libraries http://www.archive.org/details/identificationofOOathe B31 M415 ;UEWEy ! ! 1^ ^'CJ Massachusetts Institute of Technology Department of Economics Working Paper Series IDENTIFICATION OF STANDARD AUCTION MODELS Susan Athey, MIT Philip A. Haile, Univ of Wisconsin-Madison Working Paper 00-18 August 2000 Room E52-251 50 Memorial Drive Cambridge, MA 02142 downloaded without charge from the Social Science Research Network Paper Collection at http://papers.ssrn. com/paper. taf?abstract_id=XXXXXX This paper can be MASSACHUSETTS INSTITUTE OF TECHNOLOGY OCT 2 5 2000 LIBRARIES Massachusetts Institute of Technology Department of Economics Working Paper Series IDENTIFICATION OF STANDARD AUCTION MODELS Susan Athey, MIT Philip A. Haile, Univ of Wisconsin-Madison Working Paper 00-18 August 2000 Room E52-251 50 Memorial Drive Cambridge, MA 02142 This paper can be downloaded without charge from the Network Paper Collection at http://papers.ssrn. com/paper. taf?abstract_id=XXXXXX Social Science Research Identification of Standard Auction Models Susan Athey and Philip A. Haile* This version: August 8, 2000 Abstract We present new identification resiilts for models of first-price, second-price, ascending (English), and descending (Dutch) auctions. We analyze a general specification of bidders' preferences and the underlying information structure, nesting as spe- the pure private values and pure common values models, and allowing both ex ante symmetric and asymmetric bidders. We address identification of a series of such models and propose strategies for discriminating between them on the basis of observed data. In the simplest case, the symmetric independent private values model is nonparametrically identified even if only the transaction price from each auction is observed. For more complex models, we provide conditions for identification and testing when additional information of one of the following types is available: (i) one or more bids in addition to the transaction price; (ii) exogenous variation in the number of bidders; (iii) bidder-specific covariates that cial cases shift the distribution of valuations; object sold. common results include (iv) new the ex post reahzation of the value of the tests that distinguish between private and values models. Kejnvords: common Our auctions, nonparametric identification values, and testing, private values, asymmetric bidders, unobserved bids, order statistics M.I.T. and NBER (Athey) and University of Wisconsin-Madison (Haile). We are grateful to Jerry Hausman, Guido Kuersteiner, Whitney Newey, Harry Paarsch, Rob Porter, Susanne Schennach, Gautam Tripathi, and seminar participants at the Clarence Tow Conference on Auctions (Iowa) for helpful discussions. Astrid Dick, Grig- ory Kosenok, and Steve Schulenberg provided excellent research assistance. NSF grants Web addresses: Financial support from SBR-9631760 and SES-9983820 (Athey) and SBR-9809802 (Haile) is gratefully acknowledged. http://web.mit.edu/athey/www/ and http://www.ssc.wisc.edu/~phaile/. Introduction 1 This paper derives new results regarding nonparametric identification and testing of models of price sealed-bid, second-price sealed-bid, ascending (English), The theory literature has focused but not the values of his opponents. the object is spread among participation on revenues, We may be independent depend critically object, values models, information about the value of may be symmetric classes of models, bidders across bidders or correlated. market design, the optimal use of reserve issues, including and on the common In in these knows the value he places on winning the Within these bidders. asymmetric, and information and descending (Dutch) auctions. on several alternative models of the economic primitives auctions. In private values models, each bidder first- prices, and the A or variety of policy effect of increased bidder on which of these models describes the environment demand and information specific distributions characterizing the structure. consider a general specification of bidders' preferences and information, nesting as special cases the pure private values and pure common values models, and allowing correlated private information and bidder asymmetry. models and propose strategies We address econometric identification of a series of these for discriminating between them. Hence, our results address key demand empirical challenges facing researchers hoping to evaluate the structure of at auctions and to use this information to guide the design of markets. The price is types of data available from auctions vary by application. In most cases, the transaction We observed. more bids numbers consider four additional types of data that might be available: in addition to the transaction price; of bidders; (iii) (ii) auctions we may —in bidder-specific covariates that shift valuations; (iv) the realized value of In many descending auctions losing bids are not even made. lack confidence in the interpretation of losing bids even auctions, losing bids In oral "open outcry" when they In contrast, in sealed-bid auctions and ascending "button" auctions (Milgrom and the interpretation of losing bids variation in the number is of bidders clear may and records of arise when all bids al. It may also arise by design may be are observed. Weber available. (1982)), Exogenous auctions are held in diff'erent locations, internet auctions are conducted over different lengths of time, or agency) restricts entry. one or bids from auctions with exogenously varying the object for sale or, more generally, auction-specific covariates. are not recorded (i) in field when the seller (e.g., when a government experiments (Engelbrecht-Wiggans et (1999)). Bidder-specific covariates such as firm size, location, or inventories are often observed, particularly for government auctions, as are auction-specific covariates such as the appraised value or other characteristics of an object for sale. several important The realized value of the object sold is observed for government auctions, including those of offshore mineral leases (Hendricks and Porter (1988)) and timber contracts (Athey and Levin (2000)). In other cases resale prices can provide measures of realized values Most McAfee, Takacs, and Vincent (1999)). (e.g., prior research on identification of auction models has focused private values, under the assumption that mapping between observed bids and the mapping can be used and Vuong to-last bidder (if Theory predicts a one-to-one When latent private values. with all bids are observed, this to infer the distribution of values from that of the bids (Guerre, Perrigne mapping from the identity function, simplifying the problem. However, while the econometrician is may sometimes bids are observed. first-price auctions In second-price and ascending bid auctions the equilibrium (2000)). values to bids all on observe all bids in second-price auctions, ascending auctions end when the next- drops out. Hence, the winning bidder never reveals her value, implying that at best exit prices for all losing bidders are observed) the econometrician observes all but the highest value. Omitting even one order distribution of values. Similar but Our main results nonparametric identification of the statistic creates challenges for more difficult issues arise in common values models. demonstrate how different data configurations, characterized by additional information of types (i)-(iv) described above, can be used for identification and testing of alternative models. Due to the prior attention to first-price auctions (see, e.g., the recent survey and Vuong (1999)), our primary focus most common auction forms ( Lucking- Reiley (2000)).-^ on recent work on not all is on second-price and ascending auctions. These are the in practice, particularly with the rising popularity of internet auctions However, the ideas we develop first-price auctions, yielding identification and testable restrictions even when bids are observed, including the special case of a descending auction. dent private values (IPV) model is identified consider a richer data configuration, where enables testing of the symmetric IPV model. However, we trary correlation among when only the IPV model and if even one bid In spite of this lack of identification, when shown that that there there is aids identification common is observed. is We then observed. This and testing of the asymmetric arbi- unobserved in each auction (always the case we show that the is for not identified from observed bids. unrestricted private values model can be Laff^ont and Vuong (1996) have values models are observationally equivalent under the assumptions no reserve price and the number of bidders ^The prevalence is more general model allowing exogenous variation in the number of bidders. private and is transaction price also establish a negative result for a values: show that the symmetric indepen- first more than one bid from each auction ascending auctions), this unrestricted private values model of ascending auctions is well known. resources such as radio spectrum (Crandall (1998)). who enable us to build for these auctions also Beginning with second-price and ascending auctions, we tested by Perrigne is fixed. Exogenous variation in the number Second-price auctions have been used to allocate public In ascending auctions, the use of agents bid according to pre-specified cutoff prices results in an auction game (human or software) equivalent to a second-price sealed bid auction. Bajari and Hortagsu (2000) have argued that due to other features of bidding at online auctions, these are best viewed as second-price auctions. of bidders, however, provides a promising avenue for testing because the winner's curse (absent in private values auctions) causes a bidder's willingness to pay to decrease in the We number of opponents. use this fact to develop nonparametric tests of private values models that can be used when two or more bids are observed from each auction. Next, we consider a potential remedy for the non-identification result for the unrestricted private values model, relying on bidder-specific covariates. (Heckman and Honore (1990)). These (1989), Han and Hausman literatures consider situations in a random sample is We build on the literatures on competing risks (1990)) and the Roy model (Heckman and Honore which the maximal or minimal realization from observed; they show that covariates with sufficient variation can be used to These results cannot be applied directly to recover the underlying joint distribution of interest. second-price and ascending auctions when only the transaction price is reveals the second-highest realization rather than the highest. However, observed because this price we show that these results can be generalized, implying that bidder-specific covariates can give nonparametric identification in asymmetric private values models (without restriction on the correlation among bidder values) even if only the transaction price from each auction Our fourth approach is is observed. to use ex post information about the value of the object sold. This idea has previously been exploited by Hendricks, Pinkse and Porter (1999) in first-price sealed-bid auctions.^ We show can be used in cases that ex post value information for identification (or, more generally, auction-specific covariates) and testing outside the pure common values model they study, and where some bids are unobserved. Finally, we consider first-price auctions in which some bids and Vuong (1995) have shown that the symmetric IPV model are unobserved. Guerre, Perrigne, is identified from observation of the transaction price alone in first-price auctions; however, existing results for more general models have relied on observation of is identified all bids. We show, for example, that the asymmetric from the transaction price and identity of the winner. Indeed, and testing approaches based on observation of all IPV model our identification one bid or two bids from each auction extend to first-price auctions. Our work contributes to a growing literature on structural estimation of auction models.^ of this literature has focused Laffont, Ossard (1997), Deltas' Most on the IPV paradigm. Paarsch (1992b), LafTont and Vuong (1993), and Vuong (1995), Donald and Paarsch (1996), Baldwin, Marshall, and Richard and Chakraborty (1997), Bajari (1998), Donald, Paarsch and Robert (1999), Hong See also Hendricks and Porter (1988) and Athey and Levin (2000), who test implications of common values models using ex post values. ' See Hendricks and Paarsch (1995), Laffont (1997), Perrigne and Vuong (1999) and Hendricks and Porter (2000) for surveys of the recent empirical literature on auctions. and ric Shum (1999), and Haile (2000) have all proposed estimation techniques relying on paramet- distributional assumptions for identification. Nonpar am etric approaches have been proposed by Guerre, Perrigne, and Vuong (2000) for first-price auctions and by Haile and Tamer (2000) for ascending auctions. Only a few papers, including Paarsch (1992a), 1999), Hendricks, Pinkse estimation outside the Vuong (1998, and Porter (1999), Bajari and Hortagsu (2000) have considered structural IPV paradigm. Each sumptions or addresses only proposed tests Perrigne and Li, where first-price auctions for the winner's curse (1992a), Haile, Hong, and of these either relies on parametric distributional as- Shum all bids are observed. Other authors have based on variation in the number of bidders (2000), (e.g., Paarsch and Bajari and Hortagsu (2000)), although these either require parametric distributional assumptions or observation of all bids. No prior work has consid- ered nonparametric identification and testing of the standard alternative models of ascending and second-price sealed-bid auctions. identification for cases in To our knowledge, the only prior work addressing nonparametric which some bids are unobserved applies to symmetric IPV auctions (Guerre, Perrigne, and Vuong, 1995). Nonparametric first-price tests of alternative valuation and information structures have been considered only for only a few cases.^ The remainder of the paper is organized as follows. We first describe our general structural framework and review the equilibrium predictions of the theory of ascending and second-price auctions. In Section 3 we consider then takes up the case of common identification Section 5 extends values. auctions in which some bids are unobserved. bidder uncertainty regarding the price. 2 We conclude number and testing of private values models. Section 4 many of our results to first-price Section 6 discusses the robustness of our results to of opponents they face and to the seller's use of a reserve in Section 7. The Model Our model is based on that of Milgrom and Weber (1982). indivisible unit, with n > common knowledge and 2 risk-neutral bidders. there is no reserve price. U^ Other tests for common reserve prices, and Haile and Below we We consider an auction of a single In the basic model, the Each bidder z = 1, . . . number of bidders n is ,n would receive utility^ = V + A^ values are considered in Hendricks, Pinkse and Porter (1999), which requires binding Tamer IPV assumption for Enghsh auctions. depend on auction- and/or bidder-specific observables. (2000), which proposes a test of the will generalize this structure to allow {/, to from winning the object, where the random variables A= from a joint distribution -Fa,v(0-^ Let Fy and Faj(-) denote the marginal distributions A, and Ai, of V, convenience we let Each bidder X= {Xi, . . . The respectively. Fjj z's and (•) {), Fa distribution of F[/. (•) {), (^i, let Fx that the joint distribution of (A, V, . denote the induced distributions on X, that X) is is is G (L'^i, affiliated of the . M . with z, etc. When we tion Fs{-), Similarly, we denote by Fg^ S^^'^^ the j"^ For and C/,. Ui. Define paper we assume exchangeable with respect to the bidder indices, implying relax the exchangeability assumption, as asymmetric. For a sample of generic drawn {-{v). t/„) , . that the joint distributions Fa(-), ^u(-):' arid Fx(-) are exchangeable, in which case F^^ for all are Fa denoted U= X. For most () be the joint distribution of V An) G K" and , . A conditional on ^ = f private information consists of a signal ,X„), and . random variables S = we (•) = Fa will explicitly refer to the {Si, . . . , (•) model Sn) drawn from a distribu- order statistic, with ^^""^ the largest value by convention. {) denotes the marginal distribution of S^^'""'. Although bidder-specific random variables enter the bidder's utility additively in our model, the multiplicatively separable specification Ui = V and Vuong (1999)) can be incorporated framework by taking logarithms. Indeed, a natural interpretation of the well. To scale and model is that U, V, A, and we simplify the exposition, will allow the Our model in this random X Ai (see, e.g., Wilson (1998) or Li, Perrigne are in logarithms, with bids in logarithms as without reference to the logarithmic will discuss the variables variables (as well as bids) to take on negative values.^ nests a wide range of specifications of bidders' preferences and the underlying infor- mation structure. There are two main categories of models: Private Values (PV): Xi Common = U, Vi.^ Values (CV): (U,X) are affihated, and for all z,j, {U^,Xj) are strictly affiliated but not perfectly correlated. We will consider only models that fall into one of these classes. In the private values model, bidder has private information relevant to another's expected values model, bidder j would update her beliefs about her utility, Uj, to her own signal Xj. Thus, there is a "winner's curse" in the In contrast, in the utility.^ if common no common she observed Xi in addition values model: upon learning the distribution of A is unrestricted, the random variable V can be normalized to zero without loss of we include it in our notation to simplify the exposition of several special Ccises of the model. some cases, we will assume that random variables are affiliated; recall that this condition is preserved by When generality; In monotonic transformations such as the logarithm. V= with all Xi i.i.d. is also a (symmetric) private values model, although here the two private values models are observationally equivalent. Following the literature, in private values models we interchangeably. will use the terms "utility;" "valuation," and "value" more optimistic than her opponents', and that she has won, bidder j realizes that her beUefs were her behefs about her own utility become more "general affiliated values" model of to rule out pure private values model allows Milgrom and Weber (1982), CV model Common is the and guarantee that the winner's curse same for all bidders. Several of Values (Pure CV): In addition to the a special case of the is where we have restricted the model CV In general, the arises. for utilities to differ across bidders; in contrast, in the "pure the value of the object Pure The pessimistic. common values model," our results concern this special case. CV assumptions. A, = so that 0, = f/j V\/i. Further refinements of the pure PV We model. However, we have not assumed will CV model be discussed will affiliation, since many Now in Section 4. return to the of our results hold without this restriction. pay particular attention to three special cases of an Conditionally Independent Private Values (CIPV): Xi = affiliated private values = V+A^ Ui Vi, model: with {Ai,. .,An) . independent conditional on V.^^ CIPV with Independent Components (CIPV-I): X^ = Ur = V+Ar = Ui = Vi, with (yli, . . . ,/l„, V) mutually independent. Independent Private Values (IPV): ally We V= so that 0, Xi Ai Vz, with (Ai , . , . . An) mutu- independent. assume that the underlying value and signal common knowledge among (English) auctions.-'^ the bidders. distributions, and the number of bidders We initially consider second-price sealed-bid n, are and ascending In a second-price sealed-bid auction bidders submit sealed bids, with the object going to the high bidder at a price equal to the second- highest bid. For ascending auctions we assume the standard "button auction" model of Milgrom and Weber (1982), where bidders observably and irreversibly as the price rises exogenously until only one bidder remains.'^ the '" random The variable S, denote the bid variable unobserved V made by can be thought of as the random player mean z, with Hb,{-) of the private values. (to the econometrician) heterogeneity in the objects for sale. = V Vi. Below we generalize the We let distribution. In practice this can arise from Equivalently, each U, could be the common component v about which bidders have CIPV and CIPV-I models to allow private values a private component Ai and an independent E[v] its exit sum of identical expectations: that are independent conditional on a vector of auction characteristics. " We use the terms English auction and ascending auction interchangeably. forms, see Milgrom and Weber McAfee and McMillan For a review of standard auction Milgrom (1985), or Wilson (1992). (1982), This is a stylized model of an ascending bid auction that may match actual practice better in some applications than others. Ascending auctions with "activity rules" (e.g., the FCC spectrum auctions discussed in McAfee and McMillan (1996)), for example, are often designed specifically to replicate the button auction. Many of our results rely (1987a), ^ only on the interpretation of the transaction price from an ascending auction as the realization of the second-highest of the bids prescribed by the Milgrom- Weber equilibrium. In the second-price auction, each bidder b^ In the PV model = = this reduces to bi that the bid function 6(-) is = b{x,) ui = z's equihbrium bid when Xi E[U^\Xi x,. = ma.xXj CV In the = model, = Xi is^'^ Xi\. (1) strict affiliation of {Ui,Xj) implies strictly increasing. Equilibrium strategies are similar in the English auction, although bidders condition on the signals of opponents who have already dropped out, since these are inferred from their exit prices. Letting Li denote the set of bidders with signals below b^ for each bidder i. - E[U^\X^ = X, = XiVj Xi, this implies ^{iULi},Xk= XfcVfc In the case of private values, this again reduces to an equilibrium exit price G bi Li] = (2) Ui = Xi. Note that the ascending auction ends at the price 6("~^-"). Data Configurations 2.1 We consider data sets consisting of observations from the same group of n bidders We will The • independent auctions. In the basic model, participates in each of the (A, y, X) (conditional on auction-specific covariates, represents an independent T draw from if T any) is auctions. The joint distribution of fixed across auctions. Each auction this distribution.^^ assume that the following are observed by the econometrician: transaction price. This is i5("~^-") in a second-price or ascending auction, but i?("-") in a first-price or descending auction. The number • of bidders, n. In addition, the following data elements may or may not be observed: • Other bids (or exit prices), in addition to the transaction price, of the form B^'^'^K The • '' identities of some bidders, where /('"") denotes the bidder bidding fi^'"'"). Throughout the paper we focus on symmetric equilibrium when bidders are ex ante symmetric. When we allow in the private values framework we will focus on the unique equilibrium in weakly undominated bidder asymmetry strategies, in '^ which each bidder bids her valuation. Such a dataset might arise as a result of non-cooperative bidding in auctions for natural resources, where the underlying competitive environment the contracts are small from the perspective of the bidders. is some applications we might expect dependence of Examining the empirical implications Here we follow the vast majority of the theoretical and In bidders' information and/or willingness to pay on outcomes of prior auctions. of such dependence is a valuable direction for future research. procurement contracts or stationary over the relevant time period, and empirical literature by focusing on cases in which dependence across auctions is absent. • K sets of auctions, each with a different n/^ finite), where the variation in • The ex post reahzation of distribution of V V + A^+ QiiWi) arise is more or, of bidders, and possibility that the random shocks {ni, . . , . n^} [K and each exogenous. generally, auction-specific covariates W = (iyi,...,W„), where bidder Wq affecting the i's number of bidders is a pool of potential bidders and learn X,. There no reserve is noted already, exogenous variation in n can and Xi. price, so all bidders place a bid in the auction. This would lead to exogenous variation in the number who who learn Xi of bidders. As from participation restrictions by the also arise government auctions or lengths, where longer auctions may enable more potential bidders to become aware of the number is common knowledge we may wish it is In contrast, in the to the participants. Unlike first-price auctions, in private- not important that the number of bidders be CV model, the assumption plays a focus is are taken as the number of the 3 identification. number of bidders, n G . . . In Section 6.1, how many competitors they bidders. we extend face. However, when we discuss estimation, the relevant asymptotics of auctions approaches infinity. {nj, role. of When we consider auctions with a varying ,nx}, we consider asymptotics as the number of auctions in each K sets goes to infinity. Ascending and Second-Price Auctions with Private Values If all bids are values in on number common knowledge among more important the model to the case in which bidders are uncertain Our sale. to revisit the hypothesis that the value second-price and ascending auctions, bidders' strategies do not depend on the opponents, so seller in field experiments) or as the result of varying online auction considering variable numbers of bidders, of bidders receive Bidders with (e.g., in When = might vary exogenously merits comment. This could to the cost of participating, which are independent of Ui positive shocks participate by Ui utility is given an unknown function. g^{-) is (outside the formal model described above) there if n G U. • Bidder-specific covariates, The n number is observed in a second-price sealed-bid private values auction, equal to the distribution of bids. However, if any bids are unobserved (always the case an ascending auction), the problem becomes more complicated. identification in a series of nested We proceed by considering models of private values auctions, showing how richer models require richer data configurations for identification section the distribution of and testing. As discussed above, we focus in this and the next on ascending and second-price auctions, and these auction forms are assumed unless otherwise stated. 8 Symmetric Independent Private Values 3.1 We is begin with the symmetric IPV model. In this model, the underlying distribution of valuations nonparametrically identified even when only one bid per auction tested more than one bid if Proposition model 1 (i) is testable if In the (i) The i*'* observed or there IPV model, either (a) at auctions with different Proof, is variation in the is F\j{-) is identified more than one from is observed. number The model can be of bidders. the transaction price, (ii) The IPV bid per auction is .observed or (b) transaction prices numbers of bidders are observed. order statistic from an i.i.d. sample of size n from an arbitrary distribution F(-) has distribution ^'^"M' example Arnold (see for et. al („-,;(,-i), (1992, p. Fjj'^ {) is. marginal distribution (ii) f/,, Since the observed transaction price Fu (•) Fy , is identified and either [/(""•^•"^ Issues for Estimation 3.1.1 Although a Fy is {•) or t/^""^'") with ^ j,n ^ i and testing is The of ways. (3) when n) must be equal O for all u. FJj this paper, we do offer Because the right side of (3) is theorem and continuous mapping theorem (z) is substituted for F^^''^\z) is a consistent {z).^^ testing principle suggested in part A and the method. The empirical distribution a pointwise-consistent estimator of F^'"' {). imply that the value of ^(z) solving Fu [/(""-^'"^ distributions of different order beyond the scope of differentiable with respect to F{z), the implicit function estimator of strictly F\j{-), the result follows. several observations. First, note that (3) suggests an estimation function is and Testing analysis of estimation full (3) whenever any distribution equal to the order statistic is completely determines () Under the IPV assumption, values of Fu{u) implied by the statistics ([/(""'") (3) Because the right-hand side of 13)). increasing in F{z), the marginal distribution of f'"'-(i -')-<« direct Kolmogorov-Smirnov (ii) of Proposition 1 can be implemented test for equality of distributions ing bootstrap critical values (see, for example, Romano (1988)). in a number can be implemented us- Alternative possibilities include standard tests of equality of means or quantiles of the distributions. The same approaches can be taken for a number of the tests we propose below. ^^ If more than one bid observed bids using (3). is observed in each auction, more efficient estimates can be obtained by incorporating all In real-world ascending auctions, the button auction rules often free-form. Thus, some caution is may not be used; instead, bidding warranted when implementing from the same auctions. ^^ The IPV button-auction model may be is this test using different bids rejected because recorded bids understate the true willingness-to-pay of the bidders. In that case, one could compare estimates obtained from low-ranked bids to those obtained from high-ranked bids to assess the magnitude of the bias that arises from bidders' failing to bid at prices as high as their true valuations. Asymmetric Independent Private Values 3.2 If the exchangeability assumption values: relaxed, is we obtain a model bidders' valuations are independent, but valuations than others. While the previous Specifically, each Uz (Berman (1963)) for the closely related Proposition 2 In Fui (•) is The also observed. is some bidders may be more is Fir^{-). still be obtained from observation of the highest or lowest bid and the identity of the competing risks assume model with asymmetric independent that each Fu^ (•) is continuous. risks. Then each identified if either of the following hold: (a) In a second-price auction, the highest hid (B^'^'^' ) (h) have high following result follows directly from prior results IPV model, the asymm.etric likely to drawn from an idiosyncratic distribution asymmetric private values can a single bid at each auction, as long as this corresponding bidder asymmetric independent private on valuations' being identically distributed across bidders, result relied heavily identification in the case of is of and the identity of the winner are observed, In either type of auction, the lowest bid and identity of the lowest bidder are observed. Proof. See Theorem When 7.3.1 and Remark 7.3.1 in D Prakasa-Rao (1982). there are only two bidders, the lowest bid is equal to the transaction price, so part (b) provides identification from the transaction price and the identity of the loser. bidders, the transaction price an auction data set, is an intermediate order statistic, and if With more than two any bids are unobserved in the lowest bids are most likely to be missing. Unfortunately, the approaches used to establish identification based on extremal order statistics do not extend immediately to non-extremal order statistics such as the transaction bid and two more observed bidder Proposition 3 In the interior of identified if its n > and next-highest Haile and the Tamer IPV support and that bid (i.e., However, with one additional observed identities in each auction, identification follows. asymmetric 3, the identities price. model, assume that each \/i,j, {suppFir^{-) D Fi},{-) suppFu.{-)} ^ has a positive density on 0. Then each Fjj.{-) is of the top three bidders are observed, and the transaction price s("~i'"-) and i?("~2-")^ are observed. (2000) propose an alternative approach to inference and testing motivated by this observation. 10 Proof. By assumption we observe /X So (where the derivative exists) also observed. Similarly, for is / a2 C?XC?y The _ T(n-n) V > y ^ we observe r(n-2:n) • fy(n-2:Ti) y^ two expressions ratio of these x, _ T{n-l:n) lj{n-l:n) \ /"CO = \ J. .,_^ ^X lj^i,j,k is {i-FuM)fu,{y)fu,i^) {l-FuAx))fu,{x)Fu,{x) Taking the x approaches y from below (x) f yields ^'^ . , which uniquely , D identifies F„^.(-). Conditionally Independent Private Values 3.3 If limit as the IPV model rejected, is correlated. We form of the CIPV model is it natural to consider the alternative that the private values are begin with a model where the correlation or the more restrictive CIPV and CIPV-I models impose significant private values model. This question in the auctions literature (see, for theorem states that any sequences.^* Hence, without infinite if f/i, loss of generality. independence is . . . , is among private values takes the special CIPV-I model. ^' It is natural to ask whether the restrictions relative to the (affiliated of independent interest since these are example, and Vuong Perrigne, Li, and exchangeable) common assumptions The de (1999)). Finetti exchangeable sequence can be represented as a mixture of C/„ are exchangeable for all However, because we consider a a restriction on the private values model. CIPV n, the finite Even if assumption can be made number U is i.i.d. of bidders, conditional affiliated and exchangeable, conditional independence does not necessarily hold (Shaked (1979)). However, as an approximation of arbitrary affiliated distributions, even the it more can always match the mean and variance of restrictive CIPV-I model may Ui, as well as the covariances fit reasonably well: among {Ui,Uj), and it allows partial flexibility for higher moments. " To understand how CIPV-I differs from CIPV, notice that CIPV-I rules out the possibility that the distribution of Ai depends on V. '* all More 7n> 1, {X„}~=i are exchangeable random variables, there exists a a-algebra G of events such that Pr(Xi < xi, ,X„ < x„) = J TYJli^'^i^J < Xj\g)dPi{g). See, e.g., Chow and Teicher (1997). precisely, if . . . 11 for Proposition 4 The CIPV-I model (normalized so that E[At] of any exchangeable and affiliated distribution -Fu(-)the first through third Proof. = However, can 0) it places a testable restriction on moments. First, observe that the CIPV-I structure induces same for all choices of and Fv{-) such find distributions Fa(-) var(f/,)— cov(t/i, t/j) By 0. = E[V] E[Ui\, var(y)=cov([/i, = = E[V]E[A'^,] E[U^] {var{U^) if all - analogous to an tification results and var(ylj)= This possible be- is cov{U^, Uj)) all Bids in a Second-Price Auction is possible (up to a location normaliza- Vuong bids from a second-price auction are observed. Li, Perrigne, and CIPV-I model identification of the is f/j), D This section shows that identification of the CIPV-I model tion) Then we can testable. Observing 3.3.1 that: 0. cov{Ui,Uj)> affiliation, For the third moments, the CIPV-I model implies that E[UfUj] - E[U,UjUk] is and by j, distri- exchangeability, these (implying that var((7,) =var{Ai)+va.i{V), as required). cause vai{Ui) '>cov{Ui,Uj)> This restriction 7^ i For a given affihated utilities. bution Fu(-); suppose that E[Ui\, var(t/j), and cov{Ui,Uj) are given. values are the moments the first two fit in first-price sealed-bid auctions. In the measurement i.i.d. (1999) analyze CIPV-I model, each Ai error on the variable V, enabling direct application of iden- from the literature on measurement error (Kotlarski (1966), Prakasa-Rao (1992), and Vuong (1998)). In second-price and ascending auctions, observed bids correspond to Li izations of order statistics [/('"^ = yl^-'-") corresponding "measurement errors" prior results unless all Note, however, that -I- V, with j from a subset of {A^^''^^} {1, . - . ,n}. real- Therefore the are not independent. This precludes application of order statistics are observed (which when the asymmetric model is is impossible in an ascending auction).-'^ identified, it is possible to test the restriction that the Ai are identically distributed across bidders. Proposition 5 In of the random the CIPV-I model, assume variables V that\fi the characteristic functions '4)y{-) andipj^^{-) and Ai are nonvanishing. If all bids are observed in a second-price auction, then: (i) (ii) '^ The symmetric CIPV-I model is identified. The asymmetric CIPV-I model Although it is identified seems plausible that since each distribution, there may do not present such a yl'-' ") is if, in addition, all bidder identities are observed. an order statistic of an i.i.d. sample from a common be sufficient structure to identify the model from only two order statistics f/""', result here. It is interesting to note that the difference not identify Fa(-) up to location (Arnold et al. (1992, p. 143)). 12 U''^''^^ — f/^*""-' = A''^"^'' - parent f/'* "', we ^(''"' does Proof. Follows Vuong An the ex post realization of of V V is CIPV and CIPV-I models the bidders. Thus, V is V ex ante is When V observed, the is (common knowledge) CIPV model from observation of the transaction identified available if common knowledge might be some publicly observed auction-specific covariate, such as the appraised value of an item up for auction or the for a construction contract. is observed. In a second-price or ascending auction with private values, equilibrium bidding behavior does not depend on whether or not model) and D alternative approach to identification and testing in the among Li, Perrigne, (2000). Ex Post Observation 3.3.2 by as in Kotlarski (1966), or as applied to first-price auctions cost of raw materials (and thus, the nested CIPV-I price. It is testable if more than one bid is observed in each auction. Proposition 6 In (i) If (ii) one bid is the CIPV model, suppose that the realization of Given (i) is observed, observed in each auction, Fv{-) and Fa{-\v) are identified for CIPV model If at least two bids are observed in auction, the Proof, V v, a^^''^' = -u^^'"-* V then follows from Proposition 1. — v = b^^'^) — is testable. v. Identification of Fji^{-\v)ior Under the assumptions (ii) all v. of part (i), each observed a consistent estimate and standard nonparametric smoothing techniques. With of Fa.{-\v) can be constructed using (3) observation of two bids in each auction, two estimates of FAi-\v) can be constructed and their D asymptotic equality tested. 3.3.3 In Auction-Specific Covariates many private-value auctions, theory points to more than one source values; for example, in timber auctions, the species of timber However, some applications in variation in private values and CIPV-I models is it may be and tract of correlation in bidder size are typically observed. the case that conditional on such variables, the remaining independent across bidders. Formally, we can generalize the to allow for a vector of auction-specific variables. Wo, that shift CIPV bidder values and are observed to the econometrician.^'^ In a generalization of the CIPV model, we assume A is independent conditional on Wq, and analyze identification of Fa{-\wo), the joint distribution of A given Wq = wq. independent, and that Note that A more restrictive model, analogous to CIPV-I, assumes that Wq afi^ects bidder utilities through an unknown function po(')- since this structure permits V G Wo, we can assume 13 V= without loss of generahty. (A, Wq) is Proposition 7 Normalize A (i) If is V= and suppose 0, Wq and that independent conditional on Wq, then Fa{-\wo) is the transaction price are observed. identified for all wq. (li) If (A, Wo) are independent, Fa{-\wq) has density fA{'\u)o)VwQ, the unknown function gQ:supp\WQ] identified up ^ b — (i) Apply Proposition Assume QoiWo)). the case). zs 1 m each auction, for each wq. for simplicity that Then (where Wq the models described in (ii) Note that This identifies (;o(') up to a location normalization. determined through the relationship Our approaches (3), which = and (ii) = 6|Wo) the same are testable. Pr(^(-''") when this is < not -g'oi'^o)- Then, identifies F^ ('i^t'o) for FAi-\wo). , < each wq, Pr(A, b — 50(^0)) Following the same (iii) and equality can be D tested. and testing of the CIPV and CIPV-I models require potentially to identification strong independence and/or observability assumptions. we assume affiliation of observed bids can simply let and exchangeability. It is from observation of bids are unobserved? testing of the The And is straightforward to see that for fixed n, any set all Thus, the PV model bids in a sealed-bid auction, but untestable without further is the unrestricted PV model identified when some second, what additional information will permit identification and/or PV model? We PV Model Is We begin by showing that, address each of these questions below. Not Identified from Incomplete Sets of Bids without further information, the unrestricted an ascending auction, and result the CIPV-I model be rationalized in the private values framework (Laffont and Vuong (1996)): information. This raises two questions. First, is if the distribution of values be equal to the distribution of bids. identified 3.4.1 Moreover, even imphes that we cannot rule out the private values model altogether, even rejected, Proposition 4 This o-tc Unrestricted Private Values 3.4 in ^i, where the derivatives exist) approach, the additional bids can be used to identify is is (i) < Pr(5(-'"") a scalar (the argument is §'^Y>r{A(r-^)<h-go{wo)) if + Fx{-\wq) and go{') difjerentiahle, then for all wq, ^Pr(AO-)<6-(7o(«^o)) is go{Wo) to a location normalization. (Hi) If at least two bids are observed Proof, R = and Ui it is model not identified in a second-price auction unless true even under the exchangeability assumption.. This result from the literature on competing distribution of competing risks PV is risks. Cox not identified bids are observed. a generalization of a classic is (1959) and Tsiatis (1975) not identified from the lowest order 14 all is statistic. show that a We joint show that this be extended to cases in which any combination of order result can the value distribution method be exchangeable, so long as not restricted to is statistics observed, even is when are observed. Since the all we proceed by constructing a of proof of the prior literature cannot be applied directly, counter-example.-^^ Proposition 8 Consider from the vector of bids unless [0,4]" is in = Then, [/(*^""-) is l{u, G [3 -l{u^ G 5^= G S* Si < and - £,3 + e],i G [2 - e, 2 + £],i e USk-i < some k 6 for l{u, G Si} > = US'„_fc l{n, G 0, /u(-) - [1 = - [1 e, 1 and similarly permutations of the vector f ''T- " l{ui f l{ui I x) < < < x,i all x, all x,i = SG < x) 5^, and l}l{ui is > I, this result all x,i (1, . G 5fc_i} restrict U to be J ¥" ^} ' l,|5„_fc| . [4 is - is does not impose affiliated =n- k} . [4 + e], Vz G Sn-k} - e,4 + ^J, V? G S^^k} e,4 a probability density, with the to others. For example, if A; = n to the neighborhood of (1, . — 1, , . . prob- 1, 3, 4), pairs. . . . ,m — liui \} > x,i = m+ ,'"m-l,2;,1im-Hl, 1, . . . ,n} \ is •- du-m,^n) J m^k, = m+ 1, . . . ,n}c((. is . . ,Um-i,x,Um+i, ); S,£)du_rn affiliation of U. Affiliation is but not independent, eis long as constructing the counter-example. 15 = 0. replaced with /u(-)' ^-^d the probabihty D observationally equivalent to Fjj{-). potentially disturbed by small perturbations of the distribution. we = k- l{u, G c{-;S,£) 1, 2, 4) , . unchanged when /u(-) distribution based on the density /u(-) 'Note that l{«z G + e], V? •/u(«l, = l,...,m- Thus, ^Pr(y(™'") is G Sk-i} I / 1/2, positive is = )[ ^n-m/J^^_^y But, for e / \ < l,|5fe-i| some regions from the neighborhood of ^ Pr([/('"^") = +7Ssg5'= /u(-) ability weight shifts Observe that and that /x(-) = + e]yi e, 1 function c shifting probability weight from for all X {!,..., n} a subset of {U^^'"'^ {!,..., n},!^! 1/2, define c(u; 5, e) G 5i} For sufficiently small 7 n not identified is unobserved. Define a set of partitions of bidder indices {{Si,Sk-i,Sn-k) for Fu{-) not identified in a second-price auction it is the interior of the support of Suppose that throughout this region. S'' i^u(") is exchangeable. m an ascending auction; further, Suppose that observed but and assume are observed. all bids Proof. PV model the a delicate condition, and The result can if it holds weakly, it be generalized to the case where we take a smooth, "small enough" perturbation when We PV Tests of the Symmetric and Asymmetric 3.4.2 PV have shown that the unrestricted Models not identified in general. Despite this, is we now show that the model has testable implications, even in situations where the underlying value distribution is Our approach not identified. number requires exogenous variation in the as the observation of multiple adjacent bids — e.g., of bidders, as well the transaction price i5("~^-"^and the second- highest losing bid i?^""^-") in an ascending auction, or the highest and second-highest bids in the The second-price auction. Lemma Consider the 1 of the order statistics following PV model. t/(j-"), Proof. When statistics can be derived lemma The distribution ofU^^~^''^^ m— = j to I the distribution Fu(-) (e.g., central to our argument. is is n— with 1, from identified is the distribution m < n. exchangeable, the following relationship between order David (1981, 105)): p. ^fI-\u) + '-Fl;^'-\u) = Ft-'\u). from the distributions of So, U^^'''^\ jj(m-\:n-i) ^^ jj{n~2:n-i) ^gj-Qg ^t^q = j m — to 1 abovc formula. n — If (4) we can compute the 1, m= Ti — this 1, distributions of completes the argument. D Otherwise, repeating the exercise gives the result. The intuition behind (4) random) from a of the bottom n set of is When straightforward. bidders, in each auction, there r bidders, and a ^^ is dropping one bidder exogenously a - chance that the dropped bidder chance that the bidder these probabilities as weights, the distribution of ^('""^i) of i7('"+i^"^ and m > PV model bids i?(™--"), Proof. ^(m-i:m) ^(^-i^")^ 2 bidders and bids . , . 5("-") . from L/^"^") Lemma and r bidders. Using a weighted average of the distribution the transaction price ^^("-i:") . _ _ 1. [/(""^"'^ ^('"-i-'") m from auctions with n > from aucIn bidders. also sufficient to observe B^"^''^' at auctions with strategies, observing the bids equivalent to observing the order statistics [/("^"i'"), [/('"-i:'") The same argument ^('"-i^")^ . ,[/("-^i^") . . . using only the order statistics Under the private values assumption, that observed directly. to — one m bidders the n-bidder auctions. can compute the distribution of following it is Given the equihbrium bidding jg we observe is testable if the case of a second-price auction, and one of the top n is f/^'"^"). Proposition 9 The tions with is is at (i.e., this distribution . and . ^^("-i:") and t/(™-i:"^). [/(™~i-") to We jy^""-^'") must be the same applies to the case in which the order statistics as L'^f'"'") D are observed. 16 This result implies that the PV model testable whenever is second and third highest) bids from auctions with n and n symmetry that Proposition 9 relies critically on distribution of bidder values is — 1 we observe the top two bidders, holding of the bidders, i.e., Balakrishnan (1994), The variables, who m C N Vm C Vn . . Vn, and . . let Yi, . . Fjj' {) ,Ym be a sample is some If for such that n n > \VTn\ = m Tn, < then n, if m > we observe of size Then If m=n— {u) 1, (6) = Fy identities in Yi, . m . - drawn without replacement from {Uj,. . . ,Un} using are exchangeable. Define ,^771 ' . . , . Ym} + -Fu equals F^'^ = Fu (•), must satisfy (4). for r <n Since the this gives (y)- (5) ^Ft\u) + ^Ft'-^'Hu) = Ft;-'\u). (6) (y^ {y) (u), this simphfies to can be tested directly. For m < n— 1, repeated application of enables testing (6) D ^^As with Proposition and . is testable?"^ of (5). fit'"^'") of the distribution of the rth order statistic from {Ui,i G Vm}- Exchangeability —^^u '" . . and bidder asymmetric) private values model the ,5("~i-") bids B^'^~^''^\ bidders and the transaction price distribution of the rth order statistic of {Yi, F^ is 2, there is positive probability implies that the distributions of the order statistics of {Yi,...,!^} Since Vn 3 such that the probability that ,Un be the random variables corresponding to the valuations of the bidders in a discrete uniform distribution. where = with \Vn\ bidders, the (unrestricted, Proof. Let Ui, identities of (4). identities in auctions with auctions with we observe the exchangeable random variables whose distributions must obey set of participating bidders is positive. and bidder on the assumption that the following result exploits the arguments in Balasubramanian and set of Proposition 10 Take any P„ participation by every Note note that by taking random draws from samples of arbitrary random one can obtain a the recurrence relation all else fixed. exchangeable. However, we can extend the result to the case in which distributions of private values are completely unrestricted, as long as the participating bidders. (or the 9, in at auctions with the case of a second-price auction, m bidders and bids B'-"''"\ . . 17 it is also sufficient to observe ,B<"^"' from the n-bidder auctions. all bidder identities 3.4.3 The and Testing Using Bidder-Specific Covariates Identification are available, identification literatures When provided results for testing but not identification. last section on competing may also risks (e.g., be possible. ^^ To show Heckman and Honore the Roy model (Heckman and Honore the Roy model and this, (1989), bidder-specific covariates we adapt approaches from the Han and Hausman (1990)) and While Heckman and Honore (1990) point out that (1990)). the competing risk model share a similar structure, the relationship of these model to auctions has not been previously exploited. These prior is Observing extreme order observed. which either the lowest or highest order literatures are tailored to cases in (minima or maxima) creates a statistics statistic relatively simple inference problem since the distribution of an extreme order statistic provides direct information about the joint distribution of values. For example, Pr(i?(n-) < 6|w) = Fa(6 - Inference from non-extremal order statistics is if Ui = gi{Wi) 5i(u;i), ... ,6 more difficult; - + Ai^ then ff„(«;„)). however, when we observe either the highest bid in a second-price auction, or the lowest bid in an ascending auction, existing results can be applied with only minor modification. Analogous to the prior literature, we also need to observe the identity of the auction winner Proposition 11 Consider Qii^yi) for + Alii; (b) and all i,j, identified, A is the is and Proof. The proof in Theorem 12 of identified model, normalizing and {g\{Wi) up , . gn{Wn)) . . , for the bid are observed; or two-bidder case Heckman and Honore on the variation Examples Assume 0. (a) Ui = is R". Then each g,(-), i = ,n, is 1, to a location normalization if either (i) at a second-price is (ii) at either type of auction, the lowest identical to that for the two-sector The proof when n > (1990). in the bidder covariates of such covariates include the distance backlog of contracts won in 2 requires a straightforward extension, which Proposition 11 does not require exchangeability. ditions F = bid are observed. Roy model. This multi-sector ^^ continuously distributed with support R"; (c) (Ai,Wj) are independent auction, the winner's identity bidder's identity PV asymmetric (d) the support of and Fa(") (loser). It W. given identical to that for a we D omit. does, however, impose fairly stringent con- Our next from the firm previous auctions, or measures of is Roy model result relies on similarly strong to a construction site or tract of timber, a firm's demand in the home market of bidders at wholesale used car auctions. An alternative identification result can be given if bidder identities are unobserved but there is sufficient variation g{Wi) to effectively "pick" the winner (loser) through the choice of lu. More precisely, if the winner's bid and limu,j__oo 5i(ift) = — oo at a sealed bid auction, or the lowest bid is observed and Vim^^-^oo g{wi) in each gi{-) i = 1,. . . ,n is identified, and Fa(-) is identified up to a location normalization 18 is = observed, oo, then conditions on W, observed, possible to uncover the underlying joint distribution of values. it is but shows that even Proposition 12 Consider only the transaction price (the second-highest value) if PV the asymmetric model, normalizing y = 0. is Suppose hypotheses (a)-(d) of Proposition 11 hold and that Fx{-) has a differentiate density. Further assume that for each i Then = 1, . . ind each -Fa(") Proof. ,n, gi{-) is differentiahle, g[{-) is absolutely continuous, . Let = z, 6 gi{-), i — = 1, . . . and from the transaction ,n, are identified = — oo. limu,-_^_oo5i('u^i) price. gi{wi). First, observe that (abusing notation for zero-probability events to simplify exposition) n rj — < Pr(i?("-i^") 6 = |w) ^ 5] Pr (A, > / - Pr Aj = {Aj = Zi, z„ A^ < z^ Vfc ^ Ak < Zk V/c ^ i,j) n = I] X] [Pr [Aj = 2„ Ak < Zk Vfc z, J ) z^, j )] Then dwn dwi < Pr(i?("-i-) (^ V db 6 = -(n - |w )) 1) fli-g'.iw,)) f^ i=i j=i ^/a (a) Observe that db \ Thus (using = gn{uin)) dwn dwi FA{b - gi{wi), ...,b' = lim(,^_oo ^^"'"^(6) F^ib — g\{wi), 1), . . . ,b gn{iOn)) j ' F^~'^'''\b) — Qniuin)) is = = W{-g'j{wj)) /a J LL Y^ ^— Z-^ ] Qd- and lim^^(^_^^__^-) FA{b - identified (a gi{w-i), from the observable Pr(5("~-^-"^ . . .,b < 6|W) by the relation FA{b-g\{wi),...,b-gn{wn)) 1 — r I n-lj ment for . . ,b — each gn{wn)) i. = FA,{b — — and g\^w\),...,b gi{-), . . gi{wi)). Varying b Then, with knowledge of the Pr(S("-i ") < 6 \dwi---dwn Jw={-oo,...-oo) sider the separate identification of Fa{-) . ^", (- Given that the composite function F^ib gi{wi), = 5i(-)'s, — gni'Wn)) is |w)^ ) d-w. identified, we now ,5n(0- Note that limu,_,^(_oo,...__oo) •'^a(^ and Wi then we can identifies gi{-). identify Fa{-) at part of the proof shows the w^ identify FA{b — gi{wi), . . . , 6 — how "~ Repeat the argu- any point (ai, . . . , a„) D through appropriate choices of b and w. The main con- the distribution of the transaction price and variation in gn{wn)). The second 19 part of the proof show how to identify the functions gi{-), and then varying . ^gni-)- - . and b The From lUj. strategy in this second step entails taking extreme values of w_,, a practical standpoint, this strategy has disadvantages, as W. heavily on observations in the tails of the distribution of available to identify the functions gi{-) if /(j-") Pr(/("-"^ z,j, + gr{wi) > Pr {A^ We However, an alternative strategy We briefly sketch this approach. is ^^ denote the identity of the bidder making bid B^^'-^\ From the identity of the second-highest bidders, for each relies the identity of the highest and second-highest bidders are observed in addition to the transaction price. Let it Aj = /("-i-") i, + gj{wj) > ^fc = -f- j|w) is gk{wk) first and and equal to identified ij) VA: 7^ can then define the set 5^ = |w : Next, observe that the fohowing = Pr (Aj For Mv G Sui, + gj{wj) < Pr = Vi,i,Pr(/("^") (/('^'"^ = /(" -'•"^ j|w) identified for each = /("^") b i, is = i,/("-l^"^ = z,/("-i^") = j |w) is J, = Pr(/('^^"^ = i,/("-i^") = j|w)| = z,/("-i^'') = . {i,j)'- w) • Pr (/("^") constant. So for any b and any w' j |w^ 7^ w in Su,, the b' that solves p^^^(n-l:n) must up < /(«:") 5^ satisfy gj{w'.) — ^ ^j{n-l:n) gj{wj) = b' — b. ^j \y^) As long = Pr(5("-1^") < 6', G Sw} = M, varying as {wj /("^"^ =z,/("-l^") =j|w'). w and w' identifies gj{-) to location. Assuming that consistent estimators can be constructed using each PV model can be implemented by comparing them. gies in Propositions 11 and Corollary 1 Suppose that the hypotheses of Proposition 12 hold. is testable if and ^ 12,^^ a test of the each of the following is The argument follows the proof of statistics in a Then the asymmetric PV model observed: (a) the transaction price; (b) either the winner's bid identity at a second-price auction, or the lowest bidder's bid non-extreme order of the identification strate- Theorem 12 in and Heckman and Honore identity. (1990), but considers the case of model with more than two bidders (analogous to more than 2 sectors in the Roy model). We are unaware of work developing nonparametric estimation approaches model based on these identification strategies. Doing so is for competing risks beyond the scope of the present paper. 20 models or the Roy Ascending and Second-Price Auctions with 4 Testing Based on Multiple Bids and Variable 4.1 In Section 3.4.2, we established that the ber of bidders (Proposition 9). is PV of Bidders testable using exogenous variation in the num- framework. In the second-price auction, the recurrence CV replaced with a stochastic dominance relation under the distribution of transaction prices from auctions with ticular Number Values Proposition 9 relied on a recurrence relation between distributions of order statistics of bids within the relation PV model is Common n— 1 model. For example, the bidders stochastically dominates a par- A convex combination of bid distributions from n-bidder auctions. similar result holds for the ascending auction, where a recurrence relation between means of order statistics from samples of exchangeable the CV random hypothesis. variables implies a strict ordering of the Both same combination from the winner's curse, which results arise is of means under more severe in auctions with larger numbers of bidders. Proposition 13 The with m >2 bidders Proof. Assume m. CV model is and testable if bids B^^~^''^\ =n— I (the . . argument If = the transaction price B^"^~^''^> in auctions ,ij("~i-") in auctions with . auction, where (recalling (1)) each player bi we observe is m bidders. similar for other cases). First consider a second- price bids i = maxXj = E\U^\X^ n > xi] = b(x,;n). instead of b{-;n), the bidders used a symmetric monotonic bidding strategy that did not depend on n, (4) would imply - Pr(5("-2^") <b) + "^^—-^ Pr(B("-i^") n would be since bids fixed n < 6) = Pr(B"-2^"-^ monotonic transformations of exchangeable signals, were also exchangeable. However, when the equilibrium strategy b{-;n) loss of generality h) (7) implying that the bids used, taking z = 1 without and exploiting exchangeability, we have 6(xi;n) with the is < = E[U\\Xi= X2-=xi,Xj <xi, j = ?,,..., n] < E[Ui\Xi=X2 = xuXj<Xi, j = 3,...,n-l] = b{xi;n strict inequality following — (8) 1) from the fact that E[Ui\Xi, . . . ,X„] strictly increases in each Xj, due to strict affiliation of {Ui,Xi). Hence, 6(xi;n) strictly decreases in n, implying the testable stochastic dominance relation - Pr(B("-2^") < n 6) + ^^—^ p^/^in-l-n) < ^w p n 21 ,^n-2:n-l < ^x /g) instead of (7). For an ascending auction, of generality, suppose it is fix a realization of {Xi, bidder 2 who — bidders have these signals in an (n . . who has 1 l)-bidder auction, bidders 1 and 2. 3, . . . . . ,n - = is n will E [U2\Xn = Xi = X2 = X,Xj = Xj, j = 3, x\x_n)E [E . . . ,x„_i) is drop out before bidder 2's exit price having signal x. equilibrium) . . ,71 . - l] Xn, Xj=Xj,J=3,...,n-l]\ X, . . ,Xn-\. To see revealing x„. If x„ > y£ \b+ {X^'"^-''~^^ ; X3, . . . . , . . > X^""^'""^)) observe that all if x„ Xn<x] = x, bidder remaining bidders, including n, = in -EfS^^-^^")] Xn^l)] < drop out before bidder n, ,Xn-i gives the testable inequality L by exchangeabihty, Pr(JY'„ this, x, bidder 2 will based on an expectation that conditions on J since, 2, . Taking expectations over Xi, r^(n-2:n-l)l ^ 1 will (in the expected bid of bidder 2 in an auction where an additional bidder also included, given the realizations of .Yi, with = X2 = [U2 {Xy When 6+(x2;x3,...,x„_i). Here b'^{x2;xs, n — loss 1]] I + FT{Xn < Without the higher signal. ,n . x. Bidder 2 then drops out at price _ ^p^ \Xi=X2=X,Xj=XjJ = 3,...,n-l] = Ex„ [E [U2 Xn, Xi=X2 = X, X, = x„ J = 3, > Pv{Xn > X |X_„) = ,X„_i), with x^""^'""^^ has signal x and bidder drop out and reveal their signals before bidders ^(n-2:n-l) . restriction + ^^—^^[fi^""!^")] n (10) D ^. For the ascending auction, a test of the inequality (10) can be implemented using standard onesided hypothesis tests. Since (9) implies (10), the Of course, other following same test could be used for a sealed-bid auction. tests of (9) are possible, including direct tests of the stochastic McFadden dominance relation, (1989). Note that Propositions 9 and 13 imply that the observational equivalence between the and CV models noted variation in the in Li, Perrigne number and Vuong (1999) is eliminated when one CIPV observes exogenous of bidders. ^^ In addition, each of these propositions implies that the test proposed in the other has power against an important alternative. In particular, the requires that (9) and (10) hold with equality rather than inequahties. indicate that the inequality in (9) or (10) is Note PV alternative also that if the data reversed, this will suggest a violation of one of our more basic maintained assumptions. One bidders limitation of this result know the number Other approaches is that, unhke our models of private values, the assumption that of competitors they face plays an important role. for empirically distinguishing these models based on observation of auctions are given in Hendricks, Pinkse and Porter (1999) and Haile, 22 While Hong and Shum (2000). all this assumption bids at first-price does not seem severe in an ascending auction, may be it restrictive in a sealed-bid auction. We return to discuss the extent to which this assumption can be relaxed in Section 6.1. CV model. identification and testing of the pure CV model. Without some additional structure, we are unaware remainder of this section, we consider Thus, in the 4.2 Pure We begin First, Common of an approach to identify the Values by highlighting several issues that arise when attempting from an economic perspective, the scaling of the signals to identify the pure X is monotone transformations. Given 6(xj) in = maxXj = this normalization (recalling (1)), the = Xj. While the same x,n] is preserved under = x. (11) an ascending auction, all much more complex. Because bidder can be identified in change as the =x for all bidders will limit the set of data configurations under with any is However, equal to the distribution of X, and thus the distribution of if some We testing approaches developed for private values models, pursue to assume that ditional we ^* on V), not be independent in the = V + Ei Xi for CV some £, (where this additive structure Further, with pure = identified might hope to apply the identification and where the same problem may formulation. Further, although E common max^^i Bj values = bidders' first-order conditions. This is either independent of not survive the rescaling (11). consider two (nested) special cases of the pure estimates of E[V\B, is arises, and indeed, However, the assumptions required are somewhat more subtle. For example, this. X generally will X bids are unobserved, the distribution of observed bids provides in- formation only about certain order statistics of X. will model ascending auctions. distribution of bids we CV observed in a second-price auction, the normalization (11) implies that the joint If all bids are trivially. ^^ This this reason, beliefs auction progresses, no single normahzation can induce the simple strategy b{x) in the ascending auction. is higher bidders an ascending auction condition their bids on the drop-out points of lower bidders. For is bidder optimal bidding strategy in a second-price auction logic applies to the lowest bid in identification in ascending auctions is, a particularly salient normalization of signals satisfies this, E[V\X^ With model. That arbitrary. behavior and utility depends on the information content of the signal, and that CV CV may be natural or independent con- To address this concern, model:^^ and observability of ex post values, the bi,n\ based directly on the observed bids and is V it model can be tested by comparing V to those inferred from bids using the analog of the approach followed in Hendricks, Pinske, and Porter (1999) for first-price auctions. Each of these is receive signals of the a special case of the mineral rights model described in Milgrom and common value v that are independent conditional on v. 23 Weber (1982), where bidders Linear Mineral Rights (LMR): stants M {C,D) € Ui — V. In addition, R^ and random X variables {E\,. {V E,)^ ther, conditional LMR with . . with {E,V,X) non-negative; or Vz, on V, the elements of E ,En) with joint distribution Fe(-) = maxj^iXj = such that, with the normalization E'fyjXj exp(C) each n there exist two known con- for x, either = C + D{V + X^ (ii) = x, n] (i) Xi = E^) Vz. Fur- are mutually independent. Independent Components (LMR-I): In the LMR model, (V^,E) are mutually in- dependent. The LMR model is analogous to the CIPV model, is analogous to CIPV- However, the functional form assumptions are potentially more demanding I.^'^ Perrigne, and Vuong (1999) show that several plausible models satisfy LMR definition is satisfied C+ To D\n{yi). bidders see signals (Yi if known constants dent, and there exist it LMR-I model while the see this, let X, = has the same information content. C= In \ Fr^7iF E[t^^ \^i 1 Similarly, the following are and {V, We and F — max^^j Ej\ (C, L>) G M , , . . exp(C) -Yf; since The D— y„), where Yi x R4. such that LMR-I. Condition =V E^, (^, In (£'[y|yi 1] (b) there are = (i) <x two bidders, and Li, in the E) are indepen- maxj^j Yj = monotone transformation this is a following are examples: (a) f{v) yi]) = of Yi, ^^ ^ ^R^ whereby (V, E) are log-normal. j examples of case (ii); (a) the prior /v(-) (b) there are is flat; two bidders, E) are normally distributed. LMR-I model. Our main begin by analyzing identification and testing in the that our results for the CIPV-I model extend to the applies only to second-price auctions where Proposition 14 In V'a, (") . in this case. of the auction, the the random LMR-I variables LMR-I model Proof. Consider case from which we can (ii) infer f e,. 5 then applies directly. Case (i) as with Proposition 5, the result bids are observed. all and Ai are nonvanishing. is identified -I- shows model, assume that for alii the characteristic functions ipy{-) and V of the LMR-I model; result and If all bids are observed in a second-price testable. LMR-I model. By Hence, each bid follows in the (1), ijf-? ") each bidder reveals same manner V z's -\- bid bi is £(•?"). equal to C-\-D{v-\-e^), The proof of Proposition after taking logarithms. D Observing information about the ex post value of the object enables us to identify a more general pure common values model. prove identification of the fairly In particular, our results for the LMR CIPV model model. However, in the case of ascending auctions our results are weak, requiring that the lowest bid be observed. This restriction '" Identification of the See also Paarsch (1992a), can be generalized to is LMR-I model at first-price auctions has been considerd by Li, who considers several parametric specifications of the mineral 24 required because after Perrigne and Vuong (1999). rights model. the bidder drops out, the strategies of remaining bidders depend on B^^''^\ implying that no first normahzation yields bid functions equal to the identity function. Proposition 15 Consider (i) LMR the model. Suppose that the ex post realization If the transaction price is observed in a second-price auction, F^{-\v) to the normalization (11). If (a) If the lowest bid is one additional bid ofV and Fv{-) are is identified observed in each auction, the model is observed, up is testable, observed in either a second-price or ascending auction, Fe{-\v) and Fy{-) are identified up to the normalization (11). Proof. Consider case C + D(f + Ci), reveals v + (ii) in the definition of the from which we can e^""-^''^^ infer v -\- Ci. LMR model. By each bid (1), hi is equal to Hence, the transaction price in a sealed bid auction while the lowest bid in either type of auction reveals v + e^^'"). In either case, observation of v reveals one order statistic of E, so identification of Fe(-) and testing then follows directly from the arguments of Proposition First-Price Auctions 5 Most we when Some Bids are Unobserved auctions can be extended to first-price auctions, although in some require that the second-highest bid be observed in addition to the transaction price. well An PV of our results for D 6. known, the first-price auction important difference winning bid is cases As is and descending (Dutch) auction are strategically equivalent. for the econometrician, however, is that in the descending auction, the the only bid made. Hence our results below for the first-price auction that require only the transaction price apply directly to the descending auction. ^^ affiliated values in this section We will restrict attention to because this assumption guarantees monotonicity of the equilibrium bidding strategies; however, the assumption is otherwise unrelated to our identification arguments. However, here we do make the additional assumptions that Fa{-) and Fx(-) are strictly increasing on their respective supports and have associated joint densities /a(-) and /x(0- A bidder in a first-price auction views the bids of his opponents as bidder i observes X^ = max I Xi, random variables. When he solves E[Ut\Xi = Xi,ma:x.Bj < hi] — 5, | Pr(maxBj < bi\Xi = Xi). Define C(x; n) ^' Laffont and Vuong = E\Ui\Xi (1993) and Elyakime et al. = x, max JCj = x] (1994) address parametric identification and estimation of the descending auction model. 25 Then bi H first-order condition implies that a necessary condition for i's —^~ '"^"'^^Jt^' 3 '~_ '' = By standard arguments, (^{xi;n). bidding strategies are strictly increasing.''^ Thus, an equilibrium, bidder i's bid Bi has the the bids {Bi, . . — — < + ^"B^ R ^ < -^FT{ma.XJ^^Bj = h) = rv\ = b^)\^^^^^ is an equilibrium to this model, in same information content PT{maXjJ^^ Bj . ^' if to be an optimal bid bi . ,Bn) are generated from such as Xi, so that b^\Bi lo z\Bi (12) Ci^^^^) This equation, expressing the latent expectation C,{xi;n) in terms of observable bids, has been widely exploited in the literature; Elyakime et Vuong Perrigne, and for the asymmetric (1998) for the affiliated PV and Porter (1999) and models of pure When is all CV model PV is identified In the symmetric (1995) established that Fu{-) (6i) " ) , another approach If bidders are model Li, for a survey). Hendricks, Pinkse, approach to identify and estimate this is is B^; < b\Bi = z) from the distribution of bids through (12) (Laffont and Vuong IPV model Vuong of first-price auctions, Guerre, Perrigne, and also identified so the left-hand side of (12) is from the transaction price alone, since the trans- = {HB{b))"'. Thus, Pr(maxj^ji?j identified. < bi\Bi = bi) = However, when bidders are asymmetric, required. Proposition 16 Consider (i) IPV model, model, and Campo, Perrigne and Vuong (2000) and Vuong (1999) use action price identifies the distribution i^g " (6) " to analyze the auctions. observed, so Fu(-) //g it bids are observed (and bidder identities, where relevant), Pr(maxjyj (1993, 1996)). ( used first and Vuong (1999) (see Perrigne Li, Perrigne, (1994) al. the IPV model of the first-price auction. symmetric and the transaction price is observed, then Fu{-) is identified and the is testable. (a) If bidders are asymmetric and the transaction price and the identity of the winner are observed, then each Proof, (12) Fjj^ (•) is identified (i) (ii) is testable. Follows from Guerre, Perrigne, and must be restriction and the model Vuong (1995), Corollary strictly increasing in equilibrium (Laffont on the estimate of We observe the distribution of the transaction price, Pr(i?("-") In asymmetric first-price auction models, existence of equilibrium A left-hand side of this function. is is sufficient condition for such monotonicity in <b),ss well as the identity of a strictly increasing function, and not straightforward. Athey (2000) shows that as long as each bidder's best response to nondecreasing opponent strategies exists. The and Vuong (1996)), providing a testable the winning bidder, /(":"). In equihbrium, each bidder's strategy Nash equihbrium 2. a PV is itself nondecreasing, a pure strategy auction is affiliation. In CV models, the conditions required for monotone best responses are more stringent when there are more than two asymmetric bidders, but Athey (2000) shows that they are satisfied in the 26 LMR model. the bids are independent. Let Hb,{) be the marginal distribution of Bi. 2 imphes that the distributions HBi{-) are for each bidder < Pr(max,^j Bj b). Then From identified. (12) and the The proof these distributions of Proposition we can compute joint distribution of (b("^"),/("^")) identify the joint distribution of ([/("-"^ /("")). Applying Proposition 2 yields identification of each Testing follows as in (i). Note that the assumptions metric PV F[/. (•). in part (ii) contrast with existing results for identification of asym- models (Laffont and Vuong (1996)), which have relied mentioned above, Dutch auctions are strategically equivalent to feature that only the transaction price tions as well as first-price auctions be adapted from those in, e.g.. first-price auctions, all is bids. As and have the observable; thus, this result will be useful for where only the winning bid Dutch auc- recorded. Estimation strategies can Guerre, Perrigne, and Vuong (2000). Note that the tests described may in the proof of Proposition 16 is on observation of not be very powerful —one can easily construct non-equilibrium bidding rules leading to bids that satisfy the required monotonicity. However, additional tests are possible when the top two bids are observed. For example, with symmetric bidders, the approach taken in Proposition 16 can then be applied, replacing 5("-") with B("~^-"), yielding a second timate of Fu{-). equivalent. This estimate and that based on the transaction price must be asymptotically ^^ Although most bids, there may be of the top two auction data sets either contain only the transaction price or else first-price example, in procurement auctions, the top bidder bids. For is second-highest bid as "money left on the relevant to strategic bidding behavior. it is table," approach can be applied when Proposition 17 Consider (i) ^^ between the winning bid and the which has intuitive appeal as information that The next result all model. Of course, the same testing bids are observed. the affiliated and PV PV model of the first-price auction, and suppose that the 5("~^-"-)j are observed. If bidders are symmetric, then the joint distribution of ({7("-"), t/^""^'")) is identified, The case in which the lowest bid ^(""i-")) (fit"-"' top two bids iH'^~'^'"\h) + and s^H^"^"'(6) apply (12) to B<"^"' and be applied to is shows that using observations of the top her value, as in (12). This allows us to test the affiliated (^i?("-"-) disquahfied or compute the equilibrium "markdown," or gap between a bidder's bid and possible to highest two bids may be awarded, and the second-highest bidder might receive the contract instead. Further, auction participants often refer to the difference bids, all reasons for a real- world auctioneer (or auction participants) to maintain records default before the contract two es- and bidder also allows for = t/<"^"'; identity are observed is analogous. the Note that observing the an alternative estimation strategies, based on the the relation ;/^"-^="-''(6), which follows from exchangeability. then, from the observed .ff^"^"^(-) infer the distribution of Ui and from the distribution of 27 we Given H^""'^""''(6), we can infer the distribution of f/''"^"^ Finally, (3) U^"""'' can model is testable. If bidders are (ii) asymmetric and the identity of the winner distribution 0/ ([/("""^ t/^"~^-"\/("'"^) Proof, (i) Let H^{bi, . . . ,bn) and is identified, the (/("•")) is also observed, model be the joint distribution of the then the joint is testable. Taking bids. i = without loss of I generaUty ^ Pr(maxj^i B^ < bi\Bi = 61) ^ Pr(maxj^i Bj < x\Bi = b^)]^^^^ Pr(maxj^i Bj dxdy The first x, 92 dxdy x=y=z=bi Pl.(5(n-l;n) <X S(" .n) < y) x=y=bi equality follows by Bayes' rule, canceling terms from the numerator and the denominator. The second equality follows by the definitions and exchangeability; the third equality follows from differentiation and exchangeability. Since the joint distribution of (^("-i-")^^!"-")) assumption, identification of the distribution of ((/(""), [/(""' ")) follows the left-hand side of (12) must be increasing, which (ii) Bi<y) ^Pr(i?(- -) < -)x==61 92 &UaI^B(y,z,x,...,a:) Bi <y) x=y=bi y=bi 1) 61 y=bi Pr(maxj#i Bj < ^HB{y,bi,...,bi] ("~ < Extending the from logic is from <x, Bi< ^Hniy,bi,...,bi] y) y=bi 92 y) Ej#ia^a5--^B(y,22,---,2n] x=y=bi £ Pr(B("-")<x, dxdy (12). In equilibrium a testable restriction, y=bi a2 observable by now we have (i), ^Pr(maxj^aBj<5i, B^ < dxdy Pr(maxj-^i Bj is Pr(B("-i-") < /("-") x, J5("-") < = Z2 = 1)1^^^^ /("") y, = 1) x=y=bi Identification follows from (12) since the last Proposition 17 establishes that we can term above is observable; testing follows as in identify the joint distribution of the top (i). El two bidder some important policy simulations, including evaluation of alternative reserve prices. This also allows us to apply many valuations. Although this is not enough to identify F\j{-), of the results derived in Section Propositions 8, 6, 7, and Corollary 2 Consider 3. The it is sufficient for following result follows from Propositions 17, together with 11. the affiliated PV model of the first-price auction and supposesuppose that the highest two bids are observed. If bidders are allowed to be asymmetric, winner is observed as well. Then: 28 assume identity of the (i) Fu(0 not identified is (ii) Under (Hi) IfW^ beloiu the highest observed, IS — M {W\, > . . . (A,Wo) are independent, = are observed, and Ui ,Wn) Proposition 11, then each i gi{-), andUi then go{-) and Fa{-) zs differentiable, = two are unobserved.^'^ CIPV model, ifV the additional restrictions of the go:su-pp\\VQ\ (iv) If any bids if ^j(VFi) l,...,n, = is observed, then -Fa(') ^^ identified. gQ{WQ) + Ai, where the unknown function cltc + Ai, identified up to a location normalization. then under the additional restrictions of is identified, and Fa{-) is up identified to a location normalization. Further, Proposition 17 implies that the top two bids in a first-price auction give us enough PV information to test the Corollary 3 Consider model against the PV model the affiliated two bids are observed in an auction with n — auction with n (i) If (ii) CV > alternative. of the first-price auction, and suppose that the top 3 bidders, while the highest bid is observed in an bidders. 1 bidders are symmetric, the model can be tested against the If bidders are permitted to be asymmetric, the model can CV alternative. be tested if, in addition, we observe the bidder identities corresponding to each of the observed bids. Proof, (i) jj{n-i:n) ^ By Proposition 17, this information allows us to identify the distributions of [/("-i-"-i), recurrence relation Q{x\m) is PV and [/("") under the (4). Now CV consider the symmetric strictly increasing in x, so the random alternative. _ ^^ with the inequality following from unobserved this follows [j < n— 1), distribution of bids Gb(-)- = Pr (ciX^^-^^^-^); n) (8). from Proposition and Now /3 This inequality 8, is suppose that the true distribution of utihties this, observe that when player optimization problem. Since this joint distribution statistic. is Since the jth bid Proposition 8 does not impose is i < n- z) 1) < 2) (13) is F\j{-), the jth bid Fu() imply a Fu(-) and F\jO induce the same two order /3 8. But then, the bidding functions statistics of the value distribution affect her and F\]{-), player i's best affiliation; as is response bid for together with the value distribution Fu(-) imply a and Gn{-) induce the same distributions of order unobserved, Gb(-) observationally equivalent to statistics except for the Gb(). Finally, recall that discussed in footnote 21, the result can be generalized to that case. 29 /3 computes her best response to opponent bidding identical for J^u(-) unchanged. Because the bidding functions distribution over bids, Gb(-)i where Gb(-) Jth order and the set of equihbrium bidding functions. These together with is strategies /3_;, only the joint distribution of the highest is fc(X("-i^"-^); take another value distribution, F\}{-), such that form an equilibrium. To see each value of u, ar- testable. distributions of order statistics except for the jth; this exists by Proposition still The (,{X^^''^^;n), (^(X^"~-^'"^;n), > Pr ''To see how standard arguments, are identified. Then, (3) implies ^^—^Pr('c(X("^");n) <2) +ipr('c(X("~^^"^n) <2) is By satisfy the variables C,{Xi;n) are exchangeable. guments of Proposition 17 then imply that the distributions of ^(^(n-i:n-i).^ must hypothesis, in which case these distributions Using a similar argument, (ii) Pr ('[/(«-i^"-i) < /("-i^'^-i) b, = !?—1 Pr =i\ +- Pr The terms < ('[/("^") /("^") b, ft/("-^^") in this expression are identified (by Proposition 17), so <6,/("-i^'^ z— 1;ti) D equahty can be tested. Extensions 6 Bidder Uncertainty Over the 6.1 6.1.1 of Opponents Second-Price Auctions As pointed out above, our competitors they analysis of the CV models assumed that bidders know the number of may be In an ascending auction, this assumption face. auction, bidders might not know the number as long as bidders observe CV Number an informative signal model can be extended. immediately, as long as (1) of competitors ry Our rj when submitting n in the their bids.'^'^ However, of the level of competition, our results for the identification results for the replaces natural. In a sealed-bid LMR definition, LMR and and LMR-I models extend (2) in the case of a first-price auction, the econometrician can distinguish between auctions with different realizations of rj?^ following result shows that the testing approach of Proposition 13 can Proposition 18 In a public signal the and CV model rj Proof. Let argument is is strictly affiliated is testable if . 7r(r;|7i) . be applied. the second-price sealed-bid auction, suppose bidders do not that bids B^'^~^''^\ still . we observe with n (with rj unobserved n> m hi = En Given signal E[Ui\Xi ry, each player = maxXj = but observe Then m auctions with m bidders bidders. denote the conditional distribution of the signal similar for other cases). known to the econometrician). the transaction price B^'^~^'^' ,5("~i-") in auctions with The Xi = i rj. Assume m = n— \ (the bids b{xi;T]). 3+i '^ Matthews (1987) and McAfee and McMillan (1987b) provide number of bidders. theoretical analyses of auctions with a stochaistic Hendricks, Pinkse, and Porter (1999), for example, assume that bidders have the same beliefs about the number of competitors in all auctions at which the number of participating bidders falls in a certain range. 30 Taking z = 1, > the inequality (8) and strict affiliation imply that for ^ = b{xi;f]) = X2 = 77 Xj <xi, j = 3,..., n]\f]] < En[E[Ui\Xi = X2 = xuX, < xu j = 3,...,n]|r?] En[E[Ui\Xi xi, Since /oo Pr(6(X("-2^");ry) <6)d7r(r?|n) -00 strict affiliation implies /CO r?) < b)dTr{r]\n - 1). (14) Pr(6(X("-i^");77) < 6) fi7r(7?|n - 1). (15) Pr(6(X("-2^"); -00 Similarly, /oo -00 Using (4) and the fact that b{-;ri) is strictly increasing, we obtain the testable stochastic dominance relation /oo < Pr(S(X("-2-"-i);77) b)dTv{r]\n - 1) -00 /oo r 9 - Pr(6(X("-2^"); 77) < -00 77—9 -— ^ Pr(6(X("-i^"); t?) < " [" n 6) d7r(77|n - 1) J n where the inequality follows from 6.1.2 + 6) (14) and D (15). First-Price Auctions Bidder uncertainty over the number of opponents is a potentially case of a first-price auction, since bidding strategies will depend on private values, after observing signal max t] {ui each bidder — b) i more ij, difficult even in a problem PV auction. in the With solves Pr(max Bj < b\Ui = Ui^rf) j^i b giving first-order condition PT{ma.Xj^,Bj <b^\B^ "i + r) -^ ~^:. ^ Pr(maxj^, Bj < / rr; x\B^ 31 = bi,T]) n = = bi,r])\^^^^ ; — ""' (1^) If the econometrician observes a set of auctions in which and vahiations can be used the symmetric IPV in essentially the same way that winning bid case, observation of the t] (12) and Tj letting i?^'" denote the 00 00 n=2 n=2 identify Fu{-), n such common 7r(n|7/) Testing of the different values of values > 7r(n|77) PV we recover the in auctions many initial bi\r]), which b\T]), is rj is observed. equal to observed directly, Pr{Bi < b\r]) is giving (through (16)) identification the distribution of (7^"'"', (3) can be used to D F\j{-). models of PV first-price auctions can be extended hypothesis, these distributions will be identical; with distribution oi E\Ui\X^^''^'> where signal is < signal 77 = 772 > = maxfc^iXfc is observed 77 = r/j r/i is observed. = Xj,//] rather than that of will first-order stochastically auctions the seller announces a reserve price for the auction, which can be viewed as an bid entered by the of bidders' valuations. probability some For simplicity we have assumed reserve price seller. When the reserve price tq is the reserve price, ro, is If only IPV model fixed and If either (a) = i_jr that Fu{ro) ° (^ This creates a discrepancy results n. However, can be extended. '^^ of first-price, second-price, or ascending auctions, suppose that the one bid from each auction denoted F[/(-|ro) below the support and the number of participating bidders, A'', by conditioning on the participation threshold, many of our Corollary 4 In lies in the interior of the support, with positive potential bidders will be unwilling to submit a bid. between the number of potential bidders. (ii) n bidders when sufficient to is still 77 Reserve Prices 6.2 (i) with fixed in hypothesis can be achieved by comparing distributions of under the 77: < i?^ From 0. This distribution in auctions where signal dominate that In and identification results for other private values fQj. ^(j;n-)_ that which completely characterizes in similar fashion. jjii-n) b\r]) This enables construction of Pr(maxj^2 of [/("") for each Our < between bids was used above. For example, winning bid, we observe Pr(i?™^" Since (17) strictly increases in Pr(i?i identified. fixed, this relation in auctions identify Fu{-). Let Tr(n|r/) denote the probability of there being Fixing is , is is E (0, 1). observed, the distribution of Ui conditional on Ui identified on > ro, [ro,oo]. two bids from each auction are observed or (b) a single bid is observed in auctions For first-price auctions, Bajari and Hortagsu (2000) use a parametric model to simultaneously estimate a model of entry and bidding. Guerre, Perrigne, and Vuong (2000) includes a discussion of binding reserve prices in the IPV model. See also the recent paper by Li (2000). Hendricks, Pinkse and Porter (1999) estimate a model of stochastic participation and reserve prices in common value auctions. 32 IPV with different numbers of participating bidders, the (Hi) If N Proof. is fixed and number of participating the Because each potential bidder the participating bidders' valuations 1 (i) is testable. observed, is when Parts Fu{-\ro). is > x, and made Similar extensions can be Corollary 5 In a fixed and PV and bids B^^'^' in all (n — Proof. Participation _BU+i-"-) and then follow from Propositions Because the = Fu{ro), both TV and D models considered above. The models can also be applied in this case.^^ (with 2 < Both the j < PV model and the CV model are n) in all rg, is testable n-bidder auctions and bids B^^''^~^' l)-bidder auctions. is determined as type to participate in equilibrium. the recurrence relation (4) under the CV still Milgrom and Weber in Participating bidders distribution Fx{-) truncated at [xq, (13) hold distribution of or ascending auction, suppose the reserve price, first-price, second-price, in the interior of the support of each Ui. we observe common fixed. A'' is CV -Ft/(ro) are identified. of Proposition 16. for identification of the other following result shows that our tests of the and binomial with parameter A is from the distribution of n when -F[/(^o) are identified N ro, the (ii) and 16 and the remarks regarding testing following the proof participation rule for each potential bidder if bidders participates i model . , . . xq). (1982). draw their types Because exchangeability holds under the PV Let xq denote the lowest is Xj, . . . ,Xn from the preserved by this truncation, hypothesis, while the inequalities (9), (10), and D hypothesis. Conclusion 7 This paper has established new identification results for standard auction models. For second-price and ascending auctions, our main for results can be summarized as follows (where the results apply symmetric bidders unless otherwise noted): 1. The independent if 2. private values more than one bid is model observed in is identified from the transaction price, and it is testable each auction. In a second-price auction, observing the winner's identity and bid identifies the asymmetric independent private values model. In an ascending auction, this model is identified if we observe either the lowest bid and bidder identity or the bids of the top two losing bidders and the identities of the top three bidders. With a binding reserve price, one can also test the PV hypothesis (^{xo;m) CV = ro against the alternative (^(xo; model (Milgrom and Weber (1982)). This testing approach has been proposed auctions by Hendricks, Pinkse, and Porter (1999). ro implied by the 33 m) > for first-price 3. In second-price auctions, but not ascending auctions, the "conditionally independent private model values" —where bidders' private values are the sum of a common component idiosyncratic component, and idiosyncratic common component — identified is asymmetric bidders 4. If if values model is if all if in addition, the common and bids are observed in each auction. idiosyncratic This extends to identities are observed as well. common component the components are independent conditional on the and testable components are independent, and identified of values is observed ex post, the conditionally independent private from the transaction price, and if two bids are observed, it is 5. The 6. Even with potentially asymmetric bidders, the unrestricted private values model unrestricted private values model with 7. The if private values testable. not identified from incomplete sets of bids. is from the transaction price (and testable when two bids are observed) specific covariates and an if is identified there are bidder- sufficient variation. model is testable against the common values alternative, and vice-versa, the transaction price and the next-highest bid are observed in two auction data sets with and n — 1 bidders, respectively (and the variation in participation is n exogenous). This holds even though these assumptions do not guarantee identification. 8. In second-price auctions, but not ascending auctions, the "linear mineral rights model" where bidders' expected values are linear in bidder signals, signals are linear in the common value and an idiosyncratic component, and idiosyncratic components are independent conditional is on the common component is identified and testable independent of the idiosyncratic component, and extends to asymmetric bidders 9. — In a second-price auction, model is identified if if the all if, in addition, the common value bids are observed in each auction. This identities are observed as well. common from the transaction is observed ex post, the linear mineral rights price. In an ascending auction, we must observe the value lowest bid rather than the transaction price. Now consider was established how these results extend to first-price auctions with affiliated signals. Result (1) for first-price auctions (8) were established by (5) extend directly to Li, Perrigne, by Guerre, Perrigne, and Vuong (1995), and results and Vuong (1999). In this paper, first-price auctions, while results (4), (6), (7), we show that and (9) extend (3) and results (2) and if the top two bids in each auction are observed. Most of the prior literature on structural estimation in ascending has focused on parametric identification strategies. 34 and second-price auctions For first-price auctions, the prior literature provides nonparametric identification results only IPV when all bids are observed, or in the symmetric model. Our results both indicate environments in which weaker identifying assumptions can be employed and suggest cases assumptions. Our in which identification can only be obtained via functional form results further suggest strategies for the design of new data researchers are able to collect different elements of auction data (typically at Increasingly, sets. some cost); this might be the case when obtaining data from internet auctions, as the researcher might choose whether to record the entire history of an auction as opposed to just information about the total of participants and the transaction price. Our results suggest that in many number environments, just the top two or three bids provide sufficient information for identification and/or testing. This paper has focused primarily on identification. In general, identification not sufficient for existence of a consistent estimator. estimation strategies, and for estimation approaches that however, we have left many may be Many of the identified models applicable. In is necessary but of our identification proofs suggest we have pointed to nonparametric most of these cases consistency is transparent; the derivation of the asymptotic properties of these and other potential estimators for future work, along with their application to bidding data. 35 References Arnold, Barry C, N. Balakrishnan, and H.N. Nagaraja (1992). New York: John Wiley & Sons. A First Course in Order Statistics. Athey, Susan (2000). "Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information," Econometrica, forthcoming. Athey, Susan and Jonathan Levin (2000). "Information and Competition in U.S. Forest Service Timber Auctions," Journal of Political Economy, forthcoming. Hajari, Patrick (1998). "Econometrics of the First Price Auction with working paper, Stanford. Asymmetric Bidders," Bajari, Patrick and Ali Hortagsu (2000). "Winner's Curse, Reserve Prices and Endogenous Entry: Empirical Insights from eBay Auctions," working paper, Stanford University. Balasubramanian, K. and N. Balakrishnan (1994). "Equivalence of Relations for Order Statistics for Exchangeable and Arbitrary Cases," Statistics and Probability Letters, 21, 405-407. Baldwin, Laura, Robert Marshall, and Jean-Francois Richard, 1997, "Bidder Collusion in U.S. Forest Service Timber Sales," Journal of Political Economy 105 (4), 657-99. (1963). "Note on Extreme Values, Competing Risks, and Semi-Markov ProAnnals of Mathematical Statistics, 34, 1104-1106. Berman, Simon M. cesses, Campo, Sandra, Isabelle Perrigne and Quang Vuong (2000). "Asymmetry in First-Price Auctions with Affiliated Private Values," working paper, USC. Chow, Yuan Shih and Henry Teicher Martingales, New (1997). Probability Theory: Independence, Interchangeabihty, York: Springer. Cox, David R. (1959). "The Analysis of Exponentially Distributed Lifetimes with Failures," Proceedings of the Royal Statistical Society, B, 148, 82-117. W. (1998). "New Zealand Spectrum Law and Economics, 41, 821-40. Crandall, Robert Journal of David, Herbert A. (1981). Deltas, George and Idranil Order Statistics, 2nd Chakraborty (1997). Policy: edition. New A Model for the York: John Wiley "A Two-Stage Approach metric Analysis of First-Price Auctions," working paper, University of Two Types of United States?" & Sons. to Structural EconoIllinois. Donald, Stephen G., Harry J. Paarsch, and Jacques Robert (1999). "Identification, Estimation, and Testing in Empirical Models of Sequential, Ascending-Price Auctions with Multi-Unit Demand: An Application to Siberian Timber-Export Permits," working paper. University of Iowa. Donald, Stephen G. and Harry J. Paarsch (1996). "Identification, Estimation, and Testing in Parametric Empirical Models of Auctions within the Independent Private Values Paradigm," Econometric Theory, 12, 517-567. Elyakime, Bernard, Jean-Jacques Laffont, Patrice Loisel, and Quang Vuong (1994). "First-Price Sealed-Bid Auctions with Secret Reserve Prices," Annales d'Economie et Statistiques, 34: 115-141. Engelbrecht-Wiggans, Richard, John List, and David Lucking- Reiley (1999). "Demand Reduction in Multi-unit Auctions with Varying Numbers of Bidders: Theory and Field Experiments," working paper, Vanderbilt University. Guerre, Emmanuel, Isabelle Perrigne and Quang Vuong (1995). "Nonparametric Estimation of First-Price Auctions," Working Paper, Economie et Sociologie Rurales, Toulouse, no. 95-14D. Guerre, Emmanuel, Isabelle Perrigne and Quang Vuong (2000). "Optimal Nonparametric Estimation of First-Price Auctions," Econometrica, 68, 525-574. Haile, Philip A. (2000). Timber "Auctions with Resale Markets: An Application to U.S. Forest Service American Economic Review, forthcoming. Sales," and Elie T. Tamer (2000). "Inference with an Incomplete Model of English Auctions," working paper. University of Wisconsin-Madison. Haile, Philip A. Phihp A., Han Hong, and Matthew Shum (2000). "Nonparametric Tests for Values in First-Price Auctions," working paper. University of Wisconsin-Madison. Haile, Common Han, Aaron and Jerry A. Hausman (1990). "Flexible Parametric Estimation of Duration and Competing Risk Models," Journal of Applied Econometrics, 5, 1-28. Heckman, James J. and Bo E. Honore (1989). "The Identifiabihty of the Competing Risks Model," Biometrika, 76, 325-330. Heckman, James J. and Bo E. Honore (1990). "The Empirical Content of the Roy Model," Econometrica, 58, 1121-1149. Hendricks, Kenneth and Harry J. Paarsch (1995) "A Survey of Recent Empirical Auctions," Canadian Journal of Economics, 28, 403-426. Work Concerning Hendricks, Kenneth, Joris Pinkse and Robert H. Porter (1999). "Empirical Implications of Equilibrium Bidding in First-Price, Symmetric, Common Value Auctions," working paper. Northwestern University. Hendricks, Kenneth and Robert H. Porter (1988). "An Empirical Study of an Auction with Asymmetric Information," American Economic Review, 78, 865-883. Hendricks, Kenneth and Robert H. Porter (2000). "Lectures on Auctions: tive," working paper. Northwestern University An Empirical Perspec- Hong, Han and Matthew Shum (1999). "Econometric Models of Ascending Auctions," working paper, Princeton University. Kotlarski, Ignacy (1966). "On Some Characterization of Probability Distributions in Hilbert Spaces," Annali di Matematica Pura ed Applicata, 74, 129-134. Laffont, Jean-Jacques (1997). "Game Theory and Empirical Economics: The Case of Auction Data," European Economic Review, 41, 1-35. Laffont, Jean- Jacques, Herve Ossard and Quang Vuong (1995). "Econometrics of First-Price Auctions," Econometrica, 63, 953-980. and Quang Vuong (1993). "Structural Econometric Analysis of Descending Auctions," European Economic Review, 37, 329-341. Laffont, Jean- Jacques and Quang Vuong (1996). "Structural Analysis of Auction Data," American Economic Review, Papers and Proceedings, 86, pp. 414-420. Laffont, Jean- Jacques Li, Tong, (2000), "Econometrics of First-Price Auctions with Binding Reservation Prices," Mimeo, Indiana University. Li, Tong, Isabelle Perrigne and Quang Vuong (1998). "Structural Estimation of the Affiliated private values Model with an Application to OCS Auctions," Mimeo, U.S.C. Li, Tong, Isabelle Perrigne and Quang Vuong (1999). "Conditionally Independent Private Information in OCS Wildcat Auctions," Mimeo, U.S.C. Li, Tong and Quang Vuong (1998). "Nonparametric Estimation of the Measurement Error Model Using Multiple Indicators." Journal of Multivariate Analysis, 65, 139-165. Lucking- Reiley, David (2000). "Auctions on the Internet: What's Being Auctioned, and How?" Journal of Industrial Economics^ forthcoming. Matthews, Steven (1987). "Comparing Auctions for Risk Averse Buyers: A Buyer's Point of View," Econometrica, 55, 633-646. McAfee, R. Preston and John McMillan (1987a): "Auctions and Bidding," Journal of Economic Literature, 25, 669-738. McAfee, R. Preston and John McMillan (1987b). "Auctions with a Stochastic Number of Bidders," Journal of Economic Theory, 43, 1-19. McAfee, R. Preston and John McMillan (1996). "Analyzing the Airwaves Auction," Journal of Economic Perspectives,!^, 159-175. McAfee, R. Preston, Wendy Takas, and Daniel R. Vincent (1999). "Tariffying Auctions," Journal of Economics, 30, 158-179. RAND McFadden, Daniel (1989). "Testing for Stochastic Dominance," in Studies in the Economics of Uncertainty: In Honor of Josef Hadar, New York: Springer. Milgrom, Paul R. (1985): "The Economics of Competitive Bidding: A Selective Survey," in L. Hurwicz, D. Schmeidler, and H. Sonnenschein, eds., Social Goals and Social Organization: Essays in Memory of Elisha Pazner, Cambridge: Cambridge University Press, 261-289. Milgrom, Paul R. and Robert J. Weber (1982). "A Theory of Auctions and Competitive Bidding," Econometrica, 50, 1089-1122. J. (1992a). "Deciding Between Common and Private Values Paradigms Models of Auctions," Journal of Econometrics. Paarsch, Harry ical Paarsch, Harry Timber J. in Empir- (1992b). "Empirical Models of Auctions and an Application to British Columbian Research Report 9212, University of Western Ontario. Sales," Perrigne, Isabelle and Quang Vuong (1999). "Structural Econometrics of First-Price Auctions: Survey," Canadian Journal of Agricultural Economics, forthcoming. A Prakasa Rao, B.L.S. (1992). Identifiahility in Stochastic Models: Characterization of Probability Distributions, San Diego: Academic Press . Romano, Joseph of the (1988). American "A Bootstrap Revival of Some Nonparametric Distance Tests," Journal Statistical Association, 83, pp. 698-708. Shaked, Moshe (1979). "Some Concepts of Positive Dependence for Bivariate Interchangeable Distributions," Annals of the Institute of Statistical Mathematics, 31, pp. 67-84. Tsiatis, Anastasios A. (1975). "A Nonidentifiability Aspect of the Problem Academy of Sciences, USA, 72, 20-22. of Competing Risks," Proceedings of the National Wilson, Robert (1992): "Strategic Analysis of Auctions," in R. Handbook of Game Theory, Volume 1, J. Aumann and S. Hart, eds.. Amsterdam: North-Holland. Wilson, Robert (1998): "Sequential Equilibria of Asymmetric Ascending Auctions: Lognormal Distributions," Economic Theory, 12, 433-440. 5236 The Case {]k2 of Date Due Lib-26-67 MIT LIBRARIES 3 9080 01917 6095