Bidding for the future: signaling in auctions with an aftermarket

Journal of Economic Theory 108 (2003) 345–364

Notes, Comments, and Letters to the Editor

Bidding for the future: signaling in auctions with an aftermarket Jacob K. Goeree
CREED, Faculteit voor Economie en Econometrie, Universiteit van Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, Netherlands Received 14 September 2000; final version received 31 October 2001

Abstract This paper considers auctions where bidders compete for an advantage in future strategic interactions. When bidders wish to exaggerate their private information, equilibrium bidding functions are biased upwards as bidders attempt to signal via the winning bid. Signaling is most prominent in second-price auctions where equilibrium bids are ‘‘above value.’’ In English and first-price auctions, signaling is less extreme since the winner incurs the cost of her signaling choice. The opportunity to signal lowers bidders’ payoffs and raises revenue. When bidders understate their private information, separating equilibria need not exist and the auction may not be efficient. r 2003 Elsevier Science (USA). All rights reserved. JEL classification: C72; D44 Keywords: Auctions; Signaling

1. Introduction It is well known that in situations with incomplete information, players may gain by using costly signals to convey private information to others. The literature on signaling models started with Spence [29], who showed that rational workers can indicate their competence by investing in an education that has no real value. Signaling models have subsequently been applied to explain a wide variety of economic phenomena, including predatory or limit pricing to deter entry [24],


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uninformative advertising to indicate product quality [25], and strategic delays in bargaining processes [1].1 The common theme of these papers is that a player with private information may find it worthwhile to spend money or incur a cost in an attempt to change others’ future behavior by conveying some or all of her information. Such ‘‘extra spending’’ can also occur in auctions where firms’ strategic positions are at stake. Consider, for instance, a takeover battle in which there exist privately known synergies between the bidding firms and the target. By putting in a higher bid, firms not only increase their chance of winning but also signal a stronger synergy. This signal may affect others’ future actions in a way that is profitable to the winning bidder. Alternatively, consider the sale of a patented innovation that reduces a Cournot competitor’s marginal cost by some privately known amount. Imagine an English auction is used to conduct the sale, and a firm observes others dropping out at very low prices. While the firm can obtain the innovation very cheaply, it may prefer to put in a high bid (even after everyone else has dropped out) to signal a larger reduction in costs. These examples can be extrapolated to general aftermarkets in which players wish to exaggerate their private information. By bidding high and forgoing some profit in the auction, the winning bidder influences future strategic interactions in a way that is ultimately more profitable. In this paper, we consider auctions in which each bidder possesses private information that is relevant only if she wins. We assume that, after the auction has taken place, bidders engage in some kind of competition: the aftermarket. Profits in the aftermarket depend on the winner’s private information, or type, and on the type others perceive her to be but not on losers’ private information. The application we have in mind is that of a mature market where competitors have learned each other’s costs over time. This ‘‘status quo’’ situation then changes when a cost-reducing innovation (protected by a patent) is auctioned and a single firm is licensed. In this case, only the licensee’s reduction in costs will matter for future competition. The costs of all other firms remain the same and are publicly known.2 We do not make any specific assumptions about the nature of competition in the aftermarket. We require only that the winner benefits from having a higher true type (which is just a matter of convention) and from being perceived as a higher type. In other words, we consider markets in which bidders wish to overstate their private information, as in the examples outlined above. Throughout the paper we focus on separating equilibria of the auction game, in which each bidder’s strategy is a strictly increasing function of her type. In addition, there may exist (semi)pooling equilibria in which different types bid the same 1

Other examples include underpricing of IPOs to signal future profitability [2], high R & D expenditures to signal low costs [4], and congressional hearings to signal a government’s resoluteness in supervising federal agencies [8]. Riley [28] provides a comprehensive survey of the literature on screening and signaling models. 2 In other applications, it may be more natural to assume that payoffs depend on losers’ private information as well. For example, competitors may not know each other’s cost when they enter the auction. We will consider the simpler situation in which only the winning bidder’s private information matters to highlight the strategic effects of information transmission.

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amount. The reason for restricting attention to separating equilibria is that it maximizes the amount of information submitted and facilitates the study of strategic information transmission (see also [23]). Moreover, the auction will generally result in an efficient allocation only when a separating equilibrium exists. We show that the possibility of signaling via the winning bid puts an upward pressure on the equilibrium bidding strategies. The effects of signaling are most dramatic in the second-price auction. Bidders with high types know that in (a separating) equilibrium they will win almost surely and since they do not pay their own bid when they win they have a strong incentive to signal: equilibrium bids can be well ‘‘above value’’ as a result. This amount of signaling cannot occur in an English auction, which is therefore not strategically equivalent to a second-price auction. Furthermore, the winning bid in an English auction is not determined by the price level at which the second-highest type drops out. The one remaining bidder, who is the sure winner, will raise her bid to a level that maximizes the sum of auction and aftermarket profits. Signaling also occurs in a first-price auction, but again is less pronounced than in the second-price auction. While the amount of signaling varies across the different auction formats, the seller’s revenue from all three auctions is the same and exceeds the revenues when signaling is not possible (i.e. when bidders’ private information is automatically revealed). When bidders want to understate their private information, a separating equilibrium may fail to exist when the incentives to signal via a lower bid are stronger for higher types.3 This can destroy monotonicity of the bidding functions and make full separation impossible. As a result, the auction may not result in an efficient allocation. When a separating equilibrium exists, however, we show that in first- and second-price auctions the signaling bidding function lies below the nosignaling bidding function.4 Signaling is impossible in a separating equilibrium of the English auction since the winning bidder wants to reduce her bid but is constrained by the second-highest bid. The English auction may thus yield higher revenues than a sealed-bid auction when bidders have an incentive to understate their information. The literature on signaling models focuses mainly on the case where one player (e.g. a worker) possesses private information and decides whether to incur a cost to signal this information to others (e.g. employers), who use the signal to determine their optimal decision (e.g. to hire or not). In contrast, in the model discussed below, all bidders have the opportunity to signal and (signaling) decisions are made simultaneously. Our work is therefore more closely related to the literature on simultaneous signaling in oligopoly models (e.g. [23]). In these models, firms compete for multiple periods and may choose higher outputs in early periods to signal lower costs, which could improve their strategic position in later periods.5 Mailath [22] 3

This implies that the ‘‘single crossing’’ property (to be discussed below) fails to hold. This is akin to the ‘‘puppy-dog’’ underinvestment strategy of an incumbent who knows she will accommodate entry, see [12]. 5 An explicit characterization of the separating equilibrium is often not possible in these contexts (e.g. [23]), in contrast with the results reported in Section 3, where a simple characterization of the symmetric bidding functions is derived for general aftermarkets. 4

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discusses general conditions under which a separating equilibrium exists in simultaneous signaling games with a two-period structure, like the model of this paper.6 One crucial distinction is that in Mailath’s model the private information of all players matters, not just that of the winning bidder. Avery [6] provides a different rationale for signaling in English auctions. He discusses conditions under which ‘‘jump bids’’ can credibly signal a bidder’s desire to win and scare away competition, and shows that the occurrence of jump bidding (i) destroys the equivalence of second-price and English auctions, (ii) reduces the seller’s revenue, and (iii) causes the winning bid to be ‘‘path dependent.’’ In the model we consider, there is also no strategic equivalence between the English and second-price auction, but the seller may benefit from signaling, and the final bid is path independent. Our model provides an alternative answer to the question ‘‘Why would anyone raise their own bid?’’ [11]. We show that after all others have dropped out of the English auction, the winner raises her bid to signal more favorable private information. This paper is related to the literature on auctions with resale opportunities. Haile [13] studies the case where bidders have noisy signals about their private values but anticipate an improvement in their information and an opportunity for subsequent trade. Haile employs a two-stage model in which the auction in the first-stage is followed by a resale auction organized by the first-stage winner. Since the division of resale surplus depends on players’ beliefs about their opponents, a player’s secondstage resale profit depends on her bid in the first-stage auction due to the information it reveals. Depending on the resale market structure, a bidder may have an incentive to raise her bid in the first-stage auction to signal a high type or lower her bid to signal a low type.7 Haile also determines conditions under which the initial seller’s revenue is independent of the first-stage auction format given the structure of the resale market. In independent research, Das Varma [11a] and Katzman and Rhodes-Kropf [21a] also studied the possibility of signaling in auctions. Das Varma considers the special case where a first-price auction is used to allocate a cost-reducing innovation among Bertrand or Cournot oligopolists facing linear demands for differentiated goods. Das Varma shows that with Cournot competition an equilibrium always exist and he derives precise conditions for equilibrium existence with Bertrand competition. As in the current paper, Katzman and Rhodes-Kropf consider more general aftermarkets and auction formats. They show how different bid-announcement policies (e.g., conceal all bids, disclose only the winner’s bid) affect the auction’s revenue and efficiency. Katzman and Rhodes-Kropf derive general conditions for revenue equivalence of different auction formats (see also Section 3.3). Finally, this paper is closely related to an extensive literature on patent licensing, which was pioneered by Arrow [5]. Kamien and Tauman [17], Kamien et al. [19], and Katz and Shapiro [20,21] study patent licensing in oligopolistic aftermarkets when the bidding firms are ex ante symmetric and possess complete information about 6 Note, however, that the auction model of this paper does not fit Mailath’s assumptions of differentiable payoffs. A second difference is that Mailath assumes payoffs to be additively separable. 7 See also [7].

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others’ types.8 Jehiel and Moldovanu [14] allow for asymmetries among the bidding firms (but keep the assumption of complete information) and focus on the incentives to participate in an auction for a cost-reducing innovation. Jehiel and Moldovanu [16] study the effects of aftermarket competition on bids in a second-price auction when bidders’ information is private. In their model the winner’s type is automatically revealed after the auction before competition in the aftermarket takes place. Hence, there can be no signaling. This paper considers the natural extension when the winner’s information is not automatically revealed. In the aftermarket game the winner’s type is known only to the winner; others have to infer it from the winning bid. Proposition 1 reproduces Jehiel and Moldovanu’s [16, Proposition 4.1] result for the no-signaling bids in a second-price auction.9 The main focus of this paper, however, is on the effects of signaling on equilibrium bids. The paper is organized as follows. In Section 2 we discuss an example of Cournot duopolists who bid for a patented innovation that reduces their costs by a privately known amount. For ease of exposition, we consider the case of a second-price auction with uniformly distributed types. We derive the equilibrium bids with and without signaling. In Section 3 we introduce the general model and derive the symmetric equilibrium bids for the first-price, second-price, and English auctions. In Section 3.3 we show that all three auction formats yield the same revenue for the seller. In Section 4 we discuss some extensions and implications of our findings. All proofs are contained in the Appendix.

2. An example Consider two Cournot competitors who produce at constant marginal costs cX1 and face a linear demand curve P ¼ A  Q; where Q ¼ q1 þ q2 is total (industry) supply. In the Cournot–Nash equilibrium, each firm i ¼ 1; 2 produces qi ¼ q
¼ ðA  cÞ=3 and equilibrium profits are p
i ¼ q
2 : Now suppose a patented innovation, which reduces firm i’s marginal costs by a commonly known amount di A½0; 1; is sold via a second-price auction in which the highest bidder wins and pays the second-highest bid.10 When firm i acquires the patent and the rival knows di ; firm i’s profit becomes ¼ ðq
þ 23 di Þ2 : pwin i Similarly, if firm i loses and it knows the rival’s cost reduction, dj ; firm i’s profit becomes11 plose ¼ ðq
 13 dj Þ2 : i 8

See [18] for a survey. Jehiel and Moldovanu [16] also study the effects of reserve prices and show that in auctions with aftermarkets the optimal reserve price can be below the seller’s valuation. See also [15], where the effect of aftermarket competition on the seller’s revenue is studied. A crucial difference between their model and the one presented in Section 3.3 is that in their model signaling is not possible. 10 Other auction formats and more general aftermarkets are discussed in the next section. 11 To avoid (trivial) algebraic difficulties we will assume that q
X13; or equivalently, AXc þ 1: This assumption guarantees that the basic duopoly structure is preserved and both firms will produce. 9

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The value to firm i of obtaining the patent is simply the difference between the winning and the losing profit vi ¼ 49 ðdi þ 12 dj Þð3q
þ di  12 dj Þ: It is trivial to verify that vi 4vj iff di 4dj : So in the complete information case, the outcome of the auction is easy to predict: the firm with the highest cost reduction wins at a price equal to minfv1 ; v2 g: The private information case in which firm i’s cost reduction is (initially) known only by firm i; can be modeled in two different ways. First, consider the ‘‘no signaling’’ benchmark, in which the winner’s cost reduction is revealed directly after the auction before all firms choose their outputs. In this case, the winning bid will be irrelevant for play in the Cournot aftermarket and the optimal strategy is to bid the patent’s value assuming that the loser has the same private information (i.e. is of the highest possible losing type), see [26]:12 Bno

signal ðdi Þ

¼ ðq
þ 23 di Þ2  ðq
 13 di Þ2 ¼ di ð2q
þ 13 di Þ:

ð2:1Þ

Note that (2.1) is increasing in di and that the highest type, di ¼ 1; bids the maximum possible value of the patent: vmax ¼ 2q
þ 13: Alternatively, suppose the winner’s cost reduction is not revealed after the auction. Now, the only information available to the loser is the winning bid, which must somehow be used to infer the winner’s type. As we show below, this gives the winner an incentive to raise its bid in an attempt to signal a larger reduction in costs, which would yield larger Cournot profits in the aftermarket. To keep the computations simple, we assume that the distribution of types is uniform (an assumption that is relaxed in the next section). Suppose there exists a strictly increasing equilibrium bidding function BðÞ and that firm 2, say, bids its equilibrium bid Bðd2 Þ: If firm 1 wins the auction with a bid b; firm 2 infers that firm 1’s type is B1 ðbÞ; and chooses its output accordingly, i.e. q2 ¼ q
 13 B1 ðbÞ: Firm 1 anticipates this choice and responds optimally (i.e. firm 1 ‘‘best responds’’ given q2 and given firm 1’s true type d1 ) q1 ¼ q
þ 12 d1 þ 16 B1 ðbÞ; resulting in a profit pwin ðd1 ; B1 ðbÞÞ ¼ q21 : If, on the other hand, firm 1 loses the auction, firm 1’s profit is independent of its bid and type. Given that firm 2 follows To show that this is an equilibrium, suppose bidder 1 bids higher than (2.1), i.e. as if of type d01 4d1 ; while bidder 2 sticks to (2.1). If d2 od1 ; bidder 1 receives the same payoff from bidding Bðd1 Þ and Bðd01 Þ: in both cases, bidder 1 wins, pays Bðd2 Þ; and before the Cournot game starts, bidder 1’s true type d1 is revealed. If d2 4d01 ; bidder 1’s payoffs from bidding Bðd1 Þ or Bðd01 Þ are also the same, since in both cases bidder 1 receives the losing payoff based on d2 : When d1 od2 od01 ; however, bidder 1 wins the auction at a price, Bðd2 Þ; that exceeds its value, Bðd1 Þ: There is no benefit of a higher winning bid in the aftermarket since bidder 1’s true type, d1 ; is revealed before the Cournot game takes place. Hence, bidding above one’s value never pays and may result in a loss. A similar argument shows that it is not worthwile for firm 1 to bid below its value Bðd1 Þ: 12

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the equilibrium (i.e. bids Bðd2 Þ), firm 1’s losing profit is: plose ðd2 Þ ¼ ðq
 13 d2 Þ2 : Hence, firm 1’s expected payoff before it enters the auction is pe1 ðd1 ; bÞ ¼

Z 0

þ

B1 ðbÞ

Z

ððq
þ 12 d1 þ 16 B1 ðbÞÞ2  Bðd2 ÞÞ dd2

1 B1 ðbÞ

ðq
 13 d2 Þ2 dd2 :

ð2:2Þ

From (2.2) the incentive to raise the bid in case of winning is clear. Since the loser infers the winner’s type from the winning bid, a higher winning bid will reduce the loser’s output, and hence increase the winner’s profit in the Cournot game that follows. The equilibrium strategy, BðÞ; can be derived from bidder 1’s equilibrium profit p
1 ðd1 Þ; which is bidder 1’s expected payoff when also she bids according to her true type d1 : This equilibrium profit can be derived from an Envelope Theorem argument: p
1 ðd1 Þ

¼

p
1 ð0Þ

þ

Z 0

d1

dp
1 ðxÞ

¼

p
1 ð0Þ

þ

Z 0

d1

 @pe1 ðx; bÞ @x 

dx:

ð2:3Þ

b¼BðxÞ

Alternatively, the equilibrium profits simply follow from evaluating the expected payoff (2.2) at Bðd1 Þ: p
1 ðd1 Þ ¼ pe1 ðd1 ; Bðd1 ÞÞ Z d1
¼ p1 ð0Þ þ ððq
þ 23 d1 Þ2  BðxÞ  ðq
 13xÞ2 Þ dx;

ð2:4Þ

0

where we defined p
1 ð0Þ ¼

R1 0

ðq
 13xÞ2 dx: Combining (2.3) and (2.4) yields

Bsignal ðd1 Þ ¼ d1 ð73 q
þ 59 d1 Þ;

ð2:5Þ

which is strictly increasing in d1 ; as assumed.13 Note that Bsignal ðd1 Þ4Bno signal ðd1 Þ for all signals d1 40:14 Moreover, for d1 46=7; (2.5) exceeds the maximum possible value of the patent, vmax ¼ 2q
þ 13: Hence, the possibility of exaggerating one’s type results in an upward shift of the bidding function and may cause equilibrium bids to be above value. In equilibrium, losers are not fooled by the higher bids and correctly infer the winner’s type from the winning bid (i.e. equilibrium bids are type revealing). Therefore, the opportunity to signal via the winning bid does not harm the losing bidders but lowers the winner’s profit and raises the seller’s revenue. 13 The proposed bids in (2.5) constitute a global optimum, see the proof of Proposition 2 in the Appendix. 14 Signaling is too costly for the lowest possible type, d1 ¼ 0: This result also follows from a sequential equilibrium argument: since the lowest type d1 ¼ 0 is the worst type a bidder could be perceived as, there is no way to punish the lowest type. Sequentiality therefore requires that the lowest type does not signal.

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3. The general model In this section we extend the model to more general aftermarkets with N competing firms (e.g. a differentiated product oligopoly). Firm i’s type, di ; is private information and plays a role in the aftermarket only if firm i wins. Types di for i ¼ 1; y; N are assumed to be iid draws from a known distribution F ðÞ with density f ðÞ: Assumption 1. The density of types, f ðdÞ; is logconcave on ½d; d% : % Logconcavity means that the log of the density is concave. This restriction is met by many commonly used densities such as the uniform, normal, exponential, chisquare, etc.15 One consequence of Assumption 1 is that the hazard ratio F ðdÞ=f ðdÞ is non-decreasing in d:16 The aftermarket is not modeled explicitly, but is simply summarized by a winning and losing profit function, pwin and plose : These profit functions represent the present value of future discounted profits in case of winning or losing, respectively. We assume a symmetric situation: the winning profit, for example, depends on the winner’s true type, d; and on the type others perceive her to be, m; but not on the winner’s identity.17 In other words, the winning profit function does not have a firm specific subscript: pwin ðd; mÞ: Similarly, the losing profit function, plose ðd; mÞ; is firm independent.18 ð1Þ ð2Þ Let pwin and pwin denote the derivatives of the winning profit with respect to the ð1Þ

ð2Þ

winner’s true type and perceived type, respectively, i.e. pwin @pwin =@d and pwin @pwin =@m: As a matter of convention, we shall assume that, ceteris paribus, a higher ð1Þ true type (e.g. lower cost) is better for the winning bidder, i.e. pwin 40:19 The sign of ð2Þ

pwin ; however, will depend on the details of the aftermarket. In some markets it is better to be perceived as a high type (e.g. low costs with Cournot competition), while in other situations a firm can gain from convincing others its type is low (e.g. high costs with price competition). We will focus mainly on the case where bidders have an incentive to exaggerate their private information, i.e. wish to signal they are of a higher type. This means that the winning profit function is increasing in the perceived ð2Þ ð1Þ ð2Þ type: pwin 40:20 The sign of p0lose ðd; dÞ ¼ plose ðd; dÞ þ plose ðd; dÞ depends on whether 15

See e.g. [9] for a more extensive list. This follows since logconcavity of f ðxÞ implies logconcavity of F ðxÞ (e.g. [3]), so ð f ðxÞ=F ðxÞÞ0 ¼ ðlogðF ðxÞÞ00 p0; which implies that F ðxÞ=f ðxÞ is non-decreasing in x: 17 More generally, the winning profit function would depend on the expected type of the winner. As noted in the Introduction, we are mainly interested in separating equilibria for which there is a one-to-one correspondence between types and actions. 18 Since a bidder’s private information is relevant only when the bidder wins, the losing profit function does not depend on losers’ private information. 16

ð1Þ

If pwin were negative one could always redefine types to be d0 ¼ d: In Section 4 we comment on the case where bidders want to understate their private information and show that a separating equilibrium may fail to exist. 19 20

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an improved position of the winning bidder imposes a positive or negative externality on others. For instance, in a Cournot oligopoly, a reduction in the winner’s cost hurts others if goods are substitutes (p0lose o0), while it makes others better off if goods are complements (p0lose 40). Assumption 2. The difference between the winning and losing profits under complete d information is increasing in type, i.e. dd ðpwin ðd; dÞ  plose ðd; dÞÞ40: Furthermore, ð2Þ

pwin ð; Þ is positive and increasing in both arguments. Intuitively, the first condition means that a marginal increase in the winner’s type is more valuable to the winner than to a loser; for the no-signaling case it guarantees ð2Þ that higher types are willing to pay more to win. Requiring pwin ð; Þ to be increasing in its first argument is a standard ‘‘single crossing’’ condition. It means that higher types have a stronger incentive to signal, i.e. a stronger incentive to exaggerate their ð2Þ private information.21 Finally, the condition that pwin ð; Þ is increasing in its second argument is made to ensure that bidding functions (for the second-price auction) are increasing. 3.1. Second-price and English auctions We first consider the no-signaling benchmark where the winner’s true type is revealed after the auction, before competition in the aftermarket takes place. In this case, the second-price and English auction are strategically equivalent: in both auctions, it is an optimal strategy to bid ‘‘one’s value’’ because (i) higher bids never result in a gain and may cause losses when the auction is won at a price above value, and (ii) lower bids are never beneficial and may cause a bidder to lose the auction at a price for which she would have preferred to win. In the second-price auction the optimal bid is therefore given by the difference between the winning and losing profit, given one’s true type and assuming one of the opponents is of the same (or highest possible losing) type, see also Section 2. Bids in an English auction generally depend on the bid levels at which others drop out.22 Let Bfkg ðdjb1 ; y; bk Þ; k ¼ 0; y; N  1; denote the price level above which a bidder will drop out immediately given her type, d; and given the prices b1 p?pbk at which k others have already dropped out. With independent private information, bidders learn nothing about their own value from others’ drop out levels. As a result, the ‘‘exit’’ strategies Bfkg will depend only on the bidder’s type, d; except for the winning bidder. 21

A more familiar formulation of the single crossing condition is obtained by defining the utility function Uðm; d; mðmÞÞ ¼ pwin ðd; mðmÞÞ  m; where d is the player’s true type and mðmÞ her perceived type if she spends an amount m: The single crossing condition can then be stated as: @m Uðm; d; mÞ=@m Uðm; d; mÞ is increasing in d; i.e. the ratio of the marginal cost of signaling to the marginal benefit of signaling is lower for higher types. 22 The variant we consider is a continuous, irrevocable exit English auction [26]: the auctioneer continuously raises the price and bidders publicly reveal if they drop out. Bidders who have dropped out are not allowed to reenter.

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Proposition 1. Under Assumption 2, the unique symmetric no-signaling equilibrium bidding function for the second-price auction is given by Bno

signal ðdÞ

¼ pwin ðd; dÞ  plose ðd; dÞ: fkg

The exit strategies given by Bno and

fN1g Bno signal ðdjb1 ; y; bN1 Þ

signal ðdjb1 ; ?; bk Þ

ð3:1Þ ¼ Bno

signal ðdÞ

for k ¼ 0; y; N  2;

¼ bN1 constitute an equilibrium of the English auction.

Notice that under Assumption 2, all bidding functions are increasing. The nosignaling equilibrium bids are higher when an increase in the winner’s type imposes a negative externality on losers (e.g. Cournot competition with substitutes) than in the positive externality case where losers also benefit from an improved position of the winner (e.g. Cournot competition with complements), see [16]. Proposition 1 should appear familiar: when signaling is not possible, the secondprice auctions and English auctions are strategically equivalent, and in both auctions the optimal strategy is to bid one’s value. The logic behind this strategy breaks down, however, when the winner’s type is not revealed after the auction and signaling becomes possible. Since bidders gain by being perceived to be of a higher type ð2Þ (pwin 40), there is a benefit of bidding above value since rivals will infer the winner’s type from the winning bid. In the following, bwin denotes the winning bid and fðbwin Þ are the losers’ beliefs about the winner’s type after they observe the winning bid. Proposition 2. Under Assumptions 1 and 2, the unique symmetric separating (perfect Bayesian) equilibrium for the second-price auction is given by the pair ðBsignal ðÞ; fÞ; where Bsignal ðdÞ ¼ Bno

signal ðdÞ

ð2Þ

þ pwin ðd; dÞ

F ðdÞ ðN  1Þf ðdÞ

ð3:2Þ

with Bno signal given by (3.1), and losers’ beliefs are given by fðbwin Þ ¼ B1 signal ðbÞ for bwin A½Bsignal ðdÞ; Bsignal ðd% Þ; fðbwin Þ ¼ d for bwin oBsignal ðdÞ and fðbwin Þ ¼ d% for % % % bwin 4Bsignal ðd% Þ: Note that the second term on the right-hand side of (3.2) is driven primarily by the distribution of types rather than bidders’ valuations. This term disappears at the lower end d ¼ d; where the signaling and no-signaling bids coincide because it is too % costly for the lowest type to signal.23 At the upper end d ¼ d% ; this term causes the equilibrium bids to diverge when the density at the upper bound vanishes. The intuition for this result is that, in a separating equilibrium, the highest type wins for sure and never has to pay her bid. Therefore, impressing others with a higher bid bears no cost and has some benefits, resulting in an infinite bid (while the winner’s expected payment is finite). 23

As noted in Section 2, this result can also be obtained by requiring sequentiality.

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The equilibrium signaling bids in an English auction cannot be above value. The winning bid, however, will not simply be determined by the level at which the bidder with the second-best private information drops out, as is the case without signaling. To see this, suppose all bidders jai are of type d and bid accordingly: Bno signal ðdÞ: % % Since everyone else drops out at low prices, bidder i is the sure winner and only has to bid a small amount. However, bidder i gains by bidding ‘‘higher than necessary,’’ i.e. by choosing a bid that maximizes pwin ðdi ; B1 ðbÞÞ  b: The first-order condition for profit maximization yields: B0 ðdÞ ¼ pð2Þ ðd; dÞ; which together with the boundary Rd condition BðdÞ ¼ Bno signal ðdÞ implies that BðdÞ ¼ Bno signal ðdÞ þ d pð2Þ ðx; xÞ dx: % % % % Hence, even when all other bidders drop out immediately, the winning bidder will want to increase her bid. As before, let Bfkg ðdjb1 ; y; bk Þ; k ¼ 0; y; N  1; be the price level above which a bidder immediately drops out given her type, d; and given the prices b1 p?pbk at which k others have already dropped out. Proposition 3 shows that there is no signaling when more than one bidder is still active. The reason is that in a separating equilibrium the winning bidder’s type, d; is revealed. Her equilibrium profit is therefore pwin ðd; dÞ minus the price she pays. So when the price level reaches Bno signal ðdÞ; a bidder is better off dropping out and losing the auction than winning it. Once all rivals have dropped out, however, the winner has the opportunity to signal. Proposition 3. Under Assumptions 1 and 2, the unique symmetric separating (perfect fkg Bayesian) equilibrium of the English auction entails exit strategies Bsignal ðdjb1 ; y; bk Þ ¼ Bno signal ðdÞ for k ¼ 0; y; N  2; and fN1g Bsignal ðdjb1 ; y; bN1 Þ

¼ bN1 þ

Z

d B1 ðb Þ no signal N1

ð2Þ

pwin ðx; xÞ dx

ð3:3Þ fN1g

given by (3.1), and losers’ beliefs given by fðbwin Þ ¼ ðBsignal Þ1 ðbwin Þ for fN1g % fN1g % ðdÞ and fðbwin Þ ¼ d% when bwin 4B ðdÞ: bwin A½bN1 ; B with Bno

signal

signal

signal

Note that in the English auction losers know that raising the winning bid after everyone else has dropped out is done for signaling purposes, whereas in a sealed-bid auction high bids could be due to high types or to signaling. An alternative equilibrium of the English auction, in which different winning types pool, is the nosignaling equilibrium of Proposition 1 together with losers’ beliefs given by fðbwin Þ ¼ fN1g

ðBno signal Þ1 ðbN1 Þ for bwin XbN1 : Since, these beliefs are independent of the winning bid it clearly does not pay to signal by increasing the winning bid. Note, however, that the beliefs underlying this semi-pooling equilibrium are not ‘‘intuitive’’ [10] since they assume that it is the lowest possible winning type that signals even though this type would be worse off doing so (while higher winning types could possibly benefit from signaling).

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Although the equilibrium strategies for the English and second-price auctions differ substantially, we show in Section 3.3 that both auction formats yield the same revenue for the seller. First, we derive the optimal bidding function for the first-price auction in which the highest bidder wins and pays her own bid.24 3.2. First-price auctions In second-price auctions, the winning bidder expects to make a profit when she bids her value because she has to pay only the second-highest bid. In first-price auctions, the winner has to pay her own bid and should therefore bid below value (or ‘‘shade’’ her bid) to make some profit. When valuations are common knowledge, the amount of shading is obvious. The winning bid equals the second-highest value since this is the most others would bid, and the winner nets the difference between the highest and second-highest valuation. In this case, the winning bid in the first-price auction equals the second-highest bid in the second-price auction. With incomplete information the amount of shading is determined similarly. For a bidder of type d the optimal bid in the first-price auction equals her rivals’ highest expected bid in a second-price auction given that rivals’ types are less than d: Let Fy1 ðjdÞ be the distribution of the maximum of N  1 draws from F ðÞ given that the maximum is smaller than d; i.e. Fy1 ðxjdÞ ¼ ðF ðxÞ=F ðdÞÞN1 for xpd: Proposition 4. Under Assumptions 1 and 2, the unique symmetric separating (perfect Bayesian) equilibrium for the first-price auction is given by the pair ðB˜ signal ðÞ; fÞ; where Z d ð2Þ B˜ signal ðdÞ ¼ B˜ no signal ðdÞ þ pwin ðx; xÞFy1 ðxjdÞ dx ð3:4Þ d %

with B˜ no

signal ðdÞ

¼

Z

d

ðpwin ðx; xÞ  plose ðx; xÞÞ dFy1 ðxjdÞ;

ð3:5Þ

d %

˜ ˜ % and losers’ beliefs are given by fðbwin Þ ¼ B˜ 1 signal ðbÞ for bwin A½Bsignal ðdÞ; Bsignal ðdÞ; % fðbwin Þ ¼ d for bwin oB˜ signal ðdÞ and fðbwin Þ ¼ d% for bwin 4B˜ signal ðd% Þ: % % It is useful to rewrite the equilibrium bidding functions by partially integrating the no-signaling term. If Eðplose ðy1 ; y1 Þjy1 pdÞ denotes the expected value of the losing profit given that the winner’s type is less than d; then (3.4) can be written as Z d ð1Þ B˜ signal ðdÞ ¼ pwin ðd; dÞ  Eðplose ðy1 ; y1 Þjy1 pdÞ  pwin ðx; xÞFy1 ðxjdÞ dx: ð3:6Þ d %

Now the intuition behind the signaling bidding function becomes more apparent: the first two terms on the right-hand side of (3.6) represent the value of winning the 24

We do not treat the Dutch auction separately since it is strategically equivalent to the first-price auction (with and without signaling).

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auction, and the third term is how much a bidder should ‘‘shade’’ her bid. Since the latter term is positive, a bidder expects an extra gain from winning (i.e. in addition to the losing profit she receives if she does not win). In the next section we show that this additional gain is the same in all three auctions, as is the seller’s revenue. So even though the equilibrium signaling bids can be very different for the different auction formats, bidders expect the same positive profits from winning the auction. 3.3. Seller’s revenue In equilibrium, the highest-type bidder wins the auction and receives the winning profit, while all N  1 others receive the losing profit. Let Y1 ðY2 Þ denote the highest (second-highest) order statistic for N draws. The total pie to be divided in any of the three auctions is P ¼ Eðpwin ðY1 ; Y1 ÞÞ þ ðN  1Þ Eðplose ðY1 ; Y1 ÞÞ; and is independent of whether or not signaling is possible. The opportunity to signal merely results in a different allocation of the total pie among the bidders and the seller. Proposition 5. In any of the three auctions, the seller’s expected revenue is the same and can be written as R ¼ Rno signal þ Rsignal ; where Rno

signal

¼ Eðpwin ðY2 ; Y2 Þ  plose ðY2 ; Y2 ÞÞ;

ð3:7Þ

and Rsignal ¼ N

Z

d%

ð2Þ

pwin ðx; xÞ F N1 ðxÞ ð1  F ðxÞÞ dx:

d %

ð3:8Þ

Losers’ expected profits are p
lose ¼ Eðplose ðY1 ; Y1 ÞÞ and the winner’s expected profit is p
win ¼ Eðpwin ðY1 ; Y1 ÞÞ  R: This proposition shows that it is the seller who benefits from bidders’ signaling behavior. Losers’ profits are unaffected since the equilibrium is separating and the winner’s type can be correctly inferred from the winning bid. The increase in seller’s revenue is paid for by the winning bidder: to distinguish herself from others (who also signal), the winner has to spend extra to win the auction. The winner still benefits from winning the auction, however, since p
win ¼ Eðpwin ðY1 ; Y1 ÞÞ  R ¼ Eðplose ðY2 ; Y2 ÞÞ Z d% ð1Þ þ N pwin ðx; xÞ F N1 ðxÞ ð1  F ðxÞÞ dx d %

exceeds Eðplose ðY2 ; Y2 ÞÞ; her profit if she deliberately lost (e.g. by bidding as if her type is d). % The increase in seller’s revenue due to signaling may seem to contradict the usual logic underlying revenue equivalence. Equilibrium bidding functions are strictly increasing whether or not signaling is possible, so in both cases the auction is won by the same bidder (i.e. the highest type). Moreover, the expected payoff of the lowest possible type (who is a sure loser) is independent of whether or not signaling is

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possible. Hence, the standard assumptions for revenue equivalence seem to be satisfied. However, the way bidders’ signals determine their ‘‘valuations’’ changes when signaling becomes possible. When bidders’ information is automatically revealed, an increase in the winner’s true type, d; necessarily raises the winner’s perceived type, m; by the same amount. Hence it affects the winner’s payoff function, pwin ðd; mÞ; in two ways. This is not the case when signaling is possible and bidders infer the winner’s type from the winning bid. A simple Envelope Theorem argument shows that the opportunity to signal reduces bidders’ profits by Rsignal :25

4. Extensions and conclusions Here we briefly mention the case when bidders have an incentive to understate ð2Þ their private information, i.e. when pwin o0: This occurs, for instance, when Bertrand competitors bid for an innovation that reduces costs: while the winning bidder ð1Þ benefits from a larger cost reduction ðpwin 40Þ; she would like others to think the ð2Þ

reduction is small ðpwin o0Þ: Intuitively, this puts a downward pressure on equilibrium bids as bidders want to convey to others that the innovation is not that worthwhile. This underbidding is similar to strategic underinvestment by an incumbent who knows she will accommodate entry; the ‘‘puppy-dog ploy,’’ see [12]. ð2Þ When pwin o0 the bidding functions derived in the previous section need not be increasing. Consider, for instance, a second-price auction for which Eq. (3.2) is the necessary condition a separating equilibrium bidding function has to satisfy. When the density of types is zero at d ¼ d% ; the second term on the right-hand side of (3.2) tends to minus infinity as d-d% ; making it impossible for the bidding function to be everywhere increasing. A similar argument applies to the first-price auction. Hence, for some type densities the opportunity to signal destroys the existence of a separating equilibrium in the first or second-price auction. Stated differently, signaling may cause inefficiencies in these sealed-bid formats. In contrast, there ð2Þ always exists a separating equilibrium for the English auction when pwin o0: the no26 signaling equilibrium of Proposition 1. Recall that in the English auction no ð2Þ signaling occurs until only one active bidder is left. When pwin o0; however, signaling would imply lowering the winning bid which is impossible in an ascending auction. So if the seller is concerned with the auction’s efficiency she should prefer an English auction to a sealed-bid auction. 25

R d%

N1 ðxÞ þ d plose ðx; xÞ dF % ð1Þ N1 p ðx; xÞ F ðxÞ dx: Similarly, when information is automatically revealed, type d’s equilibrium d win % R d% R d ð1Þ ð2Þ profit is d plose ðx; xÞ dF N1 ðxÞ þ d ðpwin ðx; xÞ þ pwin ðx; xÞÞ F N1 ðxÞ dx: Averaging the difference over all % % possible types and multiplying by N yields Rsignal : 26

Rd

With signaling, the expected equilibrium profit for a bidder of type d is

In fact, the equilibrium of Proposition 1 is the unique separating equilibrium of the English auction ð2Þ

when pwin o0:

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359

In addition to being more efficient, the English auction may also raise more revenue. Indeed, suppose there exists a separating equilibrium for the sealed-bid formats. Such an equilibrium will be given by (3.4) for the first-price auction and by (3.2) for the second-price auction, since these solve the (necessary but not ð2Þ sufficient) first-order conditions for profit-maximization. With pwin o0 we conclude that when a separating equilibrium exists for the first-price or second-price auction, the signaling bids will be lower than the no-signaling bids. In contrast, the signaling and no-signaling bids are equal in the English auction, which may thus yield more revenue. The results of this paper can be summarized as follows. We studied the effects of competition in an aftermarket on optimal bidding behavior in first-, second-price, and English auctions. The possibility of signaling via the winning bid has dramatic consequences for equilibrium bidding and overthrows many of the standard results. In particular, when bidders have an incentive to exaggerate their private information, (i) there is no strategic equivalence between the English and secondprice auction, (ii) equilibrium bids in a second-price auction can be above value and (iii) in an English auction the winner raises her bid after everyone else has dropped out. When bidders wish to understate their information, (iv) a separating equilibrium may not exist and the auction may not be efficient. If a separating equilibrium exists, however, (vi) signaling results in a downward shift of the equilibrium bidding function in sealed-bid auctions but not in the English auction, and hence (vii) there is no revenue equivalence in this case. The optimal selling mechanism for a patented innovation thus greatly depends on the nature of competition in the aftermarket. For ‘‘Cournot-type aftermarkets’’ in which bidders want to overstate their private information, all auction formats yield the same revenues. In contrast, for ‘‘Bertrand-type aftermarkets,’’ the expected revenues from an English auction will exceed those from a first- or second-price auction. For both types of aftermarkets, the seller prefers the case where a strategic improvement for the winner imposes a negative externality on losers.

Acknowledgments I thank Glenn Ellison, Maxim Engers, Theo Offerman, Deniz Selman, John Turner, seminar participants at the University of Amsterdam and the University of Virginia, an anonymous referee and an anonymous associate editor for useful suggestions. Financial support from the Bankard Fund at the University of Virginia is gratefully acknowledged.

Appendix Proof of Proposition 2. Suppose all bidders jai bid according to their true type using some increasing bidding function BðÞ: The expected profit of firm i is

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360

given by pe ðdi ; bÞ ¼

Z d %

þ

B1 ðbÞ

Z

ðpwin ðdi ; B1 ðbÞÞ  BðdÞÞ dF N1 ðdÞ

d%

plose ðd; dÞ dF N1 ðdÞ:

B1 ðbÞ

Notice that the losing profit function is evaluated at m ¼ d; since, by assumption, all firms different from i bid according to their true type. Taking the derivative of the expected payoff function with respect to b and evaluating the result at b ¼ Bðdi Þ yields the necessary condition for the symmetric separating equilibrium bidding function:   F ðdi Þ ð2Þ Cðdi Þ  pwin ðdi ; di Þ  plose ðdi ; di Þ  Bðdi Þ þ pwin ðdi ; di Þ ¼ 0; ðN  1Þf ðdi Þ where we factored out a strictly positive term Cðdi Þ ¼ ðN  1Þf ðdi ÞF ðdi ÞN2 =B0 ðdi Þ: The unique solution to this equation is given by (3.2). Assumptions 1 and 2 guarantee that the proposed bidding function is increasing. Next we show that (3.2) corresponds to a global optimum. The derivative of the expected payoff function evaluated at b ¼ Bðd0i Þ is  pe ðdi ; Bðd0i ÞÞ ¼ Cðd0i Þ pwin ðdi ; d0i Þ  pwin ðd0i ; d0i Þ  F ðd0i Þ ð2Þ ð2Þ 0 0 0 þ ðpwin ðdi ; di Þ  pwin ðdi ; di ÞÞ : ðN  1Þf ðd0i Þ ð2Þ

Both pwin ð; Þ and pwin ð; Þ are increasing in their first arguments, so the derivative of the expected payoff is positive for d0i odi and negative for d0i 4di : This shows that bidder i’s optimal bid is Bðdi Þ when bids are restricted to the interval ½BðdÞ; Bðd% Þ: Of course, when bidder i bids less than BðdÞ her expected payoff is the same %as when she bids BðdÞ; which is inferior to bidding %Bðdi Þ: Finally, when bidder i bids an amount % R d% b4Bðd% Þ her expected payoff is pei ðdi ; bÞ ¼ pwin ðd; d% Þ  BðyÞ dF ðyÞN1 ; i.e. the d %

same payoff as when she bids Bðd% Þ: Bidding Bðd% Þ is inferior to bidding Bðdi Þ; which is thus the optimal bid. &

Proof of Proposition 3. Uniqueness follows again from the fact that (3.3) is the only solution to the necessary first-order condition for profit maximization. Suppose all bidders jai exit according to their true type using the exit functions of Proposition 3. We first consider the case where kpN  2 bidders have dropped out (at prices b1 p?pbk ) so that N  kX2 bidders are still active. Suppose bidder i’s type is di and her strategy is to exit at prices above Bðd0i Þ with BðÞ given by (3.1). It is easy to see that d0i odi cannot be optimal. (If all active rivals drop out below Bðd0i Þ; bidder i does no better than when her strategy would have been to exit at prices above Bðdi Þ:) Moreover, the latter strategy is strictly better when bidder i loses the auction and the

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winning bid is only slightly above Bðd0i Þ:) Hence, let d0i 4di : If all active rivals drop out below Bðdi Þ; bidder i does no better than when her strategy would have been to exit at prices above Bðdi Þ: The more interesting case is when all active rivals drop out 0 * where di odod * at a price BðdÞ i : Bidder i then becomes the sole active bidder and has the opportunity to further increase her bid. We next show that it is not optimal for R * * þ *m pð2Þ ðx; xÞ dx where dpmp d% ; bidder i’s her to do so. Note that by bidding BðdÞ win d profit in the aftermarket will be pwin ðdi ; mÞ: (Clearly, bidder i does not want to bid R * þ *d% pð2Þ ðx; xÞ dx since such bids cost more and do not further raise more than BðdÞ win d the aftermarket profits above pwin ðdi ; d% Þ:) Hence, bidder i faces the problem:   Z m ð2Þ *  max pwin ðdi ; mÞ  BðdÞ pwin ðx; xÞ dx : d* p m p d%

d*

ð2Þ

Since pwin is increasing in its first argument, the derivative w.r.t. m of the expression * between the brackets is negative when mXd4d i : Hence, bidder i’s optimal bid is * or, in other words, she should drop out immediately after all active rivals BðdÞ; * The payoff that results from this strategy is pwin ðdi ; dÞ *  BðdÞ * ¼ dropped out at BðdÞ: *  pwin ðd; * dÞ * þ plose ðd; * dÞ: * Since pwin is increasing in its first argument this pwin ðdi ; dÞ * * profit is less than plose ðd; dÞ; which is the profit bidder i would have obtained by exiting at prices above Bðdi Þ: Next, consider the case k ¼ N  1; i.e. when bidder i is the sole active bidder after others have dropped out at prices b1 p?pbN1 : The optimal bidding function is now determined by maximizing pwin ðdi ; B1 ðbÞÞ  b under the restriction BðB1 no signal ðbN1 ÞÞ ¼ bN1 ; since we have shown above that there is no signaling when two or more bidders drop out at the same point. The solution to this maximization problem is given by (3.3). Note that when bidder i with type di bids fN1g Bðd0i Þ ¼ Bsignal ðd0i jb1 ; ?; bN1 Þ her expected payoff is Z d0i ð2Þ 0 pwin ðdi ; di Þ  bN1  pwin ðx; xÞ dx: B1 no

signal

ðbN1 Þ

ð2Þ

Since pwin is increasing in its first argument, the derivative of this expression w.r.t. d0i is positive for d0i odi and negative for d0i 4di : This shows that bidder i’s optimal bid is Bðdi Þ when bids are restricted to the interval ½bN1 ; Bðd% Þ: Finally, if bidder i bids an amount b4Bðd% Þ her expected payoff is less than when she bids as if of type d% ; which in turn yields lower payoffs than bidding Bðdi Þ: & Proof of Proposition 4. Uniqueness follows again from the fact that (3.4) is the only solution to the necessary first-order condition for profit maximization. Suppose all bidders jai bid according to their true type using the bidding function BðÞ in (3.4). Recall that (3.4) is the integral of the bidding function of the second-price auction, with measure Fy1 ðjdÞ: Since the bidding function of the second-price auction is increasing and Fy1 ðjdÞ is stochastically increasing in d; we may conclude that (3.4) is

362

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increasing. The expected profit of firm i is given by pe ðdi ; bÞ ¼ ðpwin ðdi ; B1 ðbÞÞ  bÞ F ðB1 ðbÞÞN1 þ

Z

d%

plose ðd; dÞ dF N1 ðdÞ: B1 ðbÞ

Taking the derivative of the expected payoff function with respect to b and evaluating the result at b ¼ Bðd0i Þ yields  ð2Þ pe ðdi ; Bðd0i ÞÞ ¼ Cðd0i Þ ðpwin ðdi ; d0i Þ  B0 ðd0i ÞÞ

F ðd0i Þ ðN  1Þf ðd0i Þ  0 0 0 0 þ pwin ðdi ; di Þ  plose ðdi ; di Þ  Bðdi Þ ;

with Cðd0i Þ40 defined in the proof of Proposition 2. The slope of the bidding function, B0 ðÞ; follows from (3.4): ð2Þ

B0 ðd0i Þ ¼ pwin ðd0i ; d0i Þ þ

ðN  1Þf ðd0i Þ ðpwin ðd0i ; d0i Þ  plose ðd0i ; d0i Þ  Bðd0i ÞÞ: F ðd0i Þ

Combining the last two equations yields  pe ðdi ; Bðd0i ÞÞ ¼ C  pwin ðdi ; d0i Þ  pwin ðd0i ; d0i Þ ð2Þ

ð2Þ

þ ðpwin ðdi ; d0i Þ  pwin ðd0i ; d0i ÞÞ

 F ðd0i Þ : ðN  1Þf ðd0i Þ

ð2Þ

Both pwin ð; Þ and pwin ð; Þ are increasing in their first arguments, so the derivative of the expected payoff is positive for d0i odi and negative for d0i 4di : This shows that bidder i’s optimal bid is Bðdi Þ when bids are restricted to the interval ½BðdÞ; Bðd% Þ: Of course, when bidder i bids less than BðdÞ her expected payoff is the same %as when she bids BðdÞ; which is inferior to bidding %Bðdi Þ: Finally, when bidder i bids an amount % b4Bðd% Þ her expected payoff is pei ðdi ; bÞ ¼ pwin ðd; dÞ  b which is less than when she % would bid Bðd% Þ: pei ðd; Bðd% ÞÞ ¼ pwin ðd; d% Þ  Bðd% Þ: In turn, bidding Bðd% Þ is inferior to bidding Bðdi Þ; which is thus the optimal bid for bidder i: & Proof of Proposition 5. The revenue of the second-price auction is EY2 ðBsignal ðY2 ÞÞ; with Bsignal given by (3.2). Direct substitution gives the result in Proposition 5 if one uses the density of the second highest order statistic fY2 ðxÞ ¼ NðN  fN1g 1Þf ðxÞF ðxÞN2 ð1  F ðxÞÞ: The English auction yields revenues EY1 ;Y2 ðBsignal fN1g

ðY1 jBno signal ðY2 ÞÞjY1 XY2 Þ; with Bsignal given by (3.3). The first term on the right side of (3.3) leads to a revenue of Rno signal ; while the second term can be

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363

worked out as Rsignal ¼ EY1 ;Y2 ¼

Z

Z

d %

pð2Þ ðz; zÞ dzjY1 XY2



Y2

Z

d%

Y1

d%

Z

d %

x

pð2Þ ðz; zÞ dz fY1 ;Y2 ðx; yÞ dx dy:

y

The density fY1 ;Y2 ðx; yÞ ¼ NðN  1Þf ðxÞf ðyÞF ðyÞN2 for all xXy and zero otherwise [27]. Hence, Z d% Z d% Z x Rsignal ¼ NðN  1Þ pð2Þ ðz; zÞ f ðxÞf ðyÞF ðyÞN2 dz dx dy y

d

¼ NðN  1Þ ¼ NðN  1Þ ¼ NðN  1Þ ¼N

Z

d%

Z % d% Z Z Z

d %

d %

d %

y

d%

y

d%

Z

Z

d%

pð2Þ ðz; zÞ f ðxÞf ðyÞF ðyÞN2 dx dz dy

z d%

pð2Þ ðz; zÞ ð1  F ðzÞÞ f ðyÞF ðyÞN2 dz dy

y

d%

Z

z

pð2Þ ðz; zÞ ð1  F ðzÞÞ f ðyÞF ðyÞN2 dy dz

d %

pð2Þ ðz; zÞ ð1  F ðzÞÞ F ðzÞN1 dz;

d %

where we changed the order of integration in going from the first to the second line and in going from the third to the fourth line. Finally, revenues in the first-price auction are EY1 ðB˜ signal ðY1 ÞÞ where B˜ signal is given by (3.4). Note that B˜ signal ðdÞ can be written as B˜ signal ðdÞ ¼ EðBsignal ðY2 ÞjY1 ¼ dÞ; where Bsignal is the optimal bid for the second-price auction (3.2). Hence, the revenue of a first-price auction is EY1 ðB˜ signal ðY1 ÞÞ ¼ EY2 ðBsignal ðY2 ÞÞ; i.e. the same revenue as in a second-price auction. &

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