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Summary This paper examines identification in second‐price and ascending auctions within the private‐values framework. The first part of the paper considers an arbitrary type of dependence of bidders’ values and analyses identification under several observational scenarios, in which the highest bid is never observed. In a basic scenario, only the winner’s identity and the winning price are observed. The most informative is the scenario in which all the identities and all the bids except for the highest bid are known. Using results from Athey and Haile (2002), the joint distribution of bidders’ values in these scenarios is not identified. The paper uses the information available in auctions’ outcomes to construct bounds on the joint distribution of values for any subset of bidders. The second part of the paper takes a different tack by showing how bounds can be improved under different types of positive dependence of bidders’ values. 1. Introduction In auctions, researchers are often interested in learning model’s economic primitives, particularly the joint distribution of bidders’ values. The identification problem is the problem of learning this underlying distribution from data available from the auction’s outcomes. This paper examines identification in second‐price and ascending ‘button’ auctions within the private‐values framework. It is well known that in such auctions, a weakly dominant strategy for bidders entails submitting their true value.1 Throughout most of the paper, I consider an equilibrium where bidders employ this strategy. The identification analysis cannot be conducted without specifying what data are available from the auctions’ outcomes. I consider several observational scenarios. In the basic scenario, the only available data pertain to the winner’s identity and the winning price. In the second scenario, in addition to the winner’s identity and the winning price, an econometrician observes the identity of the highest losing bidder. In the third scenario, which is the best case scenario, all the identities and all the bids except for the highest bid are known. Athey and Haile (2002) provide a non‐identification result, which, in particular, implies that in second‐price and ascending auctions within the private‐values framework and without the assumption of independence of bidders’ values, the joint distribution of values is not identified when only a subset of bids is observed. In particular, even if all the bids except for the highest bid are known, there are many distributions consistent with the data. Athey and Haile (2002) and Komarova (2009) discuss the point identification of the joint distribution when bidders’ values are independent. The main goal of this paper is to investigate the identification issue in the absence of independence. The paper exploits the idea that even though the distribution of values is not identified, the data are informative and allow finding bounds on the distribution. These bounds can be used in the analysis of counterfactuals and other applications. Throughout the paper, I assume that in second‐price auctions, the highest bid is never observed, and this is why all the partial identification results obtained for second‐price auctions also apply to ascending auctions in which price rises continuously and exogenously. These results can be extended to allow for reserve prices in the auction, or to allow for exogenous variation in the set of participants. Many of the results obtained in this paper can be extended to dependent generalized competing risks models. In reliability theory, a classical competing risks model is a situation in which a machine breaks down as soon as one of its components reaches a failure state; the observed data pertain to the machine’s lifetime and the component that caused the failure. Generalized competing risks models relax this assumption and consider cases in which an object fails because of the cumulative failure of several of its elements rather than a single one. As noticed by Athey and Haile (2002), second‐price auctions and ‘button’ auctions within the private‐values framework is a special case of these models. For classical competing risks models, the study of identification in the absence of independence was initiated by Peterson (1976), who obtained tight pointwise bounds on the joint and marginal survival functions in classical competing risks. Crowder (1991) and Bedford and Meilijson (1997) obtained new results on bounds for those functions. For generalized competing risks, several results for bounds on survival functions were established by Deshpande and Karia (1997). The first paper that explored partial identification in auctions and constructed bounds on distributions was Haile and Tamer (2003). Haile and Tamer considered English auctions in which bidders’ behaviour satisfies the so‐called bounded rationality property. For the first‐price auctions, some contributions on partial identification and bounds are made in Gentry and Li (2010) and Armstrong (2011). Aradillas‐Lopez et al. (2011) consider inference and bounds for symmetric ascending bid auctions using data from auctions with different numbers of bidders. Section 2 of this paper considers an arbitrary type of dependence of bidders’ values. For each of the observational scenarios described above, I derive bounds on the joint distribution of values for any subset of bidders. Because the second and the third scenarios contain more information than the first scenario, and the third scenario has more information than the second one, this allows me to analyse how the bounds change and how much tighter they become when more data become available—data on the identities or the bids. The bounds on the marginal distribution functions are considered in more detail. In the basic scenario, e.g. the lower bound for the distribution of bidder i’s private value means that from the perspective of bidder i, conditional on losing the auction, the transaction price first order stochastically dominates i’s value. As for the upper bound, it gives that conditional on bidder i winning the auction, the transaction price is first order stochastically dominated by i’s value. Section 2 also addresses the question of how the equilibrium condition can be relaxed. I discuss which partial identification results in the analysed scenarios continue to hold if the equilibrium condition is replaced with the assumptions on bounded rationality introduced in Haile and Tamer (2003). This is an important issue since it is not uncommon for the auctions to be of some mixed type, in which the equilibrium condition does not hold but the bidders are bounded rational. I illustrate this with an example of stamp auctions conducted by Vance Auctions Ltd. In addition, Section 2 contains the results of a Monte Carlo experiment. It presents bounds estimated from the data obtained in a finite number of simulated auctions. This illustrates the severity of the partial identification problem in a finite sample. Section 3 analyses bounds on distributions from a different perspective. In auctions, bidders’ valuations are often considered to be positively dependent random variables. There are several notions of positive dependence. Two of them are affiliation and association, which are discussed in detail in Milgrom and Weber (1982). For simplicity, I consider only the basic scenario and show how several types of positive dependence lead to the improvement of the bounds. These results are important because, for instance, in all the scenarios in Section 2 the lower bound on the joint distribution of values of all bidders is trivial—it is 0. This happens because without the knowledge of the highest bid, an econometrician is not able to bound the value of the auction’s winner. Properties of positive dependence, however, allow constructing non‐trivial lower bounds on that joint distribution of all values. The proofs of theorems and propositions in Section 2 are collected in Appendix A. Appendix B contains figures illustrating results on the bounds from the Monte Carlo experiment. 2. Bounds under Different Scenarios Here is a brief review of sealed‐bid second‐price auctions. A single object is up for sale, and d buyers, d≥ 3, are bidding on it. The set of all bidders is known. Bids are submitted in sealed envelopes. The highest bidder wins and pays the value of the second highest bid; thus, in these auctions, the second highest bid is the winning price. Suppose that the bidders have private values and they are aware of their value. It is known that in this setting, a weakly dominant strategy for bidders is to submit their true value—and this is the equilibrium considered throughout this paper. The paper assumes that an econometrician never observes the highest bid. Within the private‐values framework, second‐price auctions are equivalent to open ascending auctions. One form of an ascending auction is a ‘button’ auction, in which bidders hold down a button as the auctioneer raises the price. When the price gets too high for a bidder, she drops out by releasing the button. The auction ends when only one bidder remains. This person wins the object and pays the price at which the auction stopped. A weakly dominant strategy for bidders is to drop out as soon as the price reaches the bidder’s value. Clearly, in ascending auctions, the winner’s value is not observed, which is equivalent to not observing the highest bid in second‐price auctions. 2.1. Notation Denote bidders’ private values as X1, … , Xd. Suppose that these values have continuous marginal distributions on a common support [t0, T]. It is also assumed that P(Xi=Xj) = 0, i≠j, so that the probability of a tie is 0.2 Let D={i1, …, ir} denote a subset that consists of bidders i1, …, ir. Define QD as the distribution function of valuations of bidders in this subset: Let Q denote QD for D={1, …, d}, Q−m denote QD for D={1, …, m− 1, m+ 1, …, d} and Fj denote QD for D={j}. The complement of set D is denoted as CD. 2.2. Basic scenario: Only the winner’s identity and the winning price are observed I first consider a scenario in which only the winner’s identity and the winning price are observed in an auction’s outcome. Then, the probability of the event is known for any t∈ [t0, T] and any i= 1, …, d. So, for each bidder i, we observe the following function Gi on [t0, T]: This function is increasing and its value Gi(T) is the probability that bidder i wins. Knowing functions Gi is not enough to identify the joint distribution or even marginal distributions of bidders’ values unless some additional assumptions on the joint distribution are imposed. Athey and Haile (2002) establish a non‐identification result which implies, in particular, that if the highest bid is not observed and the bidders’ values are not independent, then there is an infinite number of joint distributions that rationalize observable functions. Komarova (2009) proves that the joint distribution is identified when bidders’ values are independent. Here, the independence of values is not assumed, and even though the distributions are not identified, the data are informative and allow obtaining bounds on the distributions of interest. Examples of ascending auctions in which only the sale price and the identity of the winner are recorded, and the bidders are assumed to have private values, are wheat auctions in India analysed by Banerji and Meenakshi (2004) and fish auctions by Brendstrup and Paarsch (2006). Both papers treat bidders’ private values as independent but this assumption is quite strong. Examples of sealed‐bid second‐price auctions are stamp auctions, which are generally considered within the private‐values paradigm. For instance, the auction house Sam Houston Philatelics uses sealed‐bid second‐price auctions to sell stamps.3 The bids can be submitted via mail, fax, phone or e‐mail, and also can be submitted online. The data released by Sam Houston Philatelics include detailed descriptions of the stamps, sale prices, dates of auctions and lots that were being sold in a particular auction. This auction house, however, does not reveal any identities of bidders or even the number of bidders. Thus, the observations from these auctions contain even less information than what is assumed in this paper. To obtain a lower bound on QD, I use the fact that if bidder j∈CD wins, where D={i1, …, ir}, and the price does not exceed t, then the values of do not exceed t either. In other words, functions Gj, j∉D, provide information about a lower bound on QD. On the other hand, if bidder ik wins, then it is not known how large the value is, and consequently, is not helpful in finding a lower bound on QD. To obtain an upper bound on QD, I exploit the fact that if bidder ik wins and the value of does not exceed , then the price does not exceed either. If wins and we know the bounds on the values of from above, then no non‐trivial conclusion can be made about the price unless CD contains at most one bidder. The theorem below formalizes this discussion and presents bounds on QD. Theorem 2.1. Suppose the bidders play their weakly dominant strategy by submitting their true values. Also, suppose that only the winner’s identity and the winning price are observed. Then, (a) functionQDis bounded from below as follows: (b) functionQis tightly bounded from above as follows: (2.1) A tight upper bound for QD, where D={i1, …, ir}, is obtained from (2.1) by substituting tj=T for any j∉D. For instance, for any m= 1, …, d, the distribution function Q−m is tightly bounded from above as follows: If D contains at most d− 2 elements, then QD is tightly bounded from above as follows: The bounds in this theorem can be improved at the border of the support: This improvement is obvious but not helpful in the analysis on bounds. Also, it is worth noting that the lower bound on Q is simply 0, and thus completely uninformative. Bounds in Theorem 2.1 can be given an economic interpretation. As a special case, consider bounds on the marginal distribution function for bidder i: (2.2) The inequality with the lower bound can be rewritten as which means that from the perspective of bidder i, conditional on losing the auction, the transaction price first order stochastically dominates i’s private value. Because , for the inequality with the upper bound, which means that conditional on bidder i winning the auction, the transaction price is first‐order stochastically dominated by i’s private value. For non‐singleton sets D, the bounds on are more difficult to interpret because there may be different values among , … , . When , however, the bounds can be given an interpretation similar to the case of a singleton D. Namely, the lower bound means that if any bidder j∈CD wins an auction, then the sale price first‐order stochastically dominates the maximum of the values of bidders in D. The upper bound means that if somebody from D wins, then the maximum of the values of bidders in D first‐order stochastically dominates the sale price. Example 2.1. Consider an auction with four buyers. Let , , , and A be independent random variables distributed on [0, 1] with distribution functions , , , and . Let private values X1, X2, X3 and X4 of the buyers be Figure 1 shows the marginal distribution functions F1, F2, F3 and F4 (the thick lines) and the upper and lower bounds (the dotted lines) on these functions from Theorem 2.1. The upper bounds are wide for low valuations because low bids are not observed in auction’s outcomes. Figure 1. Open in new tabDownload slide Marginal distributional functions and their bounds. Figure 1. Open in new tabDownload slide Marginal distributional functions and their bounds. The form of bidders’ values in Example 2.1 shows a popular approach in representing dependent valuations. Namely, bidders’ private values are the sums of independent random variables and a common variable, which is 0.5A in this example. Moreover, if A has a density that is log‐concave, then values X1, X2, X3 and X4 are affiliated. Affiliation and other types of positive dependence are discussed in Section 3. Note that Theorem 2.1 relies on the fact that bidders submit their true values. This condition can be relaxed if we employ the following two assumptions. Assumption 2.1. Bidders do not bid more than they are willing to pay. Assumption 2.2. Bidders do not allow an opponent to win at a price they are willing to beat. These assumptions on bounded rationality were introduced in Haile and Tamer (2003). The authors were the first ones to relax equilibrium conditions in the econometric analysis of auctions and allow other types of bidders’ behaviour. One of their contributions is the construction of bounds on the distributions of interest for this limited structure in certain auction models. The proposition below shows that when the equilibrium condition is replaced with Assumption 2.1 and 2.2, the bounds on the distribution functions remain the same as in Theorem 2.1. In addition, for simplicity, it is assumed that the probability of a tie is 0. Proposition 2.1. Suppose that only the winner’s identity and the transaction price are observed andP(bi=bj) = 0fori≠j. (a)If Assumption 2.1 holds, thenQDare tightly bounded from above as in Theorem 2.1. (b)If Assumption 2.2 holds, thenQDare bounded from below as in Theorem 2.1. Relaxing the equilibrium condition means that the joint distribution of bidders’ values does not necessarily coincide with the joint distribution of their bids. Under Assumption 2.1, nevertheless, the distribution of the values is bounded from above by the distribution of the bids, and this is what gives the upper bounds and the result in part (a) in Proposition 2.1. Under Assumption 2.2, the result holds because in the auctions where j∈CD wins it remains true that the value of bidder i∈D is bounded from above by the price. It is also worth noting that part (a) of Proposition 2.1 continues to hold if the assumption on the continuous bidding is relaxed and the buyers bid in increments. The result of part (b) can be extended if more is known about the bidding policy, such as the size of required increments. Being able to use Assumptions 2.1 and 2.2 instead of the equilibrium condition is especially important in light of the fact that, in practice, it is not uncommon for the auctions to be of some mixed type, rather than to be pure second‐price or pure ascending auctions. For instance, in stamp auctions at Vance Auctions Ltd, bids can be placed by e‐mail, fax, phone or regular mail.4 A bidder can authorize the auction house to raise her bids by 10%, 20% or 30% if necessary. This additional margin is used only when the bidder would lose the lot without it. Thus, in this case, the bids do not necessarily coincide with bidders’ values, but Assumptions 2.1 and 2.2 hold. Below, I analyse how the bounds change when more information from auction’s outcomes becomes available. 2.3. The winner’s identity, the winning price and the identity of the bidder with the highest losing bid are observed In this scenario, there is additional information about the identity of the bidder with the highest losing bid. This is the bidder that submits the bid equal to the winning price. Observed are the following d(d− 1) functions: Clearly, The theorem below shows that this additional information allows improving upper bounds on the distributions of bidders’ values. The lower bounds remain the same because the knowledge that i is the loser with the highest bid does not provide any information on the bound of i’s value additional to the information that this value does not exceed the price, and, of course, does not provide any information on the bounds of the values of other losers additional to that their values do not exceed the price. For simplicity, improved bounds are presented only for the joint distribution of values of all bidders and for the marginal distribution functions. Such bounds, however, can be obtained for an arbitrary subset of bidders. Theorem 2.2. Suppose that bidders play their weakly dominant strategy by submitting their true values. Then, the joint distribution and the marginal distributions of bidders’ values are bounded from above as follows: (2.3) In Theorem 2.1, the marginal distributions are bounded from above as follows: Because Gji(T) ≥Gji(t), the upper bound on Fi in Theorem 2.2 is indeed in general tighter than that in Theorem 2.1. The improvement comes from the auctions in which i is the loser with the highest bid. In this case, an upper bound on i’s value gives a non‐trivial (smaller than T) bound on the sale price. Using the fact that , rewrite (2.3) as which has an intuitive interpretation that from the perspective of bidder i, conditional on winning the auction or submitting the highest losing bid (i.e. submitting the bid equal to the sale price), i’s private value stochastically dominates the price. In Figure 2, the thick lines depict F1, F2, F3 and F4 in Example 2.1, the dotted lines depict the bounds on these functions in the basic scenario and the dashed lines show the improved upper bounds from Theorem 2.2. Figure 2. Open in new tabDownload slide Marginal distributional functions and their bounds. Figure 2. Open in new tabDownload slide Marginal distributional functions and their bounds. The equilibrium condition in this theorem can be replaced with Assumption 2.1. 2.4. Bounds when more bids and identities are observed As Figure 1 illustrates, bounds on the marginal distributions obtained in Theorem 2.1 can be wide. Specifically, the lower bound provides a poor approximation of the distribution in proximity of T, whereas the upper bound provides a poor approximation in proximity of t0. Even though the former situation cannot be alleviated because the highest bid/value is never observed, the latter situation can be improved if more losing bids are observed. Scenarios analysed in this section deal with situations when in addition to the sale price, some lower bids and the identities of the corresponding bidders are available in auction’s outcomes. I start by considering the best case scenario, which is the situation when all the identities and all the bids except for the highest bid are observed. In reliability, such an observational scheme corresponds to the so‐called continuous monitoring models. In the auction literature, an example of the data from open ascending auctions within the private‐values framework is federal auctions of timberland in California. In those auctions, the United States Forest Service (USFS) sells logging contracts, and all the losing bids and the types of the bidders are observed. The data from these auctions are used, e.g. in Athey et al. (2011) and in Roberts and Sweeting (2011). Let Πd denote the set of all the permutations of set {1, …, d} and ρ∈Πd. Let ρ (i) stand for the ith element of permutation ρ. In the auction context, bidder ρ (i) is the ith highest bidder. The following d! functions are observed: If at least one si, i= 2, …, d, is equal to t0, then Wρ(s2, …, sd) = 0. Also, To avoid sophisticated technical expressions, the theorem below shows the bounds only for the joint distributions of values of all bidders and the marginal distributions. Theorem 2.3. Suppose bidders play their weakly dominant strategy by submitting their true values. Also, suppose that all the identities and all the bids except for the highest bid are observed. Then,QandFiare bounded from above and below as follows: The lower bound on Fi is the probability that bidder i loses, her bid does not exceed t, and all the losing bids above hers can take any values up to T (clearly, all the bids below hers are automatically bounded by t). The upper bound on Fi can be interpreted as the sum of two probabilities. One probability is simply equal to the probability in the lower bound, which considers cases when i loses. The other is the probability of i winning and the price not exceeding t. In Figure 3, the thick lines depict F1, F2, F3 and F4 in Example 2.1, the dotted lines depict the bounds on these functions in the basic scenario, and the dashed lines show the improved upper and lower bounds from Theorem 2.3. As can be seen, the bounds in the best case scenario are much tighter than those in the basic scenario, and the improvement in identification is drastic. Moreover, from Theorem 2.3, one can see that the difference between the upper and the lower bound at t is Gi(t), which is increasing in t. This means that the identification of Fi(t) becomes worse as t increases. Figure 3. Open in new tabDownload slide Marginal distributional functions and their bounds. Figure 3. Open in new tabDownload slide Marginal distributional functions and their bounds. The results on the upper bounds in Theorem 2.3 continue to hold if instead of the equilibrium play assumption, we impose Assumption 2.1. This is because under Assumption 2.1, the joint distribution of bidders’ values does not exceed the joint distribution of the bids. As for the result on the lower bound for the marginal distribution, when d > 3, the assumption of the weakly dominant strategies cannot be replaced with Assumption 2.2 because under Assumption 2.2, the order of the values of the losing bidders does not necessarily coincide with the order of their submitted bids. The analysis of the best case scenario can be extended in two directions. One direction is to consider situations when K, K≤d− 1, highest losing bids and the bidder identities are available, in addition to the identity of the winner. The case K= 1 was analysed in detail in Section 2.3. The best case scenario corresponds to K=d− 1. As K changes from K= 1 to K=d− 1, i.e. as more and more data become available, the identification of distributions gradually improves. For the distributions in Example 2.1, the overall improvement can be seen by comparing bounds in Figures 2 (K= 1) and 3 (K=d− 1). It is worth noting that if K < d− 1, i.e. the lowest bid is not observed, then upper bounds on the marginal distribution Fi takes a strictly positive value at t0.5 This is because in auctions where i is the lowest loser, the upper bound on i’s value does not provide any information about the amounts of other buyers’ bids. In other words, we can only be certain that those bids are less or equal then T. The other direction is to consider situations when N, N≤d− 1, lowest losing bids and the bidder identities are observed, in addition to the identity of the winner and the sale price. As N changes from N= 1 to N=d− 1, the identification of distributions improves, and N=d− 1 corresponds to the best case scenario.6 For the distributions in Example 2.1, the overall improvement can be seen in Figure 4 (N= 1) and 3 (N=d− 1). As can be seen, even if the lowest bid is the only additional observed bid and the identity of the lowest bidder is available, there is an improvement in how the upper bound approximates distributions in the middle of the support and in a neighbourhood of t0. The improvement around t0 is especially drastic. Specifically, the upper bound takes value 0 at t0. This happens because the minimum bid does not exceed the value of bidder i for any i, and thus, when i’s value is close to t0, the values of all observable functions are close to 0. Figure 4. Open in new tabDownload slide Marginal distributional functions and their bounds. Figure 4. Open in new tabDownload slide Marginal distributional functions and their bounds. Let us analyse the case N= 1 in more detail. Observed are the following d(d− 1) functions of two variables: Index j corresponds to the identity of the lowest bidder and s corresponds to the bound on the minimum bid. These functions have the following property: Also, is non‐decreasing in t and s. For simplicity, the following proposition presents bounds only for the joint distribution of values of all bidders and for the marginal distribution functions. They can be obtained, however, for an arbitrary subset of bidders. Proposition 2.2. Suppose that bidders play their weakly dominant strategy by submitting their true values. Then, For the marginal distribution functions, Let us first look at the lower bound. In Theorem 2.1, the marginal distributions are bounded as in (2.2). Since , then Taking into account that in general is strictly greater than , we can conclude that the lower bound on Fi in Proposition 2.2 is indeed tighter than that in Theorem 2.1. The improvement comes from auctions in which i submits the lowest bid. Since , then Taking into account that in general is strictly less than , we have that the upper bound on Fi in Proposition 2.2 is indeed tighter than that in Theorem 2.1. The improvement comes from auctions in which i does not win since now i’s value is at least as little as the minimum bid. In the basic scenario, auctions in which i loses are not informative for the upper bound on Fi. In Figure 4, the thick lines depict F1, F2, F3 and F4 in Example 2.1, the dotted lines depict the bounds on these functions in the basic scenario and the dashed lines show the improved upper and lower bounds from Proposition 2.2. As can be seen, observing the lowest bids significantly improves the bounds around the lower support point, and the upper bound takes value 0 at t0. 2.5. Monte Carlo simulations In this section, I use Monte Carlo simulations to illustrate bounds on marginal distributions obtained in a finite sample. I continue to consider the case of four asymmetric bidders with private values defined as in Example 2.1. I present the estimates of the bounds in (2.2) for the basic scenario, when only the sale price and the winner’s identity are available, and also the estimates of the bounds in Theorem 2.3 for the best case scenario, when all the losing bids are observed together with the identities of all the bidders. Let n be the number of auctions observed in a sample. First, consider the basic scenario. Let denote the sample of available prices pm and winner’s identities wm. Sub‐distribution functions Gi, i= 1, …, 4, are estimated by the empirical sub‐distribution functions The estimates of the bounds in (2.2) are constructed by replacing each Gi with . The bounds are constructed on the interval . Now, consider the best case scenario. Let denote the sample of available data. Here, b3,m stands for the second highest losing bid and q3,m stands for the identity of the corresponding bidder; b4,m denotes the minimum bid and q4,m denotes the identity of the corresponding bidder; finally, q2,m is the identity of the highest losing bidder—losing bidder who submitted the bid equal to the sale price. Functions Wρ are estimated by their empirical analogues. For instance, W1342(s2, s3, s4) is estimated by The estimates of the bounds in Theorem 2.3 are constructed by replacing each Wρ with . The bounds are obtained on the interval . Table 1 is constructed based on the simulations of outcomes in 200 auctions. It shows how often each bidder wins, or submits the highest losing bid, or the second highest losing bid or the minimum bid. The highest observed price in the simulated data is approximately 0.924, the lowest observed price is 0.121 and the lowest observed bid is 0.005. Therefore, in the basic scenario, the estimates for the bounds are constructed on the interval [0.121, 0.924], and in the best case scenario, the estimates are constructed on the interval [0.005, 0.924]. These bounds are shown in Figures B1–B4 in Appendix B. As can be seen, the bounds in the basic scenario are quite wide. On the sample support [0.121, 0.924], the underlying marginal distribution functions are within the estimated bounds. In the best case scenario, the upper and the lower bounds are much closer to each other. In the lower part of the sample support [0.005, 0.924], the bounds almost coincide for F2, F3 and F4. This is not so for F1 because in auctions with low prices, bidder 1 happens to be the winner more often than bidders 2, 3 and 4 altogether. It is also worth noting that in the best case scenario, F2, F3 and F4 do not necessarily lie within the estimated bounds. Table 1. Summary statistics for bidders' identities: n= 200. . w . q2 . q3 . q4 . Bidder 1 109 (54.5%) 57 (28.5%) 26 (13%) 8 (4%) Bidder 2 43 (21.5%) 78 (39%) 55 (27.5%) 24 (12%) Bidder 3 26 (13%) 36 (18%) 80 (40%) 58 (29%) Bidder 4 22 (11%) 29 (14.5%) 39 (19.5%) 110 (55%) . w . q2 . q3 . q4 . Bidder 1 109 (54.5%) 57 (28.5%) 26 (13%) 8 (4%) Bidder 2 43 (21.5%) 78 (39%) 55 (27.5%) 24 (12%) Bidder 3 26 (13%) 36 (18%) 80 (40%) 58 (29%) Bidder 4 22 (11%) 29 (14.5%) 39 (19.5%) 110 (55%) Note: Monte Carlo experiment for n= 200 auctions. Number of auctions in which each bidder wins (w), submits the highest losing bid (q2), the second highest losing bid (q3) and the minimum bid (q4). Open in new tab Table 1. Summary statistics for bidders' identities: n= 200. . w . q2 . q3 . q4 . Bidder 1 109 (54.5%) 57 (28.5%) 26 (13%) 8 (4%) Bidder 2 43 (21.5%) 78 (39%) 55 (27.5%) 24 (12%) Bidder 3 26 (13%) 36 (18%) 80 (40%) 58 (29%) Bidder 4 22 (11%) 29 (14.5%) 39 (19.5%) 110 (55%) . w . q2 . q3 . q4 . Bidder 1 109 (54.5%) 57 (28.5%) 26 (13%) 8 (4%) Bidder 2 43 (21.5%) 78 (39%) 55 (27.5%) 24 (12%) Bidder 3 26 (13%) 36 (18%) 80 (40%) 58 (29%) Bidder 4 22 (11%) 29 (14.5%) 39 (19.5%) 110 (55%) Note: Monte Carlo experiment for n= 200 auctions. Number of auctions in which each bidder wins (w), submits the highest losing bid (q2), the second highest losing bid (q3) and the minimum bid (q4). Open in new tab Table 2 is constructed based on the simulations of outcomes in 200 auctions. It shows how often each bidder wins, or submits the highest losing bid, or the second highest losing bid or the minimum bid. The sample supports in the case are wider than in the case of n= 200. Therefore, in the basic scenario, the estimates for the bounds are constructed on the interval [0.056, 0.9624], and in the best case scenario, the estimates are constructed on the interval [0.0023, 0.9624]. These bounds are shown in Figures B5–B8 in Appendix B. The results here are quite similar to those when n= 200. As expected, the estimated bounds, especially in the best case scenario, provide better approximations of the true bounds than the case of n= 200. Table 2. Summary statistics for bidders' identities: n= 500. . w . q2 . q3 . q4 . Bidder 1 260 (52%) 146 (29.2%) 73 (14.2%) 21 (4.2%) Bidder 2 127 (25.4%) 177 (35.4%) 143 (28.6%) 53 (10.6%) Bidder 3 70 (14%) 92 (18.4%) 168 (33.6%) 170 (34%) Bidder 4 43 (8.6%) 85 (17%) 116 (23.2%) 256 (51.2%) . w . q2 . q3 . q4 . Bidder 1 260 (52%) 146 (29.2%) 73 (14.2%) 21 (4.2%) Bidder 2 127 (25.4%) 177 (35.4%) 143 (28.6%) 53 (10.6%) Bidder 3 70 (14%) 92 (18.4%) 168 (33.6%) 170 (34%) Bidder 4 43 (8.6%) 85 (17%) 116 (23.2%) 256 (51.2%) Note: Monte Carlo experiment for n= 500 auctions. Number of auctions in which each bidder wins (w), submits the highest losing bid (q2), the second‐highest losing bid (q3) and the minimum bid (q4). Open in new tab Table 2. Summary statistics for bidders' identities: n= 500. . w . q2 . q3 . q4 . Bidder 1 260 (52%) 146 (29.2%) 73 (14.2%) 21 (4.2%) Bidder 2 127 (25.4%) 177 (35.4%) 143 (28.6%) 53 (10.6%) Bidder 3 70 (14%) 92 (18.4%) 168 (33.6%) 170 (34%) Bidder 4 43 (8.6%) 85 (17%) 116 (23.2%) 256 (51.2%) . w . q2 . q3 . q4 . Bidder 1 260 (52%) 146 (29.2%) 73 (14.2%) 21 (4.2%) Bidder 2 127 (25.4%) 177 (35.4%) 143 (28.6%) 53 (10.6%) Bidder 3 70 (14%) 92 (18.4%) 168 (33.6%) 170 (34%) Bidder 4 43 (8.6%) 85 (17%) 116 (23.2%) 256 (51.2%) Note: Monte Carlo experiment for n= 500 auctions. Number of auctions in which each bidder wins (w), submits the highest losing bid (q2), the second‐highest losing bid (q3) and the minimum bid (q4). Open in new tab 3. Bounds under Positive Dependence As we have seen in Section 2, additional information about the identities of the bidders and the bids up to the second highest bid does not improve the trivial lower bound on the joint distribution of values of all bidders, i.e. on Q(t1, …, td). This is because the absence of the knowledge of the highest bid does not allow us to bound from above the value of the auction’s winner. In auctions, it is not uncommon to think about the valuations of bidders as positively dependent random variables. There are several notions of positive dependence. The necessary relations among different notions of positive dependence are well established in the statistical literature.7 A nice review of types of dependence can be found in de Castro (2009). Milgrom and Weber (1982) describe in detail two types of positive dependence—affiliation, which is called the multivariate total positivity of order 2 in the statistical literature, and association. For instance, if bidders’ values Xi, i= 1, …, d, are defined as where A, , i= 1, …, d, are independent of each other, then Xi, i= 1, …, d, are affiliated if the densities of , i= 1, …, d, are log‐concave. Barlow and Proschan (1975) discuss total positivity properties in the context of reliability models. Second‐price auctions and ascending auctions within the private‐values framework are isomorphic to a subclass of such models. Affiliation is a strong notion of positive dependence, which implies, in particular, such properties as association, positive upper orthant dependence, positive lower orthant dependence and non‐negative correlation. Non‐negative correlation is a weak type of positive dependence implied by almost any other type of positive dependence. The example below illustrates situations of affiliation and non‐negative correlation. Example 3.1. Consider the case of three bidders whose private values X1, X2 and X3 are where Z1, Z2 and Z3 are mutually independent random variables. For now, let us ignore the assumption of the finite lower support point and suppose that Then and obviously, the variables are non‐negatively correlated iff (3.4) As indicated by Karlin and Rinott (1980), the variables in a random vector with multivariate normal distribution are affiliated iff the off‐diagonal elements of the matrix Σ−1 are non‐positive. In our case, the necessary and sufficient conditions for affiliation take the form (3.5) It can be verified that conditions (3.5) imply (3.4). Importantly, (3.5) are strictly stronger than (3.4). Indeed, consider , and ρ3= 0. Conditions (3.4) are satisfied, while (3.5) are not because Bounds on distributions in Section 2 are obtained for any type of dependence of bidders’ values. This section shows how assumptions about types of positive dependence can improve bounds on distribution functions, especially the lower bound on function Q(t1, …, td). For simplicity, I only consider the basic scenario, in which the winner’s identity and the winning price are observed. 3.1. Positive lower orthant dependence (PLOD) PLOD of random variables Xi, i= 1, …, d, is defined as It means that Xi, i= 1, …, d, are more likely to simultaneously take low values than their independent copies. In terms of copulas, this dependence means that the copula C(u1, u2, …, ud) corresponding to the joint distribution of Xi, i= 1, …, d, satisfies the property Suppose that the seller wants to introduce a reserve price r and is interested in learning the probability of the object not being sold. Using (2.1) in Theorem 2.1, the seller can find an upper bound on this probability: Theorem 2.1 gives the trivial lower bound of 0 on this probability. Under PLOD, however, it is possible to find a non‐trivial lower bound because and thus, Lower bounds on the joint distributions for subsets of bidders may also improve. The results are presented in the next theorem. Theorem 3.1. Suppose bidders’ valuesXi, i= 1, …, d, satisfy the PLOD property. Then, the joint distribution of values for bidders inD={i1, …, ir}is bounded from below as follows: (3.6) Because the highest value is not observed in auction’s outcomes, the lower bounds in Theorem 2.1 do not provide very good approximations of distribution functions around the upper support point. PLOD property may alleviate this situation. For instance, if all observed functions Gj are the same, i.e. Gj=G for any j, then If all , … , are close to the upper bound T, then the value of the first term in the maximum is close to , while the value of the second term is closer to 1 because it is equal to . Thus, the PLOD property improves the bound at the values close to the upper support point. 3.2. Positive upper orthant dependence (PUOD) PUOD of random variables Xi, i= 1, …, d, is defined in terms of survival functions: It means that Xi, i= 1, …, d, are more likely to simultaneously take high values than their independent copies. For survival functions, it is possible to obtain results analogous to the results in Theorem 2.1. The PUOD property allows us to find a non‐trivial lower bound on the joint survival function of the values of all bidders. If the seller, for instance, is interested in learning something about the probability of all bidders participating in the auction if a reserve price r is introduced, then PUOD helps to get a non‐trivial lower bound on that probability: This result also uses the upper bound on the marginal distribution functions from Theorem 2.1. 3.3. Left‐tail decreasing (LTD) Let M be a non‐empty subset of {1, …, d}. Variables Xi, i∈CM, are called LTD in Xj, j∈M, if is non‐increasing in tj for any j∈M. Suppose the LTD property holds for M={d}. Then, a non‐trivial lower bound on the joint distribution of bidders’ values is obtained in the following way: The results of Theorem 2.1 are used in the last step in these derivations. If, for instance, the LTD property holds for every singleton M={m}, m= 1, …, d, then The next theorem gives a general result on the lower bound for Q. In this theorem, denotes the collection of non‐empty subsets M such that variables Xi, i∈CM, are LTD in Xj, j∈M. Theorem 3.2. The LTD property may also help to improve the lower bound on the joint distributions for subsets of bidders. 3.4. Right‐tail increasing (RTI) Let M be a non‐empty subset of {1, …, d}. Variables Xi, i∈CM, are called RTI in Xj, j∈M, if is non‐decreasing in tj for any j∈M. Since this notion is formulated in terms of survival functions, this type of dependence is useful for improving lower bounds on the joint survival function for subsets of bidders, in particular, on the joint survival function of the values of all bidders. 3.5. Affiliation Affiliation of random variables if defined in the following way. Suppose that the distribution of bidders is continuous with respect to the Lebesgue measure. Vector X= (X1, …, Xd) is affiliated if for any x= (x1, …, xd) and y= (y1, …, yd), Affiliation is a strong condition on dependence. It is difficult to use it for the construction of bounds on the distribution functions. However, affiliation implies PLOD and PUOD, and also LTD and RTI. Based on the analysis above, this means that the affiliation property improves bounds on distributions. An important question is how much, for instance, affiliation improves bounds compared to the improvement obtained using PLOD, PUOD, LTD and RTI. This is a difficult question and is yet to be analysed. 4. Conclusion This paper addresses the identification question in auctions with incomplete data. Bidders have private values and compete in sealed‐bid second‐price auctions or open ascending auctions. It is assumed that in second‐price auctions, the value of the highest bid is never observed by an econometrician. If bidders’ values are not independent, then the distributions of bidders’ valuations can be identified only partially. One of the contributions of this paper is to obtain bounds on the joint and the marginal distribution functions for different observational scenarios. The least informative scenario is when only the price and the winner’s identity are observed in auction’s outcomes. The most informative scenario is when all the losing bids and all the identities are observed. The paper shows how much identification improves when more data become available. The second contribution of the paper is to use various notions of positive dependence of bidders’ values to improve the bounds on the joint distribution. Specifically, non‐trivial lower bounds are obtained for the joint distribution of values of all bidders. Bounds presented in this paper are not functionally sharp. For instance, for obtained pointwise bounds on the marginal distributions of values, not every collection of d distribution functions between these bounds provides a feasible collection of marginals. A problem worth exploring in future work is, e.g., finding a complete characterization of the possible marginal distributions with given observables without imposing any assumptions on the underlying joint distribution of bidders' values. Acknowledgments The author is grateful to Philip Haile, Oliver Linton, Martin Pesendorfer, Elie Tamer, two anonymous referees and the editor for helpful discussions and important comments and suggestions. The author also appreciates feedback from the participants of the 21st EC2 Conference. Footnotes 1 " See, for instance, Vickrey (1961) or Krishna (2002). 2 " Instead, it can be assumed that the joint distribution of bidders’ values is absolutely continuous, with respect to the Lebesgue measure, which implies zero probabilities of ties. 3 " http://www.shpauctions.com/ 4 " http://www.vanceauctions.com 5 " The identity of the lowest bidder, however, can be determined since the set of all bidders is known and the identities of all other bidders are observed. 6 " Because the sale price is known and the identity of highest bidder can be determined from the identities of all other bidders, the case N=d− 1 is observationally equivalent to the case N=d− 2. 7 " See, for instance, Karlin and Rinott (1980). References Aradillas‐Lopez , A. , A. Gandhi and D. Quint ( 2011 ). Identification and inference in ascending auctions with correlated private values . Forthcoming in Econometrica. Armstrong , T. ( 2011 ). Bounds in auctions with unobserved heterogeneity . 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To prove that the upper bound on Q is tight, it is necessary to show that for each bound, there exists a joint probability distribution, that has the specified observable functions Gj, and has probability mass arbitrarily close to the mass of the bound. For a given collection of observable functions Gj, consider a joint distribution that is the sum of d sub‐distributions such that the sub‐distribution j places mass at (X1, …, Xd) which is equal to xi for i≠j and α max i≠jxi+ (1 −α)T for xj according to Gj(max i≠jxi), where α < 1 is arbitrarily close to 1. This joint distribution has the specified Gj and approaches the upper bound arbitrarily closely because α can be taken arbitrarily close to 1. For D={1, …, m− 1, m+ 1, …, d}, consider a joint distribution that is the sum of d sub‐distributions such that the sub‐distribution j, j≠m, places mass at (X1, X2, …, Xd) which is equal to xi for i≠j and α max i≠j,i≠mxi+ (1 −α)T for xj and β max i≠j,i≠mxi+ (1 −β)t0 for xm according to Gj(max i≠j,i≠mxi), where α < 1 and β < 1 are arbitrarily close to 1. The sub‐distribution m places mass at (X1, …, Xd) which is equal to xi for i≠m and α max i≠mxi+ (1 −α)T for xm according to Gm(max i≠mxi), where α > 0 is arbitrarily close to 0. Similarly, for any subset D, the upper bound on QD is tight.□ Proof of Proposition 2.1: (a) According to Assumption 2.1, for any j, the event implies the event , which in its turn implies the event . Therefore, The rest of the proof for the upper bounds is the same as in Theorem 2.1. (b) Suppose that . Then According to Assumption 2.2, for j∈CD, the event implies the event . Indeed, if some was larger than , then bidder ik would not allow bidder j to win at a price less or equal than . Therefore, The rest of the proof for the lower bounds is the same as in Theorem 2.1.□ Proof of Theorem 2.2: The upper bound on Fi(t) is the upper bound on Q(t1, …, td) calculated at ti=t and Proof of Theorem 2.3: (a) The lower bound is obvious. The upper bound is obtained in the following way: (b) Without a loss of generality, consider function F1. Note that Now substitute to obtain the upper bound. To obtain the lower bound, note that and take into account that Proof of Proposition 2.2: The lower bound of 0 on Q(t1, …, td) is obvious. Let us derive the upper bound. The upper bound on Fi(t) is the upper bound on Q(t1, …, td) calculated at ti=t and Let us now derive the lower bound on Fi(t). Proof of Theorem 3.1: The first term in the maximum in (3.6) is the result of Theorem 2.1, which does not assume any dependence properties. The second term in the maximum follows from the PLOD property and the result on the lower bound for the marginal distribution functions in Theorem 2.1: Proof of Theorem 3.2: Take any . Using the LTD property, obtain Theorem 2.1 then gives Appendix B: Figures for the Monte Carlo Experiment B1 [ Bounds for F1: n= 200. ] B2 [ Bounds for F2: n= 200. ] B3 [ Bounds for F3: n= 200. ] B4 [ Bounds for F4: n= 200. ] B5 [ Bounds for F1: n= 500. ] B6 [ Bounds for F2: n= 500. ] B7 [ Bounds for F3: n= 500. ] B8 [ Bounds for F4: n= 500. ] © 2013 The Author(s). The Econometrics Journal © 2013 Royal Economic Society.
The Econometrics Journal – Oxford University Press
Published: Feb 1, 2013
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