Identi…cation in First-Price and Dutch Auctions when the Number of Potential Bidders is Unobservable Artyom Shneyerov,¤ Adam Chi Leung Wongy
CIREQ, CIRANO and Department of Economics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada School of International Business Administration, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai, China 200433 September, 2008 This Version: December, 2009
Abstract Within the IPV paradigm, we show nonparametric identi…cation of model primitives for …rst-price and Dutch auctions with a binding reserve price and auction-speci…c, unobservable sets of potential bidders. Keywords: Nonparametric identi…cation, Auctions JEL Classi…cation Numbers: C14, D44.
1
Introduction
Identi…cation in auctions has been an active area of recent research in industrial organization. Beginning with the seminal contributions of Guerre et al. (2000) and Athey and Haile (2002), the literature has explored nonparametric identi…cation of a variety of auction models under progressively weaker assumptions on observables.1 We contribute to this literature by showing nonparametric identi…cation for …rst-price auctions with a binding reserve price where the set of potential bidders varies from auction to auction and is unobservable. Those potential bidders whose valuations are lower than the reserve price do not bid (enter). We assume independent private values (IPV). The model allows for ex-ante asymmetries among bidders. Speci…cally, we assume that bidders may belong to di¤erent groups.2 We assume that only auctions that have attracted at least ¤
Corresponding author. Tel.: +1 514 848 2424 ext 5288. Fax: +1 514 848 4536. E-mail addresses:
[email protected] (A. Shneyerov),
[email protected] (A.C.L. Wong) y The authors are grateful to two anonymous referees and the associate editor for their comments. 1 See also a recent book by Paarsch et al. (2006). 2 This approach is adopted in Athey et al. (2004), Flambard and Perrigne (2006), Krasnokutskaya and Seim (2009) and Hubbard and Paarsch (2008).
1
one actual bidder are observable.3 The objects we seek to identify are (a) the distribution of valuations for each bidder, over and above the reserve price, and (b) the distribution of the sets of potential bidders. We show that these objects are identi…able under conditions that are standard in the theoretical analyses of asymmetric auctions. As in Paarsch (1997), Athey et al. (2004), Song (2005), Li and Zheng (2009), Adams (2007) and Krasnokutskaya and Seim (2009), our basic identifying assumption is that bidders’ valuations do not depend on the set of potential bidders. This is a plausible assumption in many applications. For example, in highway procurement auctions, bidders must be prequali…ed to participate in the auction based on the ability to perform the work rather than on their costs.4 Another good example is the procurement of services and materials by the US Government Printing O¢ce (GPO), where bidders are invited to participate through rotating lists.5 To illustrate the idea of our identi…cation method, consider a symmetric setting. (We allow asymmetry in our analysis.) Notice that in one special case the number of potential bidders is observable. This case occurs when the number of actual bidders is maximal, = ¹ . In this case, we can identify the distribution of valuations conditional on entry using standard methods, as in Guerre et al. (2000). Thus the identi…cation of this distribution is not a fundamental problem. But the fundamental parameter of interest is the unconditional distribution of bidders’ valuations. This requires the identi…cation of the entry probability (the probability that the bidder’s valuation exceeds ). As well, the identi…cation of the distribution of is desirable. Our trick is to notice that, when the ¡ ¢ number of actual ¹ ¡ 1 is a mixture of two bidders is = ¹ ¡ 1 so that the distribution of bids ¢j = components. The …rst component is the distribution of bids conditional on the number ¹ and the second is the distribution of bids conditional on of potential bidders = , ¹ ¡ 1. The mixture weights are the probabilities of = ¹ and = ¹ ¡ 1, = ¹ conditional on the number of actual bidders = ¡ 1. Using a theoretical result that the upper bounds of bid supports are ordered (also proved in the paper), we show that these mixture weights are identi…ed. They in turn identify the entry probability for every bidder. Once this is shown, we are able to identify the distribution of , despite the fact that only the auctions that have attracted at least one actual bidder are observable. Combining our approach with the methods in Berman (1963), and in Athey and Haile (2002), we extend our results to Dutch auctions where only the winning bids are observable. One might wonder if our approach hinges on the theoretically valid, but empirically problematic use of the bids within a maximal set and its adjacent sets. In this sense, our results would be similar in spirit to "identi…cation at in…nity" arguments whose limitations are well known. To address this limitation, we also prove identi…cation without the use of a maximal set, but under a slightly stronger, but still highly plausible set of assumptions. We prove the identi…cation of bidders’ valuations assuming that bid observations are only available for three adjacent sets, none of which is required to be maximal. It turns out that our identi…cation problem is closely related to the nonparametric identi…cation of mixtures, recently studied in Kasahara and Shimotsu (2008b) and Kasahara and Shimotsu (2008a), and we make use of some of the methods developed in those papers. 3
See Hendricks and Porter (2007) for a discussion of the empirical relevance of this assumption. Krasnokutskaya and Seim (2009). 5 See http://www.gpo.gov:80/pdfs/vendors/sfas/ppr.pdf for a description of GPO auction rules. 4
2
Hu and Shum (2009), in a paper that is closely related and was concurrently written, consider identi…cation and estimation of a model similar to ours. The main di¤erence is that they allow the distribution of valuations to depend on the number of potential bidders. (Another di¤erence is that they restrict attention to a symmetric model.) They show that identi…cation nevertheless obtains provided an instrument is available that exogenously determines the number of potential bidders.6 Their methods are based on recent results in the literature on misclassi…ed regressors and are di¤erent from ours. Several other papers in the empirical auction literature are related to our paper. Paarsch (1997), in his study of the Small Business Forest Enterprise Program (SBFEP) in British Columbia, estimates that the average number of actual bidders is about 3.29. Due to nonparticipation caused by a binding reserve price, the number of potential bidders exceeds the number of actual bidders. But if one uses a crude measure of the number of potential bidders such as the number of …rms registered in the district of the auction, the number of potential bidders could be as high as 185. Clearly, with this measure, one would substantially overestimate the level of potential competition in the majority of auctions. Paarsch (1997) adopts a clever parametric estimation strategy that is based on conditional likelihood and eliminates the need to estimate the number of potential bidders. However, his approach is limited to ascending-bid (English) auctions. Song (2005) and Adams (2007) consider identi…cation and estimation of eBay auctions with an unknown number of potential bidders. Their methods are tailored for eBay auctions and are entirely di¤erent from ours. Song (2005) shows that the joint distribution of any two order statistics identi…es the parent distribution. She then applies this result to eBay auctions, by arguing that in equilibrium, the second and third highest bidders bid truthfully. She develops a nonparametric estimator based on her identi…cation result. Adams (2007) shows that, under certain additional assumptions, observing just the transaction price is su¢cient for identi…cation. Most of the papers that estimated …rst-price auctions approached the measurement of potential competition empirically. In some cases, such a measure is readily available. For example, in highway procurement auctions conducted by state departments of transportation, the list of eligible …rms is sometimes publicly released and can serve as a good proxy for potential competition (e.g. Li and Zheng (2009), Krasnokutskaya and Seim (2009) and Marmer et al. (2007)). In other cases, researchers have used geographic proximity as a basis for …rm inclusion in the set of potential bidders (Athey et al. (2004), Hendricks et al. (2003)). Since the structural auction estimates are sensitive to the measure of potential competition (Hendricks and Porter (2007)), another approach is to treat the number of potential bidders as a parameter to be estimated, as in La¤ont et al. (1995). Ideally, this parameter would be auction speci…c, so a model for potential competition would be estimated jointly with the model of bidding. Nonparametric identi…cation of the entire model is necessary as a foundation for such an approach, and our results provide such a foundation. 6
After the revision work on this paper was completed, we have become aware of a new version of Hu and Shum (2009) where identi…cation is also shown without the instrument. More exactly, a second bid in the auction may serve this purpose.
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2
The model
We consider IPV …rst-price auctions. Bidders are ex-ante asymmetric: we assume that there are groups of bidders. Within each group the bidders draw valuations from the same distribution , but the distributions may be di¤erent across the groups. The set of groups is denoted as M ´ f1 2 ¢ ¢ ¢ g . The number of potential bidders in group is denoted as , and we sometimes use the notation ´ (1 ¢ ¢ ¢ ). We refer to such an auction as -auction. Our most important identifying assumption is that the distribution of valuations does not depend on the composition of bidder groups. (In the symmetric case, this is equivalent to the requirement that the distribution of bidders’ valuations does not depend on the number of potential bidders.) Assumption 1 The distributions of bidders’ valuations do not depend on , i.e. 8 0 2 0 0 Z + with 0 we have (j) = (j ) ´ (). This assumption rules out cases when the decision to become a potential bidder is correlated with the would-be bidder’s valuation, for example. We assume that each distribution has the same support, denoted as [ ¹], is di¤erentiable on the support, and has density which is bounded away from zero on its support.7 The set , the distributions (¢j), and the reserve price are assumed to be commonly known to the bidders. In this setting, Maskin and Riley (2000) and Lebrun (1999) have shown existence and uniqueness of Bayesian-Nash equilibrium bidding strategies (¢j ).8 These results imply that bidders from the same group must use identical bidding strategies. The support of is denoted as , i.e. () 0 if and only if 2 . Assumption 2 For every group 2 M, there exists some 2 such that ¸ 2. More succinctly, [ 2 f : ¸ 2g = M. Without this assumption, we cannot guarantee that equilibrium bidding strategies are strictly increasing on [ ¹], at least in some auction, for all groups, so that identi…cation of () for 2 [ ¹] might fail.9 Nonparticipation in the auction is due to the existence of a binding reserve price 2 ( ¹). We assume that the numbers of potential bidders in each auction are unobservable: only the bidders with valuations at least as high as the reserve price submit serious bids. We treat non-serious bids as uninformative and ignore them. From now on, it will be assumed that every bidder submits a bid only if his valuation is at least , thereby becoming an actual bidder. The decision to become an actual bidder is called the entry decision. Only the auctions that have attracted at least one actual bidder are assumed to be registered in the dataset. Assumption 3 The identities of bidders and their bids in each auction are observable by the econometrician. The reserve price is also observable and constant across auctions. 7 Identical supports is a standard assumption in the theoretical literature on asymmetric auctions. See e.g. Lebrun (1999). Little is known in general about the existence and uniqueness of equilibrium without this assumption. Also, if the supports are not identical, there exist reserve prices for which the low types are always screened and the identi…cation argument would not go through in general. 8 See also Bajari (2001). 9 See Lebrun (1999) and our Appendix for details.
4
This assumption implies that the vector ´ (1 ¢ ¢ ¢ ) of the numbers of actual P bidders in each group, is observable if =1 0. Denote the C.D.F. of bids from a group bidder, conditional on entry and the vector of potential bidders , as ¤ (¢j) ( 0). From the econometrician’s point of view, is randomly drawn from some probability distribution and is unobservable. In other words, is treated as an auction-speci…c e¤ect. Since is unobservable, the data do not reveal this C.D.F. They only reveal the C.D.F. of bids conditional on the numbers of actual bidders (j) ( 0). A bidder from group becomes active if ¸ , i.e. with probability 1 ¡ (). Since bidders draw their valuations independently, the distribution of conditional on is multinomial, with probabilities (j) =
¶ µ Y =1
[1 ¡ ()] [ ()] ¡
( · )
(1)
These probabilities are not observable. The marginal probabilities of are X () = ( ) (j) 2
Since the econometrician only observes the auctions with at least one active bidder, the marginal probabilities are also unobservable; only the conditional probabilities à ! X () ¤ () = 0 (2) 1 ¡ (0) =1
are observable.
3
Main results
The primitives that we seek to identify are (¢) for every 2 M, and ( ) for every 2 . Before we turn to our results, consider the case when is observable. Then the distribution ¤ (¢j) and the ( ) are also observable, and we can identify () from e.g. Pr f = 1j g = (1 ¡ ()) () ¡1 The distributions (j ¸ ) can be identi…ed from …rst-order equilibrium conditions following the approach of Guerre et al. (2000).10 Denote inverse bidding strategies as (j). If and 0, the inverse bidding strategies (j) can be found from the …rst-order conditions11 8 9¡1
10 11
See also the discussion in Athey and Haile (2005). For the derivation of (3), see Appendix.
5
where ( (j)) = (1 ¡ ()) ¤ (j ) + () for every 2 M This leads to the identi…cation of group ’s bidding strategy (j ) for , and consequently of the distribution of valuations (j ¸ ) = ¤ ( (j ) j). Since () is identi…able, so is () for ¸ : () = [1 ¡ ()] (j ¸ ) + ()
( )
When is unobservable, the distributions ¤ (¢j ) are in general also unobservable, but there are special cases in which they are observable. Let ¹ be the set of maximal elements of , i.e.12 © ª ¹ 2 : @ 2 s.t. ¹ ¹ ´ A typical element of ¹ for some i.e. = ¡ ¢ ¹ observable, ¤ ¢j crucial assumption
¹ When the number of actual bidders is maximal, ¹ is denoted as . ¹ ¹ 2 , obviously ¤ (¢j) = (¢j). Since the latter distribution is ¹ 2 ¹ and such that ¹ 0. We now make a is identi…able for all
¹ i.e. ¹ = M Assumption 4 All bidder types are represented in , Remark 1 This assumption holds naturally when the reserve price is lower than © the upper ª ¹ 0 bound of the support under the assumption of identical support. Clearly, [¹ 2¹ : is equal to [2 f : 0g, which is in turn equal to M. Our discussion of the observable case then implies that (j ¸ ) are identi…ed for all 2 M. It remains to show that () and () are identi…able. Denote the support£ of group ¤ ’s bid distribution in the auction with the number of potential bidders as ¹ () . (Recall that, even though bidders draw their valuations from distributions that may be di¤erent, the upper bounds of the supports are common for all bidders.) Our identi…cation proof will rely on the following lemma. Lemma 1 ¹ ( ) is strictly increasing in . It is well known that Lemma 1 always holds in a symmetric model, i.e. when bidders draw their valuations from the same distribution. In the Appendix, we prove it in general. The set ¹ is identi…able because it is also the set of maximal numbers of actual bidders. ¡ ¢ ¡ ¢ ¹ 2 ¹ are identi…able. It is because for ¹ 2 , ¹ we observe ¢j ¹ The bounds ¹ ¹¡ for ¢ ¡ ¢ ¹ . Our main and the bound ¹ ¹ is identi…ed as the upper bound of the support of ¢j result is the following proposition. Proposition 1 Given Assumptions 1-4, () and ( ) are identi…able. 12
We use the convention that: for any two vectors 1 and 2 of the same dimension, 1 2 means 1 · 2 and 1 6= 2 .
6
Proof. It is convenient to denote the conditional distribution of ¸ given as (j). By Bayes rule, (j ) () (j) = (4) () ¹ 2 ¹ such that ¹ 0. We …rst show that Fix group 2 M. Pick an ¡ an arbitrary ¢ ¹ j ¹¡ , where ¡ ¢ ¹¡ ´ ¹1 ¢ ¢ ¢ ¹¡1 ¹ ¡ 1 ¹+1 ¢ ¢ ¢ ¹ is identi…able. Notice that
1 ¡ (j) =
X
:¸
( j) [1 ¡ ¤ (j)]
¡ ¢ ¡ ¢ ¡ ¡ ¢ ¡ ¢¢ ¹ . Thus if 2 ¹ ¹¡ ¹ ¹ , we have Lemma 1 implies ¹ ¹¡ ¹ ¡ ¢ ¡ ¢£ ¡ ¢¤ ¹¡ = ¹ j¹¡ 1 ¡ ¤ j ¹ 1 ¡ j
(5)
¹ , the sum in (5) contains only one term, equal to 1 ¡ On ¡the ¢other hand, when = ¤ ¹ . It follows that j ¡ ¢ ¹¡ ¡ ¢ 1 ¡ j ¹ j ¹¡ = lim ¡ ¢ (6) ¹ ¹ ) 1 ¡ j "¹ ( is identi…able. ¡ ¢ ¹ ¹¡ . First note that We now show how to recover () from j ¡ ¢ Y ¹ ¹ ¹ j = [1 ¡ ()] =1
Y ¡ ¢ ¹ ¹¡ j ¹ = ¹ (1 ¡ ())¹ ¡1 () ¢ [1 ¡ ()] 6=
¡ ¢ ¹ () j ¹ ¹ = 1 ¡ ()
Then from (4), taking into account (1), ¡ ¢ ¡ ¢ ¹ ¹ ¡ ¢ ¹¡ j ¹ ¹¡ = ¡ ¢ j ¹¡ ¡ ¢ ¹ ¡ ¢ () ¹ ¹ ¹ ¡ ¢ = j ¹¡ 1 ¡ () We can combine this equation with
¡ ¢ ¡ ¢ ¹ j ¹ ¹ ¡ ¢ ¹ ¹ = ¡ ¢ j =1 ¹ 7
to eliminate (). This yields
From (2),
and therefore (7) implies
¡ ¢ ¹¡ () 1 ¡¹ ¹ ¢ ¡ ¢ = ¹ j¡ ¹ 1 ¡ ()
(7)
¡ ¢ ¡ ¢ ¹¡ ¹¡ ¤ ¡ ¢ = ¡ ¢ ¹ ¤ ¹
¡ ¢ ¹¡ () 1 ¡ ¹ ¹ ¢ ¤ ¡ ¢ = ¹ j¡ ¹ 1 ¡ () ¤
(8)
Since the right-hand side of this equation contains only identi…able quantities, () is identi…able for each 2 M. Finally, we can recover () from the total probability equations. For = 1 ¡ (0), the law of total probability implies the following system of linear equations for (): ¤ () ¡
X
(j) () = 0
(9)
:¸
Since () are identi…able, (j) are also identi…able; see (1). Formally, consider the above system for any 2 (0 1). Write () as ( ) to make the dependency on explicit. Since the probabilities ( ) enter the right-hand side of (9) only for ¸ , the system has a recursive structure that allows one to uniquely determine ( ) for all ¹ . To see this most easily, we can use an induction argument. Begin with those 2 , we have ¤ ( ) ( ) = (10) (j) ¹ if ( 0 ) are known for all 0 , and we can determine Next, for any given 2 , ( ) from (9) according to " # X ¡ ¢ ¡ 0 ¢ 1 ¤ 0 ( ) = £ () ¡ j (11) (j) 0 0 :
To determine , note that as a solution of a linear system, ( ) is homogeneous of degree 1 in , so that ( ) = ( 1). For = 1 ¡ (0), the law of total probability implies X (1 ¡ (0)) ( 1) = 1 2
Since ( 1) are now known, the above equation uniquely determines (0). Therefore ( ) is identi…ed: ( ) = (1 ¡ (0)) ( 1). Q.E.D. Remark 2 We have chosen to abstract from observable auction heterogeneity, a feature almost always present in auction data. But we should stress that all our results are applicable 8
under observable auction heterogeneity. In such a model, one seeks to identify (j) and ( j), where is a vector of auction characteristics that may also include . All our previous results go through if we use conditional distributions ¤ (¢j ) in place of ¤ (¢j). In particular, the distribution (j) is identi…able for ¸ .
4
Testable implications
For the completeness of exposition we provide a set of necessary and su¢cient conditions for the distribution of bids and the number of active bidders to be rationalizable by the Bayesian-Nash equilibrium of the bidding game. For simplicity, in this section we restrict attention to the symmetric case, where we can exploit some of the logic of a parallel result in GPV. As there, we consider the class of probability distributions P = f (¢) is absolutely continuous with an interval support in R+ g Also let ¢N denote the set of discrete probability distributions supported on N . And denote the joint CDF of bids of all bidders conditional on as (1 j). We then have the following analogue to Theorem 4 in GPV. (The proof is analogous to that in GPV and is available upon request.) © ª ¹ for some ¹ 2. Let 2 ¢ Proposition 2 Let N = 2 f0¹ g , and let (¢j) 2 P © ª ¹ . There exist a distribution of bidder valuations 2 P and a for each 2 2 distribution of the number of potential bidders 2 ¢N such that (a) (¢) is the distribution of the number of actual bidders and (b) (¢j) is joint distribution of bids conditional on entry of bidders, if and only if there exist a number () 2 [0 1) and a distribution ¤ (¢j ) 2 P for each 2 N such that (i) the unique distribution (see the last paragraph in the proof of Proposition 1) that solves ¹ µ ¶ X () = [1 ¡ ()] [ ()] ¡ ( ) =
satis…es ¹ µ ¶ Y 1 X ¡ (1 j) = ( ) [1 ¡ ()] [ ()] ¤ ( j) ()
(12)
=1
=
£ ¤ and (1) = (0) = 0, (ii) ¤ (¢j) 2 P with support ¹ () where ¹ : N ! ( +1) is an increasing function, and (iii) the inverse bidding strategy (¢j ) given by µ ¶ 1 ¤ (j ) () 1 (j ) = + + ¡ 1 ¤ (j ) 1 ¡ () ¤ (j) ¡ ¢ ¡ ¢ is an increasing on ¹ ( ) function, satisfying (j ) = , ¹ () j = ¹ and having a di¤erentiable inverse ¡1 (¢j).
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5
Extension to Dutch auctions
In this section, we show that our result generalizes to Dutch auctions, where only the winning bid is observable. We continue to assume that the identities of actual bidders are ¹ 2 . ¹ Restrict attention to auctions with = ¹ and groups with observable. Fix an ¹ 0. Let be the highest bid submitted from group (with ¹ 0). Let ´ max be the winning bid. And let be the identity of the winning group, i.e. = . ¹ (which Our data directly reveals the joint distribution of ( ) conditional on = ¹ also implies = ): ¡ ¢ ¹ ´ Pr = & · j = ¹ (j)
¤ ¹ ¹ The set of Begin by ©recovering on = = . ª (¢j) the C.D.F.©of ¤ conditional ª ¹ ¹ functions (¢j ) is related to the set (¢j) via the functional equations Z Y ¹ ¹ ) (j) = ¤ (j¹ )¤ (j
6=
One verify (see Berman (1963) and Athey and Haile (2002)) that the solution for © ¤ can ª ¹ is given by (¢j) 8 9 2 3¡1 < Z 1 X = ¹ )5 (j) ¹ 4 ¤ (j¹ ) = exp ¡ (j : ;
(13)
¹ is identi…able. Since the right-hand side of (13) contains only observable objects, ¤ (j) ¤ ¹ ¹ Now recall that (j ) is the probability that all bidders in group submit bids below ¹ We have , conditional on = = . £ ¤¹ ¹ ) = ¤ (j ¹ ) ¤ (j
¹ is identi…able for every ¹ 2 ¹ and every such that ¹ which proves that ¤ (j) 0. This implies that (j ) and therefore (j ¸ ) are identi…able. The rest of the identi…cation proof follows exactly parallel to that of Proposition 1.
6
Identi…cation not at in…nity
In this section, we present another identi…cation result. Under some additional (and plausible) assumptions, we show that the distribution of bidder valuations (j ¸ ) and the screening probabilities () can identi…ed only using bid observations in some, not necessarily maximal, set and one of its adjacent sets that contains an additional bidder from group . We now proceed to show that, for the identi…cation of (j ¸ ) and the screening probabilities (), one doesn’t need to use the bids in the maximal (and its adjacent) sets. We make the following linear independence assumption on ¤ (¢j ).
10
Assumption 5 For any 2 M and for any 2 ¤ with ¸ 1, with the supersets of in given by f 1 g, 91 2 R such that ¢ ¡ ¢ 3 2 ¤¡ 1 j 1 ¤ 1 j 5 ´ 4 (14) ¡ 1 ¢ ¡ ¢ ¤ ¤ j j is non-singular.
Remark 3 It can be shown that Assumption 5 is implied by the linear independence of the function f¤ (¢j) : ¸ 1g. ¹ in practice when bidders An important issue is how to determine the maximal set are asymmetric. This can be done in essentially the same way as in the symmetric setting, ¹ is usually set to be equal to the sample maximum of the actual number of bidders. where We make the following simplifying assumption about the structure of the support of the distribution of the number of potential bidders. Assumption 6 (Rectangular support) The support = ¹ .
Q
¹
=1 f g,
where 2 ·
Assumption 6 allows the econometrician to identify the support boundaries and ¹ for each separately. They can be estimated as sample maxima and minima of , essentially the same way as in the symmetric case. Note also that Assumption 6 implies that the support of the sets of actual bidders is Y ¹ g ¤ = f0 =1
In addition, we make the following technical assumption.
Assumption 7 For any 2 M, 9¤ 2 R such that the values (¤ j) ( : ¸ 1) are mutually distinct. Under these assumptions, our main result in this section shows that the distributions () ( ¸ ) are identi…able even if one only uses the information contained in some active bidder set + 2 ¤ .13 Proposition 3 For any 2 M, the distribution (¢) is identi…able for ¸ using P only (i) the joint bid distributions in any sets + + 2 2 ¤ such that ¸ 1 and ¸ 3, and (ii) the ratio of the probabilities ¤ () ¤ ( + ). Proof. Fix a bidder from group 2 M, …x a set 2 ¤ with the properties stated in the proposition, and number the supersets of in as 1 ,... . For any 2 M that are included in (i.e., , ¸ 1) use Assumptions 5 to pick bid values 1 ,..., 2 R ( = ) such that the matrices de…ned in (14) are invertible.14 13 14
Here and below denotes a vector that contains 1 in position and 0 elsewhere. We do not require 6= , i.e. the two bidders other than bidder may be from the same group.
11
Step 1: Identi…cation of ¤ (¢j) and (¢j ¸ ). Consider a £ matrix-valued function (j ) de…ned elementwise as n o (j) ´ Pr · · · j ( = 1 ). The mixture representation (12) implies
(j) =
X =1
( j) ¤ (j ) ¤ ( j )¤ ( j )
(15)
One can verify that decomposition (15) implies that the function (j) has the following factorization: (j) = ¤ (j) 0 where
Since
¤ = f(1j 1 ) (j )g ¡ ¢ ¡ ¢ (j) = f¤ j 1 ¤ j g (+1j) = ¤0
we have
¡ ¢¡1 (+1j)¡1 (¢j) = 0 (¢j) 0
(16)
It follows that the set of the eigenvalues of (+1j)¡1 (j) is equal to the P set of the diagonal elements of (¢j), i.e. to the set f¤ (¢j ) : ¸ g. Consequently, : ¸ ¤ (¢j ) is identi…able for any 2 ¤ satisfying the properties stated in the Proposition. The sets and + only di¤er by one element, an extra member of group . therefore ¤ (¢j) is identi…able as X X ¤ (¢j) = ¤ (¢j) ¡ ¤ (¢j) :¸
:¸+
A standard argument now shows that (¢j ¸ ) is identi…able. Step 2: Identi…cation of ( + j) and ( + j + ). Since (j) = ¤ (j) 0 we have parallel to (16) for any 2 :
¡ ¢¡1 (+1j)¡1 (¢j) = 0 (¢j) 0
0 (+1j)¡1 (¢j) = (¢j) 0
¡1 and therefore for any = ¤ 2 R for which the eigenvalues ¡ ¢of (+1j) (¤ j) (the ¤ ¤ diagonal elements of (¤ j), i.e. (¤ j) ¤ j ) are distinct , the columns of are identi…able as the left eigenvectors of (+1j)¡1 (¤ j), up to a permutation
12
and scaling. Recall that (¢j), the CDF of bids from group conditional on , is directly observable. In this proof only, let be Decompose the (observable) distribution (¢j) as a mixture of the distributions conditional on , (j) = ¤ (j) (j) + ¤ (j + ) ( + j) +
X =3
(17)
¤ (j ) ( j)
If we evaluate this equation at = 1 , we have = (j) 1 + ( + j) 2 +
X =3
( j)
(18)
¡ ¢ where is the (observable) column vector with components ¤ j . Step 1 implies that 1 and 2 are identi…able, while the other ’s are only identi…able up to permutation and scaling. Nevertheless, for a …xed and arbitrary permutation and scaling, system (18) uniquely determines (j) and ( + j) because is invertible. The same argument applied to the decomposition (17) with the conditioning set + instead of shows that ( + j + ) is identi…able Step 3. Identi…cation of (). Using the Bayes rule, ( + j + ) =
( + j + ) ( + ) ( + )
Furthermore, ( + j + ) = (1 ¡ ()) +1 1 ¡ () = ( + 1) () =
Y 6=
µ
(1 ¡ ())
+ 1
¶
1 ¡ () (j + ) ( + 1) ()
(1 ¡ ()) ()
Y 6=
(1 ¡ ())
Therefore 1 ¡ () (j + ) ( + ) ( + 1) () ( + ) 1 ¡ () () (j + ) ( + ) = ( + 1) () ( + ) () 1 ¡ () ¤ () = ( + j) ( + 1) () ¤ ( + )
( + j + ) =
13
Since we know from step 1 that ( + j + ) and ( + j) are identi…able, and ¤ () ¤ ( + ) is directly observable, the above equation uniquely determines (). Q.E.D. Remark 4 While we believe that Assumptions 5 and 7 will be satis…ed in most applications, the proof of Proposition 3 shows that we could alternatively assume that 91 ,..., 2 R and ¤ 2 R ( = ) such that ¡(i) the matrices (+1j) have full rank and (ii) each of ¢ ¡1 the matrices (+1j) ¤ j has distinct eigenvalues. These assumptions are on observable matrices and are potentially testable.
7
Concluding remarks
We have shown that a …rst-price IPV auction model where nonparticipation is due to a binding reserve price, and the set of potential bidders is unobservable, is nonparametrically identi…ed under weak assumptions. We do not develop a nonparametric estimation method. This may be an interesting direction for future research. On the other hand, from an empirical perspective, parametric assumptions are always used in some form. Our results provide a foundation for parametric estimation methods such as in La¤ont et al. (1995) or Donald and Paarsch (1996), but with auction-speci…c number of potential bidders. Generalization to other private value auction models, e.g. with unobserved heterogeneity, either assuming a¢liated values as in Li et al. (2002) or within the IPV paradigm as in Krasnokutskaya (2003), is also left for future research.
8
Appendix
This appendix sketches the derivations of equilibrium conditions, and proves Lemma 1. In order to simplify notations, we do not divide bidders into groups like we do in the text. The set of bidders is N with 2 · jN j 1. Each bidder draws his valuation from the C.D.F. (¢).15 From here up to the proof of Lemma 2 below, we …x an N -auction, and thus suppress the dependency of equilibrium objects on N in our notation, e.g. we write bidder ’s inverse bidding strategy as (¢) rather than (¢jN ). But when we prove Lemma 1, this dependency will become explicit. For an N -auction, bidder solves Y max( ¡ ) ( ())
The …rst-order conditions are
6=
X 1 = 0 () () ¡
(19)
6=
15 Clearly, from the theoretical point of view the setting here is equivalent to the one we use in the text, although they are di¤erent from the econometrician’s point of view.
14
where () ´ log ( ()). Formula (3) in the text follows from (19). Sum (19) over and then divide through by jN j ¡ 1:
X X 1 1 = 0 () jN j ¡ 1 () ¡
(20)
Subtract (19) from (20), we have 0 () =
1 jN j ¡ 1
2 X 4
3
1 jN j ¡ 1 5 ¡ () ¡ () ¡
The above equation holds for 2 ( ¹] where ¹ is the equilibrium maximum bid. Therefore for all 2 ( ¹] 2 3 X ( ()) 1 jN j ¡ 2 5 4 0 () = ¡ (21) (jN j ¡ 1) ( ()) () ¡ () ¡ 6=
By Lebrun (1999) Theorem 1, the equilibrium is completely characterized by di¤erential equations (21) and the following boundary conditions: (+) ¸ for all , and (+) = except possibly one bidder (¹) = ¹ for all . Lebrun (1999) also shows existence (Theorem 2) and uniqueness (Corollary 1) of the equilibrium. The proof of Lemma 1 will need the following result. Lemma 2 If jN j ¸ 3, 2 N , 2 N , and 6= , then for all 2 ( ¹], 2 3 X ( ()) 1 jN j ¡ 3 5 4 0 () ¡ (jN j ¡ 2) ( ()) () ¡ () ¡ 6=
Proof. From 6= , we can rewrite (21) and get 2 3 µ ¶ X 1 1 jN j ¡ 3 1 1 4 5 0 () = ¡ + ¡ jN j ¡ 1 () ¡ () ¡ () ¡ () ¡ 6=
From (19),
1 1 ¡ = 0 () ¡ 0 () () ¡ () ¡
15
(22)
Substitute this into (22) and solve for 0 (): 2 3 X 1 1 jN j ¡ 3 4 0 () = ¡ ¡ 0 ()5 jN j ¡ 2 () ¡ () ¡ 6=
Since 0 () 0 for all 2 ( ¹] and hence 0 () 0 as well, we get the result.16
Q.E.D.
Now we can prove Lemma 1. Proof of Lemma 1. It su¢ces to prove ¹ (N ) ¹ (N n fg) for all N with 2 · jN j 1. It is trivial if jN j = 2, so suppose jN j ¸ 3. Suppose by the way of contradiction that ¹ (N ) · ¹ (N n fg). Step 1: We claim¢ that, for small enough 0, we have (jN ) (jN n fg for all ¡ 2 ¹ (N ) ¡ ¹ (N ) and all 2 N n fg. This claim is obviously true if ¹ (N ) ¹ (N n fg). If ¹ (N ) = ¹ (N n fg) = ¹, it can be seen from ¹ ¡ ¹ ¹ ¡ ¹ 0 (¹jN ) = = 0 (¹jN n fg) (jN j ¡ 1) (¹ ) (jN j ¡ 2) (¹ ) ¡ ¢ Step 2: We claim that (jN ) (jN n fg) for all 2 ¹ (N ) and all 2 N n fg. ¹ Suppose not. Then Step 1 implies that there is a …rst ¡ going ¢ from (N ) downward, ¤ ¤ ¹ (largest) point 2 (N ) such that ¤ ( jN ) = ¤ (¤ jN n fg) for some ¤ 2 N n fg. Since ¤ is the …rst point, we also have (¤ jN ) ¸ (¤ jN n fg) for all 2 N n fg. Then it is easy to verify that Lemma 2 implies 0¤ (¤ jN ) 0¤ (¤ jN n fg). But then ¤ (¤ + jN ) ¤ (¤ + jN n fg) for small 0, contradicting to the de…nition¡ of ¤ . ¢ Step 3: It follows from Step 2 and (19) that for each 2 N n fg and each 2 ¹ (N ) , X
2N nfg
0 (jN n fg)
X
0 (jN )
2N nfg
X
0 (jN )
2N nfg
¡ ¢ Integrate over ¹ (N ) and notice that (¹(N )jN ) ¸ (¹(N )jN n fg) for all , X
2N nfg
log ( ( + jN ))
X
2N nfg
log ( ( + jN n fg)) 8 2 N n fg
Therefore, for each 2 N n fg, there is a 2 N n f g such that ( + jN ) ( + jN n fg). It follows that ( + jN ) ( + jN n fg) ¸ holds for at least two distinct ’s in N n fg, contradicting the boundary condition. Q.E.D. 16 The result that 0 () 0 is stronger than strict monotonicity of (since might be 0 at isolated points). For its proof, see Lebrun (1997) Lemma A2-2.
16
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