Econometric Models of Ascending Auctions - CiteSeerX

Econometric Models of Ascending Auctions  Han Hong Department of Economics Princeton University

Matthew Shum Department of Economics University of Toronto

April 1999

Abstract We develop general econometric models of ascending (English) auctions which allow for both bidder asymmetries as well as common and/or private value components in bidders' underlying valuations. We show that the equilibrium inverse bid functions in each round of the auction are implicitly de ned (pointwise) by a system of nonlinear equations, so that conditions for the existence and uniqueness of an increasing-strategy equilibrium are essentially identical to those which ensure a unique and increasing solution to the system of equations. We exploit the computational tractability of this characterization in order to develop an econometric model, thus extending the literature on structural estimation of auction models. An empirical example is provided which illustrates how equilibrium learning aects bidding during the course of the auction. JEL: C51,D44, D82, L96 Keywords: Ascending auctions, Asymmetric information, Simulation estimation

 We thank Takeshi Amemiya, Tim Bresnahan, Frank Diebold, Tony Lancaster, Tom MaCurdy, Steven

Matthews, George Mailath, Paul Milgrom, Peter Reiss, Jacques Robert, Frank Wolak, Ken Wolpin and participants in seminars at Stanford University, the University of Pennsylvania, the University of Toronto, the 1997 European Econometric Society Meetings in Toulouse and the 1998 North American Summer Meetings of the Econometric Society in Montreal for their comments.

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In this paper, we develop a general econometric framework for estimating structural models of asymmetric ascending (English) auctions. In these auctions, the bidding process is modeled as a multi-stage game in which bidders acquire more information during the course of the auctions as rivals drop out of the bidding. Equilibrium learning is a feature of these dynamic games, in contrast to the other familiar auction formats (including the rst- or second-price auction), which are static single-stage games which oer participants no opportunity to gain information during the course of the auction. Many real-world auction mechanisms | from art and collectibles auctions to the Japanese \button" auction cited by Milgrom and Weber1 | resemble the auctions we study, and perhaps these mechanisms arose to allow for the possibility of information acquisition. Two concerns | one theoretical, and one empirical | motivate this paper. First, the theoretical literature on ascending auctions (including the paradigmatic model presented in Milgrom and Weber (1982) 2 ) has focused primarily on symmetric models, in which the bidders' signals about the value of the object are assumed to be generated from identical distributions. However, the empirical importance of asymmetric information in applied research on auctions (eg. work by Hendricks and Porter (1988) on oshore gas auctions, and by Klemperer (1997) on the PCS spectrum auctions), indicates that symmetry may not be a realistic assumption for many real-world situations. For these reasons, we extend the theoretical results of the ascending auction model to accommodate bidder asymmetries, and develop an econometric framework which can be used in applied analyses. In the rst part of the paper, we extend the Milgrom{Weber framework by characterizing Bayesian-Nash equilibrium bidding behavior in asymmetric ascending auctions. This complements recent work (Maskin and Riley (1996), Bulow, Huang, and Klemperer (1996), Bajari (1996a), Bajari (1996b)) on asymmetric rst-price auctions. We nd that the equilibrium bid functions in each round of an ascending auction exhibit an attractive analytic property. Speci cally, the inverse bid functions3 are implicitly de ned by a system of nonlinear equations, pointwise in the bids. Therefore, conditions for the existence and uniqueness of an increasing-strategy equilibrium are essentially identical to those which ensure a unique solution to the system of equations, given primitive model assumptions about the joint distribution of the bidders' underlying valuations and private signals. We provide sucient conditions for the existence of a unique increasing-strategy equilibrium for an important subset of models, those with separable private and common value components. Furthermore, this attractive analytic property also facilitates numerical calculation of

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the equilibrium bidding strategies. This was recognized by Wilson (1998), who analytically derives the equilibrium bid functions for a log-additive asymmetric ascending auction model. In the second part of the paper, we develop an econometric model of the asymmetric ascending auction for this log-additive case. This extends the scope of the literature on the structural estimation of auction models (Paarsch (1992) and Laont, Ossard, and Vuong (1995) to multi-round and asymmetric models. We provide an empirical illustration of this model by estimating it using data from the PCS spectrum auctions run by the U.S. Federal Communications Commission (FCC). While our model accommodates the multiple-round aspect of these auctions, it does not include other essential details, such as the multiple-object aspect and the exible eligibility rules. Therefore, we view the main purpose of this example as illustrating and suggesting solutions to problems which arise in estimating this model in practice, rather than providing robust empirical ndings concerning the FCC auctions.4 We present estimated bid functions which illustrate how equilibrium learning aects bidding behavior during the course of an ascending auction. Finally, the empirical framework developed in this paper can be generalized to other auction formats, including the rst- and second-price auctions. The common element in all these auction models is that the equilibrium bid functions are described by systems of equations, which facilitates the numerical or computational algorithms required for empirical implementation of these models. We start, in the next section, by describing equilibrium bidding behavior in the asymmetric ascending auction in intuitive fashion, in part by discussing the analogies with bidding in a strategically less complex second-price auction. In Section 3, we formally characterize Bayesian-Nash equilibria in the asymmetric ascending auction, and derive sucient conditions for the important separable case. In Section 4 we develop an econometric model based on a log-normal speci cation of the auction model, and discuss estimation issues in Section 5. Section 6 contains the empirical example, and Section 7 concludes.

1 Bidding behavior in multiple-round auctions Consider an auction in which N bidders compete for the possession of a single object. Before the auction begins, each bidder i (i = 1; : : : ; N ) observes a private signal Xi concerning the value of the object up for sale. The actual value of the object to bidder i will be denoted by Vi , but this value is not known to bidder i. We assume that Vi is correlated with the rival bidders' signals Xj ; j 6= i, which implies that knowledge of Xj , bidder j 's signal, would be 3

useful to bidder i in predicting the unknown value Vi . The auction format which we focus on in this paper is an asymmetric version of the \irrevocable dropout" auction described in Milgrom and Weber (1982), pg. 1104: Initially, all bidders are active at a price of zero. As the auctioneer raises the price, bidders drop out one by one. No bidder who has dropped out can become active again. After any bidder quits, all remaining active bidders know the price at which he quit. By observing the dropout prices, remaining bidders can subsequently infer the private information possessed by the bidders who have dropped out. This information aects the behavior of the bidders only insofar as the object up for sale has a common value (CV) component, which is the same for all bidders. This common component leads to correlation among the signals which the bidders privately observe which, in turn, makes bidder j 's signal Xj useful to bidder i in estimating Vi , his valuation of the object.5 The winner's curse suggests that naive bidders who simply bid their expected valuation of the object conditional only upon their private signal will tend to gain a negative expected pro t. In equilibrium, a bidder needs to take into account the extra ex post information he would gain about losing bidders' private signals were he to win the auction.6 In this section we mainly develop the intuition of the decision problem faced by the bidders as speci ed by the equilibrium bid functions for the asymmetric ascending auctions. A formal equilibrium proof is presented in the next section. Throughout this paper, we restrict our attention to increasing strategy equilibria where each bidder i's equilibrium bidding strategy bi (Xi ) is a strictly increasing functions of Xi .7 Furthermore, we maintain two important assumptions concerning the joint distribution of the valuations and private signals (V1 ; : : : ; VN ; X1 ; : : : ; XN ) of the bidders in the auction: A1 All the bidders' valuations (Vi ; i = 1; : : : ; N ) and private signals (Xi ; i = 1; : : : ; N ) are drawn from distributions which share some region of common support. This rules out extremely asymmetric cases where, ex ante, conditional on all possible realizations of X , Vj is almost surely lower than Vi . Bidder j will never win the auction in this case and, indeed, has little incentive to participate in the auction at all.8 A2 The conditional expectation E [Vi j X1 ; : : : ; XN ] is strictly increasing in Xi . This is a weaker assumption than aliation (cf. Milgrom and Weber (1982)). Note that if (V1 ; : : : ; VN ; X1 ; : : : ; XN ) are assumed to be aliated, the conditional expectation E [Vi j X1 ; : : : ; XN ] will be nondecreasing in all Xj , j = 1; : : : ; N . 4

In the rest of this section and most of the next section, we motivate and characterize equilibrium bidding strategies in an arbitrarily general asymmetric auction where the joint distribution of (V1 ; : : : ; VN ; X1 ; : : : ; XN ) satis es assumptions A1 and A2. Later, we place additional structure on the joint distribution of (V1 ; : : : ; VN ; X1 ; : : : ; XN ) and demonstrate existence and uniqueness of equilibrium for a special but important case of the asymmetric ascending auction | that in which bidder i's valuation of an object is separable into private and common value components.

1.1 Equilibrium bidding in the ascending auction Before proceeding, let us introduce the indexing convention we will follow in this paper. The ascending auction proceeds in rounds. It enters a new round whenever another bidder drops out. N bidders are present in the auction; there will be N ? 1 \rounds" in the auction, indexed k = 0; : : : ; N ? 2. In round 0, all N bidders are active; in round k, only N ? k bidders are active. Each round ends when a bidder drops out; bidders are indexed by i = 1; : : : ; N . Without loss of generality, the ordering 1; : : : ; N indicates the order of dropout. In other words, bidder N drops out in round 0, and bidder 1 wins the auction; generally, bidder N ? k drops out at the end of round k. The dropout prices are indexed by rounds, i.e., P0 ; : : : ; PN ?2 . To sum up, bidder i drops out at the end of round N ? i, at the price PN ?i . Equilibrium bidding strategies in the ascending auction game specify, for each bidder i, bid functions ik (Xi ) for each round k, k = 0; : : : ; N ? 2, i.e., i0 (Xi ); : : : ; iN ?2 (Xi ). Given a realization of the private signal Xi , the bid function ik (Xi ) tells bidder i which price he should drop out at during round k. The collections of bid functions i0 (Xi ); : : : ; iN ?2 (Xi ) for bidders i = 1; : : : ; N are common knowledge. Consider bidder i, who is active during round k. As of round k, bidders N ? k +1; : : : ; N have already dropped out, at prices Pk ; : : : ; P0 , respectively. Since the equilibrium bid functions are common knowledge, bidder i can use this information on the identity of the dropout bidders and their dropout prices to infer the private signals XN ?k+1 ; : : : ; XN observed by these bidders by inverting these bid functions: Xj = ( jN ?j )?1 (PN ?j ), for j = N ? k + 1; : : : ; N . Bidder i observes the price level rising; at each price level P he asks himself: would I regret winning the object at price P ? Intuitively, he won't regret winning so long as his ex post expected valuation of the object exceeds P ; alternatively, his expected pro t from winning the object at price P should be nonnegative. What is his expected pro t from winning the object at price P ? In round k of the 5

ascending auction, bidder i wins at price P only when all the other active bidders in this round (bidders 1; : : : ; N ? k) simultaneously drop out at this price. But if this happens, then bidder i gains the ex post information that all the other bidders observed private signals which would have made them drop out at price P . In other words, since the equilibrium strategies are common knowledge, bidder i can infer that the other bidders' signals must be Xj = ( jk )?1 (P ), for j = 1; : : : ; N ? k and j 6= i. 9 The expected pro t to bidder i when he wins the auction at price P during round k is therefore E [V j X ; X = ( )?1 (P ) ; j = 1; : : : ; N ? k; j = 6 i; ? ? 1 X = ( ) (P ? ) ; j = N ? k + 1; : : : ; N ] ? P: If this is positive, then bidder i doesn't mind being active at price P . On the other hand, if this is negative, then bidder i shouldn't be active at price P , during round k. In equilibrium, this expected pro t is positive when P is small and negative when P is large.The price P at which he should quit the auction during round k, de ned as his bid function for round k, is the price ik (Xi )  Pk at which he will have a zero expected pro t in round k if all other active bidders simultaneously quit at the same price:   P  = (X )  P : E V j X = ( )?1 (P ); X = ( )?1 (P ); j = 1; : : : ; N ? k; j = 6 i; i o (1.1) ? X = ( )?1 (P ? ); j = N ? k + 1; : : : ; N ? P  = 0: i

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In equilibrium, Pk is unique for each bidder i, and for each round k.10 The properties of the expected bid function is illustrated in gure 1; the expected pro t function intersects the x-axis at the unique price Pk . Analogous expressions to equation (1.1) de ne the equilibrium bidding strategies for the active bidders i = 1; : : : ; N ? k during round k. These equations constitute a system of N ? k nonlinear equations, with N ? k unknowns ( ik )?1 (P ) ; i = 1; : : : ; N ? k. Therefore we can solve for the equilibrium inverse bid functions from these equations, pointwise in P . From an analytic point of view, this feature is very attractive, since it implies that the conditions for existence and uniqueness of equilibrium in this auction model are identical to those which ensure a unique solution to a system of nonlinear equations. These conditions can be constructed on a base-by-case basis, depending on the researcher's assumptions regarding the joint distribution of (V1 ; : : : ; VN ; X1 ; : : : ; XN ). Furthermore, this feature also plays an important role in the econometric model we develop below. [Figure 1 about here.] The full set of equilibrium bid functions is analogously described by sets of N ? k equations for each round k = 0; : : : ; N ? 1.11 6

From a computational point of view, the structure of these equilibrium bid functions is particularly attractive since the conditioning events are points rather than sets, which would have been the case for asymmetric second price auctions. In the case of the latter, the conditional expectations in (1.1) would involve multi-dimensional integrals, which can be dicult to compute.

2 Equilibrium in multiple-round auctions: formal characterization In this section we will provide sucient assumptions on the joint distribution of the signals X and valuations V to ensure a unique Bayesian Nash Equilibrium with strictly increasing bidding strategies in the ascending auctions. In the previous section, we labeled bidders by their dropout order, so that bidder i dropped out at the end of round N ? i. In the following characterization, however, we need to make a distinction between the (arbitrary) bidder labels and the order in which they drop out, since the dropout order is random depending on the realization of bidders' private signals. Therefore in this section let i = 1; : : : ; N , be the xed labels attached to the bidders and let i1 ; : : : ; iN denote the dropout order, with i1 indexing the rst dropout bidder and iN the winner. The random vector (i1 ; : : : ; iN ) can equal any of the possible permutations of (1; : : : ; N ). Before stating the assumptions we introduce the some notation. De ne Vi (X1 ; : : : ; XN )  E (Vi j X1 ; : : : ; XN ) ; i = 1; : : : ; N , the N expectations of the values of the object to all bidders conditional on the vector of realized private signals of bidders (X1 ; : : : ; XN ). For any round k (0  k  N ? 2) and for any permutation (i1 ; : : : ; iN ) of (1; : : : ; N ), x Xi1 ; : : : ; Xi (the private signals corresponding to the bidders who have already dropped out prior to round k) at any possible set of realized private signals. The N ? k conditional expectations for the N ? k bidders active in round k constitute a system of N ? k equations with N ? k unknowns
? V k+1 X k+1 ; : : : ; X N ; X 1 ; : : : ; X k = P (2.2) :::
? V n X k+1 ; : : : ; X N ; X 1 ; : : : ; X k = P where Xi +1 ; : : : ; Xi are the unknown variables and p is taken as the parameter. Consider the following two assumptions, k

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E1 For all 1  i; j  N , i 6= j ,

V (X; : : : ; X ) = V (X; : : : ; X )  V i

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V X; : : : ; X = V X; : : : X  V



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We refer to V ; V as the common range of bids. E2 The solution of the N ?? k unknown variables in equations (2.2) are unique and strictly
increasing for all P 2 V ; V , the common range of bids. Conditions E1 and E2 lead to an equilibrium proposition for the English auction.

PROPOSITION 1 If both assumptions E1 and E2 are satis ed, there exists a Perfect

Bayesian equilibrium for the English auction game. The equilibrium strategies are recursively de ned. At round k, let i ;k (P ; Pi1 ; : : : ; Pi ) ; k + 1  j  N solve the system of equations (2.2). De ne the inverse functions of i ;k ; k + 1  j  N with respect to their rst arguments: j (x; P 1 ; : : : ; P k ) = ?j 1 (x; P 1 ; : : : ; P k ), for k +1  j  N . The collection of bid functions i (
) ; k + 1  j  N for 0  k  N ? 2 constitute a Perfect Bayesian Nash equilibrium of the English auction game. j

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The proof is essentially based on the intuitive arguments presented in the previous section and can be found in the appendix.

2.1 Separable asymmetric ascending auction models As noted above, conditions under which the equilibrium bid functions in the ascending auction game are unique and strictly increasing are equivalently conditions which ensure uniqueness and monotonicity of the solutions to the system of equations ( 2.2). In contrast to symmetric models (such as the ascending auction model presented in Milgrom and Weber (1982)), general aliation assumptions on the joint distribution of (V1 ; : : : ; VN ; X1 ; : : : ; XN ) are not sucient to ensure (E2) for the asymmetric case. For the remainder of this paper, we restrict our attention to a class of joint distributions of (V1 ; : : : ; VN ; X1 ; : : : ; XN ) in which the value of the object to each bidder is separable into common and private value components. For example, consider the additive case Vi = Ai + V where Ai bidder i's private value and V is the component of bidder i's valuation which is common to all bidders.12. De ne for 1  i  N :

A (X1 ; : : : ; X ) = E (A j X1 ; : : : ; X ) V (X1 ; : : : ; X ) = E (V j X1 ; : : : ; X ) V (X1 ; : : : ; X ) = A (X1 ; : : : ; X ) + V (X1 ; : : : ; X ) i

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(2.3) (2.4)

In particular, The following two assumptions are imposed on this additive structure to ensure that (E2) is satis ed: F1 @ V =@X > 0, @ A =@X > 0 for 1  i  N . @ A =@X  0; j 6= i. i

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@ A > X d @ A F2 Diagonal dominance; There exist numbers d > 0; 1  j  N , such that d @X @X 6 = for j = 1; : : : ; N . j

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Assumption (F1) states that a bidder's signal is positively related to the common value component and to his own private value components. It also states that taking his signal as given, another bidder's signal is negatively related to his private value, @ A~ i =@Xj  0; j 6= i. Loosely speaking, a signal level of Xi gives bidder i an average estimate of Vi . Given this estimate of Vi , a higher signal level of X?i implies a higher level of the common value V , which in turn implies a lower level of the private value component Ai . A sucient condition for F1 is given in the following lemma, the proof of which can be found in the appendix:

LEMMA 1

Suppose A and V are independent, A is independent from A for i 6= j , the signals A is conditionally independent conditional on A + V . If the (log) conditional density of X given A + V is concave in A + V , then @ A =@X  0; j 6= i. i

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In particular, if conditional on Ai + V , Xi is normally distributed with mean Ai + V , then log fX jA +V (
) is concave in Ai + V . Assumption (F2) is a condition of diagonal dominance. If we let D be the diagonal matrix with Dii = di , and let A be the Jacobian matrix [@ Ai =@Xj ; 1  i; j  n], then each column of DA sum to a positive number. We interpret this as saying that the eect of Xi on his own private evaluation Ai dominates its eect on other bidders' private evaluations Aj .13 In particular, this assumption is satis ed by the log normal model, which is described in the next section. i

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PROPOSITION 2 Assumptions F1 and F2 implies E2. Proof: See appendix.

3 Log normal asymmetric ascending auction model An econometric ascending auction model would use observed dropout prices to recover the parameters of the equilibrium bid function, which in turn depend on the preferences for the objects being sold, which vary across bidders. Formulating such a model involves rst and 9

foremost specifying distributions on the latent signals and valuations X1 ; :::; X ; V1 ; : : : ; V such that the resulting mapping (i.e., the equilibrium bid function) from observed dropout prices to unobserved valuations is tractable enough to apply data to and estimate. Diculties arise in doing this because the updating process introduces a large amount of recursivity into the de nition of the bid function. For example, assume four bidders (A,B,C,D) and assume the rst three drop out in rounds 1,2, and 3, respectively. After bidder A drops out, the remaining bidders (B,C,D) invert the equilibrium bid function for bidder A, in order to obtain his private signal XA . In round three, bidder C and D must invert bidder B's bid function during round two, which is her expected value of the object in round two, conditional not only on her private signal but also on xA which she inferred by inverting bidder A's conditional expectation function from the previous round. The recursive structure which results (involving conditional expectations | in computational terms, integrals | functions which have as arguments inversions of other conditional expectation functions which are themselves inversions of other conditional expectation functions) quickly becomes intractable if the conditional expectations derived during the updating process become too complicated. Thus the challenge in implementing this model econometrically lies in picking a conjugate family of distributions for the latent variables (V; X ) such that the resulting conditional expectation functions have tractable functional forms which facilitate inversion. Fortunately, if we assume that bidders' valuations are log-normally distributed, we can derive closed-form formulas for the conditional expectation functions. we use as the basis of our empirical model the analytic formulas for the equilibrium bid functions for an multiplicative log-normal model, which generalizes Wilson (1998)'s noninformative prior model to an informative prior model. This section contains the main steps in the derivation of these dropout prices given the log-normality assumption. 14 Suppose N bidders are contending for an single object in an ascending auction. The value of the object to bidder i is assumed to be Vi = Ai  V . Ai is a bidder-speci c private value for bidder i. V is the common market value which is not known to any of the bidders. In other word, Vi is the product of a common value part and a private value part. The V and the Ai 's are independently log normally distributed. If we let v  ln V denote the natural log of V , and similarly let ai  ln Ai . Then N

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v = m+v  N (m; r02) ai = ai +a  N (ai; t2i ): i

Each bidder is assumed to have a noisy signal of the value of the object, Xi , which 10

has the form Xi = Ai  Ei . Here Ei is a noisy estimate of the common market value V . Ei = V  expfsi i g in which i is an (unobserved) error term that has a normal distribution with mean 0 and variance 1. If we let vi  ln Vi 15 and xi  ln Xi , then conditional on vi , xi = vi + e  N (vi ; s2i ). Note that Bidder i observes Xi , which is also known to other bidders after bidder i drops out. p Finally, de ne ri  ti + si and denote the variance for v by r0 . The joint distribution of (Vi ; Xi ; i = 1; : : : ; N ) = exp(vi ; xi ; i = 1; : : : ; N ) is fully characterized by fm; a; t; s; rg where a is the collection of ai 's, similarly for t, s, and r. These are all common knowledge among the bidders. From bidder i's perspective, it is useful to know xj because xj ? aj + ai is normally distributed with a mean of vi , which is the unknown value which bidder i wants to estimate. In this way, bidder i obtains an unbiased estimate of vi from observing bidder j 's private signal xj . i

3.1 Deriving the equilibrium bid functions Before describing the process whereby we derive the equilibrium bid function, note that given the log-normality assumptions on Vi and the Xi 's, i = 1; : : : ; N , the conditional expectation functions for Vi take the following form:   1 E [Vi j X1 ; : : : ; XN ] = exp E (vi j x1 ; : : : ; xN ) + 2 V ar(vi j x1; : : : ; xN ) ; (3.5) for i = 1; : : : ; N . Given that (vi ; x1 ; : : : ; xN ) are jointly normal, it turns out that these conditional expectation functions for Vi  exp(vi ) are log-linear in xi  log(Xi ).16 . In period k, bidders N ? k + 1; : : : ; N have dropped out, revealing XN ?k+1 ; : : : ; XN . The equilibrium conjectures stated above mean that, essentially, bidder i conjectures that Xj = ( jk )?1 ( ik (Xi )); 8 j = 1; : : : ; N ? k; j 6= i. Let ( jk )?1 (P ) indicate the inverse of jk (
) evaluated at an arbitrary price P , i.e., what bidder j 's type ( jk )?1 (P ) would be if he chooses to drop out at price P . Recall from the discussion above following equation (1.1), that the equilibrium inverse bidding strategies for the N ? k bidder active in round k are de ned by the following system of nonlinear equations: P = E [V1 j X1 = ( 1 )?1 (P ); ( 2 )?1 (P ); : : : ; ( ? )?1 (P ); X ? +1 ; : : : ; X ] P = E [V2 j ( 1 )?1 (P ); X2 = ( 2 )?1 (P ); : : : ; ( ? )?1 (P ); X ? +1 ; : : : ; X ] k

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P = E [V ? j ( 1 )?1 (P ); ( 2 )?1 (P ); : : : ; X ? = ( ? )?1 (P ); X ? +1 ; : : : ; X ]: N

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As we noted earlier, (3.6) de nes a set of (N ? k) nonlinear equations with ( 1k )?1 (P ); ( 2k )?1 (P ); : : : ; ( Nk ?k )?1 (P ) as the (N ? k) \unknowns". Using these equations, we can solve for the set of (N ? k) inverse bid functions for round k, pointwise in P. Using the conditional mean formula for log-normal variates (3.5) and the conditional mean and variance formulas for jointly normal variates, the formulas in (3.6) take the following form (where p  log P and b  log ): NX ?k (bk )?1 (p) ? aj + ai 2 2 2 2 s =r + t h s j i i i i p= xi + 2 2 2

ri h

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N 2 X (3.7) xl ? al + ai + 1 ri2 s2i t2i h + s4i ; m + a s i i + + 2 2 ri h r02 rl2 ri4 h l=N ?k+1 i = 1; : : : ; N ? k: This constitutes a system of N ? k linear equations in (bkj )?1 (p); j = 1; : : : ; N ? k, the log-inverse bid functions of the remaining bidders in round k. In other words, if we invert this (N ? k) system of equations, we can solve for (bkj )?1 (p) as a function of p and xk+1 : : : xN , for j = 1 : : : N ? k, in each round k of the auction. After deriving the log-inverse

bid functions in this manner, we can easily invert them to get the log bid functions, which we will use in deriving the likelihood function for the observed bids. In order for equilibrium to exist (and for the inverse bid functions to be well-de ned), we require that the bid function are increasing in x. For the log-normal case considered in this paper, it turns out that we can directly verify (from analytically inverting the matrix in the above system of equations) that the inverse bid functions are increasing in p. This in turns implies that the bid functions themselves are increasing in x; therefore, the monotonicity conditions required for existence of equilibrium are satis ed.17

3.2 Deriving the joint density of dropout price vector The likelihood function for a given auction is the (multivariate) density of the observed set of dropout prices from that auction. Given distributional assumptions on the x's and the nature of equilibrium in the ascending auction game, the \regression equation" which we want for a given dropout price is

pk = f (xN ?k ; xN ?k+1 : : : xN ); (3.8) in other words, the dropout price in round k as a function of bidder N ?k's valuation xN ?k ,18 as well as all the information he has in round k, which are the valuations (xN ?k+1 : : : xN ) 12

of the bidders who have dropped out before round k. This \regression equation" speci es the transformation from the unobserved latent variables (the x's) which we have made distributional assumptions about to the observed variables (the dropout prices). However, the system of equations in 3.7 give pi as a function of all of the x's. Therefore, in order to derive the regression equations in 3.8, we must go through two steps: rst, we solve for xk as a function of (p; xk+1 ; : : : ; xN ), where p is a generic p, for rounds k = 1; : : : ; N ? 119 ; secondly, we invert these (N ? 1) functions to get pk as a function of (xk ; : : : ; xN ), as desired. We go into each step in more detail.

Step 1 In the rst step, we loop over each round k of the auction in order to solve for xk ,

bidder k's valuation, as a function of pk and xk+1 : : : xN , the valuations of the bidders who have already dropped out (here we continue the indexing convention that bidder N drops out rst). This is done using the methodology described in the previous section. In round 0, for example, none of the x's are known (since no one has dropped out yet) so using the entire system of equations in 3.7 (N equations in N unknowns) we can solve for (b1j )?1 (p) as a function of just a generic p, for j = 1 : : : N . From these equations, we keep only the one for bidder N | (b1N )?1 (p) | the bidder who drops out during round 0. This equation is the inverse bid function for bidder N during the round that he drops out. We keep only one equation from each round because we only observe dropout prices. Similarly, in round k, we solve for (bkj )?1 (p) as a function of generic p and xk+1 : : : xN , for j = 1 : : : N ? k, and keep the equation for bidder N ? k. So after looping through all rounds k = 0 : : : N ? 2, we are left with the (N ? 1) equations: namely, xk as a function of p and xk+1 : : : xN , for k = 0; : : : ; N ? 2.

Step 2 Secondly, we invert each of these equations singly in order to get pk , the logdropout price during round k, as a function of xk : : : xN , for k = 0 : : : N ? 2. Given our

assumptions about the means and variances of the x's, we can analytically derive the vector of means ~p (; z ) and the covariance matrix p(; z ) for the vector of N ? 1 log-prices observed in the course of the auction (let P denote this vector).  is shorthand for the parameter vector and z are the included covariates (both described below).

13

3.3 Equilibrium consistency conditions

At this point, it would appear that, since (x1 ; : : : ; xN ) are assumed multivariate normal, P would also be distributed multivariate normal, with density: n f (P ; ; z ) = (2)? 2 j (; z ) j? 21 exp p





? 12 (P ?  (; z ))0  (; z )?1 (P ?  (; z )) : p

p

p

(3.9)

This is not completely true, however, since the multivariate normal density in equation 3.9 places positive probabilities on events which are inconsistent with an observed vector of dropout prices. The joint density of the dropout prices in equation 3.9 is derived assuming the observed dropout order, i.e., the rst element of p (; z ) is derived under the assumption that bidder 1 is the observed winning bidder, the second element is derived under the assumption that bidder 2 is the observed second-highest bidder, etc. However, there are regions in the support of P from which a vector of prices (call this P~ , with elements p~i ; i = 1; : : : ; N ? 1) could be drawn which imply a dropout order dierent than the one which is observed.20 We must rule out all such vectors P  . These consistency considerations impose a set of truncation conditions on the joint distribution of the observed sequence of dropout prices in a given auction. These conditions ensure that, in each round of the auction, given the parameters , the targeted dropout prices of the remaining bidders for that round must be higher than the dropout price at that round. Speci cally, the multivariate normal density (3.9) only imposes the condition that a bidder who drops out during round k, at a price pk (; z ), has an expected valuation of the object equal to pk (; z ):

E [vN ?k j xN ?k ; (bk1 )?1 (bkN ?k (xN ?k )); : : : ; (bkN ?k?1 )?1 (bkN ?k (xN ?k )); xN ?k+1 ; : : : ; xN ] = pk (; z) = bkN ?k (xN ?k ; ; z): In what follows, we suppress z as an argument in pk (
) and bkj (
). To ensure that the \correct" dropout order occurs, we need to impose that, at the given parameter values, all remaining bidders j = 1; : : : ; N ? k ? 1 have expected valuations greater than pk (): b () = E [v j x ; (b )?1 (b (x )); l = 1; : : : ; N ? k ? 1; l 6= j ; x ? +1 ; : : : ; x ] > p (); (3.10) k j

j

j

k l

k j

j

N

k

N

k

for all j = 1; : : : ; N ? k ? 1, the bidders who are left in the auction after round j . 14

In other words, bkN ?k (xN ?k ; ) < bkj (xj ; ), for j = 1; : : : ; N ? k ? 1; k = 0; : : : ; N ? 2, where bkj (xj ; ) denotes the targeted (log) dropout price of bidder j in round k, at the current values of the parameters . We will de ne A as the region of the support of the vector of dropout prices P over which all the consistency conditions (3.10) hold. Then the likelihood function is the sum of NOBS terms, where NOBS represents the number of auctions conducted. Each of the NOBS terms is the multivariate normal density of the observed dropout prices truncated to the region A: 8 f (P ; ; z) Xi0 +1 ; : : : ; Xi > Xi0 . Therefore i +1 ;k Xi +1 ;


> Pi , : : : , i ;k (Xi ;


) > Pi , which implies that the equilibrium price path must be increasing. It is essential to require that the price path does not have decreasing parts. k

k

k

k

k

k

N

k

k

j

k

k

N

k

k

N

k

k

N

k

k

N

k

N

N

k

k

k

N

k

N

k

21

k

k

Now suppose that bidder i stays over his bidding limit in round k, and during round k bidder ik+1 drops out at price level Pi +1 . If we de ne Xi00 by i;k (Xi ; Pi1 ; : : : ; Pi ) = Pi +1 , then Xi00 > Xi . Applying this argument inductively, assume bidder i later wins the auction at round k0 > k, because bidders  ik0+1 ; : : : ; iN ?1 drop out  instantaneously at price level Pi 0 +1 . If we de ne Xi by i;k0 Xi ; Pi1 ; : : : ; Pi0 = Pi 0 +1 , then Xi > Xi . That is, bidder i plays as if he has observed a signal Xi = Xi . Let X? i be the vector ?of revealed signals of other
bidders, then bidder i will have to pay Pi0?+1 = E Vi j Xi = X
i ; X?i = X? i , which is smaller than his expected gain, E Vi j Xi ; X?i = X? i , since Xi < Xi . This also implies that once bidder i drops out in some round k, he will never be interested in being active in the auction again. Q.E.D. Remark: It is clear from the proof that the crucial conditions for the equilibrium construction are: 1) Vi (
) is strictly in its ith arguments (assumption A2); 2) There exists strictly increasing solutions to (2.2) (assumption E2). In particular, it is not neccessary that Vi (
) be increasing in its j th arguments, j 6= i. k

k

k

k

k

k

k

B Proof of lemma (1) Note that the additive structure of the model (speci cally, the presence of the common component V ) implies that the private signals (X1 ; : : : ; XN ) will be aliated. For conditionally independent signals, aliation means that fX jV (Xi j v) satis es the monotone likelihood ratio property, in that f (Xi j V1 )=f (Xi j V2 ) is increasing in Xi whenever V1 > V2 , and is decreasing whenever V1 < V2 . The most important implication of MLRP is that for any increasing function g(V ), E (g(V ) j Xi ) is increasing in Xi . See Theorem 2.1 of Milgrom (1981). Without loss of generality consider i = 1. We can write the conditional expectation E (A1 j X1 ; : : : ; XN ) by rst conditioning on, and integrating out with respect to, V , the common value: i

Z

E (A1 j X1 ; : : : ; X ) = E (A1 j V; X1 ) f j N

1

V X ;::: ;X

N (V

j X1 ; : : : ; X ) dV N

Note that E (A1 j V; X1 ; : : : ; XN ) = E (A1 j V; X1 ) because of our assumptions of conditional independence. Lemma (1) will hold if we can show that E (A1 j V; X1 ) is a decreasing function of V . This will be true if, conditional on X1 , f (V j A1 ; X ) satis es the reverse MLRP in that for A11 > A21 and V1 > V2 : f (V1 j A1 ; X1 ) f (V2 j A1 ; X1 ) f (V1 j A2 ; X1 ) < f (V2 j A2 ; X1 ) 22

By writing f (V j A1 ; X1 ) as f (V )f (A1 ) f (X1 j V + A1 ) =f (A1 ; X1 ) and after some simplication, this is equivalent to requiring: f (X1 j A1 + V1 ) f (X1 j A1 + V2 ) f (X1 j A2 + V1 ) < f (X1 j A2 + V2 ) This will be true if log f (X1 j A1 + V ) is concave in A1 + V .

C Proof of proposition (2) Proof : Suppose (X10 ; : : : ; Xn0 ) solves (2.2) for P = P0 , then we can linearize (2.2) around

P0 and write it as

P ? P0 =

X ~
? @ V =@X  (P ) ? X 0 ; i = 1; : : : ; n n

j

i

=1

j

j

j





Solve this system of linear equations for j (P ) ? Xj0 ; j = 1; : : : ; n,

0 " # 1 ? (P ) ? X 0
= @X  @V ?1 A (P ? P ) ; i = 1; : : : ; n 0 @X n

i

i

h? @V
? i

j

=1

i;j

where we use @X 1 i;j to denote the i; j th element of the inverse matrix of @V @X ; i = 1; : : : ; N; j = 1; : : : ; N . The condition that i (P ); 1  i  n are increasing in P is equivalent to requiring that: X " @V ?1# > 0; 81  i  n =1 @X i.e., Each row of the inverted Jacobian matrix must sum to a positive number. The Jacobian matrix of (2.2) is the sum of the jacobian matrix for the corresponding private value and common value parts: [@V =@X ] = [@ A =@X ] + [@V =@X ] ; 1  i; j  N: N ? k; 0  k  N ? 2 principal submatrix of the left hand side Any @V j =@X k ; k + 1  i ; i  N can be rewritten as D + xy0 , where D = D [i +1 ; : : : ; i ; i +1 ; : : : ; i ] is the corresponding principal submatrix of D [@ Ai =@Xj ], x is a N ? k column vector of 1's, and y is the N ? k vector of @V==@X  k+1 ; : : : ; @V =@X N . The dominant diagonal assumption implies that D is a nonsingular matrix. Therefore @V j =@X k ; k + 1  i ; i  N can be uniquely inverted because it is the sum of a nonsigular matrix and a rank one matrix. The following matrix inversion formula 28   1 1 ? 1 0 ? 1 ? 1 0 ? 1 ? 1 0 ? 1 (D + xy ) = D ? 1 + y0 D?1 x D xy D = D I ? 1 + y0 D?1 x xy D : (C.16) i

j

n

j

i

j

i

i

k

i

N

i;j

j

i

j

j

k

k

N

i

i

j

i

23

k

is applicable here, with D replacing the role of D. Because D has a dominant diagonal with positive diagonal elements and non-positive o-diagonal elements, D?1 is a nonnegative matrix.(Theorem 4, McKenzie (1959)) Therefore y0 = y0 D?1 is a nonpositive vector. after (C.16) to show that  We can1 follow0 the same arguments as those ? 1 ? 1 the row sums of I ? 1+ 0 ?1 xy D are positive. Since D 's elements are also   nonnegative, it follows that the row sums of D?1 I ? 1+ 0 1 ?1 xy0 D?1 must also be positive. Assumption E2 is satis ed. A unique equilibrium can be constructed through the arguments used in Proposition 1. y D

x

y D

x

D Details of dropout price density derivation In this appendix we provide a detailed derivation of the joint density of the dropout prices, given in equation 3.11 in the main text. Let (V1 ; : : : ; VN ; X1 ; : : : ; XN ) be lognormally distributed. Let vi  ln Vi and xi  ln Xi , 8i = 1; : : : ; N . Assume that (v1 ; : : : ; vN ; x1 ; : : : ; xN ) are distributed as jointly normal with mean vector U = (u1 ; : : : ; uN ; u1 ; : : : ; uN )  (;  ) and variance-covariance matrix ^ = ((; 12 ); (012 ;  ))0 . Explicit formulas for the elements of the vector U and the matrix ^ can be derived from the distributional assumptions made at the beginning of section 3. As before, we have   1 E [Vi j X1 ; : : : ; XN ] = exp E (vi j x1 ; : : : ; xN ) + 2 V (vi j x1 ; : : : ; xN ) If we denote the marginal mean-vector and marginal variance-covariance matrix of (vi ; x1 ; : : : ; xN ) by

i  (ui;  ) and i 

i2 i 0 i 

!

also let x0  (x1 ; : : : ; xN )

Then, using the conditional mean and variance of joint normal random variables (see, for example, Amemiya (1985), pg. 3):





E (vi j x) = ui ? 0 ?1 i + x0 ?1 i and

V (vi j x) = i2 ? i 0 ?1 i : At round k,29 let xkd  (xN ?k+1 ; : : : ; xN ) denote the vector of k valuations for the bidders who have dropped out prior to round k, and xkr  (x1 ; : : : ; xN ?k ) denote the 24

vector of (N ? k) valuations for the bidders who have not yet dropped out as of round k. 1 ?1 0 ?1 Analogously, partition ?1 into (? k;1 ; k;2 ) where k;1 is a ((N ? k)  N ) matrix and 1 ? k;2 is a (k  N ) matrix. Then the conditional mean function can be re-written as:

?




1  k 0 ?1  E (vi j x) = ui ?  ?1 i + xkr 0? k;1 i + xd k;2 i :

After substituting the conditional mean and variance formulas into the equations in (3.5) and taking the log of both sides, we get the following set of (N ? k) linear equations for the N ? k bidders active in round k: ?
 0 ?1 0 k 1 ? 2 10 k  0 ?1 
(D.17) p = ui ?  ?1 i + i 0 ? k;2 xd + i k;1 xr + 2 i ? i  i for i = 1; 2;


; N ? k. Equations (3.7) in the main text are derived from (D.17) by 1 ?1 substituting in the explicit formulas for (ui ;  ; ?1 ; i ; ? k;1 ; k;2 ) which we can derive from the distributional assumptions made at the beginning of section 3.

Deriving pk = f (xN ?k ; xN ?k : : : xN ), the \regression equation" for round k dropout price pk If we let lk be the (N ? k)  1 vector of 1's, k = (u ; : : : ; uN ?k )0 , ?k = +1

1

( ; : : : ; N ?k )0 , k = (1 ; : : : ; N ?k )0 , then we could rewrite the above system of linear 2

2 1

equations (D.17) as   ?1 0 k 10 k ?1  p  lk = 21 ?k ? diag(k ?1 0k ) + k ? x k;2 d + k ? k   + k k;1 xr : (D.18) Solving out for the xkr , we obtain the set of (N ? k) log-inverse bid functions at round k:

xkr =



?1  ? 10 lk  p k k;1 1  ?1 0 ?1 



?1  10 k ?k ? diag(k ?1 0k ) + k ? k;2 xd + 2k ? 2k k  : (D.19)

? 2 k k;

1

Next let us introduce some shorthand notation:



0 Ak  k ? k;

and



0 Bk  12 k ? k; 1 1

?  1

1 1

?

1

lk



10 k ?1  ?k ? diag(k ?1 0k ) + k ? k;2 xd + 2k ? 2k   ;

25

both of them (N ? k)-dimensional vectors. Then the inverse bid functions in (D.19) can be rewritten:

xkr = Ak  p ? Bk ; or, each equation singly:

xkri = Aki  p ? Bik ;

for i = 1; : : : ; N ? k;

(D.20)

where Aki and Bik are the ith elements of the vectors Ak and Bk .30 As described in the main text, in round k we keep the (N ? k)-th equation from the system in (D.20); this is the equation xkr N ?k in the notation of (D.20). This equation gives us xN ?k as a function of p and xN ?k+1 ; : : : ; xN . As described before, we do this for rounds k = 0; : : : ; N ? 2, and end up with N ? 1 analogous equations for x2 ; : : : ; xN : we can denote these by x0rN ; : : : ; xkr N ?k ; : : : ; xrN ?2 2 . From the data, we observe the vector of dropout prices P  (p0 ; p1 ; : : : ; pN ?2 ). We can invert each of these equations described in the previous paragraph singly in order to derive an equation for pk , the log dropout price for round k, as a linear function of xN ?k ; : : : ; xN , for k = 0; : : : ; N ? 2. We will get

pk =

xN ?k + BNk ?k

AkN ?k

; for k = 0; : : : ; N ? 2:

(D.21)

Since the P vector is a linear transformation of the x vector, it is also multivariate normal. For purposes of deriving the joint density, it is useful to look at these equations describing the dropout prices in some more detail. To simplify notation, label bidders with N; N ? 1; : : : ; 1 in the order they drop out. Further decompose Bk into ?1 ? 
10 C k  21 k ? ?k ? diag(k ?1 0k ) + 2k ? 2k ?1  k;1 and ?1  ?10   10 k k;2 : Dk  12 k ? k;1 Then the equations for P in terms of the x's in (D.21) can be rewritten as 0 p0 = ACN0 + AxN0 N N i C Di (x ; : : : ; xN )0 pi = ANi ?i + AxNi ?i + N ?i NA?ii+1 i = 1; : : : ; N ? 2

N ?i

N ?i

N ?i

26

(D.22)

Again, introduce more shorthand notation:

0

1

2 0 F = A0 ; : : : ; AC 2 ?2 ; G = @|0; :{z: : ; 0}; A 1 ; AD ? A ; i = 0; 1; : : : ; N ? 2 ?2 ? ?

 C0

N

N

N

i

N

N

Let G = (G00 ; : : : ; G 0 ?2 )0

? ?1 i

i N

i

i N

i

i N

i

N

Given this notation, the system of equations describing the dropout prices (D.22) can be very succinctly written as:

P = F + G (x ; : : : ; xN )0 : 2

We denote the model parameters by . Suppose we also have a set of covariates z . The above mean and variance of observed dropout prices, together with the coecients and constant terms in the truncation conditions, are all functions of the parameter vector  and the covariates z . Mean of the bidder evaluation x  = (; z) Var-Cov of the bidder evaluation x  = (; z ) Jacobian of transforming x into P G =G (; z) Constant terms in the transformation of x into P F =F (; z ) The mean vector p and variance-covariance matrix p of the vector of dropout prices are:  (; z ) = F (; z ) ? G (; z )  (; z )  (; z ) = G (; z )  (; z ) G (; z )0 and therefore the joint density of these prices without incorporating the truncation conditions is p

p



f (P ; ; z )  (2)? 2 j (; z ) j? 21 exp ? 12 (P ?  (; z ))0  (; z )?1 (P ?  (; z )) : n

p

p

p



p

Equilibrium truncation conditions The equilibrium truncation conditions can also be

explicitly expressed in terms of the shorthand notation introduced above. At each round k of the auction, the targeted dropout prices of the remaining bidders for that round must be higher than the dropout price at that round, in other words, pk  pkN ?k < pki , for i = 1; : : : ; N ? k ? 1, k = 0; : : : ; N ? 2, where pki denotes the targeted dropout price of bidder i in round k. Note that x = G ?1 (P ?F ). Let eNi be the ith column 27

of a N  N identity matrix, and let EkN = (eNN ?k+1 ; : : : ; eNN )0 , then xi = eNi 0 G ?1 (P ?F ), and i k (xN ?k+1 ; : : : ; xN )0 = EkN G ?1 (P ? F ). Since pki = Cki + xik + Di (xN ?k+1k; : : : ; xN ) , then

Ak Ai Ai 0 N ? 1 k k ? 1 k pki = Cik + ei G (Pk ? F ) + Di Ei G k(P ? F ) , the constraint that pk  pkN ?k Ai Ai Ai

thus be rewritten as

0 G ?1 D E G ?1 ! 0 ?1 ?1 e e ? ? A ? A P < AC ? e AG F ? D EAG F ; ! i = 1 ; : : : ; N ? k ? 1 For k = 0; : : : ; N ? 2 N N

k

0

N i

k i

k i

k i

k i

k i

k i

N i

k i

< pki can

k i

k i

k i

(D.23)

These truncation conditions, expressed in equation (D.23), involve model parameters. Both the coecients and the constant terms in equation (D.23) are functions of parameters and covariates, i.e.,

Aki = Aki (; z) Dik = Dik (; z) Cik = Cik (; z) : This violates regularity conditions for maximum likelihood estimation.

Joint density Given these truncation conditions, the joint density of the observed vector of dropout prices P , given the parameter vector  and the covariates z , is given by 8 f (P ; ; z) 0 for all P < P  (Xi ), and < 0 for all P > P (Xi ). From a decision-theoretic point of view for each individual bidder, the structure of the equilibrium bid function shares some interesting similarities with the optimal policy in an optimal stopping problem, in which the \state variable" is the conditional expectation of the object's value (EVi ) and time is measured by the (deterministically rising) price level. 11

Note, however, that the \law of motion" for the state variable EVi is not stationary, i.e., functionally independent of our time measure P : how EVi evolves as P rises depends on the number (and identities) of the bidders who have already dropped out before price P , as well as the number and identities of the remaining bidders. Because of this, the optimal \stopping policy" would also be non-stationary, in the sense that the \critical value" of the state under which bidder i should drop out | this is his bid function | is not constant over the course of the auction. Alternatively, we could have a multiplicative model Vi = Ai  V , with the following conditions adjusted correspondingly 12

13

In the terminology of input-output analysis, [@ Ai =@Xj ; 1  i; j  n] is a Leontief matrix.

14

The notation below, however, diers slightly from the original notation in Wilson (1998).

15

Recall from above that Vi is the true value of the object for bidder i. 34

This is because for (vi ; x1 ; : : : ; xN ) jointly normal, the conditional mean function E (vi j x1 ; : : : ; xN ) is linear in x1 ; : : : ; xN and the conditional variance V ar(vi j x1 ; : : : ; xN ), is constant for all x1 ; : : : ; xN . This follows from the familiar conditional mean and variance 16

formulas for jointly normal random variables (eg. Amemiya (1985), pg. 3)

In fact, this model satis es all the conditions of lemma (1) in the last section for models with both private value and common value parts, after the log-transformation. Use 12 to denote the covariance matrix between P = (A1 ; : : : ; AN ) and X = (X1 ; : : : ; XN ), and use 22 to denote the variance-covariance matrix of X = (X1 ; : : : ; XN ). Then the jacobian matrix of the private values becomes [@ A =@X ]1  = 12 ?221 . By assumption Ai and X?i are independent, so 12 is a diagonal matrix with t2i as the diagonal elements. On the other hand, aliation of (X1 ; : : : ; XN ) implies that ?221 has a dominant diagonal. It then follows easily that 12 ?221 also has a dominant diagonal by simply taking di to be t?j 2 in condition F2. The existence of a unique equilibrium follows directly from Propositon 2. 17

i

j

i;j

n

Recall that by our indexing convention, bidder N ? k drops out in round k, at the (log) price of pk . 18

19

since we don't observe the \dropout price" for the winning bidder

For example, if p~N ?k < p~N ?k+1, the vector P  implies that bidder N ? k drops out before bidder N ? k + 1. This contradicts the observed dropout order but, furthermore, is inconsistent with the likelihood function (3.9), which we derived assuming that bidder N ? k drops out after N ? k + 1 and therefore had the opportunity to condition his dropout price on XN ?k+1 . 20

In fact, the MLE is super-consistent, converging at rate T to a mixture of exponential distributions. 21

For a xed value of S , the probability limit of the SNLS objective function QS;T is not equal to QT , the NLS objective function. Therefore the estimates of  will be inconsistent. Unfortunately, in our case, we cannot introduce a bias-correction term into the SNLS objective function, as in Laont, Ossard, and Vuong (1995) (pg. 959), because the corresponding bias-correction term in our case (i.e., the function
S;T () which sets plimT !+1 (QS;T () +
S;T () ? QT ()) = 0 for xed S ) is dicult to calculate. 22

Respectively, major trading area and basic trading area. This designations are from Rand McNally. 23

As explained in the very beginning of the paper, this is because, in equilibrium, bidder 1 bids a price 1 (X1 ) in which his expected revenue from winning is just equal to 1 (X1 ). If he in fact wins at price 1 (X1 ), this must mean that bidder j has dropped out at that price, implying that bidder j 's private signal Xj = j?1 ( 1 (X1 )). For the symmetric case, j (
) = (
); 8 j , so that Xj = j?1 ( 1 (X1 )) = X1 . 24

35

This is clear from equation 3.7, in which 2all the coecients associated with bidders' private signals (for bidder i, these would be r2shr2 for all bidders j 6= i) are positive. 25

i

i

j

For example, empirical studies of open-call timber auctions in the structural vein by Paarsch (1991) and Haile (1998) have used the independent private values framework. Asymmetries are potentially important in these auctions, arising from both geographical locational dierences among the rms as well as collusive behavior. 26

From a computational point of view, the equilibrium bidding strategies in a second-price auction with common values are more dicult to calculate than those from an ascending auction. 27

28

See pp. 39 Dhrymes (1984).

29

k = 0; : : : ; N ? 2: x0d = ;.

30

 The original bid function in its level can then be written as bi (Xi ) = exp x A+B 1 i

  exp AB XiA : k i k i

k i

k i



=

k i

To our knowledge, a discrete version of the ascending auction model has not been developed. 31

36

References Amemiya, T. (1985): Advanced Econometrics. Harvard University Press. Ausubel, L., P. Cramton, R. McAfee, and J. McMillan (1997): \Synergies in

Wireless Telephony: Evidence from the MTA auction," Journal of Economics and Management Strategy, 7.

Bajari, P. (1996a): \Properties of the First Price Sealed Bid Auction with Asymmetric

Bidders," Mimeo, Stanford University.

(1996b): \A Structural Econometric Model of the First Price Sealed Bid Auction: With Applications to Procurement of Highway Improvements," Mimeo, Stanford University. Bikhchandani, S., and J. G. Riley (1991): \Equilibria in Open Common Value Auc-

tions," Journal of Economic Theory, 53, 101{103.

Bulow, J., M. Huang, and P. Klemperer (1996): \Toeholds and Takeovers," Stanford

Graduate School of Business working paper, # 1393.

Bulow, J., and P. Klemperer (1997): \The Winner's Curse and the Failure of the Law

of Demand," Mimeo.

Cellular Telephone Industry of America (1996): Wireless Market Book. Cramton, P. (1997): \The PCS Spectrum Auctions: An Early Assessment," Journal of

Economics and Management Strategy, 7, 431{495.

DeGroot, M. (1970): Optimal Statistical Decisions. McGraw-Hill Book Company. Dhrymes, P. J. (1984): Mathematics for Econometrics. Springer-Verlag. Gourieroux, C., and A. Monfort (1996): Simulation-based econometric methods. Ox-

ford University Press.

Haile, P. (1998): \Structural Estimation of Oral Ascending Auctions," University of Wis-

consin, Madison mimeo.

Hendricks, K., and R. Porter (1988): \An Empirical Study of an Auction with Asym-

metric Information," American Economic Review, 78, 865{883. 37

Hong, H. (1998): \Non-regular Meximum Likelihood Estimation in Auction, Job Search

and Production Frontier Models," mimeo.

Hong, H., and M. Shum (1999): \Structural Estimation of Auction Models," in Game Practice, ed. by I. Garcia-Jurado, S. Tijs, and F. Patrone. Kluwer, Forthcoming. Klemperer, P. (1997): \Auctions with Almost Common Values: The "Wallet Game" and

its Applications," mimeo.

Laffont, J. J., H. Ossard, and Q. Vuong (1995): \Econometrics of First-Price Auc-

tions," Econometrica.

Maskin, E., and J. Riley (1996): \Asymmetric Auctions," mimeo. McFadden, D. L. (1996): \Lectures on Simulation-Assisted Statistical Inference," papers

presented at the EC-Squared Conference, Florence, Italy, December.

McKenzie, L. (1959): \Matrices with Dominant Diagonals and Economic Theory," Math-

ematical Methods in the Social Sicences, pp. 47{61.

Milgrom, P. (1981): \Rational Expectations, Information Acquisition, and Competitive

Bidding," Econometrica, 49, 921{943.

Milgrom, P., and R. Weber (1982): \A Theory of Auctions and Competitive Bidding,"

Econometrica, 50, 1089{1122.

Newey, W., and D. McFadden (1994): \Large Sample Estimation and Hypothesis Testing," in Handbook of Econometrics, Vol. 4, ed. by R. Engle, and D. McFadden. North

Holland.

Paarsch, H. (1991): \Deriving and Estimate of the Optimal Reserve Price: An Application

to British Columbian Timber Sales," Univ. of British Columbia working paper, upcoming in Journal of Econometrics.

(1992): \Deciding between the common and private value paradigms in empirical models of auctions," Journal of Econometrics, 51, 191{215. Pakes, A., and D. Pollard (1989): \Simulation and the Asymptotics of Optimization

Estimators," Econometrica, 57(5), 1027{1057.

Spiller, P., and P. Moreton (1996): \What's in the air: Synergies in the FCC Spectrum

Auctions," Forthcoming, Journal of Law and Economics. 38

Takayama, A. (1985): Mathematical Economics. Cambridge University Press. Vickrey, W. (1961): \Counterspeculation and Competitive Sealed Tenders," Journal of

Finance.

Wilson, R. (1998): \Sequential Equilibria of Asymmetric Ascending Auctions," Economic

Theory.

39

List of Figures 1 2 3 4

Bidder i's decision problem during round k . . . Simulated Nonlinear Least-Squares Estimates . . Estimated bid functions: Using Model 1b results Summary statistics for data variables . . . . . . .

40

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

41 42 43 44

E(V i Xi = X, βj ( Xj , Ωk)=P,∀ j {i,i1 ,...,ik }, Ω k ) -P k

∋

P

P*k

Figure 1: Bidder i's decision problem during round k

n

k  Xj = ( jN ?j )?1 (PN ?j ); j = N ? k + 1; : : : ; N

41

o

Figure 2: Simulated Nonlinear Least-Squares Estimates M (number of simulation draws): 100; H (bandwidth)=0.01

Model 1a

Model 1b

Coecient Estimate Std. Err Estimate Std. Err Components of log s: Constant 2.515 0.354 1.627 0.604 POP (mills) 1.124 0.824 INCOME (per cap.,$'000) 5.852 1.549 Components of log t: Constant

3.004

0.315

2.999

0.237

log r0 a : Constant

0.793

0.318

0.813

0.183

Components of m: Constant POP (mills) POP CHANGE (%)

10.762 -0.035 0.212

8.306 4.301 0.328

6.458 -0.036 0.483

14.069 0.081 0.511

Components of a: Constantb CELL PRES

0.009

0.069

0.008

0.019

NLS Obj.Fun.c Num of Obs

23002.29 91

21955.38 91

 m (the mean of the log common value distribution for a given object) should be a function

of MTA-level demographic variables which capture the across-license variation in values. We use POP (population) and POP CHANGE (population change).  a (the publicly-known mean of the private value component of V ) is a function of rm-andobject speci c covariates. We only use CEL PRES, an indicator of cellular presence in the surrounding area. More precisely, this regressor is a tally of the total number of the BTA's surrounding32 a particular MTA in which a given rm has cellular presence.33  (s; t; r0 ) (the standard deviations for the private and common value estimates, and for the common value distribution, respectively) are parameterized dierently across the two speci cations. Both speci cations restrict these quantities to be the same over all bidders; Model 1b, however, allows s to vary over objects as a function of POP and INCOME. i

i

variance of common value distribution Not separately identi ed from constant in m c simulated

a

b

42

Figure 3: Estimated bid functions: Using Model 1b results auction 57

auction 76

4

3.5

3.5

3

3

2.5 round 1

round 1

round 2

2.5

round 2

2

round 3

round 3

round 4

2 1.5

1

1

0.5

0.5

0

0

−0.5

−0.5 −2

−1.5

−1

−0.5

0

0.5

1

1.5

round 4

1.5

−1 −2

2

MTA 31, Block B: Indianapolis

−1.5

−1

−0.5

0

0.5

1

1.5

2

MTA 41, Block A: Oklahoma City

auction 88

auction 89

5

5

4.5

4.5

4

4 round 1

round 1

round 2

3.5

round 2

3.5

round 3

round 3

round 4

3

round 4

3

round 5

round 5

2.5

2.5

2

2

1.5

1.5

1

1

0.5 −2

−1.5

−1

−0.5

0

0.5

1

1.5

0.5 −2

2

MTA 47, Block A: Hawaii

43

−1.5

−1

−0.5

0

0.5

1

1.5

2

MTA 47, Block B: Hawaii

Figure 4: Summary statistics for data variables

Variable

N mean StdDev min max

Winning prices ($mill) 911 75.87 Population (millions) 91 5.15 Pop'n change (1990-95, %) 91 6.00 Per capita income ('000) 91 15.86 Dropout prices ($mill) 423 53.18 Cell. pres 423 0.61

89.71

4.39 493.5

4.14

1.15 26.78

3.53

0.40 12.80

3.71

11.96 20.70

69.28 1.28

0.89 493.5 0 8

1 : We omitted the observations for: Puerto Rico, Guam, Samoa

44