Ranking sealed high-bid and open asymmetric auctions

Journal of Mathematical Economics 42 (2006) 471–498

Ranking sealed high-bid and open asymmetric auctions Harrison Cheng∗ Department of Economics, University of Southern California, University Park, Los Angeles, CA 90089-0253, United States Received 16 November 2005; received in revised form 30 May 2006; accepted 30 May 2006 Available online 11 July 2006

Abstract For an important family of asymmetric auctions, we show that the seller’s expected revenue is higher in the sealed high-bid auction than in the open auction. This is true for arbitrary numbers of weak and strong buyers. We establish many interesting properties of the linear asymmetric auction model. We show how the linear model can also be useful for non-linear models. Revenue comparisons for the two auction formats are performed using data observed in U.S. forest timber auctions. We show that the revenue difference is minimal with a fixed number of participants, but can be as high as 14% when the difference in participation is taken into account. The revenue difference predicted by the linear model is quite similar to the empirical results of Athey et al. (2004). © 2006 Elsevier B.V. All rights reserved. JEL classification: C72; D02; D44; D82 Keywords: Revenue ranking; Open and sealed-bid auctions; Linear bidding model; Lumber auctions

1. Introduction The analysis of auctions with identical buyers has been extensively studied. Many of the standard results in such symmetric auctions are often false in asymmetric auctions with different buyers. For example, the well-known revenue equivalence theorem applies mainly to symmetric auctions. In asymmetric auctions, sealed high-bid (first price) auctions and open (second price) auctions usually yield different revenues for the seller. Empirical studies of the different revenues in the two types of auctions of the U.S. Forest timber can be found in Johnson (1979), Hansen (1986), Schuster and Niccolucci (1994), Stone and Rideout (1997), and Athey et al. (2004). ∗

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Despite the importance of going beyond symmetric auctions, the theory of asymmetric auctions is much less developed. Asymmetric auctions have been analyzed in the seminal papers of Maskin and Riley (2000a,b, 2003), as well as in Bajari (1997), Lebrun (1999), Pesendorfer (2000), Brannman and Froeb (2000), Arozamena and Cantillon (2004) and Cantillon (2001). It is clear from these important contributions that in asymmetric auctions we often do not have closed-form solutions of the equilibrium bidding strategies. As a result, we have relatively little understanding of the properties of the asymmetric auctions. Comparisons of the revenues between different auction formats remain difficult if not impossible. It is desirable to have a simple benchmark model in which asymmetric auctions can be more easily analyzed. We provide such a model in this paper. In our model, buyers’ valuation distribution functions are well-known power functions. Such valuation distributions are used often in symmetric auctions, and they also arise naturally in asymmetric auctions. A uniform (cumulative) distribution of valuation over an interval, such as F (v) = v over [0, 1], describes a situation in which a buyer has little prior knowledge of where the valuation lies within the interval. When two such buyers merge, and pool their private information together, the new buyer formed from the merger has the valuation distribution F (v) = v2 . Thus, a power function represents the valuation of a merger of several such buyers who collude by bidding jointly for the object. The model we study here focuses on a situation in which the strength of the buyers differ due to joint bidding or collusion rather than to prior knowledge. In our model, equilibrium behavior is much easier to understand because the strategies are linear functions of the valuations. We believe that linear asymmetric models provide an important class of tractable models for the study of asymmetric auctions. To obtain linear equilibrium strategies, we need to impose restrictions on the valuation intervals of the buyers. We thus see the linear models here as a testing ground for ideas and possible results in asymmetric auctions. It is important to know whether the results in linear models extend to non-linear models. Our main purpose here is to establish important properties and closed-form solutions of the linear models, and to also provide evidence of the usefulness of linear models for the analysis of non-linear ones. In the asymmetric auction with linear equilibrium strategies, a buyer’s bidding strategy depends only on the total “strength” of his opponents. The strength is measured by the power of the cumulative distribution function. Each buyer has a measure of strength, and the strength of a group of buyers is the sum of the individual measures of strength. When a buyer’s strength changes, it changes the opponents’ bidding behavior but not his own. This simplified pattern of interaction makes it easier to understand the properties of the model. Another useful property is that a buyer with valuation below the maximum valuation of the weak buyer always bids an amount equal to the expected second highest valuation of the losing buyer. This means that the price a buyer pays when he or she wins is the same in both auctions. This property is usually reserved for symmetric auctions but is true in our model. The difference in expected payment, then, is due to the difference in winning probabilities. When a strong buyer has valuation higher than the maximum valuation of a weak buyer, we can also show that the strong buyer pays more when he wins in the sealed-bid auction than in the open auction. To demonstrate the usefulness of the model, we investigate the revenue ranking problem in asymmetric auctions. The classic result in the literature is in Maskin and Riley (2000a). They are able to rank the sealed high-bid and open auctions only in cases of “first order asymmetry”. Specifically, they show that if the strong buyer’s valuation distribution is a “shift” or a “stretched” version of the weak buyer’s distribution, then the sealed high-bid (first price) auction yields higher revenue to the seller than does the open (second price) auction. In our model, differences in valuations arise from joint bidding and are not covered by their study. We show that in our

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model with linear equilibrium bidding strategies, the seller always prefers the sealed auction over the open auction. Furthermore, the result holds for an arbitrary number of strong and weak buyers. The asymmetric auctions with more than two buyers is generally a difficult model to analyze. As far as we know, this is the first systematic result of the revenue ranking between the two auction formats in asymmetric auctions with many buyers. Although our linear model is parametric and the result can be thought of as another parametric class of ranking results beyond those of Maskin and Riley (2000a), we do suggest, in Section 5, how our analysis may be extended to non-linear models and how the merger effects can be analyzed with more general distributions. One main idea is that it is possible to make comparisons between a linear model and a non-linear one so that we know what results are likely to hold and what kinds of conditions are needed for the ranking result. We use the term “weak Getty effect” to explain the intuition of our result. “Getty effect” refers to the case in which the strong buyer’s bid dominates the weak buyer’s bid. When the Getty effect is active, the sealed-bid auction yields higher revenue to the seller. This is because the strong buyer pays the maximum of the weak buyer’s valuation in the sealed auction, but pays the expected value of the weak buyer’s valuation in the open auction. Since the maximum is higher than the expected value, the seller gets a higher revenue in the sealed-bid auction. The weak Getty effect refers to the case in which the strong buyer is relatively more likely to be in the high range than in the low range. In asymmetric auctions, a weak buyer has higher winning probabilities; hence the seller receives more revenue from the weak buyers in sealed-bid than in open auctions. The opposite is true for the lower-valued strong buyers. This delicate balance is broken by the higher-valued strong buyers who tend to pay more in the sealed-bid auction. This weak Getty effect results in higher revenue in sealed-bid auctions. A more detailed explanation is presented in Section 2 using a simple example with two buyers. When the Getty effect is present, a strong buyer bids high for the object, and the sealed-bid auction is superior to the open auction. This contrasts with the low-balling strategy in which a strong buyer bids low for the object, and the open auction is superior to the sealed-bid auction. A natural question to ask is when to adopt the high bidding strategy and when to adopt the low-balling strategy. This issue is discussed in Section 7. The weak Getty effect is an intuitive but vague statement. As a strategy of proof, it quickly becomes too complicated. The proof of our main ranking result is based on a property called the “bundling effect”. When several weak buyers are merged into one weak buyer with the same total strength, they behave as a collusive block. We call this bundling rather than collusion because the valuation interval of the weak buyer needs to be “stretched” to maintain the linearity of the model after the weak buyers are bundled into one. We show that bundling does not affect the seller’s revenue in the sealed-bid auction but raises the seller’s revenue in the open auction. This bundling effect makes it possible for us to prove the ranking result. A more detailed explanation is provided in the next section through an example. The effect of bundling can be understood intuitively as follows. Bundling is the combined effect of a merger and a stretch of the weak buyer’s distribution to the right. While a merger tends to decrease revenue because of reduced competition, a stretch of the distribution to the right tends to raise the revenue. For the sealed bid auction, the two effects cancel each other, while for the open auction, the merger effect is smaller. This explains why the revenue stays the same in the sealed-bid auction, but increases in the open auction. Another useful idea is that a merger may have a bigger effect on revenue in the sealed bid auction than in the open auction. A merger does not change the bidding behavior of the buyers in the open auction, but changes the bidding behavior of the buyers in the sealed bid auction. The bidding is less aggressive (for the weak buyers) and hurts the revenue beyond the effect of a reduced number of bidders. In Section 5, we show how our analysis in the linear model sheds light on the problem of merger effects.

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We perform numerical computations with our linear model using the summary data from the U.S. forest timber auctions in Athey et al. (2004). In the lumber auctions, a weak buyer is a logger who has a single source of valuation for the product. A strong buyer is an integrated mill who can exploit many different independent sources of valuation of lumber (for use in construction, furniture or paper making). We want to get a quantitative measure of the size of the revenue gap between sealed-bid auctions and open auctions using realistic parameters. By choosing parameters of the model to fit the pattern observed in the data, the linear model predicts that the revenue difference between the two auction formats is very small (less than 1%) with a fixed number of participants. Since there is a difference in participation by buyers, with weak buyers preferring sealed-bid auctions, we also consider the effect of such difference in participation on the revenue. When the participation pattern is taken into account, the theoretical prediction of the revenue difference in the linear model is about 14%, which is quite similar to the range obtained by Athey et al. (2004) in their empirical analysis with a more complicated specification.1 Our model adds more families of auctions in which the sealed auction performs better than the open auction in revenues. Graham and Marshall (1987) consider collusion in asymmetric auctions. Klemperer (1998) considers auctions with almost common values. Each paper points to cases in which the seller’s revenues are higher in the sealed auction. Empirical evidence in the timber auctions cited earlier also points to the higher revenues in the sealed-bid auctions. There are related works on revenue comparisons between symmetric and asymmetric auctions. Cantillon (2001) argues that asymmetry reduces the seller’s revenue when compared with a symmetric auction with the same social surplus, while Kaplan and Zamir (2002) make the comparisons under different beliefs (or information structure). Section 2 gives a simple summary of the important properties of the linear asymmetric auction model. Section 3 states these properties formally and proves several of them (with many auxiliary results proved in the Appendix A). In Section 4, we show that the sealed-bid auction yields higher expected revenue for the seller than the open auction. We also give an example to show the difference between merger and bundling. In Section 5, we discuss possible extensions of the ranking result to procurement auctions with more general cost intervals. We also discuss how our analysis is relevant for the analysis of merger effects and how to go beyond the class of power function distributions. In Section 6, we perform numerical computations using typical data observed in timber auctions. In the concluding remarks of Section 7, we raise the issue of the tension between the Getty effect and the low-balling strategy, and we also discuss the usefulness of the model for empirical testing. Many of the proofs are provided in Appendix A. 2. Main properties This section illustrates the main properties of the linear asymmetric auctions. In the first example, we show that the equilibrium bidding strategy of a buyer in the sealed-bid auction depends on the strength of the opponent, not on the opponent’s strategy, or his own strength. Example 1. There is one weak buyer and one strong buyer. Let the weak buyer’s c.d.f. of the valuation be F1 (v) = 43 v with the support [0, 43 ], and that of the strong buyer be F2 (v) = v2 1 They also consider issues of collusion and other relevant factors. We do not consider these matters, because we just want to get a feel of the revenue gap between the two auction formats under competitive conditions. They use Weibull bid distributions while our equilibrium bid distributions are power functions.

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with the support [0, 1]. It will be shown later in Proposition 4 that we have the following linear equilibrium bidding strategies for the buyers in a sealed-bid auction 2 1 b2 (v) = v b1 (v) = v, 3 2 Note that if the valuation interval of the weak buyer were [0, 21 ], and F1 (v) = 2v, then we don’t have linear equilibrium strategies. Assume that the strength (the exponent 2 in the power function v2 )
3 of the strong buyer is now changed to 3 with F2 (v) = av over the interval [0, a]. To maintain the linear property of the model, we need to take a = 98 . In this new auction, the equilibrium strategies are 3 1 b2 (v) = v. b1 (v) = v, 4 2 The chosen value for a insures that the highest bids of the two buyers are the same and both are equal to 3 3 9 1 9 = × = × . 16 4 4 8 2 Subject to the maintenance of the linear equilibrium strategies, the strategy of the weak buyer is now more aggressive (closer to the true valuation) because of the increased strength of the strong buyer. The equilibrium strategy of the strong buyer remains the same as before, because the strength of the weak buyer is unchanged. Notice also that the strategy of the weak buyer has changed, but this does not affect the strategy of the strong buyer. The property illustrated by Example 1 holds when there are many weak and strong buyers. A buyer’s equilibrium bidding strategy depends only on the total strength of all the other buyers
(the r sum of all the exponents) not on his or her own strength. The strength of a buyer with F (v) = vc over [0, c] has two parameters r and c. In a linear model, both r and c vary in the same direction, and c is determined by r. Therefore r alone can represent the strength of a buyer. We shall now illustrate the second important property of our model. A buyer with valuation less than the maximum valuation of the weak buyer pays the same (expected) price as a winner in both the sealed-bid and open auctions. We shall refer to this price as the winner’s price (payment conditional on winning). This property holds in symmetric auctions, but is generally false in asymmetric auctions. It is true in our linear asymmetric model, and plays an important role in our analysis and computations of the revenues. The expected payment in equilibrium of a buyer with valuation v is his or her winning probability times the winner’s price. Hence in our model, the difference of the expected payments in the sealed-bid and open auctions is explained by the difference in winning probabilities as the winner’s price is the same in both auctions. Example 2. Take the same example we considered in Example 1 with one weak buyer and one strong buyer and F1 (v) = 43 v over [0, 43 ], F2 (v) = v2 over [0, 1]. Take any weak buyer with v ≤ 43 . In the sealed-bid auction, the equilibrium bid is 23 v, and the winner’s price 23 v is not stochastic. In the open auction, the equilibrium bid is v, and the winning probability is F2 (v) = v2 . The winner’s price is stochastic, with the expected price equal to   1 v 2 1 v 2 2 v3 2 = v, x d(x ) = 2x dx = 2 2 2 v 0 v 0 v 3 3 which is the same the winner’s price in the sealed-bid auction. The same property can be shown in the same way for a strong buyer with v ≤ 43 . For the strong buyer with v > 43 , the winning

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probability is 1 in the open auction, and the payment is the same as the strong buyer with valuation 3 4 . In the sealed-bid auction, the winner’s price is higher than that of the strong buyer with valuation 3 4 . Hence the winner’s price is higher in the sealed-bid auction than in the open auction. 2.1. The bundling effect An important property called the “bundling effect” is a central component of our analysis and the key to the proof of our main result. We shall now illustrate the bundling effect with an example. r Assume that there are n weak buyers each with a cumulative valuation distribution F1 (v) = vcr over [0, c]. Assume that there is only one strong buyer with the cumulative valuation distribution function F2 (v) = v10 over the interval [0, 1]. All valuation distributions are assumed to be independently distributed. A linear equilibrium model requires that 10 + (n − 1)r nr c= 10 + (n − 1)r + 1 nr + 1

(1)

with the equilibrium strategies: b1 (v) =

10 + (n − 1)r v, 10 + (n − 1)r + 1

b2 (v) =

nr v nr + 1

for the weak and strong buyers, respectively. For example, let n = 2, r = 1, then (1) says that we 8 2 11 11 8 must have c = 23 12 11 = 11 . If we have n = 1, r = 2, then we must have c = 3 10 = 15 > 11 . The strong buyer’s bidding behavior depends only on the total strength of the weak buyers nr = 2. Assume that the two weak buyers are bundled into one weak buyer with the same total strength 2. We have just seen that c becomes larger by bundling. The new distribution for the weak buyer
2 after bundling is F1 (v) = vc over [0, c], c = 11 15 . Before bundling, the equilibrium strategy of a weak buyer is b1 (v) =

11 v. 12

After bundling, the weak buyer’s equilibrium strategy becomes b1 (v) =

10 v. 11

The equilibrium strategy of the strong buyer is not affected and is b2 (v) = 23 v before and after bundling. Therefore the bid distribution of the strong buyer is the same as before. The cumulative bid distribution of a weak buyer before bundling is   12 3 2 11 = x, for x ∈ 0, x× 8 2 3 11 and the cumulative distribution of the highest bid from the two weak buyers is F (x) = ( 23 x)2 . After bundling, there is only one weak buyer with the cumulative bid distribution:    2 11 15 2 3 x× x = 10 11 2 which is the same as F (x). This means that the bid distribution of the highest bid from the weak buyers is the same before and after bundling, and therefore the seller’s revenue is unchanged

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before and after bundling in the sealed-bid auction.2 This is a general property, as can be seen in Corollary 8. For the open auction, bundling has more complicated effects on winning probabilities, and payments upon winning. If c does not change, there is no effect from bundling on the strong buyer’s winning probabilities or payments upon winning. However bundling does increase c. For a weak buyer, bundling reduces competition, and increases the winning probabilities, but also reduces payments upon winning. Fortunately, the impact on the seller’s revenue is unambiguous from the following demonstration which is a simplified version of the general proof in the main Theorem 11. A weak buyer’s winning probability when he bids x is  x (n−1)r x10 c and the expected payment upon winning, by the second property, is 10 + (n − 1)r x. 10 + (n − 1)r + 1 Hence the seller’s revenue from a weak buyer is  c  x (n−1)r 10 + (n − 1)r  x r r 10 + (n − 1)r xd c11 x10 = c 10 + (n − 1)r + 1 c 10 + (n − 1)r + 1 nr + 11 0

(2)

and the seller’s revenue from all the weak buyers is nr nr nr 10 + (n − 1)r c11 = c10 . 10 + (n − 1)r + 1 nr + 11 nr + 1 nr + 11 Note that in the above expression for the revenue, bundling affects the revenue through its effect on c only. This is because bundling changes r while keeping nr unchanged. Similarly, the strong buyer’s winning probability, when she bids x < c, is  x nr c and the expected payment upon winning is nr x. nr + 1 The strong buyer bidding x > c pays the seller the following amount upon winning  c  x nr nr xd =c . c nr +1 0 Hence the seller’s revenue from the strong buyer is  c x nr nr 10 nr nr nr x d(x)10 + c (1 − c10 ) = c11 + (c − c11 ) c nr + 1 nr + 1 nr + 1 nr + 11 nr + 1 0   nr + 1 11 nr . c− c = nr + 1 nr + 11 2 I want to thank an anonymous referee for suggesting this idea to me. It is a more intuitive argument than the one used in an earlier version of the paper.

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The seller’s revenue in the open auction is therefore:     nr 2 nr + 1 11 3 nr 2 = c10 + c − c c + c10 − c11 nr + 1 nr + 11 nr + 11 3 13 13 To show that bundling increases the seller’s revenue, it is sufficient to show that the above expression is increasing in c, as c increases after bundling. We want to show that the function: 2 10 3 c − c11 13 13 is increasing in c when c ≤ 1. The derivative with respect to c is   20 9 33 10 20 10 33 10 20 33 10 10 1+ c − c ≥c + c − c = 1+ − c = 0. 13 13 13 13 13 13 c+

Hence we have shown that bundling increases the seller’s revenue in the open auction. 2.2. Bundling and revenue ranking To show the implications of the bundling effects on revenue ranking between the sealed-bid and open auctions, consider the following variation of the example in the last section. There are five weak buyers n = 5 and one strong buyer. The strong buyer has the same valuation distribution in the last section. Assume that r = 2 so that the total strength of the weak buyers is nr = 10. When we bundle all five weak buyers into one with the strength 10, the auction becomes a symmetric one with two identical buyers. In this new auction, the revenue equivalence theorem says that the seller’s revenue from the sealed-bid and open auctions are the same. Since bundling increases the seller’s revenue in the open auction, and does not affect the seller’s revenue in the sealed-bid auction, we conclude that the seller’s revenue from the sealed-bid auction must be higher than that from the open auction before bundling. Hence in this example, bundling effects allow us to show that the sealed-bid auction is superior to the open one in the original asymmetric auction. Consider now the auction with six weak buyers instead. In other words, assume that n = 6, r = 2 while other things are kept the same as before. The total strength of the weak buyers is now nr = 12 which is higher than the strength of the strong buyer 10. We cannot bundle the six weak buyers into one, and apply the bundling effect, as it only applies when the merged buyer is weaker than the strong buyer. However, we can take r = 10, n = 1.2. The fractional number of the weak buyers n = 1.2 can be interpreted as follows. There are two identical weak buyers each with r = 10. One of them only shows up in the auction with probability 0.2. In other words, we can allow probabilistic participation of the weak buyers. After bundling, all buyers are the same and there are 2.2 of them. In this symmetric auction, the revenue equivalence theorem applies and, combined with the bundling effects, we conclude that the sealed-bid auction is superior to the open auction in the original asymmetric auction. When we have n = 2, r = 1, the total strength of the weak buyers is less than the strength of the strong buyer. We can bundle the weak buyers into a fractional number of buyers with n = 0.2, r = 10. Bundling effect will then apply, and we get the same conclusion on the ranking of the sealed-bid and open asymmetric auctions. After we derive the formulas for the seller’s revenues later, the revenue ranking result we want to show becomes a mathematical inequality. These inequalities make sense when n takes on real numbers rather than integers. Therefore, the model with probabilistic participation need not be used in the rigorous proof. It is however a good motivation for the inequalities we will show later.

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2.3. Weak Getty effects We now use the second property to give an intuitive explanation of the higher revenue of the sealed-bid auction. This is only an intuitive explanation using the concept of “weak” Getty effects. Take the buyers we considered in Example 1 with one weak buyer and one strong buyer and F1 (v) = 43 v over [0, 43 ], F2 (v) = v2 over [0, 1]. We have the following equilibrium bidding strategies in the sealed-bid auction: 2 1 v, b2 (v) = v. 3 2 Let qi (v), qi∗ (v) be the equilibrium winning probabilities of buyer i ∈ {1, 2} in the sealed-bid auction and the open auction, respectively. We have     4 3 16 2 q1 (v) = F2 q2 (v) = F1 v = v , v = v; q1∗ (v) = F2 (v) = v2 , 3 9 4 b1 (v) =

4 v. 3 It is clear that the weak buyer has higher winning probabilities in the sealed-bid auction than in the open auction, while the reverse is true for the strong buyer. The payments in both auctions for the buyer as a winner are 23 v, 21 v for the weak and strong buyers, respectively, if their valuation is v ≤ 43 . The weak buyer’s expected payment to the seller in the sealed-bid auction is  3/4 2 q1 (v) v dF1 (v). 3 0 q2∗ (v) = F1 (v) =

For easier comparisons with the open auction, we shall change the variable from the valuation variable v to the payment variable p = 23 v. The integral after the change of variable is written as    1/2   3 3 1 q1 p p dF1 p = = 0.125. (3) 2 2 8 0 In the open auction, the weak buyer’s expected payment to the seller can be written as  3/4 2 q1∗ (v) v dF1 (v). 3 0 We can change to the (expected) payment variable p = 23 v as well, and it becomes    1/2   3 3 9 q1∗ p p dF1 p = = 0.070313. 2 2 128 0

(4)

We now compare the weak buyer’s payments in the two auctions. Since the strong buyer shades his bid more than the weak buyer, we must have q1 (v) > q1∗ (v). From (3) and (4), we know that the weak buyer’s payment to the seller must be higher in the sealed-bid auction than in the open auction. The difference is 0.125 − 0.070313 = 0.054687. Similarly, the strong buyer’s expected payment to the seller in the sealed-bid auction is written as  1/2 q2 (2p)p dF2 (2p) = 0.25. (5) 0

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In the open auction, the strong buyer’s expected payment to the seller is  3/4 1 q2∗ (v) v dF2 (v). 2 0

(6)

A strong buyer with valuation v ∈ [ 43 , 1] has the following interim payment:    3/4 41 3 2 3 v dF1 (v) = = . 32 4 8 0 This is a constant. When v > 43 , a winning v-valued buyer pays the same amount as the buyer with valuation 43 . Note that this interim payment is smaller than that in the sealed-bid auction given by 1 3 v> . 2 8 When we change to the payment variable p = 21 v, (6) becomes     3/8 3 3 q2∗ (2p)p dF2 (2p) + 1 − F2 . 8 4 0 We can compare (7) to (5), which is rewritten as  1/2  3/8 q2 (2p)p dF2 (2p) + q2 (2p)p dF2 (2p). 0

(7)

(8)

3/8

Note that q2∗ (v) = 43 v > v = q2 (v), and therefore the first integral in (7) is larger than the first integral in (8). The second expression in (7) can be easily dominated by the second integral in (8) if the probability that the strong buyer has valuation higher than 43 is large enough. We can compute their values as      3 3 3 9 1 − F2 = 1− = 0.16406; 8 4 8 16

 4   1/2  1/2 3 1 3 = 0.17090. − q2 (2p)p dF2 (2p) = 16p dp = 4 16 8 3/8 3/8 Hence in our example, the second integral of (8) is indeed slightly larger3 than the second expression of (7). We also compute the values of the first integrals as follows:    3/8  3/8 64 3 16 3 4 q2∗ (2p)p dF2 (2p) = = 0.10547; p dp = 3 3 8 0 0  4  3/8 3 16p3 dp = 4 = 0.07910. 8 0 The payment to the seller by the strong buyer is higher in the open auction than in the sealed-bid auction by the amount 3 This is not always true. However the difference of the two integrals is not large enough to affect the ranking of the two auctions.

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0.10547 + 0.16406 − (0.07910 + 0.17090) = 0.01953. This amount is dominated by the difference for the weak buyers 0.054687. Therefore in our example, the seller has higher revenue in the sealed-bid auction than in the open auction. The superiority of the sealed-bid auction in our model is explained by two factors: (1) the higher payment by the higher-valued stronger buyer in the sealed-bid auction than in the open auction which offsets the higher revenue from the lower-valued strong buyer in the open auction; (2) the strong buyer has a high probability of being in the higher end of the valuation interval. This explanation has connections to the Getty effect. The Getty effect in Maskin and Riley (2000a) refers to a situation in which the strong buyer’s valuation is so strong that his bid dominates that of the weak buyer with certainty. We shall use the term “strong Getty effect” to refer to a situation in which the strong buyer’s bid dominates that of the weak buyer with probability close to 1. The strong Getty effect occurs when the c.d.f. of the strong buyer has a very high power F2 (v) = vn , n → ∞. When this is true, the strong buyer’s payment to the seller will be higher in the sealed-bid auction. If the strong buyer’s valuation is very likely to be above the maximum valuation of the weak buyer, then the strong buyer’s payment to the seller in the open auction would not be much larger, if it is larger, than his payment in the sealed-bid auction. We loosely refer to this as the weak Getty effect. 3. The model We assume that the auction is a standard private valuation auction with a single object for sale. The buyer’ valuation for the object is independent of each other. There are two kinds of buyers: weak and strong ones. Let n1 be the number of weak buyers and n2 the number of strong buyers. Each buyer has a valuation distributed according to a power function over some interval. For simplicity, assume that the seller has no value for the single object for sale. Let r1 > 0, r2 > 0 be two numbers with r1 < r2 . Let the c.d.f. of the valuation of the weak buyer F1 (v) be distributed over [0, c1 ]. The c.d.f. is assumed to be a power function of the form kvr1 . r The requirement F1 (c1 ) = 1 means that F1 (v) = vr11 over [0, c1 ]. Let the c.d.f. of the valuation of the strong buyer be F2 (v) = F1 (v) =

vr2 r c22

vr1 vr2 r1 > r2 = F2 (v), c1 c2

c1

over [0, c2 ]. Moreover, we have for all 0 < v < c2 .

Hence F2 first-order stochastically dominates F1 . It is also easy to see that the property of conditional stochastic dominance is also satisfied. This is stated in the following lemma. Lemma 3. The conditional first-order stochastic dominance of F2 over F1 holds. In our model, the strength of a buyer is represented by the powers r1 , r2 . The higher the power, the stronger the buyer is. A strong buyer also has a larger support of the valuation distribution. It will turn out that the equilibrium strategy of each buyer is linear. When a buyer faces many other buyers each with a power function distribution, it is as if the buyer is facing a single buyer with a power function distribution under the assumption of independent private valuation. Moreover, the strength of all the opponents can be summed up by a single number which is the sum of all the powers of the opposing buyers. Let r2∗ = n1 r1 + (n2 − 1)r2

(9)

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be the strength of all the opponents of a strong buyer, and r1∗ = (n1 − 1)r1 + n2 r2

(10)

be the strength of all the opponents of a weak buyer. We have r2∗ − r1∗ = r1 − r2 < 0. r∗ (r ∗ + 1) . Since Let m = 2∗ 1∗ r1 (r2 + 1) r∗ (r ∗ + 1) m = 2∗ 1∗ < 1. r1 (r2 + 1)

(11) 1+

1 1 < 1 + ∗ , we must have r1∗ r2

r1∗

+1 < r1∗

r2∗

+1 , and r2∗ (12)

For weak and strong buyers, the usual assumption is c1 ≤ c2 . We further assume that c1 = mc2 < c2 . This is necessary to preserve the structure of linear equilibrium bidding strategies in the sealed-bid auction in our model. Note that in the two-buyer case, we have r1∗ = r2 , r2∗ = r1 . Let bi (v), i ∈ {1, 2} denote the equilibrium bidding strategies of the buyers in the sealed-bid (or first price) auction. We will show that the following pair of bidding strategies: b1 (v) =

r1∗ v, r1∗ + 1

b2 (v) =

r2∗ v r2∗ + 1

(13)

is a Bayesian Nash equilibrium and the equilibrium is unique. Proposition 4. The pair of strategies in (13) is a Bayesian Nash equilibrium in the sealed-bid auction, and the equilibrium is unique. Proof. When a weak buyer with valuation v bids x, the winning probability is  ∗  ∗       ∗ r + 1 n1 −1 r + 1 n2 (r1 + 1)x (n1 −1)r1 (r2∗ + 1)x n2 r2 ∗ F1 1 ∗ x F2 2 ∗ x = = kxr1 r1 r2 r1∗ c1 r2∗ c2 where (r1∗ + 1)(n1 −1)r1 (r2∗ + 1)n2 r2 . (r1∗ c1 )(n1 −1)r1 (r2∗ c2 )n2 r2

k=

Hence the profit function is ∗

k(v − x)xr1 .

(14)

The first order condition can be written as ∗

∗

−xr1 + r1∗ (v − x)xr1 −1 = 0 or −x + r1∗ v − r1∗ x = 0. We have the solution: r∗ x = ∗ 1 v. r1 + 1 If r1∗ > 1, the profit function is not a concave function of x. However, it is still true that the first order condition is sufficient for a bid x to be optimal. The reason is that the derivative of ∗ g(x) = (v − x)xr1 is ∗

∗

g (x) = vr1∗ xr1 −1 − (r1∗ + 1)xr1

(15)

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and the second derivative is given by ∗

∗

g (x) = vr1∗ (r1∗ − 1)xr1 −2 − r1∗ (r1∗ + 1)xr1 −1 =

r1∗ ∗ xr1 −2 ∗ r1 + 1



 r1∗ − 1 v − x . r1∗ + 1

483

(16)

r1∗ −1 r1∗ +1 v. This occurs below the critical point r∗ −1 ≥ r1∗ +1 v. Furthermore, g(x) is an increasing 1

The second derivative becomes positive when x < x=

r1∗ r1∗ +1 v.

The function g(x) is concave when x r∗ −1

r∗

1 v is the optimal bid for the weak buyer. The function within the range x < r1∗ +1 v; hence x = r∗ +1 1 1 same argument applies to the strong buyer as well. Note that the end values c1 , c2 have been lined r1∗ r2∗ up so that r∗ +1 c1 = r∗ +1 c2 . 1 2 To show uniqueness, we need some adjustments in the arguments of Maskin and Riley (2003), referred to as M-R. Using the arguments in Lemma 10 of M-R, we can show that all the strong buyers have the same maximum bid. Then a minor change also shows that all weak buyers have the same maximum bid. From Lemma 11 in M-R, we know that all the weak buyers have the same connected interval as the support of their bid distribution, and the same is true also for all the strong buyers. The differential equations in (2.9) of M-R are symmetric with respect to all weak buyers. If two weak buyers have different bidding strategies, we can switch their strategies to get a different solution to the system of differential equations. This violates the fundamental theorem of the ordinary differential equations and hence all weak buyers have identical bidding strategy. The same is true for all strong buyers. The problem is now reduced to the simpler “two-buyer case”, one strategy for the weak buyers, and another for the strong buyers. We have a system of two differential equations. Now Proposition 1 of M-R is applicable, and we have the uniqueness of equilibria.


The following result states the equilibrium winning probabilities of the strong and weak buyers in either types of auctions. Let qi (v), qi∗ (v) be the equilibrium winning probabilities of buyer i in the sealed-bid and open auctions, respectively. Lemma 5. The equilibrium winning probabilities of the buyers with valuation v in the sealed-bid auction are given by ∗

qi (v) =

vri

r∗

for i = 1, 2.

,

ci i

In the open auction, they are given by ∗

q1∗ (v) =

mn2 r2 vr1 r∗

∗

,

c11  (n2 −1)r2 v ∗ , q2 (v) = c2

q2∗ (v) =

vr2

r∗

mn1 r1 c22

,

for v ≤ c;

for v > c.

Another interesting property of our model is that in equilibrium a buyer with valuation v ≤ c1 pays the same expected price as a winner in both sealed-bid and open auctions. This property simplifies our analysis of the comparisons between the revenues from the two auctions. A strong buyer with valuation v > c1 pays more in the sealed-bid auction than in the open auction when she wins. The interim payment is a constant plus the interim payment in an auction (sealed-bid or open) with only the strong buyers. This is shown in the following proposition.

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Proposition 6. When any v-valued buyer with v ≤ c1 wins, the (expected) payment to the seller is the same in the sealed-bid and the open auctions. If v > c1 , then the (strong) buyer pays more in the sealed-bid auction than in the open auction. The expected payment of a winner with v > c in the open auction is given by  (n −1)r2 +1  r2∗ c1 2 (n2 − 1)r2 (n2 − 1)r2 v + (n −1)r − , for v > c1 . 2 (n2 − 1)r2 + 1 v 2 r2∗ + 1 (n2 − 1)r2 + 1 and the interim payment is q2∗ (v)

(n2 − 1)r2 (n2 −1)r2 v+m c1 (n2 − 1)r2 + 1



r2∗ (n2 − 1)r2 − ∗ r2 + 1 (n2 − 1)r2 + 1

 ,

for v > c1 .

We are now ready to compare the revenues from the two auctions. With the simple equilibrium strategy in Proposition 4, we can compute the seller’s revenue in a sealed-bid auction very easily. This is our next result. Theorem 7. The seller’s expected revenue from the sealed-bid auction is 2 ri∗ ci r2∗ c2 n i ri n 1 r1 + n 2 r2 = . ∗ ∗ ri + 1 n1 r1 + n2 r2 + 1 r2 + 1 n1 r1 + n2 r2 + 1

(17)

i=1

Proof. Let d=

r1∗ c1 ∗ r1 + 1

=

r2∗ c2 . ∗ r2 + 1

The cumulative distribution of the highest bid of the weak buyers is given by  ∗  r1 + 1 x n1 r1  x n1 r1 F (x) = = , for x ≤ d; r1∗ c1 d while that of the strong buyers is given by  ∗  r2 + 1 x n2 r2  x n2 r2 G(x) = = , r2∗ c2 d

for x ≤ d.

Hence the cumulative distribution of the winning bid is given by  x n1 r1 +n2 r2 . H(x) = F (x)G(x) = d The seller’s revenue is given by  d n1 r1 + n2 r2 n1 r1 + n2 r2 r2∗ c2 x dH(x) = d= , n 1 r1 + n 2 r2 + 1 n1 r1 + n2 r2 + 1 r2∗ + 1 0 which is also equal to 2 n i ri ri∗ ci . ∗ ri + 1 n1 r1 + n2 r2 + 1




i=1

One important corollary of Theorem 7 is that the seller’s revenue depends on n1 only through its dependence on the total strength of the weak buyers n1 r1 . Note that in our linear model, a

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different n1 usually means a different c1 and a different distribution for the weak buyer. The following says that bundling has no revenue effect on the seller’s revenue in the sealed-bid auction. Corollary 8. Given the number of strong buyers n2 fixed, the seller’s revenue depends only on the total strength of the weak buyers n1 r1 not n1 alone. Therefore in two auctions with the same number of strong buyers and the same total strength of the weak buyers, the seller’s revenue is the same. Proof. In the second expression of (17), we have r2∗ = n1 r1 + (n2 − 1)r2 . Therefore n1 always appears jointly with r1 in n1 r1 in the expression for the seller’s revenue.
We now compute the seller’s revenue in the open auction. Because of different valuation intervals of the weak and strong buyers, the algebraic expression is quite complicated. A simpler formula for the special case with one strong buyer is stated as a corollary. and will be used for computations later. Theorem 9. The seller’s expected revenue in the open auction is given by   (n2 − 1)r2 n2 r ∗ n2 (n2 − 1)r2 n 2 r2 c2 + c2 m(n2 −1)r2 +1 ∗ 2 − − c2 mn2 r2 +1 R (n2 − 1)r2 + 1 n2 r2 + 1 r2 + 1 (n2 − 1)r2 + 1 where R=

(n2 − 1)n2 r2 r∗ n1 r1 n2 r2∗ − − ∗ 1 . n 1 r1 + n 2 r2 + 1 n 2 r2 + 1 r1 + 1 n1 r1 + n2 r2 + 1

Corollary 10. When there is only one strong buyer, the seller’s revenue in the open auction is given by   n 1 r1 n1 r1 c1 − c1 mr2 . n1 r 1 + 1 ((n1 − 1)r1 + r2 + 1)(n1 r1 + r2 + 1) 4. The ranking We now provide a general result on the ranking of the sealed-bid auction and the open auction for our linear asymmetric auction models with strong and weak buyers and an arbitrary numbers of them. The sealed-bid auction raises higher revenue for the seller. The proof is based on the bundling effects. Theorem 11. With any combination of the weak and strong buyers in the linear asymmetric auctions, the seller’s revenue is higher in the sealed-bid auction than in the open auction. Proof of Theorem 11. In this proof, we will change r1 while keeping n1 r1 fixed. We will also keep n2 , c2 , r2 fixed throughout the proof. The number r1 will vary between 0 and r2 . Since     r∗ r1∗ + 1 r2∗ r2∗ 1 1 m= ∗ 2 = , = 1 + 1 + r2 + 1 r1∗ r2∗ + 1 r1∗ r2∗ + 1 n 1 r1 − r 1 + n 2 r 2 we know that m is an increasing function of r1 (when n1 r1 is fixed). Let RV h denote the seller’s revenue in the sealed-bid auction. By Corollary 8, RV h stays constant while r1 changes. From (17), we have RV h =

c2 r2∗ n 1 r1 + n 2 r2 . + 1 n 1 r1 + n 2 r 2 + 1

r2∗

(18)

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From Theorem 9, the seller’s revenue in the open auction, denoted by RV o , is given by (n2 − 1)r2 n 2 r2 c2 + c2 m(n2 −1)r2 +1 Q − c2 mn2 r2 +1 R (n2 − 1)r2 + 1 n2 r2 + 1

(19)

where Q=

n2 r2∗ n2 (n2 − 1)r2 − ; ∗ r2 + 1 (n2 − 1)r2 + 1 − −

R=

n2 r2∗ (n2 − 1)n2 r2 − n 1 r1 + n 2 r 2 + 1 n 2 r2 + 1

n1 r1 r1∗ n2 r2∗ (n2 − 1)n2 r2 = − (r1∗ + 1)(n1 r1 + n2 r2 + 1) n 1 r1 + n 2 r2 + 1 n2 r2 + 1 (r2∗

n1 r1 r2∗ 1 . + 1)(n1 r1 + n2 r2 + 1) m

Note that in the last expression for R, the only term that will change with r1 is the last one, and it changes only through the dependence of m on r1 . Substitute this expression for R into (19), the seller’s revenue in the open auction can now be written as RV o =

(n2 − 1)r2 n 2 r2 c2 + c2 Qm(n2 −1)r2 +1 − c2 R1 mn2 r2 +1 + c2 R2 mn2 r2 (n2 − 1)r2 + 1 n2 r2 + 1

where R1 =

n2 r2∗ (n2 − 1)n2 r2 − ; n 1 r 1 + n 2 r2 + 1 n 2 r2 + 1

R2 =

n1 r1 r2∗ . (r2∗ + 1)(n1 r1 + n2 r2 + 1)

Note that the expressions Q, R1 , R2 do not depend on r1 as n1 r1 is fixed. The only dependence of the revenue on r1 is through the term m. Since m and r1 are positively related, RV o is increasing in r1 if and only if it is increasing in m. To show the increasing property in m, it is sufficient to show that the following function is increasing in m: Qm(n2 −1)r2 +1 − R1 mn2 r2 +1 + R2 mn2 r2 . Take the derivative with respect to m, we have the derivative ((n2 − 1)r2 + 1)Qm(n2 −1)r2 − (n2 r2 + 1)R1 mn2 r2 + n2 r2 R2 mn2 r2 −1 ≥ [((n2 − 1)r2 + 1)Q − (n2 r2 + 1)R1 + n2 r2 R2 ]mn2 r2 .

(20)

We have ((n2 − 1)r2 + 1)Q − (n2 r2 + 1)R1 = =−

n2 r2∗ (n2 r2 + 1) n2 r2∗ ((n2 − 1)r2 + 1) − r2∗ + 1 n 1 r1 + n 2 r2 + 1

n2 r2 r2∗ n1 r1 = −n2 r2 R2 . (r2∗ + 1)(n1 r1 + n2 r2 + 1)

Hence the expression in (20) is 0. We conclude that the revenue RV o is increasing in m. Note that when r1 takes the maximum value r2 , we have m = 1. The revenue in the open auction becomes RV o =

(n2 − 1)r2 n 2 r2 c2 + c2 (Q − R1 + R2 ). (n2 − 1)r2 + 1 n2 r2 + 1

(21)

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Since Q − R 1 + R2 =

(n2 − 1)n2 r2 n2 (n2 − 1)r2 n2 r2∗ n2 r ∗ − − + ∗ 2 n 2 r2 + 1 (n2 − 1)r2 + 1 n1 r1 + n2 r2 + 1 r2 + 1 +

n1 r1 r2∗ −(n2 − 1)n2 r22 = (r2∗ + 1)(n1 r1 + n2 r2 + 1) (n2 r2 + 1)((n2 − 1)r2 + 1)

+

r2∗ (n1 r1 + n2 r2 ) , (r2∗ + 1)(n1 r1 + n2 r2 + 1)

(22)

we can substitute (22) into (21). We have RV o =

n 1 r 1 + n 2 r2 c2 r2∗ = RV h , + 1 n 1 r1 + n 2 r2 + 1

r2∗

when m = 1.

The increasing property of RV o in m then implies that RV o < RV h ,

when m < 1 (or r1 < r2 )

and our proof is complete.




An important property used in the above proof is that the revenue increases from bundling in the open auction. The bundling effect in the linear model is not a satisfactory representation of the merger effect because the support of the distribution shifts in bundling several weak buyers together. This is necessary in maintaining the linear property of equilibrium, and makes the analysis simpler. With a fixed valuation interval, we would like to know what happens to the seller’s revenue when merger and joint bidding occur. The increase of the seller’s revenue in the open auction from bundling does not imply that the seller’s revenue will increase from merger. The following is a simple example in which a merger reduces the seller’s revenue in the open auction. Example 12. Let there be two weak buyers, each with a valuation described by a c.d.f. F1 (v) = v over [0, 1]. Let there be a strong buyer with a valuation described by F2 (v) = v2 over the same interval. When the two weak buyers merge into a single buyer, the model becomes a symmetric auction with two identical buyers. Our ranking result in the linear model is not applicable, as the asymmetric model with three buyers is not a linear model. Before the merger, a weak buyer with valuation v has the winning probability v3 , and upon winning pays the expected price  v  x 3 3v . xd = v 4 0 Hence the seller’s revenue from the two weak buyers is  1 3v 3 2 v3 dv = . 4 10 0 Similarly, the seller’s revenue from the strong buyer is  1  v  1  x 2  4 2v 2 2 dv = . v xd v2 2v dv = v 3 15 0 0 0 4 , which is also After the merger, the seller’s revenue from the strong buyer remains the same 15 3 4 the revenue from the merged buyer by symmetry. Since 10 > 15 , the seller’s revenue decreases 8 . after the merger. The seller’s revenue after the merger (the symmetric case) is 15

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5. Possible extensions Consider an example with one weak buyer and one strong buyer with the valuation distributions given by F1 (v1 ) = v0.5 1 , F2 (v2 ) = v2 , respectively, over [0, 1]. It is a non-linear model. None of the existing results allows us to rank the two auctions. If we modify the model by changing the distribution of F1 to 3v21 over [0, 23 ], then it becomes a linear model. We want to show how our analysis with the linear model does provide helpful insights into the question, and suggests a very promising approach to the solution of the ranking problem. Let φi = bi−1 be the inverse of the bidding function. Let J1 (v1 ), J2 (v2 ) be the virtual values of the weak and strong buyers, respectively, and T (v1 ) = J2−1 (J1 (v1 )). Let Q be defined by the relationship φ2 (b) = Q(φ1 (b)). Let T l , Ql be the corresponding functions for the linear model. The function T can be easily computed as T (v1 ) =

3 1 √ v 1 + − v1 2 2

and T (0) = 21 , T (1) = 1, T  (1) = 1. Furthermore, the function T is convex. The corresponding T l is

2v1 3 1 l ; T (v1 ) = v1 + − 2 2 3 hence T l ≥ T for v1 ≤ 23 . The graphs of the functions T and Q are plotted in Fig. 1. Fig. 1 has been carefully drawn to reflect the delicate trade off in the analysis. The two convex curves with the labels T, T l and the concave curve with the label Q represent the graphs of the corresponding functions. It is known that the area between the graph of T and the diagonal OH is the area in which the open auction deviates from the optimal auction. The area between the graphs of Q and T (the shaded area plus the region ODG) is the area in which the sealed bid auction deviates from the optimal auction (see the explanations for Fig. 1 in the Maskin and Riley (2000a) paper4 ). Our ranking result holds if the loss of revenue due to the misallocation in the region OGH is bigger than the shaded area. The box OABC is for the corresponding linear model, and the graph of the Ql function for the linear model is the straight line OB. Hence the area of misallocation for the sealed bid auction is the triangle BEF plus the region ODE, while the region ODFK is the area of misallocation for the open auction. In the linear model our result shows that the misallocation of the open auction causes more loss of revenue than that of the sealed bid auction. This means that the loss of revenue due to the misallocation in area BEF causes less damage to the revenue than the area OEFK. To plot the graph of Q, we need to know some of its properties. It can be shown that in the non-linear case, the buyers initially bid as if it is linear, i.e. when v is small, b1 (v), b2 (v) are close to 21 v, 13 v, respectively, and we have φ1 (0) = 2, φ2 (0) = 3. The system of differential equations satisfied by the inverse bidding functions are φ1 (b) 2 = , φ1 (b) φ2 (b) − b

φ2 (b) 1 = φ2 (b) φ1 (b) − b

(23)

4 The explanation for Fig. 2 in their paper is incorrect. The correct version is offered in the explanation for Fig. 1. The graph of our paper looks more similar to the graph of Fig. 2 in their paper.

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489

Fig. 1. Revenue comparisons.

Using chain rule, we have φ2 (b) = Q (φ2 (b))φ2 (b). Combine this with (23), we have Q (0) = 3 1  2 , Q (1) = 2 . Furthermore, it can be shown that Q is a concave curve (using the property of the differential equations). Thus the graph of Q lies to the right of the line OB. To show that the same ranking holds for the non-linear case, we need to show that the loss of revenue due to the misallocation in OGH is greater than the loss of revenue due to the misallocation in the shaded area. There are three reasons why a good result in this direction may be possible: (1) the small shaded area involves high-valuation buyers who have relatively small differences in virtual values (marginal revenues) compared to the low-valuation buyers, so that for the same area a lower region has a greater impact on the loss of revenue; (2) the distribution of F1 has a decreasing density that makes the impact of the shaded area even smaller; (3) the ranking result of the corresponding linear model is robust to parameter specifications and holds in great generality as we show in this paper. Thus it is reasonable to expect that the ranking of the two auctions may remain the same as that of the linear case. These of course need to be analyzed carefully, as the factor (2) need not hold for all parameters. There is clearly room for generalization here. Such extensions are important for the procurement auctions in which the objective is to reduce the cost of the procurement. The range of possible costs is a more general interval [a, b], a > 0. Note that the linear property is easily lost, unless v−a 2 we also shift the distribution accordingly as in F (v) = ( b−a ) . The above discussions also have implications for the effects of merger. Suppose we add another identical weak buyer into the example above. Note that when the two identical weak buyers merge into one, we have a symmetric model with the same seller’s revenue for both auctions. If the merger has a more damaging effect on the revenue in the sealed-bid auction than in the open auction, then we also have the same ranking result. The converse is also true. Thus the ranking result has an important connection to the merger effects.

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Once we relax the restrictions on interval ranges, the next step is to consider the more general distribution function F (v). The power function arises from the merger of buyers with little information on the valuation. If the buyers have some prior information about the valuation represented by a particular function F, then a merger leads to a distribution of the form F n , and we can ask whether there are useful ranking results for certain types of distributions. The above discussions suggest that if the distribution is more focused in areas in which virtual values diverge, then a merger is going to have a bigger impact on the revenues. This is an interesting area of further investigation. Another important property is the convexity of the function T. When T is convex, the misallocation of the open auction is more serious for the low-valuation buyers. This tends to have the dominant effect on the seller’s revenue as their virtual values differ more. Thus we also want to know whether the ranking results can be established when the distribution F possesses this property. 6. Applications to forest timber auctions We shall use our model to compare the revenue differences between sealed-bid and open auctions. We take the typical data that we observe in the timber auctions, using the tables in Athey et al. (2004) as a reference. The U.S. Forest Service timber program provides an excellent test case for comparisons in revenues as it has historically used both open and sealed-bid auctions, at times even randomizing the choice. There are two kinds of participants in Forest Service timber auctions. One type is the individually-owned logging companies, called loggers. The second type is the large vertically integrated forest products conglomerates, called mills. Loggers are weak buyers, while mills are strong buyers. Mills, who have manufacturing capacity, tend to have higher values for a given contract than loggers, who have to re-sell the timber. For a selected sample, the average numbers of logger and mill bidders are 3.42 and 1.14, respectively, in the sealed-bid auctions. In the open auctions, they are 2.84 and 1.40, respectively. Hence there are more logger bidders in the sealed-bid auctions and more mill bidders in the open auctions. Another important difference between the two types of auctions in the sample is the track size. The average open auction has an estimated 2893 mbf (thousand board feet) of timber, while the average sealed-bid sale has only 1502 mbf. These differences in track size and the composition of weak and strong buyers may cause revenue differences which need to be separated from the effects of the auction formats. Athey et al. (2004) found that for a fixed set of participants, their calibrated model predicts relatively small discrepancies between sealed-bid auctions and open auctions. Sealed-bid auctions raise more revenue, but the effects are small (less than 1%). We shall apply our linear model to predict the revenue difference between the two auction formats. Before performing any comparisons, we need to determine the parameters r1 , r2 . We adopt two criteria in determining these parameters. First we want the winning probabilities of the loggers to be consistent with the data. This winning probability is 0.69 in Table 1.A for Northern forest sealed-bid auctions in Athey et al. (2004). Secondly, we want the ratio of the average mill bids and logger bids to be consistent with the data. Athey et al. (2004) estimated that the mill bids are 24% higher than the logger bids on average, conditional on the sale characteristics such as tract size. We choose these two criteria for the simplicity of computations. By assuming the linear model, we have the following simple result on the allocation of the good for sale in the sealed-bid auction.

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r2 1 Proposition 13. The equilibrium winning probabilities are n1 r1r+n , for a weak and 2 r2 n1 r1 +n2 r2 strong buyer, respectively, in the sealed-bid auction. The probability that a winner is a weak buyer is given by n1 r1 . n 1 r 1 + n 2 r2

Proof. Using Lemma 5, the winning probability of a weak buyer is  c1  c1  c1 r∗  r1 v r1 r1 v1 ∗ q1 (v) dF1 (v) = d = r∗ +r vr1 +r−1 dv = ∗ r1∗ 1 c r 1 1 0 0 c 0 1 + r1 c1 1 r1 = . n 1 r1 + n 2 r2 Similarly, it is r2 n 1 r 1 + n 2 r2 for a strong buyer. The probability that a weak buyer wins is then equal to n1 r1 .
n 1 r 1 + n 2 r2 We also use the following simple result on the comparisons of the average bids of the weak and strong buyers in fitting our model to the data. Lemma 14. The average equilibrium bid of a buyer i = 1, 2 in the sealed-bid auction is ri ri∗ ci ; ∗ ri + 1 ri + 1 hence the ratio of the average bid of the strong and weak buyers is r2 r1 + 1 . r2 + 1 r1

(24)

Proof. We compute the equilibrium average bid of a weak buyer as follows  r1  c1  c1  c1 r1∗ r1∗ r1 r ∗ c1 v r1 b1 (v) dF1 (v) = = vr1 dv = ∗1 v d . r1 ∗ ∗ r + 1 c r + 1 r + 1 r c 1 1+1 0 0 0 1 1 1 1 Similarly, we can derive the result of the strong buyer in the same way. Since of i, we have (24).


ri∗ ci ri∗ +1

is independent

In the sealed-bid auctions, we assume that there are three logger bidders and one mill bidder. By Proposition 13, the probability that the winner is a logger is 3r1 n1 r1 = . n 1 r 1 + n 2 r2 3r1 + r2 Hence we have the first condition 3r1 = 0.69 3r1 + r2

(25)

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and we have 0.69 (26) r2 = 0.74194r2 . r1 = 0.93 The second condition says that the average mill bids are 24% higher than the average logger bids. By Lemma 14, we have the condition r1 r2 = 1.24 . (27) r2 + 1 r1 + 1 From (26) and (27), we get the solution r1 = 0.33329,

r2 = 0.44921.

Hence r1∗ = 2r1 + r2 = 1.1158,

r2∗ = 3r1 = 0.99987

and 0.99987 2.1158 = 0.94805. 1.99987 1.1158 Using these numbers, we can compute the seller’s revenue in the sealed-bid auction as m=

3r1 + r2 c1 r1∗ = 0.31203c1 . ∗ r1 + 1 3r1 + r2 + 1 By Corollary 10, the seller’s revenue in the open auction is   3r1 3r1 c1 r2 − c1 m = 0.31158c1 . 3r1 + 1 (2r1 + r2 + 1)(3r1 + r2 + 1) The revenue from the sealed-bid auction is 0.31203 − 0.31158 = 0.14% 0.31158 higher than that from the open auction. This is consistent with the findings in Athey et al. (2004) with very different specifications. We now consider the possible effect of entry pattern on the revenue difference between the two formats. The findings in Athey et al. (2004) say that mill entry in both auction formats is roughly the same in the Northern forest region, but there is more entry by loggers in the sealed-bid auctions. They measured the revenue difference to be in the range of 12–18% higher in sealedbid auctions than in open auctions. We now want to compare these results with the theoretical predictions of the linear model. In this comparison, we use the same r1 , r2 above. First assume that the number of loggers is reduced to 2 in the open auction, and there is still one mill bidder. We assume that c1 remains the same with some downward adjustment for c2 to maintain the linear model. We have r1∗ = r1 + r2 = 0.7825,

r2 = 2r1 = 0.66658,

m=

0.66658 1.7825 = 0.91111 1.66658 0.7825

and the seller’s revenue in the open auction is   2r1 2r1 c1 − c1 mr2 = 0.23046c1 . 2r1 + 1 (r1 + r2 + 1)(2r1 + r2 + 1)

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The revenue difference is therefore 0.31203 − 0.23046 = 35.39% 0.23046 higher in sealed-bid auctions. The average increase of the loggers in the sealed-bid auction in the Northern forest region is about 0.5. If we allow for fractional numbers of loggers, and take n1 = 2.5, we have r1∗ = 1.5r1 + r2 = 0.94915,

r2∗ = 2.5r1 = 0.83323,

0.83323 1.94915 = 0.93338 1.83323 0.94915 and the seller’s revenue in the open auction is   2.5r1 c1 2.5r1 r2 − c1 m = 0.27293c1 . (1.5r1 + r2 + 1)(2.5r1 + r2 + 1) 2.5r1 + 1 m=

Therefore, the revenue difference is 0.31203 − 0.27293 = 14.3% 0.27293 higher in the sealed-bid auction. This is also quite consistent with the result of Athey et al. (2004) with participation decisions. 7. Conclusion Maskin and Riley (2000a) suggest one reason to explain why the art auctions are often conducted openly rather than in sealed bids. A buyer with high valuation may think that he or she is quite unique in such tastes, and may think that others’ valuation may be quite low. In sealed-bid auctions, low-balling may be a good strategy in such environments. The open auction is a format that may safeguard against such a strategy. The Proposition 4.5 in their paper supporting such a result assumes that the weak buyer’s valuation has a concentration on the very low end of the interval. For linear models, this reasoning does not seem to work well. Even if the valuation of the weak buyer is highly concentrated around the origin (with r close to 0), the sealed-bid auction is still better than the open auction. We don’t know whether the same holds for non-linear models. Our result suggests another important consideration for choosing between the sealed-bid and open auctions. This consideration focuses on the Getty effect rather than the low-balling strategy. Assume that the strong buyer is likely to be on the high end of the valuation interval. In a sealedbid auction, it may encourage him or her to bid aggressively and outbid the other buyers so that a win is guaranteed. This is the Getty effect we discussed earlier. In an open auction, Mr. Getty may just sit out the bidding process comfortably and coasts to the end to pay the second highest price without much effort on his part. This will allow him to pay only the expected second highest price in an open auction. Hence in an auction with a high probability of the strong buyers’s high valuation, the Getty effect is an important consideration, and the sealed-bid auction is better. There is a tension here. The Getty effect leads to high bidding strategy, while the low-balling strategy is just the opposite. Which consideration is more important in a particular context? An answer to this question requires an investigation of the Getty effect with more general distributions. Hopefully, the extensions in Section 5 may allow us to address this question in the future.

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Athey et al. (2004) provides evidence that collusion is perhaps a primary factor that may cause substantial revenue gaps between the two auction formats. We have not explored this issue here. However, the study of merger effect and joint bidding is related to our investigation here, and we have taken a small step in this direction. There is also a great potential for using the linear model in empirical studies. The linear equilibrium auction model enhances our understanding of the different benefits in the choice between the sealed-bid and open auctions. We also hope that that it is useful in providing a simple model for theoretical investigations as well as empirical estimation, testing, and experiments. Acknowledgements I want to thank Guofu Tan, Wei Li, and Eric Maskin for suggestions and references. I also want to thank Jonathan Levin for his help in using the data from the forest timber auctions. Appendix A Proof of Lemma 3. Let v˜ 1 , v˜ 2 be the random variables for the valuation distributions of the weak and strong buyers, respectively. For all 0 < x < y ≤ c1 , we have Pr[˜v1 < x|˜v1 < y] =

xr1 xr2 > = Pr[˜v2 < x|˜v2 < y]. yr1 yr2

For 0 < x ≤ c1 < y < c2 , we have Pr[˜v1 < x|˜v1 < y] =

xr1 xr2 > = Pr[˜v2 < x|˜v2 < y]. yr2 c1r1

For all 0 < c1 < x < y < c2 , we have Pr[˜v1 < x|˜v1 < y] = 1 >

xr2 = Pr[˜v2 < x|˜v2 < y]. yr2

The inequality is satisfied for all possible ranges of x and y. Therefore the conditional stochastic dominance condition is satisfied.
Proof of Lemma 5. We have b1−1 b2 (v) = mv. In the sealed-bid auction, the equilibrium winning probability of a weak buyer is ∗  v n2  v (n1 −1)r1  v n2 r2 vr1 n1 −1 F2 = = r∗ . q1 (v) = F1 (v) m c1 mc2 c11 For a strong buyer, it is q2 (v) = F1 (mv)n1 F2 (v)n2 −1 =



mv c1

n1 r1 

v c2

(n2 −1)r2

∗

=

vr2

r∗

.

c22

In the open auction, a strong buyer with v ≤ c1 wins with probability  n1 r1  (n2 −1)r2 ∗ v v vr2 = , q2∗ (v) = F1 (v)n1 F2 (v)n2 −1 = r∗ c1 c2 mn1 r1 c22

(28)

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and a weak buyer wins with probability  (n1 −1)r1  n2 r2 ∗ v v mn2 r2 vr1 ∗ n1 −1 n2 F2 (v) = = . q1 (v) = F1 (v) ∗ r c1 c2 c11

495

(29)

For a strong buyer with v > c, the winning probability is  (n2 −1)r2 v ∗ n2 −1 q2 (v) = F2 (v) = .
c2 Proof of Proposition 6. To compute the winner’s expected payment in the open auction, first consider the case v ≤ c1 . Conditional on winning, his (or her) probability of paying a price p is the conditional probability of all opponents bidding lower than p, and is given by 1

F1 (p) q2∗ (v)

n1

F2 (p)n2 −1 .

From Lemma 5, we have ∗

q2∗ (v)

=

vr2

r∗

;

mn1 r1 c22

hence the expected price to pay is given by 

v

0



r∗

p mn1 r1 c22 n1 n2 −1 d(F (p) F (p) ) = 1 2 ∗ q2∗ (v) vr2 r∗

=

mn1 r1 c22 ∗

(n −1)r2

c1n1 r1 vr2 c2 2

 0

v

r2∗

p d(p ) =

r2∗

∗

vr2



v

pd 0

 0

v

p c1

n1 r1 

p c2

(n2 −1)r2 

∗

r ∗ vr2 +1 r∗ p dp = r2∗ ∗ = ∗ 2 v, r2 + 1 v 2 r2 + 1 r2∗

which is exactly the equilibrium bid of the strong buyer in the sealed-bid auction. Similarly, we prove the same result for a weak buyer. For a strong buyer with valuation v > c1 , the winning probability is  (n2 −1)r2 v ∗ . (30) q2 (v) = c2 The probability of paying a price p conditional on winning is 1

F1 (p) q2∗ (v)

n1

F2 (p)n2 −1 ,

when p ≤ c1 ;

1

F2 (p) q2∗ (v)

n2 −1

,

when v ≥ p > c1 .

The expected price to pay is   c1  v 1 n1 n2 −1 n2 −1 p d(F (p) F (p) ) + p d(F (p) ) 1 2 2 q2∗ (v) c1 0

   

 

    (n −1)r2 c1 v p n1 r1 p (n2 −1)r2 p (n2 −1)r2 c2 2 + pd pd = (n −1)r 2 v 2 c1 c2 c2 c1 0    c1  v 1 1 ∗ = (n −1)r p d(pr2 ) + p d(p(n2 −1)r2 ) 2 v 2 c1n1 r1 0 c1

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= = =



1 v(n2 −1)r2 1 v(n2 −1)r2



r2∗



c1n1 r1

c1



∗

pr2 dp + (n2 − 1)r2

0

v



p(n2 −1)r2 dp

c1

(n −1)r +1

2 r2∗ c1 2 r2∗ + 1



(n2 − 1)r2 (n −1)r2 +1 ) + (v(n2 −1)r2 +1 − c1 2 (n2 − 1)r2 + 1

(n −1)r +1

2 (n2 − 1)r2 c 2 v + 1 (n −1)r 2 (n2 − 1)r2 + 1 v 2



r2∗ (n2 − 1)r2 − ∗ r2 + 1 (n2 − 1)r2 + 1

 .

(31)

Note that the first term in (31) is the payment in an auction with only the strong buyers. The second term in (31) gives us the interim payment  (n −1)r2 +1  r2∗ c1 2 (n2 − 1)r2 ∗ q2 (v) (n −1)r − 2 v 2 r2∗ + 1 (n2 − 1)r2 + 1  ∗  r2 (n2 − 1)r2 (n2 −1)r2 =m c1 ∗ − . r2 + 1 (n2 − 1)r2 + 1 This payment is a constant. The payment of a winning strong buyer in a sealed-bid auction is the equilibrium bid r2∗

r2∗ v. +1

We have  (n −1)r2 +1  r2∗ r2∗ (n2 − 1)r2 c1 2 (n2 − 1)r2 v− v − (n −1)r − 2 r2∗ + 1 (n2 − 1)r2 + 1 v 2 r2∗ + 1 (n2 − 1)r2 + 1   ∗  (n2 −1)r2 +1 (n −1)r2 +1 r2 − c1 2 (n2 − 1)r2 v = > 0. − r2∗ + 1 (n2 − 1)r2 + 1 v(n2 −1)r2 Hence in equilibrium, a winning strong buyer with v > c1 pays more in the sealed-bid auction than in the open auction.
Proof of Theorem 9. Using Lemma 5 and Proposition 6, the interim payment of a weak buyer is q1∗ (v)

mn2 r2 r1∗ r1∗ ∗ v = vr1 +1 . ∗ ∗ ∗ r r1 + 1 c11 r1 + 1

(32)

The expected revenue from the weak buyers is  c1 n2 r2  c1 n2 r2 m r1∗ m r1∗ r1 ∗ r1∗ +1 n1 dF (v) = n v vr1 +1 r1 vr1 −1 dv 1 1 ∗ ∗ r1 r ∗ + 1 r1 r ∗ + 1 c1 0 0 1 1 c1 c1  ∗ c1 ∗ mn2 r2 n1 r1 r1∗ n 1 c1 r1 +r1 n2 r2 r1 r1 = v dv = m ∗ ∗+1 ∗ + 1 r∗ + r + 1 r1 +r1 r r 1 0 1 1 1 c1 = mn2 r2

r1∗ c1 ∗ r1 + 1

r ∗ c2 n1 r 1 n1 r1 = mn2 r2 +1 ∗1 . n 1 r1 + n 2 r 2 + 1 r1 + 1 n1 r1 + n2 r2 + 1

(33)

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497

For the strong buyers, we first compute the revenue for the range v ∈ [0, c1 ]. By Proposition 6, we have the interim payment q2∗ (v)

r2∗

r2∗ 1 r2∗ ∗ v= vr2 +1 , ∗ ∗ r +1 mn1 r1 c22 r2 + 1

and the revenue  c1  c1 1 r2∗ r2∗ r2 n2 ∗ r2∗ +1 v dF2 (v) = vr2 +r2 dv n2 ∗ ∗ ∗ ∗ r2 r r r + 1 r + 1 c 2 2 0 mn1 r1 c 0 2 2 mn1 r1 c2 2 2 =

n2

r∗ +r2

mn1 r1 c22

r∗ +r2 +1

n 2 r2 r2∗ r2 c11 r ∗ c2 = mn2 r2 +1 ∗2 . ∗ ∗ r2 + 1 r2 + r2 + 1 r2 + 1 n1 r1 + n2 r2 + 1

(34)

Again, by Proposition 6, the expected revenue from the strong buyers in the range v > c1 is  c2  (n2 −1)r2 n2 (n2 − 1)r2 v v dF2 (v) (n2 − 1)r2 + 1 c1 c2   n2 (n2 − 1)r2 n2 r2∗ − c1 m(n2 −1)r2 (1 − F2 (c1 )) + r2∗ + 1 (n2 − 1)r2 + 1 (n2 − 1)r2 n 2 r2 c 2 (1 − mn2 r2 +1 ) (n2 − 1)r2 + 1 n2 r2 + 1   n2 r2∗ n2 (n2 − 1)r2 c2 + − m(n2 −1)r2 +1 (1 − mr2 ). r2∗ + 1 (n2 − 1)r2 + 1 =

(35)

Hence the seller’s expected revenue is the sum of (34), (33) and (35), given by   n2 r ∗ n2 (n2 − 1)r2 c2 n 2 r2 (n2 − 1)r2 c2 + m(n2 −1)r2 +1 ∗ 2 − − mn2 r2 R; (n2 − 1)r2 + 1 n2 r2 + 1 r2 + 1 (n2 − 1)r2 + 1 where r2∗ (n2 − 1)r2 n 1 r1 n 2 r2 n2 r2. r1∗ − + r1∗ + 1 n1 r1 + n2 r2 + 1 r2∗ + 1 n1 r1 + n2 r2 + 1 (n2 − 1)r2 + 1 n2 r2 + 1   n2 r2∗ n2 (n2 − 1)r2 n2 r2∗ r2 + ∗ − = ∗ 1− r2 + 1 (n2 − 1)r2 + 1 r2 + 1 n 1 r1 + n 2 r2 + 1

R=−

− −

r1∗

n1 r1 (n2 − 1)n2 r2 n2 r2∗ (n2 − 1)n2 r2 r1∗ − = − + 1 n 1 r1 + n 2 r2 + 1 n 2 r2 + 1 n 1 r1 + n 2 r2 + 1 n2 r2 + 1

r1∗

r1∗ n1 r1 , + 1 n 1 r1 + n 2 r2 + 1

and the proof is complete.




Proof of Corollary 10. When there is only one strong buyer, we have n2 = 1, and r1∗ = (n1 − 1)r1 + r2 , R=

r2∗

r2∗ = n1 r1 ;

r2∗ r∗ n 1 r1 − ∗ 1 + 1 r1 + 1 n1 r1 + r2 + 1

(36)

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H. Cheng / Journal of Mathematical Economics 42 (2006) 471–498

− =

r2∗

r2∗ r2 r∗ n 1 r1 + 1 r∗ n 1 r1 = ∗ 2 − ∗ 1 + 1 n1 r 1 + r 2 + 1 r2 + 1 n1 r1 + r2 + 1 r1 + 1 n1 r1 + r2 + 1

r2∗ (r1∗ + 1) − n1 r1 (r1∗ − r2∗ ) n 1 r1 = ∗ . ∗ ∗ (r1 + 1)(r2 + 1)(n1 r1 + r2 + 1) (r1 + 1)(n1 r1 + r2 + 1)

Hence the seller’s revenue is   r2∗ c1 n1 r1 r2∗ c1 r2 r2 m R = m − c − c 1 1 r2∗ + 1 r2∗ + 1 (r1∗ + 1)(n1 r1 + r2 + 1)   n1 r1 c1 n 1 r1 = − c1 mr2 . n 1 r1 + 1 ((n1 − 1)r1 + r2 + 1)(n1 r1 + r2 + 1)




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