Nonparametric Identification of First-Price Auction with Unobserved Competition: A Density Discontinuity Framework
NNonparametric Identification of First-Price Auction with Unobserved Competition: A Density
Discontinuity Framework
Emmanuel GuerreSchool of EconomicsUniversity of KentUnited Kingdom Yao LuoDepartment of EconomicsUniversity of TorontoCanadaAugust 2019
Acknowledgments : The authors would like to thank Lu Han, Xavier d’Haultfœuille, QuangVuong, together with conference and seminar participants for stimulating discussions and usefulsuggestions. Cheok in Fok and Jiaqi Zou provided excellent research assistance. The second authorthanks SSHRC for financial support. All remaining errors are our own. a r X i v : . [ ec on . E M ] A ug bstract We consider nonparametric identification of independent private value first-price auction models, in which the analyst only observes winning bids. Ourbenchmark model assumes an exogenous number of bidders N . We show that,if the bidders observe N , the resulting discontinuities in the winning bid densitycan be used to identify the distribution of N . The private value distribution canbe identified in a second step. A second class of models considers endogenously-determined N , due to a reserve price or an entry cost. If bidders observe N ,these models are also identifiable using winning bid discontinuities. If bidderscannot observe N , however, identification is not possible unless the analystobserves an instrument which affects the reserve price or entry cost. Lastly,we derive some testable restrictions for whether bidders observe the number ofcompetitors and whether endogenous participation is due to a reserve price orentry cost. An application to USFS timber auction data illustrates the useful-ness of our theoretical results for competition analysis, showing that nearly onebid out of three can be non competitive. It also suggests that the risk aversionbias caused by a mismeasured competition can be large. Keywords : Auction models, unobserved competition, nonparametric identifi-cation, density discontinuities, endogeneous participation, unobserved hetero-geneity, discrete mixture models
JEL classification : C14, C57, D44
Introduction
There exists a large literature on nonparametric identification of auction models;see, e.g., Athey and Haile (2007) or Hendricks and Porter (2007) for a review. Inthe case of sealed-bid first price auction, a vast majority of work assumes that theanalyst can observe all of the bids, or both the winning bid and the number ofcompetitors. This may not always be observed. Lamy (2012) mentions that, inFrench timber auctions, only the winning bid may be available to researchers topreserve anonymity of the bidders. It is common practice in many markets that canbe treated as auctions for only the winning bid (i.e. the transaction price) to berecorded. For instance, a company soliciting price quotes for a task to be completedis implicitly organizing a first-price auction. While the company may not record allquotes or the number of responses, the price charged to the winning bidder is likelyto appear in accounting records. As noted in Han and Strange (2015), “bidding wars”are becoming commonplace in housing markets, where houses are sold through acompetitive bidding process resembling an informal first-price auction. Governmentsmay offer subsidies to attract firms as recently considered by Kim (2018) and Slattery(2019) using an auction framework. Observing all the subsidy offers is again unlikely.Hence, in many economic situations of interest, the records only contain the finalwinning bid. Therefore, the ability to identify auction primitives solely from winningbid data may enlarge the scope of auction theory applications.A second motivation stems from misspecification considerations, because the ob-served number of bids can be a misleading indicator of competition. For instance,Laffont, Ossard and Vuong (1995) consider an application where bidders are agentsacting for several buyers such that the number of bids underestimates competition.Alternatively, some buyers may enter an auction simply to gain information, in whichcase their bids will be dominated and never impact the winning bid; counting these1ids would overestimate competition. When the observed number of bids as a mea-sure of competition is at doubt, using only winning bids provides a robust approachfor identifying primitives of interests. Our USFS timber application illustrates thebias in estimating risk-aversion when we use the number of bids as the measure ofcompetition as in Lu and Perrigne (2008) or Campo, Guerre, Perrigne and Vuong(2011). Bidders are much more likely to be risk neutral when we are agnostic aboutthe distribution of competition.Last, the distribution of competition is a parameter of interest in itself. We view itas a latent variable and recover it from the winning bid. As noted above, the numberof observed bids is often different from the number of bids actually contributing tothe winning bid. In this case, the more meaningful measure is the latter, which isuseful for auction design such as choosing an optimal reserve price. In our USFStimber application, it is found that nearly one bidder out of three does not bid in acompetitive way. As noted by Bulow and Klemperer (1996), increasing competitiveparticipation would give a higher seller expected revenue than choosing an optimalreserve price. Our finding therefore suggests that, among other things, the sellershould consider a bidder training program, as studied in De Silva, Li and Zhao (2019),or should investigate whether the dominated bid is due to collusion, see eg Abrantes-Metz and Bajari (2009).This paper therefore studies the identification of model primitives using only dataon winning bids. Specifically, we develop a new approach for first-price auction modelidentification that exploits discontinuities in the density function of the winning bid.First, we identify the distribution of unknown competition. In particular, we build onan important restriction that first-price auction models impose on the data: the bidquantile function must be strictly increasing with respect to the number of bidders.Therefore, the upper boundary of the bid distribution, conditioning on the numberof bidders, is strictly increasing, as well. We show that this creates discontinuities, or2umps, in the winning bid density function at these upper boundaries. A novel resultof the paper is that these jumps identify the distribution of the number of bidders.Second, we identify the value distribution function by iteratively exploiting twoequilibrium mappings relating the value and bid quantile functions. Based on thelocation of winning bid density discontinuities, we create a sequence of expandingquantile intervals over which the private value quantile is identified. For every itera-tion, we start by identifying the bid quantile function in the most competitive auction,which has the largest number of bidders. This information can then be used to iden-tify the value quantile function in the same quantile interval and further calculate thecorresponding bid quantile for other competition levels.We then extend our results to endogenous participation due to the presence of areserve price or entry cost. When active buyers observe the number of participants,identification extends naturally. On the other hand, identification fails due to a lack ofdiscontinuity if active buyers cannot observe the number of participants. We restoreidentification assuming that the analyst can instrument for the reserve price or entrycost, as in Gentry and Li (2014). In addition, it is assumed that the analyst observesauctions where the object is not sold or there is only one participant. Identification isthen established exploiting the simple binomial distribution of the number of bidders.Lastly, we derive testable restrictions of whether or not bidders observe the numberof competitors, and if participation is constrained by a reserve price or entry cost.In models of endogenous participation, it is usually assumed that the buyers do notobserve the number of active buyers. See, among others, Guerre, Perrigne and Vuong(2000), Marmer, Shneyerov and Xu (2013) or Gentry and Li (2014). As known sinceMcAfee and McMillan (1987), the expected payoff of a first-price auction with reserveprice or entry cost does not depend upon buyers information regarding competition.Hence, under risk neutrality of buyers and sellers, both kinds of buyer competitioninformation are likely to hold. Whether buyers observe competition can be a known3arket characteristic, as in the case where active buyers sit in the same room orin housing market “bidding wars” where real estate agents often decide to revealthe number of offers. But it can also be a parameter of interest, as in the case of acompany contacting contractors in a public or private way unobserved by the analyst.
Related literature
Auctions with unobserved competition.
Allowing for unknown competitionstarted early in the empirical auction literature. Laffont et al. (1995) estimate thenumber of buyers N as a parameter that they take to be constant across auctions.Paarsch (1997) treats unknown competition as a nuisance parameter, which is elim-inated using conditional likelihood estimation. For ascending eBay auctions, Song(2004) shows that the private value distribution and a constant number of buyersare identified from winning and second highest bids, but not from winning bids alonewhen N is random. More pertinent to our paper is the misclassification approach ofAn, Hu and Shum (2010), who achieve identification from the winning bid using aproxy N ∗ ≤ N for the number of buyers and an instrument that can be a discretizedsecond bid. Shneyerov and Wong (2011) suppose that only winning bids and thenumber of active bidders are observed. Recent work for ascending auctions includeQuint (2015) and Freyberger and Larsen (2017). Mixture distribution.
The present paper contributes to the literature on non-parametric identification of finite mixtures; see for instance the review of Compianiand Kitamura (2016). Existing identification results require either exclusion restric-tions or multiple independent measurements. A first-price auction example of thelatter is d’Haultfœuille and Février (2015), who recover the distribution of an un-observed continuous auction characteristic from three bids. Hu and Sasaki (2017)4btain identification from two measurements in a model with discrete unobservedheterogeneity. In our setting, the number of buyers can be viewed as unobserved het-erogeneity while the winning bid is a unique bid. Identification is however possiblebecause the mixture components are the bid distribution given N = n , issued fromthe same private value distribution and constrained by an optimal bidding condition.When the buyers do not observe the number of competitors, this is restrictive enoughto ensure identification in presence of a reserve price or entry cost instrument, withoutthe exclusion restrictions of Compiani and Kitamura (2016). Discontinuity design.
The discontinuity design (DD) literature has expandedrapidly in recent years; interested readers are encouraged to refer to review papers byImbens and Lemieux (2008), Kleven (2016) and Jales and Yu (2017). Recent auctionapplications include Coviello and Marinello (2014), and Choi, Neisheim and Razul(2016). As in the DD literature, this paper employs jump sizes for identificationpurposes - more specifically, to identify the probability that N = n . However, thispaper departs from the DD literature by considering an unknown number of densitydiscontinuities with unknown location, which identify the support of N .The remainder of the paper is organized as follows. In Section 2, we describethe benchmark model. In Section 3, we extend our analysis to auction models withendogenous competition generated by reserve price or entry cost, for when buyerscan or cannot observe competition. Section 4 reports the results of our empiricalapplication. Section 5 concludes. We also include a proof section that contains allproofs omitted in the main text. A discontinuity detection algorithm, which alsocompute a discontinuous density estimator, is presented in the appendix.5 The benchmark model
In this section, we start by describing the benchmark auction model and introduce twoequilibrium mappings that are convenient for describing our discontinuity identifica-tion strategy. Next, we derive the restrictions that the model imposes on the observedwinning bids, especially with respect to the formation of discontinuities. Finally, wedescribe our identification strategy in two steps. First, we identify the distributionof the number of buyers from the discontinuities in the winning bid density func-tion. Second, we identify the value distribution function using the two equilibriummappings iteratively.
Suppose there is a single item for sale with N active symmetric buyers bidding for theitem. All buyers observe N . In contrast, the analyst does not observe N , which causesauction-specific unobserved heterogeneity. Each buyer i also observes her privatevaluation V i , which is unknown to other buyers. The private values V i are i.i.d. drawsfrom a distribution F ( · ) , which is known to all the buyers and is independent of N .The buyers are risk neutral and their bids B i are formed according to a symmetricbest-response strategy. In sum, the primitives are the distribution of the number ofbuyers N and the private value distribution.We assume that the analyst only observes the winning bid W , i.e., the maximumbid among the N buyers in the set N of active buyers W = max i ∈N B i . Hence, the analyst observes draws from the unconditional cumulative probabilitydistribution of the winning bid G ( · ) , which is a mixture of the conditional winning6id distributions given N : G ( b ) = + ∞ (cid:88) n =2 P (cid:18) max ≤ i ≤ n B i ≤ b (cid:19) × P ( N = n ) = + ∞ (cid:88) n =2 G nn ( b ) P ( N = n ) , (1)where G n ( · ) is the conditional bid distribution given N = n .The two next assumptions introduce some additional conditions for the distribu-tion of N and for the private value distribution F ( · ) . Assumption N . The number of active buyers N is a discrete random variablewith support { n, . . . , n } for some integers ≤ n ≤ n , i.e., p n = P ( N = n ) > for n = n, n + 1 , . . . , n with (cid:80) nn = n p n = 1 . Assumption IPV . Buyers’ private values V i are unknown to competitors and i.i.d. draws from a common knowledge distribution F ( · ) . The cumulative distributionfunction F ( · ) has a compact support [ v, v ] . Its probability density function f ( · ) iscontinuous and strictly positive over [ v, v ] .Both theoretical and empirical literatures adopt the assumption of a private valuedistribution with compact support. In particular, it rules out multiple asymmetricequilibria; see Maskin and Riley (1984, Remark 2.3), who also establish that symmet-ric Bayesian Nash Equilibrium bids are given by a strictly increasing and continuouslydifferentiable function of private values.For our discontinuity approach, the compact support assumption ensures the ex-istence of discontinuities in the density of unconditional winning bids that we exploitin this paper. In particular, the winning bid densities g n ( · ) given N = n stay boundedaway from at its upper boundary; see (7) below.7 .2 Bid and value quantile equilibrium mappings In this subsection, we describe two equilibrium mappings that are repeatedly usedin our identification procedure. Specifically, there is an equilibrium mapping fromthe value distribution and the bid distribution, and vice versa. Our discontinuityidentification strategy is conveniently described using the quantile framework as inGuerre, Perrigne and Vuong (2009), Liu and Luo (2016), and Guerre and Gimenes(2019), that we recount below.Let V ( α ) = F − ( α ) represent the private value quantile function, where α ∈ [0 , is the quantile level. Let B n ( α ) denote the bid quantile function given that n buyersparticipate in the auction. Following Milgrom (2001)’s exposition of the identificationstrategy of Guerre, Perrigne and Vuong (2000), the private value quantile function V ( · ) can be viewed as the common valuation function of buyers who receive indepen-dent uniform private signals A i = F ( V i ) , which determine their private values V i = V ( A i ) . By Assumption IPV, B i = β n ( A i ) for all i , where β n ( · ) is strictly increasing and continuously differentiable. It followsthat for any b in the range of β n ( · ) , G n ( b ) = P ( β n ( A i ) ≤ b ) = P (cid:0) A i ≤ β − n ( b ) (cid:1) = β − n ( b ) because A i is uniformly distributed over [0 , . Hence the best-response strategy isthe bid quantile function β n ( α ) = B n ( α ) for all α ∈ [0 , . Now, let us relate the bid and private value quantile functions. Suppose that buyer i receives signal α but makes a suboptimal bid B n ( a ) for some a ∈ [0 , . Since heropponents bid B n ( A j ) , the probability that her bid B n ( a ) wins the auction is given8y P (max j (cid:54) = i A j ≤ a ) , which is equal to a n − as the signals of the n − opponents A j , where j (cid:54) = i , are independent and uniform. It follows that the expected payoff ofbuyer i is ( V ( α ) − B n ( a )) a n − , which is maximized when a = α . Since ∂∂a (cid:2) ( V ( α ) − B n ( a )) a n − (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) a = α = V ( α ) ( n − α n − − ∂ [ B n ( α ) α n − ] ∂α = ( n − α n − (cid:32) V ( α ) − B n ( α ) − αB (1) n ( α ) n − (cid:33) , setting this derivative to gives V ( α ) = B n ( α ) + αB (1) n ( α ) n − . (2)This constitutes the equilibrium mapping from the bid quantile function to the valuequantile function, which is the basis of the identification of V ( · ) with knowledge of B n ( · ) .Now, let us consider the inverse of the mapping (2). Indeed, (2) is equivalent to ∂ [ B n ( α ) α n − ] ∂α = V ( α ) ( n − α n − and, it follows, B n ( α ) = n − α n − (cid:90) α t n − V ( t ) dt. (3)For convenient of identification that will be clarified later on, let us introduce theconditional bid upper bound b n = B n (1) = ( n − (cid:90) t n − V ( t ) dt, which gives B n ( α ) = n − α n − (cid:20) b n n − − (cid:90) α t n − V ( t ) dt (cid:21) . (4)This constitutes the equilibrium mapping from the value quantile to the bid quantilefunction. The two mappings represented in (2) and (4) are repeatedly used in ouridentification procedure. 9 .3 Structure of the winning bid distribution The structure of winning bid distributions compatible with a first-price auction wherebuyers observe N follows from the mixture expression of G ( · ) in Equation (1) andthe best-response differential equation (2). Proposition 2.1
A c.d.f. G ( · ) is rationalized by a first-price auction model satisfy-ing Assumptions N, IPV, and observability of N by buyers if and only if(i). The c.d.f. G ( · ) has a mixture structure G ( · ) = n (cid:88) n = n p n G nn ( · ) , (5) where the G n ( · ) are c.d.f., ≤ n ≤ n and the positive p n satisfy (cid:80) nn = n p n = 1 .(ii). The quantile functions B n ( · ) = G − n ( · ) are continuously differentiable over [0 , and satisfy the compatibility conditions B n ( α ) + αB (1) n ( α ) n − B m ( α ) + αB (1) m ( α ) m − for all n ≤ n, m ≤ n and all α ∈ [0 , . Moreover the function V ( α ) = B n ( α ) + αB (1) n ( α ) n − is continuously differentiable over [0 , with V (1) ( · ) > . Proof of Proposition 2.1 : It remains to be shown that (i) and (ii) are sufficient. Thefunction V ( · ) in (ii) is a quantile function associated with a c.d.f. F ( · ) satisfying therequirements of Assumption IPV while the mixture weights p n define a distributionfor N as in Assumption N. These { p n , n ≤ n ≤ n } and private value quantile function V ( · ) generate a distribution for N and best response bidding strategy functions B n ( · ) by (3), with G ( · ) as a winning bid c.d.f. (cid:3) In short, a c.d.f. G ( · ) as in Proposition 2.1 is a mixture with components con-strained by compatibility conditions driven by the best response differential equation102). The compatibility conditions of Proposition 2.1-(ii) reflects that the mixturecomponents G n ( · ) are generated by the same private value distribution, an importantfeature for identification. In particular, our identification results rely on the con-straints it imposes on the extremities of the conditional bid p.d.f. g n ( · ) , as illustratedin the next corollary. Recall b n = B n (1) , v = V (0) = b n and v = V (1) . Corollary 2.1
Suppose that the compatibility conditions of Proposition 2.1-(ii) hold.Then for all n = n, . . . , n , b n < v , and g n ( v ) = nn − f ( v ) , with (6) g n (cid:0) b n (cid:1) = 1( n − v − b n ) . (7) Proof of Corollary 2.1 : The compatibility conditions imply that (3) holds, andintegrating by parts gives B n ( α ) = 1 α n − (cid:90) α V ( t ) d (cid:2) t n − (cid:3) = V ( α ) − (cid:90) α (cid:18) tα (cid:19) n − V (1) ( t ) dt. Hence b n = v − (cid:82) α t n − V (1) ( t ) dt < v as V (1) ( · ) > . Note that this also gives B n ( α ) < V ( α ) for all α > , and then B (1) ( α ) > by (2). When α goes to , thefollowing holds B n ( α ) = V (0) + V (1) (0) α + o ( α ) − (cid:90) α (cid:18) tα (cid:19) n − (cid:0) V (1) (0) + o (1) (cid:1) dt = V (0) + n − n V (1) (0) α + o ( α ) , which shows that B (1) n (0) = n − n V (1) (0) . As B (1) n ( · ) > , the conditional bid p.d.f. g n ( · ) satisfies g n ( b ) = 1 B (1) n ( G n ( b )) for all b ∈ (cid:2) v, b n , (cid:3) . (8)11ence g n ( v ) = 1 /B (1) n (0) = nn − /V (1) (0) = nn − f ( v ) , which is (6). For (7), (2) and(8) give g n (cid:0) b n (cid:1) = G n (cid:0) b n (cid:1) ( n − (cid:0) V (cid:0) G n (cid:0) b n (cid:1)(cid:1) − b n (cid:1) = 1( n − (cid:0) v − b n (cid:1) as G n (cid:0) b n (cid:1) = 1 , so that (7) holds. (cid:3) Equation (7) implies that g n (cid:0) b n (cid:1) is strictly positive. It turns out from (1) thatthis causes discontinuities in the winning bid p.d.f. g ( · ) at each b n , as studied inthe next section. As it follows that b n is identified, (7) shows that g n (cid:0) b n (cid:1) , where n ∈ { n, . . . , n } , are determined by the common unknown parameter v . We employthis consequence of the compatibility conditions of Proposition 2.1-(ii) later on toidentify the distribution of N . In this subsection, we introduce a numerical example to illustrate the discontinuityfeatures of the winning bid p.d.f that follows from Corollary 2.1. This example willalso be useful for introducing our identification procedure. A general lemma completesthe example.
Consider the private value c.d.f. F ( v ) = v for all v in [0 , and a number of buyers N = { , } with equal probability. As V ( α ) = α / , it follows that B n ( α ) = n − α n − (cid:90) α t n − dt = n − n − α / . Hence b n = n − n − and (8) yields the conditional bid p.d.f g n ( b ) , given N = n , is equal to b/b n on [0 , b n ] and vanishes outside this interval. Figure 1 displays the conditionalc.d.f. and p.d.f. of the winning bid when N = { , } . Note that the support of12 G n ( b )1 v b b (a) c.d.f. G nn ( · ) bnG n − n ( b ) g n ( b ) v b b G ( b ) g ( b )3 G ( b ) g ( b ) (b) p.d.f. nG n − n ( · ) g n ( · ) Figure 1: Conditional winning bid distribution, where N = { , } and V ( α ) = √ α the conditional density function increases with the number of buyers. Both densitiesjump to zero at their upper boundaries as expected from (7).Let us now turn to the winning bid, the observation of the analyst. As expectedfrom Figure 1b, the unconditional p.d.f. g ( b ) = · G ( b ) g ( b ) + · G ( b ) g ( b ) displayed in Figure 2b is discontinuous at b and b , with jump sizes ∆ and ∆ respectively. The resulting winning bid c.d.f. exhibits kinks at these values, as il-lustrated by Figure 2a. In this example, Figure 2b exhibits two discontinuities (andFigure 2a exhibits two kinks) because N takes two potential values here. The increasing support property observed in Figure 1 and the winning bid p.d.fdiscontinuities in Figure 2b are generic, as shown in the upcoming lemma. Lemma2.1-(i) recalls more generally that bids increase with competition — a key feature of13 G ( b )1 v b (1 + G ( b )) b (a) c.d.f. G ( · ) bg ( b ) v b b ∆ ∆ (b) p.d.f. g ( · ) Figure 2: Winning bid distribution ( V ( α ) = √ α and P ( N = 2) = P ( N = 3) = 1 / )first-price auctions that does not hold in ascending or second-price ones. Lemma2.1-(ii) focuses on the winning bid p.d.f. discontinuities and its jumps.
Lemma 2.1
Suppose Assumptions N and IPV hold. Then, all of the following hold.(i). For all α in (0 , , B n ( α ) < · · · < B n ( α ) < V ( α ) with B n (0) = V (0) for all n .In particular, for b n = B n (1) , b n < · · · < b n < v .(ii). The c.d.f. G ( · ) has a p.d.f. g ( · ) with g ( v ) = 0 , which is continuous over thestraight line with the exception of the n − n +1 discontinuity points b n < · · · < b n ,the interval (cid:2) v, b n (cid:3) being the support of G ( · ) . For n ≤ n ≤ n , the jumps This feature also does not hold when buyers do not not observe N , in which case the modelprimitives are not identified. For the sake of brevity, we do not establish here that observation ofcompetition by buyers is essential for purposes of identification. This can be done as in Proposition3.2-(ii), which considers a reserve price model wherein active buyers cannot observe competition.Haile, Hong and Shum (2003) have also used support variation to test a common value settingagainst a private value one. n = lim t ↓ (cid:0) g ( b n − t ) − g ( b n + t ) (cid:1) satisfy ∆ n = np n ( n − v − b n ) . (9) (iii). It holds that n = lim t ↓ G ( v + t )log t . Proof of Lemma 2.1: As B n ( α ) = 1 α n − (cid:90) α V ( t ) d (cid:2) t n − (cid:3) = V ( α ) − (cid:90) α (cid:18) tα (cid:19) n − V (1) ( t ) dt differentiating with respect to n gives ∂B n ( α ) ∂n = − (cid:90) α (cid:18) tα (cid:19) n − log (cid:18) tα (cid:19) V (1) ( t ) dt ≥ , the inequality is strict except when α = 0 , in which case B n (0) = v for all n . Itfollows that the bid c.d.f. G n ( · ) given that N = n has a support [ v, b n ] , with an upperbound b n = B n (1) which is strictly increasing with respect to n and strictly smallerthan v = lim n ↑∞ b n . Hence, this proves Part (i). For part (ii) The expression forjumps (9) follows from (5), which shows that the winning bid p.d.f. is g ( b ) = n (cid:88) k = n p k kG k − k ( b ) g k ( b ) , (10)with g k ( b ) = 0 for b > b n when k ≤ n by Lemma 2.1-(i). This gives g ( b n − t ) − g ( b n + t )= n (cid:88) k = n p k kG k − k (cid:0) b n − t (cid:1) g k (cid:0) b n − t (cid:1) − n (cid:88) k = n +1 p k kG k − k (cid:0) b n + t (cid:1) g k (cid:0) b n + t (cid:1) → np n g n (cid:0) b n (cid:1) = ∆ n when t goes to . The equality (7) for g n (cid:0) b n (cid:1) then gives (9). For part (iii), continuityof B (1) n ( · ) , which is bounded away from and infinity, and (8) shows that g n ( · ) is15ontinuous with g n ( v ) > by (6). This gives when t goes to G ( v + t ) = n (cid:88) n = n p n (cid:18)(cid:90) v + tv g n ( u ) du (cid:19) n = n (cid:88) n = n p n g nn ( v ) t n (1 + o (1))= p n g nn ( v ) t n (1 + o (1)) as p n g nn ( v ) > , which implies n = lim t ↓ G ( v + t )log t . (cid:3) Lemma 2.1 is an important building block for identifying the competition distri-bution. Part (iii) is a tail identification result for n as in Hill and Shneyerov (2013).Lemma 2.1-(ii) shows that the jumps in the winning bid p.d.f. identify P ( N = n ) upto the unknown v . Here, we first illustrate our identification procedure using the numerical example ofSection 2.4.1. We then turn to the identification of competition and the private valuedistribution.
By Lemma 2.1-(iii), n = 2 is identified. The winning bid c.d.f. of Figure 2a has twokinks, which identify the upper bounds b < b , and then n = 3 . Consider now thejumps ∆ and ∆ of the winning bid p.d.f. in Figure 2b. Expression (9) for the jumpsallows for the identification of p = p = ; see Lemma 2.2.Let us now turn to the identification of the private value distribution, which isbased on the winning bid c.d.f. G ( b ) = 12 (cid:0) G ( b ) + G ( b ) (cid:1) displayed in Figure 2a. Since G ( b ) = 1 on [ b , b ] , G ( b ) = (2 G ( b ) − , b ∈ [ b , b ] . B n ( α ) V ( α ) n = 2 n = 3 b b α = G ( b ) β α β α β α β Figure 3: Iterative identification of B n ( α ) and V ( α ) from G ( · ) , as in Figure 2.It follows that B ( · ) is identified on [ α , , where α = G ( b ) , using the top portionof the winning bid distribution; see Figure 2a when G ( b ) ∈ [ (1 + G ( b )) , . Usingthe mapping (2) from the bid quantile function to the private value one gives V ( α ) = B ( α ) + αB (1)3 ( α ) , and V ( · ) is identified on [ α , . Additionally, using the mapping (4) from the privatevalue quantile function gives B ( α ) = 1 α (cid:20) b − (cid:90) α V ( t ) dt (cid:21) so that B ( · ) is also identified on [ α , . The identified B ( α ) , B ( α ) , and V ( α ) ,where α ∈ [ α , , are displayed in blue in Figure 3.17ext, we enlarge the interval [ α , over which V ( · ) is identified. For this purpose,let β = B ( α ) < b and observe that G ( b ) is identified for b ≥ β . Since G ( b ) = (cid:0) G ( b ) − G ( b ) (cid:1) ,G ( b ) is identified for b ≥ β , as B ( α ) is identified for α ≥ α = G ( β ) . Figure 3shows that α < α and arguing as above gives us identification of V ( · ) and B ( · ) on [ α , . Three portions of V ( · ) , B ( · ) , and B ( · ) are identified through three iterationsand plotted in purple, red and orange, respectively, in Figure 3. Furthermore, Figure3 suggests that additional iterations of this identification procedure should allow usto recover any V ( α ) . N In this subsection, we describe the identification of the support of N and its distribu-tion using the discontinuity points and jump sizes. To identify the support, we exploittwo implications of Lemma 2.1: (1) the minimum number of buyers n is identifiedfrom the winning bid distribution tail near the lower boundary; (2) each number ofbuyers n generates a discontinuity in the winning bid distribution, which identifiesthe difference n − n . More specifically, Lemma 2.1-(ii) identifies n and n through n = lim t ↓ G ( v + t )log t and n = n + Card { b ; g ( · ) is discontinuous at b } − . This also identifies the support of the distribution of N as P ( N = n ) > for all n with n ≤ n ≤ n by Assumption N.Next, we exploit the jumps in the p.d.f. to identify p n = P ( N = n ) . Recall thatEquation (9) identifies p n up to the private value upper bound v , p n = n − n ∆ n (cid:0) v − b n (cid:1) . (cid:80) nn p n = 1 implies v = 1 + (cid:80) nn = n n − n ∆ n b n (cid:80) n = nn n − n ∆ n . (11)Hence p n satisfies p n = n − n ∆ n (cid:80) nk = n k − k ∆ k + n − n ∆ n (cid:32) (cid:80) nk = n k − k ∆ k b k (cid:80) nk = n k − k ∆ k − b n (cid:33) , n = n, . . . , n, (12)and is identified because the discontinuity points b k and jump sizes ∆ k are. Wesummarize these identification results in the next lemma. Lemma 2.2
Suppose Assumptions N and IPV hold. Then v , n , n , b n < · · · < b n , v and the probabilities p n , n = n, . . . , n , are identified. The identifying equations (11) and (12) can also be used to derive inequalityconstraints satisfied by the jumps sizes ∆ n , discontinuity locations b n , the lowest andlargest numbers of bidders n and n . Indeed v > b n and ≤ p n ≤ are equivalent tothe equations n (cid:88) k = n k − k ∆ k (cid:0) b n − b k (cid:1) ≤ , n (cid:88) k = n k − k ∆ k (cid:0) b k − b n (cid:1) ≤ (cid:80) nk = n k − k ∆ kn − n ∆ n , n = n, . . . , n given that it must also holds that ∆ n > by Lemma 2.1-(ii). A violation of any ofthese inequalities indicate that the model is not correct. We now obtain identification of the value quantile function by iteratively exploitingthe two equilibrium mappings in (2) and (4). We proceed in three steps: It is easily seen that b n < v , which is equivalent to the first inequality, implies ≤ p n , n = n, . . . , n . The second inequalities are equivalent to p n ≤ , n = n, . . . , n . tep 1. Note that the winning bid distribution satisfies G ( b ) = 1 − p n + p n G nn ( b ) for all b in (cid:2) b n − , b n (cid:3) so that G n ( · ) is identified over (cid:2) b n − , b n (cid:3) as follows G n ( b ) = (cid:18) G ( b ) − (1 − p n ) p n (cid:19) /n for b in (cid:2) b n − , b n (cid:3) . Set α = G n (cid:0) b n − (cid:1) . It follows that B n ( · ) is identified on [ α , , i.e., B n ( α ) = W (cid:2) (1 − p n ) + p n α n (cid:3) , where W ( · ) = G − ( · ) is the winning bid quantile function.Using the mapping from the bid quantile function to the value quantile function(2) shows that the private value quantile function satisfies, for all α ∈ [ α , , V ( α ) = B n ( α ) + 1 n − αB (1) n ( α )= W (cid:2) (1 − p n ) + p n α n (cid:3) + np n n − α n W (1) (cid:2) (1 − p n ) + p n α n (cid:3) , and V ( · ) is identified over [ α , .Using the mapping from the value quantile function to the bid quantile function(4) shows that the bid quantile functions B n ( · ) , n = n, . . . , n − are also identifiedover [ α , . Hence { B n ( α ) , α ∈ [ α , } and (cid:8) G n ( b ) , b ∈ [ B n ( α ) , b n ] (cid:9) are identified,for all n = n, . . . , n . Step 2.
We now expand the interval over which G n ( · ) is identified using aniterative argument. Define β = B n − ( α ) , β < b n − whenever α > because,by Lemma 2.1-(i), β = B n − (cid:2) G n (cid:0) b n − (cid:1)(cid:3) < B n (cid:2) G n (cid:0) b n − (cid:1)(cid:3) = b n − . The definition of G ( · ) implies that G n ( b ) = (cid:32) G ( b ) − (cid:80) n − n = n p n G nn ( b ) p n (cid:33) /n , (13)where G ( · ) and p n are identified, and G n ( · ) are identified on [ B n ( α ) , b n ] for all n = n, . . . , n − . Since B n ( α ) < . . . < B n − ( α ) = β , [ β , b n ] ⊆ [ B n ( α ) , b n ] for all n . Therefore, the conditional bid distribution G n ( b ) is identified on [ β , b n ] . Step 3.
We now identify V ( · ) on a growing interval [ α k , using an inductionargument and the identified V ( · ) on [ α , . For an integer k ≥ , define α k = G n ( β k − ) = G n [ B n − ( α k − )] , β k = B n − ( α k ) . Identification of V ( · ) on the growing interval [ α k , is established in Lemma 2.3 below. Lemma 2.3
Suppose Assumptions N and IPV hold. Then(i). the sequences { α k , k ≥ } and { β k , k ≥ } are decreasing sequences with lim k →∞ α k = 0 . (ii). { α k , k ≥ } is identified. For any integer number k ≥ , { V ( α ) , α ∈ [ α k , } isidentified if { V ( α ) , α ∈ [ α k − , } is identified. Proof of Lemma 2.3:
See Section 6.1 in the proof section.Combining Lemmas 2.2 and 2.3 shows that the joint distribution of private valuesand the number of active buyers is identified.21 heorem 2.1
Suppose Assumptions N and IPV hold and that the buyers observe thenumber of active buyers N . Then F ( · ) and the distribution of N are identified. Proof of Theorem 2.1 : Lemma 2.2 shows that the distribution of N is identified.Since N is independent of private values, it remains to be shown that the privatevalue quantile function V ( · ) is identified. Take α > . By Lemma 2.3-(i), there exists k such that α > α k and Lemma 2.3-(ii) yields identification of V ( α ) . Since V (0) = v is identified by Lemma 2.2, the Theorem is proven. (cid:3) So far, we have assumed exogenous participation, i.e., the number of buyers beingindependent of private values. In many cases of interest, participation arises endoge-nously, such as when the seller imposes a reserve price below which bids will notbe considered, or when buyers face entry costs. This section extends our results toendogenous competition due to a reserve price or entry cost.In these cases, the breadth of possible outcomes is richer than in our benchmarkmodel. In particular, the auctioned object may not be sold when buyers decide notto enter the auction, as accounted in the next Definition of the outcome distribution . Definition 3.1
A distribution G ( · ) is an outcome distribution if and only if itassigns probabilities solely to the events in the σ -field generated by { the object is not sold } , { the object is sold at a price less than b } , b ∈ R . In the sequel, these two events will be abbreviated as { Not Sold } = { N = 0 } , alsorefereed a failed auction later on, and { W ≤ b } , respectively, where W stands for thewinning bid or, more generally, the transaction price. The winning bid distribution of22he benchmark model, which attributes zero probability to { Not Sold } is an exampleof a continuous auction outcome distribution.Additionally, Definition 3.1 allows for the following two kinds of discrete compo-nents. First, the object may remain unsold with positive probability. Second, G ( · ) can have discrete components even when the object is sold. For instance, if a uniquebuyer enters an auction with a reserve price R and observes an absence of competitors,she can win the auction by bidding R , such that P ( W = R ) > if P ( N = 1) > . Inthis reserve price example, as in many others, the (renormalized) continuous compo-nent of G ( · ) , denoted as G ( · ) , is the distribution of the winning bid given that thereis at least two participants.That the model allows for these additional outcomes raises the issue of econometricselection. Consider, for instance, winning bids data derived from firms’ accountingbooks. A failed auction with an unsold object will be absent from the records asno transaction took place. Furthermore, a seller may be tempted to cancel auctionsthat are attended by a sole buyer who wins simply by bidding the reserve price. Inthis case, identification of the model should be based not on G but on a conditionaloutcome distribution given that the events above do not happen. In the reserve priceand entry cost examples considered below, the latter conditional outcome distributioncoincides with the continuous component G ( · ) of G ( · ) . In this subsection, we consider endogenous competition due to the presence of areserve price. Assume that the seller will not sell the item if the maximum bidis lower than a reserve price R , which is known to the buyers. Only buyers with V i ≥ R , or equivalently A i ≥ F ( R ) can win, and are called participants or activebuyers. We assume that the analyst only observes the winning bid max i ∈N B i when23he auctioned good is sold, does not know the reserve price R , the number n ≥ ofpotential buyers or the number N R of participants (except when failed auction with N R = 0 are observed).The number of participants N R has a binomial distribution with parameter ( n, − F ( R )) ;that is P ( N R = n ) = n ! n !( n − n )! [1 − F ( R )] n F n − n ( R ) . Because the buyers participating in the auction receive a signal A i ≥ F ( R ) , it isconvenient to renormalize the signal as A i,R = A i − F ( R )1 − F ( R ) (14)assuming v ≤ R < v from now on. Given N R = n ≥ , the auction participants’signals A i,R are i.i.d. draws from the [0 , uniform distribution. Let F R ( · ) and V R ( · ) be the conditional c.d.f. and quantile function of the private values V i given V i ≥ R ,such that F R ( v ) = if v < R F ( v ) − F ( R )1 − F ( R ) if R ≤ v ≤ v if v > v ,V R ( α ) = V [ F ( R ) + (1 − F ( R )) α ] . As discussed for the benchmark model, the winning bid p.d.f is discontinuouswhen the participants observe N R before submitting bids, a property that plays animportant role for identification. We therefore consider separately the cases whereparticipation is known or unknown to active buyers. There is no need to identify the minimal number of participants n among the parameters as thelower bound of N R is known to be . .1.1 Participation known to active buyers First, let us recall the expression of the best-response bidding strategy when theparticipants observe N R = n . If n = 0 , the auctioned object is not sold. If n = 1 , itis optimal for the unique participant to bid the reserve price R . If n ≥ , the optimalbidding strategy is a strictly increasing function of A i,R under Assumption IPV, whichis equal to the bid quantile equation B n,R ( · ) = B ( ·| N = n, R ) , given N R = n ≥ .Arguing as in (2), except using A i,R and V R ( · ) in place of A i and V ( · ) , yields the bestresponse differential equation B n,R ( α ) + αn − B (1) n,R ( α ) = V R ( α ) . (15)It follows that B n,R ( α ) = n − α n − (cid:90) α t n − V R ( t ) dt = V R ( α ) − (cid:90) α (cid:18) tα (cid:19) n − V (1) R ( t ) dt. (16) Structure of the outcome distribution.
We describe the constraints on thedistribution of observables imposed by the reserve price. The continuous component G ( · ) of the outcome distribution is the conditional winning bid distribution given W > R , such that G ( b ) = G ( W ≤ b | W > R ) . (17) Proposition 3.1
An outcome distribution G ( · ) is rationalized by a first-price auctionmodel satisfying Assumptions IPV where buyers observe the number of participantsand with a reserve price R in ( v, v ) if and only if(i). There is a probability q ∈ (0 , , an integer number n ≥ with G ( Not sold ) = q n , and a unique R > such that G ( W = R ) > and is equal to nq n − (1 − q ) . Equivalently, N R ≥ . Using the event { W ≥ R } instead is more convenient for identificationpurposes, as W is observed by the analyst and R is identified as the lower bound of the support of G ( · ) . ii). Let G ( · ) be as in (17). There exists some c.d.f. G n ( · ) such that G ( · ) = n (cid:88) n =2 p n G nn ( · ) , p n = n ! n ! ( n − n )! q n − n (1 − q ) n − q n − nq n − (1 − q ) . (iii). The quantile functions B n ( · ) = G − n ( · ) satisfy the compatibility conditions ofProposition 2.1-(ii) with B n (0) = R . Proof of Proposition 3.1 : See Section 6.2 in the proof section.Proposition 3.1 is very similar to Proposition 2.1 of the benchmark model, up tothe mixture weights p n which are now given by a binomial distribution. Identification results.
The compatibility condition of Proposition 3.1-(iii) inducesdiscontinuities in the p.d.f. of G ( · ) . This gives the next identification corollary, whichmostly follows from Theorem 2.1. Corollary 3.1-(ii) shows that the model primitivesare identified from well functioning auctions with more than two participants providedthe number of potential buyers is greater than or equal to three. Corollary 3.1
Consider a first-price auction satisfying Assumption IPV with a re-serve price R in ( v, v ) and N R participants, where N R is known to the buyers but notto the analyst. Then(i). the reserve price R , number of potential buyers n and conditional private valuedistribution F R ( · ) are identified from the c.d.f. G ( · ) defined in (17).(ii). if n ≥ , the private value c.d.f. { F ( v ) , v ≥ R } is identified from the c.d.f. G ( · ) .(iii). if n = 2 , the c.d.f { F ( v ) , v ≥ R } is identified from the outcome distribution G ( · ) but not from G ( · ) . roof of Corollary 3.1: See Section 6.3 in the proof section.Corollary 3.1-(i) follows from Theorem 2.1 as F R ( · ) is the updated distributionof the private value given participation. Identification of F ( R ) from G ( · ) requiresat least three potential buyers. This discrepancy between n = 2 and n ≥ can beunderstood using Lemma 2.2, which shows here that, for n ≥ , the probabilities p n ( R ) = P ( N R = n | N R ≥ n ! n !( n − n )! F n − n ( R ) [1 − F ( R )] n − F n ( R ) − nF n − ( R ) [1 − F ( R )] , n = 2 , . . . , n are identified through the discontinuities of the p.d.f. g ( · ) of G ( · ) . This is sufficientto identify F ( R ) and then to identify F ( v ) for v ≥ R .This contrasts with n = 2 , in which case g ( · ) is continuous over its inner support.The c.d.f. G ( · ) is helpful for recovering F R ( · ) but not informative enough regardingthe screening level F ( R ) due to the lack of density discontinuities. As Corollary 3.1-(i)shows that n is identified, F ( R ) can be recovered from G ( Not sold ) , and observingfailed auctions is necessary to identify { F ( v ) , v ≥ R } . More generally, the proof of Corollary 3.1-(ii,iii) shows that G ( · ) overidentifies F ( R ) when n ≥ , while G ( · ) overidentifies F ( R ) for all n ≥ . This can be used totest the null hypothesis that a reserve price restricts participation. Winning bid density discontinuities arise because buyers observe the number of par-ticipants N R before submitting their bids. If they do not observe N R , the winningbid density becomes continuous on its inner support. However, it is shown that iden-tification still holds if the analyst observes whether the object is left unsold, whether N R = 1 or if an instrument for the reserve price is available. Since G ( W = R ) = 2 F ( R ) (1 − F ( R )) only identifies the set { F ( R ) , − F ( R ) } when n = 2 , ,auctions with a good sold at the reserve price to a unique bidder cannot be used to recover F ( R ) .
27e first recall the quantile expression of the optimal bidding strategy. Becausebuyers participate in the auction if and only if A i ≥ R , the optimal bidding strategy B R ( · ) is a strictly increasing function of the normalized uniform signals A i,R in (14),which is also the bid quantile function. Suppose now that Buyer 1 participates anddecides to place a bid B R ( a ) given her signal A ,R = α . Given Buyer 1’s participation,the number of participants in N \ { } , N R − , has a Binomial distribution withparameter ( n − , − F ( R )) and is independent of participant signals A i,R . It followsthat the probability that Buyer 1 wins is P (cid:18) max i ∈N \{ } A i,R ≤ a | (cid:19) = n − (cid:88) n =0 a n ( n − n !( n − − n )! [1 − F ( R )] n F n − − n ( R ) which is equal to [ F ( R ) + (1 − F ( R )) a ] n − . Hence the expected payoff of Buyer 1 is ( V R ( α ) − B R ( a )) [ F ( R ) + (1 − F ( R )) a ] n − . Arguing as for the best response differential equation (2) shows that B R ( α ) + (cid:18) α + F ( R )1 − F ( R ) (cid:19) B (1) R ( α ) n − V R ( α ) , B R (0) = R, (18)of which the unique solution is B R ( α ) = V R ( α ) − (cid:90) α (cid:18) F ( R ) + [1 − F ( R )] tF ( R ) + [1 − F ( R )] α (cid:19) n − V (1) R ( t ) dt. (19) Structure of the outcome distribution.
Proposition 3.2 and Lemma 3.1 belowsummarize the model implications for the auction outcome distribution. In theseresults, G R ( · ) stands for B − R ( · ) and G ( · ) for the continuous component G ( ·| Sold ) ofthe outcome distribution. Lemma 3.1 extends a result of Guerre et al. (2000) on thedivergence of the bid p.d.f. at the vicinity of the reserve price.28 roposition 3.2 An outcome distribution G ( · ) is rationalized by a first-price auctionmodel satisfying Assumptions IPV, with a reserve price R in ( v, v ) , and buyers whoobserve the number of participants if and only if(i). There is a probability q ∈ (0 , , and an integer number n ≥ with G ( Not sold ) = q n . The c.d.f. G R ( b ) = [(1 − q n ) G ( W ≤ b | Sold ) + q n ] /n − q − q has a support [ R, b R ] for some R < b R < ∞ .(ii). B R ( · ) = G − R ( · ) is continuously differentiable with B (1) R (0) = 0 . The function B R ( α ) + (cid:18) α + q − q (cid:19) B (1) R ( α ) n − has a continuous derivative which is strictly positive over [0 , . Proof of Proposition 3.2:
See Section 6.4 in the proof section.
Lemma 3.1
The c.d.f. G R ( b ) and G ( b ) have p.d.f. g R ( b ) and g ( b ) , respectively,which are continuous over ( R, b R ] and diverge when b goes to R with < lim b ↓ R (cid:8) ( b − R ) / g R ( b ) (cid:9) < ∞ , < lim b ↓ R (cid:8) ( b − R ) / g ( b ) (cid:9) < ∞ . Proof of Lemma 3.1:
See Section 6.5 in the proof section.Compared to the case where the participants observe N R , the outcome distribution G ( · ) puts no mass at the reserve price, G ( W = R ) = 0 . Proposition 3.2-(ii) impliesthat the winning bid p.d.f. g ( · ) and g R ( · ) are continuous on their inner support.Lemma 3.1 shows, however, that these winning bid p.d.f. are infinite at the reserveprice. These features can be used to test whether or not the buyers observe the levelof competition. 29 dentification results. Proposition 3.2-(i) shows that the winning bid c.d.f. G ( · ) has a mixture structure; that is, since q = F ( R ) , G ( b ) = [ F ( R ) + (1 − F ( R )) G R ( b )] n − F n ( R )1 − F n ( R )= n (cid:88) n =1 n ! n !( n − n )! [1 − F ( R )] n F n − n ( R )1 − F n ( R ) G nR ( b ) . The c.d.f. mixture components are powers of the same c.d.f. G R ( · ) , so that thiswinning bid distribution looks simpler than it counterpart (17) of the case wherebuyers observe participation. However, due to the absence of support variation acrossmixture components, it lacks the p.d.f discontinuities that are crucial to identifyingthe distribution of N R when buyers observe participation. This explains the nonidentification result of Proposition 3.3-(ii). Proposition 3.3
Consider a first-price auction satisfying Assumption IPV with areserve price R in ( v, v ) and a number N R of participants unknown to buyers and theanalyst. Then(i). the reserve price R is identified from the winning bid c.d.f. G ( · ) .(ii). the private value c.d.f. { F ( v ) , v ≥ R } and the maximal number n of partici-pants are not identified from the outcome distribution G ( · ) .(iii). { F ( v ) , v ≥ R } and n are identified from G ( · ) and P ( N R ≥ if the analystobserves the auction outcomes and whether there were at least two participants. Proof of Proposition 3.3:
See Section 6.6 in the proof section.As seen from the simple mixture structure of the distribution G ( · ) , G R ( · ) canbe recovered as soon as F ( R ) and n are identified, which ensure identification of30he quantile function V R ( · ) by (18). This is used in Proposition 3.3-(iii) to restoreidentification as F ( R ) and n can be identified from F n ( R ) = P ( N R = 0) = G ( Not sold ) ,nF n − ( R ) [1 − F ( R )] = P ( N R = 1) = 1 − P ( N R ≥ − G ( Not sold ) . Alternatively, identification also holds under an exclusion restriction, which con-siders an instrument z that affects the reserve price but not the other primitives ofthe model, as assumed later on for the cost of an entry model. Assumption R . The reserve price is a non-constant function R ( z ) of the instru-ment z ∈ Z . The private value c.d.f. F ( · ) and n do not depend upon z . Proposition 3.4 shows that, thanks to variation in the instrument, model primitivescan be identified from the conditional outcome distribution { G ( ·| z ) , z ∈ Z} , withoutadditional observation on N R as in Proposition 3.3-(iii) but with the assumption thatfailed auctions where the object remains unsold are observable to the analyst. Proposition 3.4
Consider a first-price auction satisfying Assumptions IPV and Rwith a reserve price R ( z ) ∈ ( v, v ) for all z ∈ Z and N R participants, where N R isunknown to the buyers and the analyst. Then(i). R ( · ) is identified from the conditional winning bid distribution G ( ·|· ) .(ii). the private value c.d.f { F ( v ) , v ≥ inf z ∈Z R ( z ) } and the maximal number n ofparticipants are identified from the conditional outcome distribution G ( ·|· ) . Proof of Proposition 3.4:
See Section 6.7 in the proof section.As Theorem 2.1 shows that the updated private value c.d.f. { F R ( z ) ( · ) , z ∈ Z} isidentified, it also follows that F ( · ) is identified from the collection { G ( ·| z ) , z ∈ Z} ofconditional winning bid c.d.f. in auctions with at least two bidders if inf z ∈Z R ( z ) = v .31 .2 First-price auction with entry costs Endogenous variation of the number of buyers can arise from entry costs. This sec-tion considers the affiliated-signal entry model of Ye (2007) considered in Marmer,Shneyerov and Xu (2013) and Gentry and Li (2014). Each auction has two stages.In the first stage, buyer i observes a private signal S i for her unknown private value, i = 1 , . . . , n . Each buyer decides whether to enter the auction. Entry involves pay-ment of an entry cost c . In the second stage, the N c buyers who decide to enterobserve their private values V i , which are independent draws from the conditionalc.d.f. F ( ·| S i ) . The active buyers then submit bids B i in a first-price auction.Gentry and Li (2014) assume for identification purposes that the entry cost de-pends upon an observed auction-specific continuous variable, c = c ( Z ) , and that Z affects neither the signals nor the private values. The signals S i are assumed to be i.i.d. , from a standard uniform distribution without loss of generality. Given S i , theprivate value V i is independent of N c , Z , and the other private values and signals.Following Gentry and Li (2014), we will use the following assumption: Assumption E.
The number of potential entrants n is greater than or equalto and does not depend upon z . The support of F ( ·| s, z ) = F ( ·| s ) is [ v, v ] forall s ∈ [0 , . F ( ·|· ) is continuous over [ v, v ] × [0 , with a continuous p.d.f. f ( ·|· ) bounded away from over this set. For any v ∈ [ v, v ] , F ( v | s ) decreases with s ∈ [0 , . The cost function c ( · ) is non-constant and continuous over Z , which is a closedconnected set with non-empty interior .Gentry and Li (2014, Proposition 1) show that any symmetric Nash equilibriumof the two-stage auction game has a payoff equivalent equilibrium in which Stage 1entry decisions involve a signal threshold s ( z ) = σ n ( c ( z )) that we detail now. Let32 c ( v | s ( z )) be the updated private value distribution given that the buyer participates, F c ( v | s ) = P ( V i ≤ v | S i ≥ s ) = 11 − s (cid:90) s F ( v | t ) dt. The signal threshold is characterized through backward induction. For the mo-ment, suppose that buyers do not observe the number of entrants N c . Since thesymmetric second-stage bidding strategy is increasing under Assumption IPV, theprobability that a buyer with private value v wins the auction is ( s + (1 − s ) F c ( v | s )) n − , where s = s ( z ) is a threshold below which buyers do not enter the auction. A buyer’spost-entry expected profit is (cid:90) v ( s + (1 − s ) F c ( t | s )) n − dt ; see Riley and Samuelson (1981, Eq. 8). Thus, the first-stage expected profit of abuyer with a signal S is Π n ( S ; s ) = (cid:90) v [1 − F ( v | S )] ( s + (1 − s ) F c ( v | s )) n − dv. (20)This expression is also valid when buyers observe the number of entrants.By Gentry and Li (2014)’s Proposition 1, all buyers enter if c ( z ) ≤ Π n (0; 0) andnone enter if c ( z ) ≥ Π n (1; 1) . If Π n (0; 0) ≤ c ( z ) ≤ Π n (1; 1) , the threshold s ( z ) solves the break-even condition Π n ( s ; s ) = c ( z ) ; that is c ( z ) = (cid:90) v [1 − F ( v | s ( z ))] × [ s ( z ) + (1 − s ( z )) F c ( v | s ( z ))] n − dv, (21)a condition which is useful for identifying the cost. Under the conditions imposedon Z in Assumption E, it follows from Gentry and Li (2014) that s ( Z ) is a closedinterval of [0 , , which is not a singleton if < s ( z ) < , as will be assumed for theremainder of this paper. 33et us now study the identification of model primitives for each of when buyerscan and cannot observe the number of entrants N c . The analyst observes the winningbid but does not observe the number of entrants N c . The distribution of N c given Z = z is binomial ( n, − s ( z )) such that P ( N c = n | Z = z ) = n ! n !( n − n )! [1 − s ( z )] n s n − n ( z ) . Buyer signals are A i,c = F c ( V i | s ( Z )) , i = 1 , . . . , N c (22)which are i.i.d. from the [0 , uniform distribution and independent of N c and Z .The updated second stage private value quantile function is V c ( ·| s ) = F − c ( ·| s ) . Finding the best response bidding strategy when buyers observe N c = n follows thesame steps as those of the reserve price case with observed participation. When n = 0 the object is not sold and when n = 1 , the unique entrant can win the object with abid of . When n ≥ , the optimal bidding strategy is a function of n , Z , and A i,c from (22), which is strictly increasing with respect to the latter. The optimal biddingstrategy is equal to the conditional bid quantile function B n,c ( ·| z ) = B ( ·| N c = n, Z = z ) . Arguing as for (2) yields B n,c ( α | z ) + αB (1) n,c ( α | z ) n − V c ( α | s ( z )) , B n,c (0 | z ) = v, (23) B n,c ( α | z ) = n − α n − (cid:90) α t n − V c ( t | s ( z )) dt = V c ( α | s ( z )) − (cid:90) α (cid:18) tα (cid:19) n − V (1) c ( t | s ( z )) dt. (24) Structure of the outcome distribution.
The next proposition summarizes theimplications of this entry model for the outcome distribution. For the remainder of34his section, G ( · ) will be the continuous component of the outcome distribution, suchthat G ( b | z ) = G ( W ≤ b | W > , Z = z ) . (25) Proposition 3.5
A conditional outcome distribution { G ( ·| z ) , z ∈ Z} is rationalizedby a first-price auction model with entry cost satisfying Assumption E, where buyersobserve N c , with a continuous threshold s ( · ) such that s ( Z ) ⊂ (0 , , if and only if(i). there is a continuous q ( · ) , where q ( Z ) ⊂ (0 , , and an integer number n ≥ such that G ( Not sold | z ) = q ( z ) n and G ( W = 0 | z ) = nq ( z ) n − (1 − q ( z )) .(ii). Let G ( ·|· ) be as in (25). There exists some conditional c.d.f. G n ( ·| q ( · )) suchthat G ( ·| z ) = n (cid:88) n =2 p n ( z ) G nn ( ·| q ( z )) , p n ( z ) = n ! n !( n − n )! q ( z ) n − n (1 − q ( z )) n − q ( z ) n − nq ( z ) n − (1 − q ( z )) . (iii). The quantile functions B n ( ·| q ( z )) = G − n ( ·| q ( z )) are continuously differentiablewith B n ( α | q ( z )) + αB (1) n ( α | q ( z )) n − B m ( α | q ( z )) + αB (1) m ( α | q ( z )) m − V c ( α | q ( z )) for all ≤ n, m ≤ n and all α ∈ [0 , , where V c ( ·|· ) belongs to the class V q,v of V ( ·|· ) satisfying: (1) inf ( α,z ) ∈ [0 , ×Z ∂V ( α | q ( z )) ∂α > , V (0 | q ( z )) = v over Z ;(2) The function − ∂∂q [(1 − q ) V − c ( v | q )] decreases with q over the closed interval q ( Z ) ⊂ (0 , and is a conditional c.d.f. with positive density on its supportclosure. Proof of Proposition 3.5:
See Section 6.8 in the proof section.The probability q ( z ) of Proposition 3.5 is an entry probability, or equivalently, athreshold signal. Proposition 3.5-(ii) reveals that entry costs constrain the dependence35etween auction outcomes and entry probability: G ( ·| z ) is equal to the conditionaloutcome distribution given q ( z ) . A similar sufficiency property recently appearsin Liu, Vuong and Xu (2017), who characterize Bayesian Nash equilibrium in entrygames.
Identification results.
The identification results in the next corollary follows fromthe density discontinuities generated by the compatibility condition of Proposition3.5-(iii). Corollary 3.2 parallels Corollary 3.1, which holds for reserve price first-priceauction where buyers observe participation. As in the latter result, Corollary 3.2bases identification on the winning bid c.d.f. G ( ·| z ) of first-price auctions that attractat least two buyers. Corollary 3.2
Consider a first-price auction model with entry cost satisfying As-sumption E with s ( Z ) ⊂ (0 , and N c number of entrants, where N c is observable tobuyers but not the analyst. Then(i). the number of potential buyers n and { F c ( ·| s ( z )) , z ∈ Z} are identified fromthe winning bid c.d.f. G ( ·| z ) in (25).(ii). if n ≥ , { s ( z ) , z ∈ Z} , { c ( z ) , z ∈ Z} and { F ( ·| s ) , s ∈ s ( Z ) } are identifiedfrom the winning bid c.d.f. G ( ·| z ) (iii). if n = 2 , c ( z ) and s ( z ) are identified for all z ∈ Z and F ( ·| s ) is identified forall s ∈ s ( Z ) from the outcome distribution G ( ·| z ) . Proof of Corollary 3.2:
See Section 6.9 in the proof section.Corollary 3.2-(i) follows from Theorem 2.1 and the fact that the second stage ofthis auction is a first-price auction with private values drawn in F c ( ·| s ( z )) . Part (ii,iii), A similar result holds in presence of a reserve price; see Proposition 3.4 and the discussion atthe end of Section 3.3 F ( ·|· ) and c ( · ) when s ( Z ) = [0 , ,follows from (i) and Gentry and Li (2014, Section 3.2), who have noted that the costfunction can be identified from F c ( ·| s ( z )) and (21). If the buyers submit their bids without observing N c , arguing as for reserve price withunknown participation shows that the bid quantile B c ( ·| z ) satisfies B c ( α | z ) + (cid:18) α + s ( z )1 − s ( z ) (cid:19) B (1) c ( α | z ) n − V c ( α | s ( Z )) , B R (0) = V c (0 | s ( z )) = v, (26)for which the unique solution is B c ( α | z ) = V c ( α | s ( z )) − (cid:90) α (cid:18) s ( z ) + [1 − s ( z )] ts ( z ) + [1 − s ( z )] α (cid:19) n − V (1) c ( t | s ( z )) dt. (27) Structure of the outcome distribution.
The next proposition and lemma sum-marize the implications of this entry model for the outcome distribution.
Proposition 3.6 { G ( ·| z ) , z ∈ Z} is rationalized by a first-price auction model withentry cost satisfying Assumption E, buyers do not observe N c , and there is a contin-uous threshold s ( · ) such that s ( Z ) ⊂ (0 , , if and only if(i). There is a continuous q ( · ) , where q ( Z ) ⊂ (0 , , and an integer number n ≥ such that G ( Not sold | z ) = q ( z ) n and G ( ·| z ) = G ( ·| q ( z )) . The c.d.f. G c ( b | q ( z )) = [(1 − q n ( z )) G ( W ≤ b | Sold , z ) + q n ( z )] /n − q ( z )1 − q ( z ) has a support [ v, b c ( z )] for some v < b c ( · ) < ∞ .(ii). B c ( ·| z ) = G − c ( ·| z ) is continuously differentiable with B (1) c (0 | z ) = 0 . The func-tion V c ( α | q ( z )) = B c ( α | z ) + (cid:18) α + q ( z )1 − q ( z ) (cid:19) B (1) c ( α | z ) n − belongs to V q,v . emma 3.2 Let the c.d.f. G c ( ·| z ) and G ( b | z ) = G ( W ≤ b | Sold , Z = z ) be as inProposition 3.6. Then G R ( ·| z ) and G ( ·| b ) have p.d.f. g R ( ·| z ) and g ( ·| z ) , respectively,which are continuous over ( v, b R ( · )] and diverge when b goes to v with < lim b ↓ v (cid:8) ( b − v ) / g R ( b | z ) (cid:9) < ∞ , < lim b ↓ v (cid:8) ( b − v ) / g ( b | z ) (cid:9) < ∞ . The proofs of these results are omitted as they involve arguments similar to thoseof Propositions 3.5, 3.2 and Lemma 3.1.
Identification results.
As for reserve price, variation of the instrument is sufficientfor identification of the entry model when entrants do not observe competition.
Proposition 3.7
Consider a first-price auction model with entry cost satisfying As-sumption E, where the buyers and analyst do not observe the number of entrants N c .Suppose in addition that the maximal bid B c (1 | z ) varies with z ∈ Z .Then the maximal number of entrants n , the threshold signal { s ( z ) , z ∈ Z} , thec.d.f { F ( ·| s ) , s ∈ s ( Z ) } , the signal and cost function { c ( z ) , z ∈ Z} are identifiedfrom the outcome distribution { G ( ·| z ) , z ∈ Z} . Proof of Proposition 3.7:
See Section 6.10 in the proof section.The variation condition on B c (1 | z ) is testable. While the proof of Proposition 3.4establishes that the maximal bid B R (1 | z ) under reserve price R ( z ) is not constant,showing that B c (1 | z ) satisfies such a requirement seems out of reach in full generality. Interestingly, the important role played by the upper bound of the bid distribution inPropositions 3.4 and 3.7 parallels the role of the conditional bid distribution’s upperbounds for achieving identification when the buyers observe competition. This amounts to studying the variations of B c (1 | z ) with respect to s ( z ) , as in Li and Zheng(2009,2012). But these authors consider different entry models, where, in particular, at least twobidders enter. .3 Information and entry models identification The preceding sections considered four auction models whether competition is re-stricted by a reserve price or an entry cost and whether or not active buyers observethe number of competitors. We now show that the winning bid distribution char-acterizations obtained above can be used to decide which of these models hold. Todecide whether entry is restricted by a reserve price or an entry cost, it is convenientto rely on an instrument satisfying the following assumption.
Assumption ER . Buyers are restricted either by a reserve price R ( · ) as inAssumptions IPV and R, or an entry cost c ( · ) , as in Assumption E. The instru-ment z ∈ Z satisfies Assumption R with R ( Z ) ⊂ ( v, v ) or Assumption E with c ( Z ) ⊂ (Π n (0 , , Π n (1 , and v > , respectively. The instrument z can stack instruments specific to reserve price and entry cost,i.e. z = ( z R , z c ) with R ( z ) = R ( z R ) and c ( z ) = c ( z c ) . Our first result deals with buyer information. Corollary 3.3-(i,ii) allows to identifywhether buyers observe or not competition from the continuous component of theconditional outcome distribution G ( ·| z ) , that is using auction data with at least twoactive buyers. Corollary 3.3 also holds in the absence of instrument. Corollary 3.3
Under Assumption ER, active buyers observe the number of partici-pants if any of the three following conditions hold:(i). If n > and for each z ∈ Z , the continuous component of G ( ·| z ) has a p.d.f. g ( ·| z ) which has at least one discontinuity on its inner support. Assumption ER restricts to cases where reserve price or entry cost is binding, so that the twoparticipation models have an empty intersection. Although it is feasible to assume a nonbindingreserve price or entry cost, we have not done so here for the sake of brevity. ii). For each z ∈ Z , the p.d.f. g ( ·| z ) of the winning bid distribution is bounded atits support lower bound w ( z ) .(iii). There is a (cid:37) ( z ) ≤ w ( z ) such that G ( W = (cid:37) ( z ) | z ) > for each z ∈ Z . If so,entry is restricted by a reserve price (cid:37) ( z ) . Proof of Corollary 3.3: (i,ii) follows from Lemma 2.1-(ii) and the compatibilityconditions in Propositions 3.1-(iii) and 3.6-(ii), which give that g ( ·| z ) is continuouson its support, except possibly, at a finite number of discontinuity points when n > ,and when buyers observe competition, while Lemmas 3.1 and 3.2 give that g ( ·| z ) diverges at its lower boundary. (iii) is a consequence of Propositions 3.1-(i), 3.2-(i),3.5-(i) and 3.6-(i). (cid:3) The next corollary investigates whether entry is restricted by a reserve price orentry cost using instruments variations. In this result, w ( z ) stands for the lowersupport bound of the continuous component G ( ·| z ) of the outcome distribution G ( ·| z ) . Corollary 3.4
Suppose Assumption ER holds. Then { G ( ·| z ) , z ∈ Z} is rationalizedby an entry cost if any of the following conditions hold:(i). w ( · ) does not depend upon on the instrument, in which case w ( · ) = v .(ii). Either G ( W = 0 | z ) > for all z ∈ Z , or for v > independent of z as in (i), g ( ·| z ) diverges at v with < lim b ↓ v (cid:8) ( b − v ) / g ( b | z ) (cid:9) < ∞ for all z ∈ Z . Proof of Corollary 3.4:
This follows from Propositions 3.1-(ii,iii), 3.1-(i), 3.5-(ii,iii),3.5-(i), and Corollary 3.2, observing that, for a reserve price, w ( z ) = R ( z ) > mustdepend upon z by Assumption R, that G ( W = 0 | z ) = 0 , and either G ( W = R ( z ) | z ) > or g ( ·| z ) diverges at R ( z ) by Corollary 3.1. (cid:3) G ( ·| z ) in entry games only dependsupon the entry probability π ( z ) , i.e. G ( ·| z ) = G ( ·| π ( z )) . As noted after Proposition3.5, a similar sufficiency property holds under entry cost or reserve price. Using theidentified entry updated private value c.d.f. and the identified probability of entrygive functional forms that can be tested. Under a reserve price model, the updatedprivate value c.d.f. is, for v ≥ R ( z ) and π ( z ) = F ( R ( z )) , F R ( v | z ) = F ( v ) − π ( z )1 − π ( z ) , and (1 − π ( z )) F R ( v | z )+ π ( z ) does not depend upon z . This contrasts with the updatedprivate value distribution under an entry model, for which π ( z ) = s ( z ) as F c ( v | π ( z )) = 11 − π ( z ) (cid:90) π ( z ) F ( v | s ) ds. Hence observing (1 − π ( z ) F c ( v | π ( z )) + π ( z ) depending upon z is evidence against areserve price model. This section illustrates applications of our identification results to USFS timber auc-tions. These auctions have been studied extensively in the literature. However, noexisting work allows the actual competition differ from the observed number of bids.We study the first-price auction data used in Lu and Perrigne (2008). The data con-tain 107 two-bid auctions and 108 three-bid auctions, and report the appraisal valueand timber volume of each auctioned lot. See Lu and Perrigne (2008) for furtherinformation on this dataset. Since the literature argues that the USFS reserve pricesare too low, we also assume that they are nonbinding.41ur first exercise illustrates detection of pdf discontinuities and estimation of thedistribution of competition N from winning bids. Hence N should be interpreted asthe number of bidders contributing to the winning bid, or the number of competitive,or equivalently equilibrium bids, which may differ from the distribution of the numberof bids if, for instance, some bids are dominated. A robustness analysis using onlythree-bid auctions confirms this finding. Our second exercise studies bidders’ riskattitude as in Lu and Perrigne (2008). Treating competition as a latent variable, weprovide a robust approach for bounding the CRRA parameter. Let W (cid:96) be the winning bid in auction (cid:96) and X (cid:96) = ( X (cid:96) , X (cid:96) ) be the associated auc-tioned lot covariate, which includes the appraisal value and volume. Being agnosticabout the competition, we consider only a winning bid sample computed from the twosamples with two and three bids. The number of bidders is considered as unobservedheterogeneity and the sample size is L = 215 . We first estimate the lowest numberof bidders n and then focus on discontinuities in the conditional winning bid pdf andon the conditional distribution of N . The lowest number of bidders n is assumed to be independent of the covariate X ,but the presence of X complicates its identification and estimation. Without auction-specific covariates X , G ( b ) = g nn (0)( b − v ) n (1 + o (1)) when b decreases to v . We canestimate n as a lower tail index of the cdf G ( · ) , see Lemma 2.1-(iii). In this case, theHill estimator as in Hill and Shneyerov (2013) can be applied. With auction-specificcovariates X , n can be identified as a conditional lower tail index when b decreases42o the common conditional lower bound v ( X ) G ( b | X ) = g nn (0 | X ) ( b − v ( X )) n (1 + o (1)) . However, the standard Hill estimator may not be consistent except when v ( · ) isFigure 4: Hill estimators (cid:101) n : two-bid and three-bid auctionsconstant. To address this issue, we first estimate the lower bound v ( X ) and use it tonormalize the winning bids. Applying to the normalized sample of winning bid, theHill estimator is consistent.Note that auction theory predicts that the lower bound of value v ( X ) is the lowerbound of the conditional bid distribution. Recall that auction-specific covariatesconsists in the appraisal value and volume, say X (cid:96) = ( X (cid:96) , X (cid:96) ) . For a growinginteger number K = K L and k = 0 , . . . , K , let ˆ X ( k/K ) ˆ X (cid:96) ( k/K ) be the samplequantiles of order k/K of X (cid:96) and X (cid:96) . Let X (cid:96)K be the set ( ˆ X (( k − /K ) , ˆ X ( k/K )] × ( ˆ X (( k − /K ) , ˆ X ( k/K )] to which X (cid:96) belongs and set (cid:98) v (cid:96) = min (min { B ij , i = 1 , and X j in X (cid:96)K } , min { B ij , i = 1 , , and X j in X (cid:96)K } ) Note that the lower boundary estimator (cid:98) v (cid:96) depends upon the bids and not the winning bids,which is permitted because the aim here is to analyze competition and not to recover primitives
43e use these lower bound estimates to normalize the winning bids as W † (cid:96) = W (cid:96) / (cid:98) v (cid:96) min ≤ (cid:96) ≤ L ( W (cid:96) / (cid:98) v (cid:96) ) − . The proposed estimator of the lowest number of bidders (cid:98) n is a rounding of the tailindex Hill (cid:101) n estimator of Hill and Shneyerov (2013) using W † (cid:96) (cid:101) n = ln W † ( M ) − M − M (cid:88) m =2 W † ( m ) , M = M L ≥ with M L = o ( L ) . (28)The lowest number of bidders estimator (cid:98) n is equal to the integer number k whenever k − . < (cid:101) n ≤ k + . . Figure 4 reports Hill estimation values (cid:101) n when M ranges from to and K = 2 in solid line, with confidence interval bounds in dashed line.The most common estimate of the lowest number of bidders (cid:98) n is (cid:98) n = 2 , with (cid:98) n = 1 for only small M yielding noisy (cid:98) n , or large M which gives biased estimates. Hence itseems very likely that n = 2 for this sample. The presence of conditioning variables also affects de-tection discontinuities. In principle, the conditional winning pdf g ( ·| X ) can be esti-mated using observations with covariate X (cid:96) close to X . In view of our small samplesize, we adopt a data-driven approach and consider three subsamples labelled "Low","Medium" and "High" defined as follows: • Low: auctions with appraisal value and volume both smaller than their medianvalues (45 auctions, among which 23 have two bidders); • Medium: auctions with appraisal value and volume both between their and quantiles (53 auctions, among which 28 have two bidders); from winning bids. Using the winning bids to estimate (cid:98) v (cid:96) gives normalized winning bids which aretoo concentrated. Alternative Hill estimation procedures for conditional tail index, which does notuse such normalization, can be found in Gardes and Stuffler (2014) and the references therein. High: auctions with covariates above their median values (44 auctions, amongwhich 20 have two bidders).Although these subsamples are small, detecting discontinuities and their locationscan be done with super efficient rates, see Oudshoorn (1998) and Gayraud (2002).
Detection of discontinuities.
We propose in the appendix a multiple testingapproach, inspired by Chu and Cheng (1996) and Oudshoorn (1998), for detectingdiscontinuities, locate them and compute a corresponding discontinuous conditionalpdf estimator. To cope with an irregular distribution of the winning bids acrossthe straight line, we adopt a k-nearest neighbor (k-NN) approach in place of Chuand Cheng (1996) kernel approach. The idea is to conclude a point of discontinuitywhen the difference between the densities on the two sides of this point is largerthan a threshold. Naturally, this threshold depends on the magnitude of the densityfunction in the neighborhood.Implementing the discontinuity detection algorithm gives a unique discontinuityinside the subsample support for each subsample, so that (cid:98) n = 3 in all cases. Figure 5reports the discontinuous conditional density estimators obtained for each subsample.In each case, the estimated discontinuity (cid:98) ∆ , in the middle of the subsamples, is quitelarge compared to the end discontinuity (cid:98) ∆ . Table 1 gives, for each subsample, theestimation of the probability that N = 2 derived from the discontinuities in thewinning bid pdf using the conditional version of (12) (line "From W ") and using thenumber of bids observed in the sample (line "Observed"). The value inferred fromthe estimated winning bid distribution is much higher than the one obtained from thebid distribution, consistently across subsample. This suggests that, for most auctionswith three reported bids, only two bidders contribute to the winning bid while theremaining one makes a dominated bid. 45igure 5: Conditional density estimation for ’Low’, ’Medium’ and ’High’ subsamples We provide further evidences on the lack of competition by repeating the above esti-mation procedure on the sample of three-bid auctions. First, we estimate the lowestnumber of bidders. Figure 6 displays the results of the Hill estimation procedure forthis sample. Despite the maximum bids W (cid:96) are now computed using only three-bidauctions, Figure 6 supports an estimate (cid:98) n = 2 for the lowest number of bidders. Thisis a first evidence suggesting that one bidder may not contribute to the winning bidin this dataset. Second, we estimate the probabilities of N = 2 from pdf disconti-nuities obtained as in the previous section for each subsample ’Low’, ’Medium’ and’High’. Table 2 reports the estimates. The results are qualitatively similar to theones in Table 1, except for the ’Medium’ sample for which the estimated probability46ow Medium HighFrom W .95 .90 .92Observed .51 .53 .45Table 1: Estimated P ( N = 2 | A ) using both two-bid and three-bid auctionsFigure 6: Hill estimators (cid:101) n : three-bid auctionsof having two bidders contributing to the winning bid is lower but still higher than . . However, the variance of these estimations are likely to be very high due to theirnonparametric nature and small sample sizes. Therefore, the estimation (cid:98) n = 2 forthe lowest number of bidders derived from Figure 6 may be a stronger evidence. Thisfinding can be explained by the presence of a passive or dominated bidder in manyauctions. More advanced explanations are however outside the scope of this paper.47ow Medium HighFrom W .98 .75 .91Table 2: Estimated P ( N = 2 | A ) using three-bid auctions Bidders’ risk aversion behaviors in auctions have attracted much attention in the lit-erature. A commonly adopted specification is the Constant Relative Risk Aversion(CRRA) U ( x ) = x θ . While Campo et al. (2011) showed that θ is not identified whenparticipation is endogenous, Guerre et al. (2009) obtained risk-aversion identificationwhen the exogenous participation is observed. Using USFS timber auction data withboth first-price and ascending auctions, Lu and Perrigne (2008) obtained estimatesof θ in the range of . . Using semiparametric restrictions for the bid distribution,Campo et al. (2011) obtained estimates of θ between . and . from a sampleof USFS first-price auctions. All these works assume that all bids are equilibriumoutcomes, whose failure leads to biased estimates of risk aversion parameters. Allow-ing for auction unobserved heterogeneity may however lead to conclude in favor ofrisk-neutral bidders, as in Grundl and Zhu (2019) who also consider USFS first-priceauctions.Hereafter, we assume the competition level N is exogenous and unobservable tothe analyst. Following the same lines as in our benchmark case, we derive lowerbounds for the CRRA parameter θ using the winning bid distribution. Suppose that Relaxing exogeneity is possible with entry models as seen from Gentry, Li and Lu (2017). Theseauthors assume that all bids are equilibrium outcomes, which, as argued below, is an assumptionthat can be questioned for the considered samples. Indeed, under CRRA, the above estimation (cid:98) n = 2 in the sample with three bids auctions remains valid, suggesting that one bid is not competitive inmany auctions. = n and observe that the expected utility generated by a bid B n ( a ) for a privatevalue V ( α ) is ( V ( α ) − B n ( a )) θ a n − = (cid:104) ( V ( α ) − B n ( a )) a n − θ (cid:105) θ when the bidder’s utility function is CRRA. The latter expression shows that, forequilibrium bids, α must maximize a (cid:55)→ ( V ( α ) − B n ( a )) a n − θ . Therefore, introducingrisk-aversion here amounts to change n − , i.e., the number of opponents of the risk-neutral case, into ( n − /θ . Arguing as for (3), it follows that the equilibrium bidquantile function now becomes B n ( α ) = ( n − /θα n − θ (cid:90) α t n − θ − V ( t ) dt = V ( α ) − (cid:90) α (cid:18) tα (cid:19) n − θ V (1) ( t ) dt. The last expression of B n ( · ) shows that the maximum bid B n (1) increases with n . As(2) becomes V ( α ) = B n ( α ) + θαB (1) n ( α ) / ( n − , the conditional bid pdf g n ( · ) is con-tinuous and bounded away from . Arguing as for Corollary 2.1 yields that g n ( v ) > ,implying that our Hill estimator is still consistent under risk-aversion. Hence the es-timate (cid:98) n = 2 obtained in the last subsection still applies under risk-aversion. In otherwords, one shall not treat all bids as equilibrium outcomes. Similarly, discontinuitiesof the winning bid identifies the largest number n of bidders, which amounts to under risk-neutrality.We now show how to derive bounds for the CRRA parameter θ , despite beingagnostic about N . Accounting for risk-aversion in formula (7) of Corollary 2.1 and inthe jump formula (9) yields that g n (cid:0) b n (cid:1) = θ ( n − (cid:0) v − b n (cid:1) and ∆ n = nθp n ( n − v − b n ) . v and p n gives v ( θ ) = θ + (cid:80) nn = n n − n ∆ n b n (cid:80) n = nn n − n ∆ n ,p n ( θ ) = n − n ∆ n (cid:80) nk = n k − k ∆ k + 1 θ n − n ∆ n (cid:32) (cid:80) nk = n k − k ∆ k b k (cid:80) nk = n k − k ∆ k − b n (cid:33) , n = n, . . . , n. Using that v ( θ ) ≤ b n and ≤ p n ( θ ) ≤ for n = n, . . . , n gives bounds for θ thatcan be estimated. For the estimated values of n , n , b n and ∆ n , only the constraint p ( θ ) ≤ is informative. This gives, for each considered subsamples, estimated lowerbounds for θ which are reported in the next table.Low Medium HighTwo and three bids .78 .48 .44Three bids .90 .25 .65Table 3: Lower bound for the CRRA parameter θ inferred from (cid:98) p ( θ | A ) ≤ Assuming that the risk aversion parameter is constant across subsamples, weobtain the best lower bound for the CRRA coefficient θ . , which is quite highrelative to the estimates reported in the literature, such as Lu and Perrigne (2008)and Campo et al. (2011). This suggests that accounting for the possibility that somebids are not necessarily equilibrium outcomes may affect risk-aversion estimation. Risk-neutrality becomes more plausible when assuming that N is not observed, ie Note also that the CRRA lower bounds of Table 3 could be, in theory, improved. Indeed, anestimated upper bound (cid:98) p ( A ) can be obtained by dividing the number of winning bids below theestimated discontinuity (cid:98) b , as reported in Figure 5 by the considered subsample size and by usingthe inequality (cid:98) p ( θ | A ) ≤ (cid:98) p ( A ) instead of (cid:98) p ( θ | A ) ≤ to obtain a lower bound for θ . This wouldlead to replace the lower bound . by . , but, when providing a meaningful bound, would give arisk-aversion larger than in many other cases. Such finding is not so surprising in view of the smallsample sizes. Guerre and Gimenes (2019) also report that risk-aversion can be difficult to estimate. This paper shows that, under the independent symmetric private value paradigm, thefirst-price auction winning bid is sufficient to identify model primitives when buyersobserve competition. The case where buyers do not observe competition is moredifficult, but still feasible with an instrument when participation is restricted by areserve price or entry cost. Whether entry is constrained by a reserve price imposedby the seller, or by an entry cost can be tested.An empirical application on USFS timber first-price auction winning bids has il-lustrated the usefulness of our theoretical results for competition analysis. A disconti-nuity detection procedure has found discontinuities in the winning bid pdf, suggestingthat bidders observe competition according to our theoretical findings. Investigatingfurther the distribution of the number of competitive bidders reveals that most auc-tions with three bids include a dominated bidder, who does not contribute to thewinning bid. Relaxing risk-neutrality by assuming a Constant Relative Risk Aver-sion (CRRA) utility function leads to a similar conclusion. Deriving bounds for theCRRA parameter from the winning bid illustrates the impact of ignoring dominatedbidding on risk-aversion estimation: risk-neutrality looks much more plausible usingour approach which tackles with this issue than found in previous studies.Our theoretical findings could be the source of many further developments. Dis-criminating the various possible explanations for dominated bidding, such as a poorunderstanding of the auction rules or collusion among others, would be very inter-esting. Econometric methods for modeling the participation decision based upon the51inning bid can also be useful for data that can be analyzed through the auctionmachinery. The statistical methods used in this paper for competition analysis canprobably be refined. Estimating the private value distribution from winning bidsnecessitates to extend estimation methods for irregular models as proposed in Hi-rano and Porter (2003) or Chernozhukov and Hong (2004) to allow for many densitydiscontinuities.Further, our theoretical results shed light on new identification arguments for dis-crete mixture models, which are widely used in economic applications, in particularwhen unobserved heterogeneity is plausible. In our model, the mixture componentsare generated by the same function. The components are ordered according first-orderstochastic dominance and their supports are nested. These two features may appearin other relevant economic mixtures. See for instance An (2017) who studied nonequilibrium bids from heterogeneous agents whose beliefs follow from level k think-ing, and where k is unobserved. A key ingredient is that these support componentscan be identified, here through discontinuities of the mixture pdf but many othercharacteristics can be used for such a purpose. Consider (i) first. Since α k = G n [ B n − ( α k − )] with B n − ( α ) ≤ B n ( α ) , α k = G n [ B n − ( α k − )] ≤ G n [ B n ( α k − )] = α k − , which implies that α k decreases. Moreover, β k = B n − ( α k ) decreases because B n − ( · ) is strictly increasing. Since α k ≥ , α k converges to a limit α which satisfies α = G n [ B n − ( α )] under Assumption IPV. In other words, the limit α satisfies B n ( α ) = B n − ( α ) . This gives α = 0 as B n ( α ) > B n − ( α ) except for α = 0 .52ow, consider (ii). That α k is identified for all k will follow from an inductionargument, observing α is identified. Suppose then that α k and { V ( α ) , α ∈ [ α k , } are identified. Recall α k +1 = G n ( β k ) = G n [ B n − ( α k )] , β k +1 = B n − ( α k +1 ) . Then (4) and Lemma 2.2 give that { B n ( α ) ; α ∈ [ α k , } , for all n = n, . . . , n − areidentified, as β k . Now (13) and Lemma 2.2 show that G n ( b ) is identified for all b ≥ β k ,and then α k +1 = G n ( β k ) is identified. (2) then gives that { V ( α ) ; α ∈ [ α k +1 , } isidentified. This ends the proof of the Lemma. (cid:3) Suppose that the transaction price is generated from a first-price auction with areserve price R and N R known by the n buyers and such that Assumption IPVholds. Then q in (i) satisfies q = F ( R ) , R is the unique real number such that G ( W = R ) > , and N R has a Binomial distribution with parameter ( n, F ( R )) . Thisgives G ( Not sold ) = P ( N R = 0) = q n , G ( Object sold at price R ) = P ( N R = 1) = nq n − (1 − q ) . Now, for G n ( · ) = B − n,R ( · ) , it holds for W = max i ∈N B i that G ( b ) = G ( W ≤ b | N R ≥ n (cid:88) n =2 G (cid:18) max i =1 ,...,n B i ≤ b | N R = n (cid:19) P ( N R = n | N R ≥ n (cid:88) n =2 G nn ( b ) n ! n !( n − n )! F n − n ( R ) [1 − F ( R )] n − F n ( R ) − nF n − ( R ) [1 − F ( R )] , which is (ii). G n ( · ) = B − n,R ( · ) and (15) gives (iii).53or the reverse implication, observe that (i)-(iii) identify a number of potentialbuyers n , a screening level F ( R ) , a reserve price R which is uniquely defined when < F ( R ) < , and a private value quantile function V ( α ) , where α ≥ F ( R ) , whichcan be extended to [0 , so that Assumption IPV holds. It can be easily seen from(16) that the distribution of the outcome distribution of this first-price auctions is G ( · ) . (i) follows from Theorem 2.1, which gives identification of V R ( · ) and n . For (ii),observe that the expression of G ( · ) in Proposition 3.1-(ii) holds with q = F ( R ) . Since G ( · ) is the winning bid distribution of a first-price auction with i.i.d. private valuesdrawn from F R ( · ) and independent from the number of participants N R , Lemma 2.2shows that p n ( R ) = (1 − F ( R )) n − F ( R ) n − nF ( R ) n (1 − F ( R )) ,p n − ( R ) = n (1 − F ( R )) n − F ( R )1 − F ( R ) n − nF ( R ) n (1 − F ( R )) , are identified if n ≥ . Thus the ratio np n ( R ) /p n − ( R ) is identified, and thereby F ( R ) , since F ( R ) = 11 + np n ( R ) /p n − ( R ) . This together with the expression of F R ( · ) shows that F ( v ) is identified for v ≥ R .For (iii), G ( Not sold ) = F ( R ) identifies F ( R ) and then F ( v ) is also identified for v ≥ R by (i). When n = 2 , F ( R ) cannot be recovered from G ( · ) . Since v < R , itcan be easily seen that a given F R ( · ) can be generated by two distinct { F ( v ) , v ≥ R } ,which generates the same G ( · ) . Hence { F ( v ) , v ≥ R } cannot be identified when n = 2 . (cid:3) .4 Proof of Proposition 3.2 Suppose that G ( · ) has been generated by a first-price auction satisfying AssumptionIPV, with a reserve price R but with the number of participants N R not observed bybuyers. The probability that N R = 0 is F ( R ) n ∈ (0 , if v < R < v , which is alsoequal to G ( Not sold ) . Since the B i ≥ R are given by B R ( A i,R ) , W has a continuousdistribution with support (cid:2) R, b R (cid:3) , b R = B R (1) . Since all A i,R and N R are independentand because N R has a binomial distribution with parameter ( n, − F ( R )) , the c.d.f. G ( b ) = G ( W ≤ b | Sold ) of W is given by G ( b ) = (cid:80) nn =1 P (max i =1 ,...,N R B R ( A i,R ) ≤ b | N R = n ) P ( N R = n )1 − P ( N R = 0)= n (cid:88) n =1 n ! n !( n − n )! (1 − F ( R )) n F ( R ) n − n − F ( R ) n G nR ( b )= ( F ( R ) + (1 − F ( R )) G R ( b )) n − F ( R ) n − F ( R ) n , where G R ( · ) = B − R ( · ) has support (cid:2) R, b n (cid:3) . Setting q = F ( R ) gives (i) and (ii) followsfrom (18) and (19).For the converse, observe that the reserve price is identified as B R (0) , set F ( R ) = q gives V R ( α ) = B R ( α ) + (cid:18) α + q − q (cid:19) B (1) R ( α ) n − ,F R ( · ) = V − R ( · ) and F ( v ) = (1 − F ( R )) F R ( v ) + F R ( v ) for v ≥ R . Choose v < R and note that F ( · ) can be extended to [ v, R ] to obtain a continuously differentiablec.d.f. with bounded support [ v, v ] . It can be easily seen that these primitives generatefirst-price auction best response bids compatible with the outcome distribution G ( · ) . (cid:3) .5 Proof of Lemma 3.1 It holds that g R ( b ) = 1 B (1) R [ G R ( b )] for all b in ( R, b R ) which diverges when b goes to R as B (1) R (0) = 0 and G R ( R ) = 0 . Let V R ( · ) be thecontinuously differentiable quantile function of Proposition 3.2: V R ( α ) = B R ( α ) + (cid:18) α + q − q (cid:19) B (1) R ( α ) n − . Hence, for α > , B (2) R ( α ) = ( n − V (1) R ( α ) − B (1) R ( α ) α + q − q − V R ( α ) − B R ( α ) (cid:16) α + q − q (cid:17) = ( n −
1) 1 − qq V (1) R (0) + o (1) when α → .It follows that, when α goes to and b to RB (1) R ( α ) = ( n −
1) (1 − q ) q V (1) ( R ) α + o ( α ) ,B R ( α ) = R + n −
12 (1 − q ) q V (1) ( R ) α + o (cid:0) α (cid:1) ,G R ( b ) = (cid:18) n − q (1 − q ) V (1) ( R ) (cid:19) / ( b − R ) / + o (cid:16) ( b − R ) / (cid:17) ,g R ( b ) = (cid:18)
12 ( n − q (1 − q ) V (1) ( R ) (cid:19) / ( b − R ) − / + o (cid:16) ( b − R ) − / (cid:17) , which shows < lim b ↓ R ( b − R ) / g R ( b ) < ∞ . Now, for b in ( R, b R ] , G ( b ) = [ q + (1 − q ) G R ( b )] n − q n − q n ,g ( b ) = n (1 − q ) [ q + (1 − q ) G R ( b )] n g R ( b )1 − q n , which gives the divergence result for g ( · ) . (cid:3) .6 Proof of Proposition 3.3 For (i), note that the lower bound of the support of G ( · ) is R . For (ii), consider n = 2 ,reserve price R = 1 / and private values uniform over [0 , . The next computationshows that the outcome distribution can also be rationalized with n = 4 , R = 1 / ,and a private value distribution distinct from uniform.We first compute the distribution G ( · ) generated by n = 2 , R = 1 / , privatevalues uniform over [0 , and a reserve price R = 1 / , so that F ( R ) = 1 / and F n ( R ) = 1 / . The private value quantile function is V ( τ ) = τ , V R ( α ) = (1 + α ) / and the best-response bid quantile function is B R, ( α ) = 1 + α α ) = (1 + α ) + 14 (1 + α ) , α ∈ [0 , . The corresponding continuous component G ( b ) = G ( W ≤ b | Sold ) of G ( · ) is G ( b ) = (cid:0) + B − R, ( b ) (cid:1) − − , setting B − ,R ( b ) = 0 for b ≤ / and B − ( b ) = 1 for b ≥ / . The outcome distributionis G ( not sold ) = 14 , G ( W ≤ b | sold ) = G ( b ) . If G ( · ) can be rationalized by first-price auctions with reserve price / , n = 4 and F ( R ) n = 1 / , which gives F (1 /
2) = 1 / √ , there must exist a bid quantile function B ,R ( · ) satisfying B ,R (0) = 1 / , such that G ( b ) = (cid:16) √ + (cid:16) − √ (cid:17) B − ,R ( b ) (cid:17) − − and a private value quantile function as in Proposition 3.2-(ii). Solving (cid:0) + B − ,R ( b ) (cid:1) − − = (cid:16) √ + (cid:16) − √ (cid:17) B − ,R ( b ) (cid:17) − − B ,R ( α ) = B ,R (cid:34) (cid:18) √ (cid:18) − √ (cid:19) α (cid:19) − (cid:35) = (cid:16) √ + (cid:16) − √ (cid:17) α (cid:17) (cid:16) √ + (cid:16) − √ (cid:17) α (cid:17) − , which is such that B ,R (0) = R = 1 / . This also gives B (1)4 ,R ( α ) = (cid:18) − √ (cid:19) × (cid:18) √ (cid:18) − √ (cid:19) α (cid:19) − (cid:16) √ + (cid:16) − √ (cid:17) α (cid:17) − , which satisfies B (1)4 ,R (0) = 0 . To check that B ,R ( · ) is compatible with a private valuequantile function, it remains to be checked that v ( α ) = B ,R ( α ) + √ + (cid:16) − √ (cid:17) α − √ B (1)4 ,R ( α )3= 56 (cid:18) √ (cid:18) − √ (cid:19) α (cid:19) + (cid:16) √ + (cid:16) − √ (cid:17) α (cid:17) − increases with α . Note that dv ( α ) dα = 2 (cid:18) − √ (cid:19) × (cid:18) √ (cid:18) − √ (cid:19) α (cid:19) − (cid:16) √ + (cid:16) − √ (cid:17) α (cid:17) − = 2 (cid:16) − √ (cid:17)(cid:16) √ + (cid:16) − √ (cid:17) α (cid:17) − (cid:40) (cid:18) √ (cid:18) − √ (cid:19) α (cid:19) − (cid:41) with, for all α ∈ [0 , , (cid:18) √ (cid:18) − √ (cid:19) α (cid:19) − ≥ × −
124 = 16 . It follows that v (1) ( · ) > . Recall v (0) = 1 / . Consider a c.d.f. F ( · ) over [0 , suchthat, for α ∈ (cid:2) / √ , (cid:3) , F − ( α ) = v (cid:32) α − √ − √ (cid:33) , so that F (cid:18) (cid:19) = 1 √ .
58 proper definition of F ( · ) on [0 , / ensures that F ( · ) satisfies Assumption IPV. ByProposition 3.2, G ( · ) is rationalized by a first price auction with a reserve price / and: 1) uniform private values with n = 2 ; 2) non-uniform private values with c.d.f. F ( · ) , n = 4 , and F ( R ) = 1 / √ . This establishes the non-identifiability statement in(ii).For (iii), note that < F ( R ) < . It is sufficient to show that P ( N R = 0) = F ( R ) n and P ( N R = 1) = nF ( R ) n − (1 − F ( R )) identify n and F ( R ) , as G ( · ) will in turn identify G R ( · ) and F ( v ) for all v ≥ R .Since n = log P ( N R = 0)log F ( R ) and P ( N R = 1) P ( N R = 0) = n − F ( R ) F ( R ) , it follows that F ( R ) ∈ (0 , solves − F ( R ) F ( R ) log F ( R ) = P ( N R = 1) P ( N R = 0) log P ( N R = 0) ; in other words, ϕ ( F ( R )) is identified, where ϕ ( x ) = (1 − x ) / ( x log x ) . Since, ∂ϕ ( x ) ∂x = − x log x − (1 − x ) (1 + log x )( x log x ) = − log x − ( x − x log x ) ≥ , where the inequality is strict for all x in [0 , except for x = 1 , ϕ ( · ) is a one-to-onemapping over, (0 , and F ( R ) is identified so that n = log P ( N R =0)log F ( R ) is identified. (cid:3) As R ( z ) is the lower bound of G ( ·| z ) , R ( · ) is identified. Since G ( not sold | z ) = F ( R ( z )) , n Using ∂∂x [log x − ( x − x − ≥ gives log x − (1 − x ) ≤ log 1 = 0 over [0 , . ( R ( · )) n is identified. The upper bound b R ( · ) = B R (1 | z ) of G ( ·|· ) is identified and,for all b and all z ∈ Z , ( F ( R ( z )) + (1 − F ( R ( z ))) G R ( b | z )) n is also identified. Since, when t > goes to , (cid:0) F ( R ( z )) + (1 − F ( R ( z ))) G R (cid:0) b R ( z ) − t | z (cid:1)(cid:1) n = (cid:2) − F ( R ( z ))) (cid:8) G R (cid:0) b R ( z ) − t | z (cid:1) − (cid:9)(cid:3) n = 1 − n (1 − F ( R ( z ))) g R (cid:0) b R ( z ) | z (cid:1) t + o ( t ) , the function γ ( z ) = n R (1 − F ( R ( z ))) g R (cid:0) b R ( z ) | z (cid:1) is identified over Z . Since g R (cid:0) b R ( z ) | z (cid:1) = 1 /B (1) R (1 | z ) , taking α = 1 in (18) gives g R (cid:0) b R ( z ) | z (cid:1) = 1( n −
1) (1 − F ( R ( z ))) (cid:0) v − b R ( z ) (cid:1) , and thus γ ( z ) = nn − v − b R ( z ) . Since, by (19), b R ( z ) = v − (cid:90) [ F ( R ( z )) + (1 − F ( R ( z ))) t ] n − V (1) R ( t ) dt = v − (cid:90) [ F ( R ( z )) + (1 − F ( R ( z ))) t ] n − × V [ F ( R ( z )) + (1 − F ( R ( z ))) t ] (1 − F ( R ( z ))) dt = v − (cid:90) F ( R ( Z )) u n − V (1) ( u ) du, with V (1) ( · ) > , R ( z ) (cid:54) = R ( z ) implies that b n ( z ) (cid:54) = b n ( z ) and then γ ( z ) (cid:54) = γ ( z ) . This identifies v and n as v = γ ( z ) b R ( z ) − γ ( z ) b R ( z ) γ ( z ) − γ ( z ) ,n = γ ( z ) (cid:0) v − b R ( z ) (cid:1) γ ( z ) (cid:0) v − b R ( z ) (cid:1) − . F n ( R ( · )) is identified, F ( R ( · )) is identified. Hence { G R ( b | z ) , b ≥ R ( z ) , z ∈ Z} is identified, since { B R ( α ) , α ≥ F ( R ( z )) , z ∈ Z} is identified. (18) implies that F ( v ) is identified for any v such that there is a z satisfying v ≥ R ( z ) . Then { F ( v ) , v ≥ inf z ∈Z R ( z ) } is identified by continuity of F ( · ) . (cid:3) Suppose that an entry model with entry cost satisfying Assumption E generates G ( ·| z ) . That (i)-(iii) hold can be established as in Proposition 3.1. Suppose now that(i)-(iii) hold. Let n and s ( z ) = q ( z ) be as in (i). For s in q ( Z ) , let V c ( α | s ) be as in(iii). Set F c ( ·| s ) = V − c ( ·| s ) and F ( v | s ) = − ∂∂s [(1 − s ) F c ( v | s )] which is a conditional c.d.f. over [ v, v ] × q ( Z ) , that can be extended over [ v, v ] × [0 , .Defining c ( · ) through (21) gives a cost function which rationalizes G ( ·|· ) . (cid:3) (i) follows from Theorem 2.1, observing that n is identified by the number of discon-tinuities of the p.d.f. g ( ·| z ) . The proof of (ii) starts as the proof of Corollary 3.1-(ii),using the probabilities p n ( z ) = (1 − s ( z )) n − s ( z ) n − ns ( z ) n (1 − s ( z )) ,p n − ( z ) = n (1 − s ( z )) n − s n ( z )1 − s ( z ) n − ns ( z ) n (1 − s ( z )) , which are identified from G ( ·|· ) if n ≥ to identify s ( z ) . Then F ( ·| s ) can beidentified over the identified interval s ( Z ) using F ( v | s ) = − ∂∂s [(1 − s ) F c ( v | s )] andthe identification of c ( · ) over Z follows from (21). The proof of (iii) is similar to theproof of Corollary 3.1-(iii), identifying s ( z ) from G ( not sold ) = s ( z ) . (cid:3) .10 Proof of Proposition 3.7 The proof proceeds as for Proposition 3.4. Repeating the arguments of Section 6.7gives that s n ( z ) and γ ( z ) = nn − v − b c ( z ) are identified over Z . As in the proof of Proposition 3.4, this holds provided that b c ( z ) = B c (1 | s ( z )) is a non-constant function to identify s ( · ) and n as assumed inProposition 3.7. (cid:3) Appendix: discontinuity detection algorithm
Consider a subsample A = Low, Medium, or High. Hereafter, we omit A for con-venience. Let W (cid:96) , (cid:96) = 1 , . . . , L be winning bids. First, we estimate a tentativediscontinuity at each data point as the difference between the density estimates onits left and right sides. Consider a first "small" bandwidth h , set to . in the appli-cations. The tentative discontinuity at W ( (cid:96) ) is estimated using the difference of leftand right k-NN density estimators (cid:98) δ h ( W ( (cid:96) ) ) = (cid:96) − max ( (cid:96) − h L/ , L (cid:0) W ( (cid:96) ) − W ( (cid:96) − h L/ (cid:1) − min ( (cid:96) + h L/ , L ) − (cid:96)L (cid:0) W ( (cid:96) + h L/ − W ( (cid:96) ) (cid:1) , where (cid:96) − h L/ is truncated to if negative and (cid:96) + h L/ to L if larger than L . Second, we estimate the magnitude of the density at each point and calculate a62hreshold. Define also the k-NN pdf estimator and the critical value (cid:98) g h ( W ( (cid:96) ) ) = min ( (cid:96) + h L/ , L ) − max ( (cid:96) − h L/ , L (cid:0) W ( (cid:96) + h L/ − W ( (cid:96) − h L/ (cid:1) ,C ( (cid:96) ) ( (cid:15) ; h ) = (cid:98) g h ( W ( (cid:96) ) ) c ( (cid:15) ; h ) with c ( (cid:15) ; h ) = (cid:112) ln(1 /h ) + ln ln(1 /h ) − ln( π ) + 2 (cid:15) (cid:112) ln(1 /h ) and where (cid:15) = (cid:15) L goes to with L , and is set to . here.We now use the tentative discontinuity estimates and thresholds to estimate lo-cations of jump points and jump sizes. If (cid:98) δ h ( W ( (cid:96) (cid:63) ) ) = max ≤ (cid:96) ≤ L (cid:98) δ h ( W ( (cid:96) ) ) is smaller than the critical value C ( (cid:96) (cid:63) ) ( (cid:15) ; h ) then the conditional winning bid pdf g ( · ) has no discontinuities. Otherwise a discontinuity is found at W ( (cid:96) (cid:63) ) , with an estimatedjump (cid:98) δ h ( W ( (cid:96) (cid:63) ) ) . The next jump will be searched using the same procedure butexcluding the indexes (cid:96) between (cid:96) (cid:63) − h L/ and (cid:96) (cid:63) + h L/ . The procedure is theniterated until iteration (cid:98) q , such that the potential jump is smaller than C ( (cid:96) (cid:63) (cid:98) q ) ( (cid:15) ; h ) .The number of jumps is then (cid:98) q − , so that the estimation of the largest number n of bidders is (cid:98) n = (cid:98) n + (cid:98) q . Ordering the jumps locations W ( (cid:96) (cid:63)q ) gives an estimation ofthe conditional bid support boundary (cid:98) b (cid:98) n + q and of the discontinuity jumps (cid:98) ∆ (cid:98) n + q . Aconditional winning bid density estimator incorporating discontinuities is then, for (cid:96) (cid:63)q < (cid:96) ≤ (cid:96) (cid:63)q +1 with (cid:96) (cid:63) = 1 , (cid:98) g dh ( W ( (cid:96) ) ) = min (cid:0) (cid:96) + h L/ , (cid:96) (cid:63)q +1 (cid:1) − max (cid:0) (cid:96) − h L/ , (cid:96) (cid:63)q + 1 (cid:1) L (cid:16) W ( min ( (cid:96) + h L/ ,(cid:96) (cid:63)q +1 )) − W ( ( max ( (cid:96) − h L/ ,(cid:96) (cid:63)q +1 )) (cid:17) , where the bandwidth h > h is set to . in our application. Other values of (cid:98) g dh ( · ) are then obtained by linear interpolation. Note that the critical value C ( (cid:96) ) ( (cid:15) ; h ) is proportional to the estimated pdf, as standard for k-NNestimation. This contrasts with the square root estimated pdf used in Chu and Cheng (1996) fortheir kernel approach. eferences Abrantes-Metz, R. & P. Bajari (2009). Screens for conspiracies and their mul-tiple applications.
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