Entry coordination and auction design with private costs of information acquisition.

Lu, Jingfeng

I. INTRODUCTION

In most auction literature, bidders are passively endowed with private information about their valuations. The analysis then focuses on optimal elicitation of private information. On many occasions, bidders may instead have to incur costs to collect this information. (1) Auction design in these cases has to balance between information acquisition and information elicitation, which are interdependent. The performance of an auction depends not only on the bidding equilibrium but also crucially on the information acquisition equilibrium. (2) As a salient feature, auction design with information acquisition costs has been complicated by entry coordination among bidders due to multiple entry equilibria issue. In a symmetric independent private value (IPV) setting of McAfee and McMillan (1987) with fixed information acquisition cost, Levin and Smith (1994) note that the ex ante efficient and revenue-maximizing auction (a second-price auction with no entry fee and no reserve) induces many asymmetric entry equilibria other than the targeted symmetric one. The existence of asymmetric entry equilibria in this setup fundamentally lies in bidders' constant marginal cost of entry. (3)

Information acquisition costs could be private information of bidders just as with their private information about values. For example, in the cases of construction procurements or U.S. timber auctions, many aspects of prebid information acquisition and analyzation are private knowledge of bidders. (4) Clearly, when information acquisition costs are private information of bidders, auction design has to additionally take into account information elicitation at the information acquisition stage. This aspect of analysis has yet to be reflected in the literature, while the case with fixed information acquisition costs has been thoroughly studied by Milgrom (1981), McAfee and McMillan (1987), Engelbrecht-Wiggans (1987, 1993), Harstad (1990), Levin and Smith (1994), McAfee, Quan, and Vincent (2002), Ye (2004, 2007), and Cremer, Spiegel, and Zheng (2009) among others. A widely recognized insight of these studies is that ex ante efficiency can be achieved through a second-price auction while setting the reserve at the seller's valuation. If ex ante entry fees can be used to extract all the expected surplus of bidders then there is a congruence between the revenue and total surplus. (5) This article advances this line of research by studying the implications of private acquisition costs on auction design with an emphasis on bidders' coordination at the information acquisition stage. Specifically, we consider the IPV setting of McAfee and McMillan (1987) and Levin and Smith (1994) while allowing the information acquisition costs to be bidders' private information, which follow a continuous distribution. In light of our previous discussion, this article attempts to answer the following questions: (1) How are ex ante efficient and revenue-maximizing auctions affected by this additional dimension of private information? In other words, how does information rent extraction at the entry stage affect the auction designs? (2) Can this private information alleviate (rather than aggravate) the problem of entry coordination? With the dispersion in private information acquisition costs, the multiplicity of entry equilibria arising from the constant marginal cost in Levin and Smith (1994) might be avoidable. The types with higher information acquisition costs must have less incentive to enter, which could reduce the number of entry equilibria. The questions are: Will sufficient dispersion coordinate bidders and induce a unique entry? If yes, how much dispersion is enough?

Due to the potential multiplicity of entry equilibria for any given mechanism, it is rather difficult to compare performances across varieties of mechanisms. To overcome this difficulty, we come up with an alternative approach. Our analysis begins with characterizing efficient and revenue-maximizing auctions for any given feasible entry pattern, which can be described through a vector of bidders' entry thresholds of acquisition costs. (6) For a given feasible entry pattern, we cannot do better than ex post efficient allocation to maximize the expected total surplus of the seller and bidders. While achieving ex post efficient allocation through a second-price auction with a reserve price equal to the seller's valuation, appropriate ex ante entry fees (or subsidies) are sufficient to make sure that the given entry pattern is indeed induced. (7) Thus, a second-price auction with a reserve price equal to the seller's valuation and appropriately set entry fees (subsidies) is ex ante efficient for the given entry pattern. (8) The same auction is also revenue maximizing as participating types enjoy the smallest possible information rents, which equal the differences between the entry thresholds and the information acquisition costs of participants. The optimality of a revenue-maximizing reserve that equals the seller's value thus extends to a setting with private acquisition costs. This result clearly relies on the availability of ex ante entry fees that can be used to extract the expected surplus of the entrants. (9) The availability of ex ante entry fees guarantees the optimality of ex post efficiency, which in turn calls for an optimal reserve equal to seller's valuation.

These observations facilitate our search for ex ante efficient and revenue-maximizing auctions. What is essential is to characterize the entry patterns that maximize the expected total surplus and seller's expected revenue and then pin down the entry fees that implement these entry patterns. Convenient expressions of the optimal expected total surplus and seller's expected revenue as functions of bidders' entry thresholds are discovered. These expressions allow us to establish useful connections between the first-order conditions characterizing the optimal (efficient or revenue maximizing) entry thresholds and the expected payoff of these threshold types in a second-price auction with appropriate entry fees and a reserve price equal to the seller's valuation. Based on these connections, we find that (1) a second-price auction with no entry fee and a reserve price equal to the seller's valuation is ex ante efficient and (2) a second-price auction with the same reserve price and appropriate ex ante entry fees is revenue maximizing. Specifically, these entry fees equal, respectively, the hazard rates of the information acquisition cost distribution, which are evaluated at the bidders' entry thresholds.

The intuition is clear why the simple second-price auction is ex ante efficient. When ex ante entry fees are zero, a type of bidder enters this auction if and only if his marginal contribution to the total expected surplus is nonnegative. Our result extends the findings of Engelbrecht-Wiggans (1993), Levin and Smith (1994), and Ye (2004) to a more general setting. Revenue maximization, on the other hand, requires optimal balance between ex ante efficiency and rent extraction at the entry stage. We find that the contribution of the threshold type to the revenue equals its contribution to the total efficiency minus the corresponding hazard rate of the cost distribution. The hazard rate measures the impact of the threshold on rent extraction at the entry stage. Given the hazard rate is nonnegative, an entry threshold is revenue maximizing only if its contribution to the total surplus is nonnegative. In this case, the threshold type enjoys a positive payoff (which equals his marginal contribution to the total surplus) in a second-price auction with a reserve price equal to the seller's valuation. A nonnegative entry fee that equals the threshold type's contribution to the total efficiency thus in demand to extract the threshold type's surplus for the purpose of revenue maximization. (10)

The entry coordination problem does not dissappear naturally even with dispersion in acquisition costs. First, like the case of fixed information acquisition costs (Levin and Smith 1994), the issue of multiple entry equilibria still prevails with the identified efficient and revenue-maximizing auctions. (11) An important issue is whether conditions on dispersion of acquisition costs can be identified to ensure a unique implementation of ex ante efficient entry as well as revenue-maximizing entry. Second, we find that it could be an asymmetric entry rather than a symmetric one that maximizes the expected total surplus or the seller's expected revenue. As one may argue that symmetric entry can be a focal point of the entry game, an interesting issue is whether conditions can be identified to ensure the desirable entry to be symmetric across bidders. To address these two issues, we show that when the distribution of acquisition costs is more disperse than a particular uniform distribution according to the Biekel-Lehman dispersive order (Bickel and Lehman 1976), the efficient entry must be symmetric across bidders and it is the unique entry equilibrium of the proposed efficient auction. (12) If further the hazard rate of the information acquisition cost distribution is increasing, that is, a higher entry threshold must associate with a higher entry fee in the revenue-maximizing auction, then the revenue-maximizing entry must also be symmetric and it is the unique entry equilibrium. The intuition behind this is as follows. When the distribution of acquisition costs is highly disperse, a type with high acquisition cost has no incentive to enter even when it is incentive compatible for his rivals to adopt low entry thresholds because the increase in his winning chance cannot justify his high information acquisition cost plus the higher entry fee that is additionally required by the revenue-maximizing auction.

Our article is closely related to a parallel line of auction literature on information acquisition, which includes Dasgupta (1990), Tan (1992), Bag (1997), Persico (2000), and Bergemann and Valimaki (2002) among others. They consider the case where the value distribution depends on the bidders' endogenous investment. The efficiency of the simple second-price auction in our new environment is consistent with the finding of Bergemann and Valimaki (2002), while our revenue-maximizing auction with ex ante entry fees echoes the insight of Bag (1997). (13) The entry coordination problem in this alternative setting has been noted by Tan (1992) (Proposition 3), who finds that multiple investment equilibria exist for a second-price auction when bidders are equipped with constant return to scale technology. Tan (1992) (Proposition 4) further provides sufficient conditions for a unique entry equilibrium. The key conditions include decreasing return to scale investment technology and increasing marginal investment cost. Our finding that sufficient dispersion in acquisition cost resolves the entry coordination problem in our setting shares the same spirit of Tan (1992).

This article is organized as follows. In Section II, we introduce a symmetric IPV setting where potential bidders share the same distributions on valuations and information acquisition costs. In Section III, the ex ante efficient entry and revenue-maximizing entry are characterized, and the ex ante efficient auction and revenue-maximizing auction are established. Section IV studies entry coordination among bidders. In this section, we provide sufficient conditions for the symmetry of the desirable entry (efficient or revenue maximizing) and the unique implementation of them. Section V provides a concluding remark.

II. SYMMETRIC IPV SETTING

There are N potential bidders who are interested in a single item, where N is public information. Denote this group of potential bidders by N = {1,2,..., N}. The seller's valuation is [v.sub.0], which is public information. Bidder i has to incur an information acquisition cost [c.sub.i] to discover his private value [v.sub.i]. These [c.sub.i] are sunk costs, which differ from entry fees that the seller can collect from bidders as revenue. We assume that a bidder does not enter without discovering his value. (14) Both [c.sub.i] and [v.sub.i] are assumed to be private information of bidder i. The cumulative distribution function of [c.sub.i] is G(x) with density function g(x), while the cumulative distribution function of vs is F(x) with density function f(x). The support of G(x) is [[c.bar], [bar.c]] and the support of F(x) is [[v.bar], [bar.v]]. We assume g(x) > 0 on its support. The distributions of [c.sub.i] and [v.sub.i], i [member of] N are assumed to be public information. The information acquisition costs can be interpreted as the bidders' efficiencies in discovering their valuations. In this article, we study a setting where the bidders' valuations do not depend on their efficiencies in discovering their valuations. Specifically, we assume [c.sub.i] and [v.sub.j], [for all]i, j [member of] N are mutually independent. The seller and bidders are assumed to be risk neutral. The timing of the auction is as follows:

Time 0: The group of potential bidders Af, the seller's valuation [v.sub.0], and the distributions F(x) and G(x) are revealed by nature as public information. Every bidder i observes his private cost [c.sub.i], i [member of] N.

Time 1: The seller announces the rule of the auction.

Time 2: The bidders simultaneously and confidentially make their entry decisions. If they do not enter, they take the outside option, which gives them zero payoff. If they enter, they incur their private information acquisition costs and observe their private values. (15)

Time 3: Every participant bids. (16)

Time 4: The payoffs of the seller and all the participating bidders are determined according to the announced rule at Time 1.

We study the ex ante efficient auction rule and the revenue-maximizing auction rule announced at Time 1. Here, the ex ante efficient auction refers to the auction that maximizes the expected total surplus of the seller and bidders and the revenue-maximizing auction refers to the auction that maximizes the expected revenue of the seller. The induced entry equilibria corresponding to the maximal expected total surplus are called ex ante efficient entry, while the induced entry equilibria corresponding to the maximal expected revenue are called revenue-maximizing entry.

III. AUCTION DESIGN

The setting of Section II involves private information on information acquisition costs and valuations. However, bidders are endowed with private information about their values if and only if they incur the private information acquisition costs. Clearly, auction design in this setting differs from a typical one-or two-dimensional screening problem where we can apply the revelation principle and thus focus on truthful revelation mechanisms. A particular way must be developed to approach auction design in our setting. Our strategy is as follows. We first characterize all feasible entry patterns in terms of every bidder's entry threshold of information acquisition costs. We then show that the highest feasible expected total surplus and seller's expected revenue associated with a given feasible entry pattern are attainable through a particular auction. For further analysis, we develop convenient expressions for the optimal expected total surplus and seller's expected revenue as functions of entry thresholds of bidders. These expressions enable us to characterize efficient entry and revenue-maximizing entry. Finally, these characterizations of the desirable entry enable us to describe the efficient and revenue-maximizing auctions. (17)

The focus of this article was to study the impact of private information acquisition costs on auction design with simultaneous entry. (18) Following the above strategy, we first characterize all feasible simultaneous entry equilibrium patterns. Here, entry equilibrium refers to information acquisition equilibrium at Time 2. At Time 2, bidders are informed about their information acquisition costs. Clearly, if a type enters with a positive probability, the types with lower information acquisition costs must strictly prefer to enter. This observation leads to the following characterization of a feasible entry equilibrium.

LEMMA 1. Any simultaneous entry equilibrium can be described through a vector of entry thresholds [C.sup.e] = ([c.sup.e.sub.1],..., [c.sup.e.sub.N]) that satisfy the following properties: (i) [c.sup.e.sub.i] [member of] [[c.bar], [bar.c]], [[for all].sub.i] [member of] N; (ii) if [c.sub.i] < [c.sup.e.sub.i], bidder i participates with probability 1 and if [c.sub.i] > [c.sup.e.sub.i], bidder i participates with probability 0.

Proof. See Appendix.

Given thresholds [C.sup.e], where [c.sup.e.sub.i] [member of] [[c.bar], [bar.c]], [for all]i [member of] N, without loss of generality, we assume (i) if [c.sup.e.sub.i] > [c.bar], bidder i participates if and only if [c.sub.i] [less than or equal to] [c.sup.e.sub.i] and (ii) if [c.sup.e.sub.i] = [bar.c] no type of bidder i participates. (19)

As pointed out before, since the revelation principle does not apply here, one cannot look at auction design through simply considering a particular class of truthful direct revelation mechanisms. The performance of an auction is determined by both entry equilibrium induced at Time 2 and equilibrium bidding strategy at Time 3. There are innumerous auctions that induce different entry and bidding equilibria. How should we compare across all auctions to identify the efficient and revenue-maximizing ones? A plausible approach is as follows. We first identify the restricted optimal efficient and revenue-maximizing ones among all auctions that induce any given entry thresholds. The corresponding total expected surplus and revenue can then be written as functions of entry thresholds. We then use the first-order conditions for the desirable entry to identify the efficient and revenue-maximizing auctions.

We first establish the following results regarding the restricted efficient auction and revenue-maximizing auction, which implement given entry thresholds [C.sup.e] = ([c.sup.e.sub.1], ..., [c.sup.e.sub.N],), where [c.sup.e.sub.i] [member of] [[c.bar], [bar.c]], [for all] i [member of] N. For convenience, we use [A.sub.0] to denote the second-price auction with no entry fee and a reserve price equal to the seller's valuation [v.sub.0].

PROPOSITION 1. (i) Among all auctions implementing given entry threshold [C.sub.e] where [c.sup.e.sub.i] [member of] [[bar.c], [for All] i [member of] N, a second-price auction with a reserve price equal to the seller's valuation and appropriately set ex ante entry fee (or subsidy) for each bidder provides the highest expected total surplus and the highest seller's expected revenue; (ii) the entry fees (or subsidies) are charged upon entry at Time 2 before the valuations are learned by the entrants and are set at levels such that the threshold-type entrants get zero-expected payoff; (iii) the expected surplus of participating bidder i with information acquisition cost [c.sub.i] [less than or equal to] [c.sup.e.sub.i] is [c.sup.e.sub.i] - [c.sub.i].

Proof See Appendix.

The results of Proposition 1 are rather intuitive. For a given entry pattern, we cannot do better than ex post efficient allocation to maximize the expected total surplus of the seller and bidders even if we temporarily ignore the entry incentive of bidders. While ex post efficient allocation can be achieved through a second-price auction with a reserve price equal to the seller's valuation regardless of the value distribution, appropriate ex ante (at Time 2) entry fees (or subsidies) are sufficient to make sure that the given entry pattern is indeed an entry equilibrium for the proposed auction. We only need to set the entry fees (subsidies) properly to make the entry threshold types indifferent between participating and not participating. Since the entry fees (subsidies) are transfers between the seller and the bidders, they do not affect the expected total surplus. Thus, the proposed auction provides the highest expected total surplus for the given entry pattern. The same auction also provides the highest revenue as participating types enjoy the smallest possible information rent for the given entry. For any auction, the expected payoff of bidder i with information acquisition cost [c.sub.i] [less than or equal to] [c.sup.e.sub.i] cannot be smaller than [c.sup.e.sub.i] - [c.sub.i] since a type [c.sub.i] of bidder i can always adopt the bidding strategy of the entry threshold type [c.sup.e.sub.i] and get at least a payoff of [c.sup.e.sub.i] - [c.sub.i]. For the proposed auction, bidder i's expected surplus is exactly [c.sup.e.sub.i] - [c.sub.i] if his information acquisition cost is [c.sub.i], which reaches the minimum. Thus, the Proposition 1 auction is revenue maximizing among all auctions that induce the given entry.

In Proposition 1, the seller is allowed to charge ex ante entry fees before buyers check the object. This has been pointed out by McAfee and McMillan (1987): "... the seller must extract these payment ... before any bidders learns his valuation: this could be done, for example, by demanding a fee before potential bidders inspect the item for sale." In addition, Engelbrecht-Wiggans (1993) and Levin and Smith (1994) also include ex ante entry fees in their analysis. We will show later that ex ante entry fee turns out to be redundant for the ex ante efficient auction we will establish. For revenue maximization, the unrestricted mechanism would instead require ex ante entry fee. Nevertheless, in Section IVB, we will further discuss revenue maximization when ex ante entry fee is not allowed.

Proposition 1 reveals the optimality of a reserve that equals the seller's value for both the efficient and the revenue-maximizing auctions regardless of the entry to be implemented. This result clearly relies on the availability of ex ante entry fees that can be used to extract the expected surplus of the entrants. The availability of ex ante entry fees guarantees the optimality of ex post efficiency, which in turn calls for an optimal reserve equal to seller's valuation. (20) When ex ante entry fees are not allowed, then a more sophisticated reserve is in demand. This issue is further discussed in Section IVB.

It is the insight of Proposition 1 that makes it feasible to begin our analysis on efficient and revenue-maximizing auctions. For given entry thresholds [C.sup.e] where [c.sup.e.sub.i] [member of] [[c.bar], [bar.c]], [for all]i [member of] N, we denote the highest expected total surplus and the highest seller's revenue attainable through the auction identified in Proposition 1 by S([C.sup.e]) and R([C.sup.e]), respectively. We next introduce a convenient way of writing S([C.sup.e]) and R([C.sup.e]). The advantage of these expressions is that they allow us to characterize the desirable entry. These characterizations through the first-order conditions deliver insightful economic interpretations.

We first introduce some notations. We define set K = {([k.sub.1], [k.sub.2], ..., [k.sub.N])|[k.sub.i] [member of] {0, 1}, i [member of] N}, where [k.sub.i] denotes bidder i's ex post entry status. Specifically, [k.sub.i] = 1 stands for participation of bidder i, while [k.sub.i] = 0 represents nonparticipation of bidder i. In addition, let [k.sub.0] = 1 symbolize participation of the seller for convenience. For each ex post entry status vector k = ([k.sub.1], [k.sub.2], ..., [k.sub.N]) [member of] K, we use [v.sub.k] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to denote the highest valuation of all participants including the seller. The c.d.f, and p.d.f. of [v.sub.k] are denoted by [F.sub.k]([v.sub.k]) and [f.sub.k]([v.sub.k]), respectively. Furthermore, [V.sub.k] denotes the expectation of [v.sub.k]. Using the above notations, S([C.sup.e]) and R([C.sup.e]) can then be written as the following:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the probability that the entry status denoted by k happens. (21) The first term [[summation].sub.{k [member of] K} Pr(k) [V.sub.k] in Equation (1) is the contributidn of the valuations of ali participants (including the seller) to the expected total surplus if potential bidders participate according to thresholds [C.sup.e]. The second term [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in Equation (1) is the (negative) contribution of the information acquisition costs of participants to the expected total surplus if the potential bidders participate according to thresholds [C.sup.e]. Following Proposition l(iii), we know that the expected information rent of bidder i is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This leads to the seller's expected revenue in Equation (2), which is the difference between the expected total surplus and all bidders' expected information rent.

A. Ex Ante Efficient Auction

The ex ante efficient auction can be shown through two steps. First, we derive the firstorder conditions for the optimal threshold vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which maximizes the expected total surplus S(*). These conditions will be shown to be closely related to the payoff of threshold types in auction [A.sub.0]. Second, we apply Proposition 1 to show that auction [A.sub.0] implements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] while achieving the highest attainable efficiency.

Let us consider bidder i's optimal threshold [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Define [K.sub.-i] = {([k.sub.1], ..., [k.sub.i-1], [k.sub.i+1], ..., [k.sub.N])|[k.sub.j] [member of] {0, 1}, j [not equal to] i}, [for all] i [member of] N. [K.sub.-i] is the set of entry status of bidders other than i. For any [k.sub.-i] = ([k.sub.1], ..., [k.sub.i-1], [k.sub.i+1], ..., [k.sub.N]) [member of] [K.sub.-i], we use [k.sub.1]([k.sub.-i]) to denote the N-element vector in K where the ith element is 1 and other elements are same with [k.sub.-i], while we use [k.sub.0]([k.sub.-i]) to denote the N-element vector in K where the ith element is 0 and other elements are same with [k.sub.-i]. We then have:

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the probability that the entry status denoted by [k.sub.-i] happens. Define Si([C.sup.e]) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By definitions of [V.sub.k1]([k.sub.-i]) and [V.sub.k0]([k.sub.-i]), [S.sub.i]([C.sup.e]) is the marginal contribution of bidder i with information acquisition cost [c.sup.e.sub.i] to the expected total surplus, given that other bidders participate in auction [A.sub.0] according to [C.sup.e].

S(x) is a differentiable function that is defined on a compact support. The threshold [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that maximizes S(x) must satisfy the following first-order conditions. For all i [member of] N,

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For convenience, the following result of Levin and Smith (1994) is put into the following Lemma. (22)

LEMMA 2. [V.sub.k1]([k.sub.-i]) - [V.sub.k0]([k.sub.-i]) is the expected payoff of bidder i with zero information acquisition cost from participating in auction [A.sub.0] if all other entrants are those with [k.sub.j] = 1 in vector [k.sub.-i] (Levin and Smith 1994).

It follows from Lemma 2 that [S.sub.i]([C.sup.e]) is the expected payoff of bidder i with cost [c.sup.e.sub.i] when he participates in auction [A.sub.0] if all other potential bidders participate according to [C.sup.e]. This insight together with Equation (4) leads to the following proposition, which addresses the ex ante efficient auction.

PROPOSITION 2. The second-price auction [A.sub.0] with a reserve price equal to the seller's valuation and no entry fee is ex ante efficient.

Proof. The design of [A.sub.0] follows the spirit of Proposition 1. Since g(*) > 0, it is clearly a Nash equilibrium that every bidder participates in auction [A.sub.0] according to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , as Equation (4) is satisfied for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is a weakly dominant strategy for bidders to bid their true values when participating, thus ex post efficiency is achieved by [A.sub.0]. Therefore, [A.sub.0] achieves the highest possible expected total surplus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The intuition for the efficiency of [A.sub.0] is pretty clear. When ex ante entry fees are zero, a type of bidder enters a second-price auction with a reserve price equal to the seller's valuation if and only if his marginal contribution to the total expected surplus is nonnegative. The efficiency of auction [A.sub.0] when information acquisition costs are private information of bidders is consistent with the literature (e.g., Bergemann and Valimaki 2002; Engelbrecht-Wiggans 1993; Levin and Smith 1994; Ye 2004). This result holds since bidders' information rents arising from their private information acquisition costs have no impact on expected total surplus of the seller and bidders.

The result in Proposition 2 accommodates the flexibility of an optimal entry at the corner solution, as indicated by Equation (4). An example of a corner solution is provided in the following symmetric setting, where [v.sub.0] = 0, N = 2, F([v.sub.i]) = [v.sub.i], [for all] [v.sub.i] [member of] [0, 1], and G([c.sub.i]) = 10([c.sub.i] - 0.4), [for all] [c.sub.i] [member of] [0.4, 0.5]. In this setting, S([C.sup.e]) takes the maximum of 0.05 when [c.sup.e.sub.1] = 0.5 and [c.sup.e.sub.2] = 0.4. This example also illustrates that the efficient entry can be asymmetric even the bidders are symmetric. The source of asymmetry in the desirable entry will be discussed in Section IIIC.

Note that in the setting of the above example, there exists another symmetric entry equilibrium where [c.sup.e.sub.1] = [c.sup.e.sub.2] = 0.4231 for [A.sub.0]. Thus, an issue of multiple entry equilibria for the efficient auction [A.sub.0] arises. In Section IV, we will address this issue of multiple entry equilibria.

B. Revenue-Maximizing Auction

We now turn to the revenue-maximizing auction, which will be derived in a similar procedure to that of Section IIIA. First, we derive the first-order conditions for the revenue-maximizing entry. Second, we use these conditions and Proposition l to pin down the revenue-maximizing auction.

From Equation (2), we have:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The term G([c.sup.e.sub.i])/g([c.sub.e.sub.i]) captures the marginal impact of entry threshold [c.sup.e.sub.i] on bidder i's information rent [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Expected revenue is the difference between the expected total surplus and the information rent of bidders. The marginal impact of entry threshold [c.sup.e.sub.i] on expected revenue is thus the difference between its marginal impact on total surplus (i.e., [S.sub.i]([C.sup.e])) and that on information rent of bidders (i.e., G([c.sup.e.sub.i])/g([c.sub.e.sub.i])).

Define [R.sub.i]([C.sup.e]) = [S.sub.i]([C.sup.e]) - G([c.sup.e.sub.i])/g([c.sub.e.sub.i]). Suppose that [C.sup.e[dagger]] = ([c.sup.e[dagger].sub.1], ..., [c.sup.e[dagger].sub.N]) maximizes R([C.sup.e]), then we have the following characterization for [C.sup.e[dagger]]. For all i [member of] N,

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Suppose that all other potential bidders participate according to [C.sup.e[dagger]]. Based on Lemma 2, we have that [R.sub.i]([C.sup.e[dagger]]) is the expected payoff of bidder i with cost [c.sup.e[dagger].sub.i] if he participates in a second-price auction with a reserve price equal to [v.sub.0] and a Time 2 ex ante entry fee of G([c.sup.e.sub.i])/g([c.sub.e.sub.i]) for bidder i. This insight together with Equations (5) and (6) and Proposition 1 lead to the following proposition that addresses the revenue-maximizing auction.

PROPOSITION 3. Suppose that [C.sup.e[dagger]] maximizes R(x), then a second-price auction with reserve price equal to the seller's valuation and Time 2 ex ante entry fees [E.sub.i] for bidder i defined below leads to the highest expected revenue for the seller. [E.sub.i], i [member of] N are defined as:

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof The result follows Proposition 1 immediately. Since g(*) > 0, it is clearly a Nash equilibrium that every bidder participates in the above defined auction according to [C.sup.e[dagger]] while Equation (6) is satisfied. The entry fees are set according to Proposition 1. When [c.sup.e[dagger].sub.i] [member of] ([bar.c], [bar.sub.] [E.sub.i] is set as [S.sub.i]([C.sup.e[dagger]]) to extract the total expected surplus of the entry threshold type of bidder i. Note that when [c.sup.e[dagger].sub.i] [member of] ([c.bar], [bar.c]), we have [S.sub.i]([C.sup.e[dagger]]) = G([c.sup.e.sub.i])/g([c.sub.e.sub.i]) from Equation (6). When [c.sup.e[dagger].sub.i] = [bar.c], we have [S.sub.i]([C.sup.e[dagger]]) [greater than or equal to] G([bar.c])/g([bar.c]) = 1/g([bar.c]).

From Proposition 3, if information acquisition costs are private information of bidders then essentially, the revenue-maximizing auction involves personalized ex ante entry fees. (23) The entry fees that equal the hazard rates of the cost distribution optimally balance between social efficiency and rent extraction at the entry stage. These entry fees implement the revenue-maximizing entry while extracting the expected surplus of the entry threshold types. Similar to the case of the efficient auction, there might exist multiple entry equilibria for the revenue-maximizing auction of Proposition 3, and the revenue-maximizing entry can be asymmetric across bidders. In Section IV, we will address this issue.

When information acquisition costs are private information of bidders, the optimal entry patterns that maximize the expected total surplus and the seller's expected revenue generally differ from each other. The intuition behind this difference is as follows. When the information acquisition costs are private information of the bidders, the seller has no way of extracting all the surplus of the participants according to Proposition 1(iii). Thus, revenue-maximizing entry must optimally balance between the expected total surplus and the information rent associated with entry, while ex ante efficient entry focuses only on total expected surplus. It would be interesting if we can show revenue-maximizing auction induces less entry or less expected information acquisition costs than efficient auction. (24) The comparison is not an easy task if the desirable entry is allowed to be asymmetric across bidders. The complication lies in that adding a personalized positive ex ante entry fee for every bidder to efficient auction [A.sub.0] does not necessarily lead to an entry equilibrium with lower entry threshold for every buyer. (25) We will come back to this issue in Section IIID where entry is restricted to be symmetric. (26)

C. Source of Asymmetry in Desirable Entry

As we have mentioned in Sections IIIA and IIIB, the efficient and revenue-maximizing entry patterns are generally asymmetric across bidders. Recall the example of Section IIIA, where [v.sub.0] = 0, N = 2, F(v) = v, [for] v [member of] [0, 1], and G(c) = 10(c - 0.4), [for all] c [member of] [0.4, 0.5]. Direct calculations using Equations (1) and (2) give the following results. S([C.sup.e]) takes the maximum of 0.05 when [c.sup.e.sub.1] = 0.5 and [e.sup.e.sub.2] = 0.4, and R([C.sup.e]) takes the maximum of 0.025 when [c.sup.e.sub.1] = 0.45 and [c.sup.e.sub.2] = 0.4. If we restrict [c.sup.e.sub.1] = [c.sup.e.sub.2] then we have S([C.sup.e]) takes the maximum of 0.023 when [c.sup.e.sub.1] = [c.sup.e.sub.2] = 0.4231, and R([C.sup.e]) takes the maximum of 0.01875 when [c.sup.e.sub.1] = [c.sup.e.sub.2] 0.4187. Thus, the optimal entry patterns maximizing the expected total surplus and the seller's expected revenue are asymmetric. What is the source of the asymmetry?

Define [W.sub.n], n [greater than or equal to] 0 as the expectation of the highest valuation of the seller and n([greater than or equal to] 0) bidders. The following lemma provides some useful properties of series [W.sub.n], n [greater than or equal to] 0.

LEMMA 3. Both [W.sub.n + 1] - [W.sub.n] and ([W.sub.n + 1] - [W.sub.n]) - ([W.sub.n + 2] - [W.sub.n + 1]) decrease with n ([greater than or equal to] 0).

Proof. See Appendix.

The results of Lemma 3 are rather intuitive. From Lemma 2, [W.sub.n] - [W.sub.n - 1] is the expected payoff of a representative bidder with zero information acquisition cost when there are altogether n bidders in auction [A.sub.0]. Lemma 3 says that this expected payoff decreases with the number of bidders n, and the rate of decrease also decreases with n.

The reason why the desirable entry (efficient and/or revenue maximizing) can be asymmetric lies in that there might not be sufficient dispersion in information acquisition costs. We next provide the intuitions on why sufficient dispersion in the information acquisition costs is essential for the optimality of symmetric entry.

For simplicity, let us consider the case with two potential bidders (N = 2). From Equations (1) and (2), which express the total expected surplus and expected revenue as functions of entry thresholds, the common component [[summation].sub.{k[member of]K}] Pr(k) [V.sub.k] of S([c.sup.esub.1], [c.sup.e.sub.2]) and R([c.sup.e.sub.1], [c.sup.e.sub.2]) can be written as:

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From Lemma 3, [W.sub.2] - [W.sub.1] < [W.sub.1] - [W.sub.0] as [W.sub.n + 1] - [W.sub.n] decreases with n. Thus, for a given summation [2.summation over (i=1)] G([c.sup.e.sub.i]), we want to maximize G([c.sup.e.sub.1]) - G([c.sup.e.sub.2]) to maximize Equation (8). Thus, any symmetric entry where the entry threshold belongs to ([c.bar],[bar.c]) can be asymmetrized to increase Equation (8) while keeping the sum of entry probabilities unchanged.

Suppose the restricted symmetric entry threshold that maximizes Equations (1) or (2) belongs to ([c.bar],[bar.c]) as in the example at the beginning of this Section IIIC. This entry can thus be asymmetrized to increase component (8) while keeping the sum of entry probabilities unchanged. If there is not much dispersion in information acquisition costs (i.e., [c.bar] - [bar.c] is rather small), doing so does not change much the other terms in Equations (1) and (2). Thus, creating asymmetry in entry thresholds leads to higher expected total surplus and seller's expected revenue. The above arguments can be generalized to the case where N > 2 by focusing on the entry probabilities of any two bidders while assuming that the entry probabilities of all other bidders are fixed.

The remaining questions are: Will enough dispersion is sufficient to coordinate bidders and induce a unique entry? If yes, how much dispersion is enough? Section IV will further address these issues.

D. Auction Design within Symmetric-Entry Class

"Symmetric" entry across bidders means that entry thresholds [c.sup.e.sub.i] are the same across all potential bidders. Though symmetric entry is generally restrictive for auction design as illustrated in Section IIIC, in many cases, the seller may not be allowed to discriminate against some bidders. Moreover, one may argue that symmetric entry can be a focal point of the entry game. For this reason, symmetric entry makes coordination easier among bidders and thus is a quite realistic assumption. Here, we thus focus on auctions within the symmetric-entry class. In Section IV, we further provide sufficient conditions to ensure that this restriction of symmetric entry leads to no loss of generality for auction design.

Suppose [c.sup.e.sub.i] = [c.sup.e] [member of] [[c.bar], [bar.c]], [for all] i [member of] N. We define [S.sub.s]([c.sup.e]) = S([C.sup.e]) and [R.sub.s]([c.sup.e]) = R([C.sup.e]), where [c.sup.e.sub.i] = [c.sub.e] in vector [C.sup.e], [for all]i [member of] N. Under this restriction, we have:

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Suppose that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] maximizes [S.sub.s]([c.sup.e]) and [c.sup.e[dagger]] maximizes [R.sub.s]([c.sup.e]). Define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [C.sup.e[dagger].sub.s] = ([c.sup.e[dagger]], ..., [c.sup.e[dagger]]). Then, we have the following characterizations for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [c.sup.e[dagger]] from Equations (9) and (10). For all i [member of] N,

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the expected payoff of bidder i with cost when he participates in auction [A.sub.0] if all other potential bidders participate according to threshold [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. [R.sub.i]([C.sup.e[dagger].sub.s]) is the expected payoff of bidder i with cost [c.sup.e[dagger]] when he participates in a second-price auction with a Time 2 ex ante entry fee of G([c.sup.e[dagger]])/g([c.sup.e[dagger]]) and a reserve price equal to [v.sub.0], if all other potential bidders participate according to threshold [c.sup.e[dagger]]. Thus, Equations (11) and (12) lead to the following proposition.

PROPOSITION 4. (i) The second-price auction [A.sub.0] with a reserve price equal to the seller's valuation and no entry fee is ex ante efficient among the symmetric-entry class; (ii) suppose [c.sup.e[dagger]] maximizes [R.sub.s]([c.sup.e]), then a second-price auction with a reserve price equal to the seller's valuation and a Time 2 ex ante entry fee, [E.sub.0], defined below, maximizes the seller's expected revenue among the symmetric-entry class. [E.sub.0] is defined as:

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. The results follow from Proposition 1. First, since g(x) > 0, clearly it is a Nash equilibrium that every bidder participates in auction [A.sub.0] according to threshold [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], as Equation (11) is satisfied for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Second, clearly, it is a Nash equilibrium that every bidder participates in the Proposition 4(ii) auction according to threshold [c.sup.e[dagger]], while Equation (12) is satisfied for [c.sup.e[dagger]]. The entry fee [E.sub.0] is set at the level to extract all the expected surplus of entry-threshold type in light of Proposition 1.

There are two remarks on optimal entry. First, clearly [c.sup.e[dagger]] is lower than [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] according to Proposition 4, as a positive entry fee [E.sub.0] must lead to less entry compared to a zero entry fee. In this sense, the revenue-maximizing auction induces less entry than the ex ante efficient auction. Second, from Equations (11) and (12), we have that the restricted maximum [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [C.sup.e[dagger].sub.s] may be locally efficient and revenue maximizing, respectively. In Section lV, we will provide sufficient conditions for them to be globally efficient and revenue maximizing, respectively.

IV. ENTRY COORDINATION

The performance of an auction crucially depends on the entry equilibrium induced. We have shown in Sections IIIA to IIIC that the desirable entry may be asymmetric and the efficient and revenue-maximizing auctions may induce multiple entry equilibria. Some of them are not efficient or revenue maximizing. In this section, we provide sufficient conditions that resolve this entry coordination problem. We will first establish sufficient conditions for the optimality of symmetric entry. This is the first step to cope with the entry coordination problem since symmetric entry makes coordination easier among bidders and thus can be a focal point of the entry game. Furthermore, as pointed out in Sections IIIA to IIIC, the efficient auction [A.sub.0] and revenue-maximizing auction may induce multiple entry equilibria. We will provide conditions for the proposed auctions in Section IIID to uniquely implement the desirable symmetric entry. These results justify the convention of looking at only the symmetric entry for the efficient or revenue-maximizing auction.

A. Unique Implementation of Optimal Entry

In Section IIIC, we have pointed out that symmetric entry is generally restrictive for auction design. We now provide sufficient conditions to guarantee that this restriction leads to no loss of generality. Recall that [W.sub.n], n [greater than or equal to] 0 denotes the expectation of the highest valuation of the seller and n([greater than or equal to] 0) bidders.

LEMMA 4. If G(c) - G(c')/c-c' [less than or equal to] 1/([W.sub.1] - [W.sub.0])-([W.sub.2]-[W.sub.1]), [for all] c, c' then there exists a unique entry equilibrium for a second-price auction with reserve equal to the seller's valuation and any uniform ex ante entry fees E. The unique entry equilibrium is symmetric.

Proof. See Appendix.

The condition G(c) - G(c')/c-c' [less than or equal to] 1/([W.sub.1]-[W.sub.0]) - ([W.sub.2]-[W.sub.1]), [for all]c, c' will be repeatedly used. It can be interpreted as that the information acquisition cost is more Bickel-Lehman disperse than a uniform distribution on [0, ([W.sub.1] - [W.sub.0]) ([W.sub.2] - [W.sub.1])]. (27) Let X and Y be two real-valued random variables with distributions H(*) and Z(x), respectively. X is said to be more Bickel-Lehman disperse than Y, if [for all] p, p' [member of] [0, 1], p > p', [H.sup.-1](p) - [H.sup.-1](p') [greater than or equal to] [Z.sup.-1](p) - [Z.sup.-1](p'), that is, the difference between any two quantiles of Z is smaller than the difference between the corresponding quantiles of H (Bickel and Lehman 1976). Let [xi](x) denote the cumulative distribution function of the uniform distribution on [0, ([W.sub.1] - [W.sub.0]) - ([W.sub.2] - [W.sub.1])]. Let p = G(c) and p' = G(c'). Condition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be equivalently written as [G.sup.-1](p) - [G.sup.-1](p') [greater than or equal to] [[xi].sup.-1](p) - [[xi].sup.-1](p'), [for all] 0 [less than or equal to] p' [less than or equal to] p [less than or equal to] 1.

Lemma 4 provides a sufficient condition to guarantee that a second-price auction with reserve equal to the seller's valuation and uniform ex ante entry fees must induce a unique entry, which is symmetric. The intuition behind this is as follows. Suppose bidders [i.sub.1] and [i.sub.2] have different entry thresholds with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Without loss of generality we assume [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are in ([c.bar], [bar.c]). If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are rather close then the difference between the expected payoffs of these two threshold types is mainly determined by the difference in their information acquisition costs since when they enter their winning probabilities are rather close. Therefore, when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is relatively small compared to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the difference between the expected payoffs of threshold types [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] must be negative, as it has to share the same sign with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. However, this conflicts with the fact that the expected payoffs of these two threshold types must be zero. (28)

The globally efficient entry is always implemented through [A.sub.0] according to Proposition 2. Lemma 4 implies that [A.sub.0] has a unique entry equilibrium, which is symmetric. We thus have the following result summarized in Proposition 5.

PROPOSITION 5. If G(c) - G(c')/c-c' [less than or equal to] 1/([W.sub.1] - [W.sub.0])-([W.sub.2]-[W.sub.1]), [for all] c, c' then (i) the efficient entry must be symmetric; (ii) the efficient entry is uniquely implemented by auction [A.sub.0].

Since G(x) belongs to [0, 1], the condition G(c) - G(c')/c-c' [less than or equal to] 1/([W.sub.1] - [W.sub.0])-([W.sub.2]-[W.sub.1]), [for all] c, c' can easily be satisfied. A sufficient condition for G(c) - G(c')/c-c' [less than or equal to] 1/([W.sub.1] - [W.sub.0])-([W.sub.2]-[W.sub.1]) is g(x) < 1/([W.sub.1] - [W.sub.0])-([W.sub.2]-[W.sub.1]). If the information acquisition costs follow a uniform distribution then as long as the range of the information acquisition costs is sufficiently big, there exists no asymmetric efficient entry equilibrium. In this sense, sufficient dispersion in information acquisition costs resolves the issue of asymmetric efficient entry equilibria. In the Section IIIC example where asymmetric entry emerges for auction [A.sub.0], the range of the information acquisition costs is rather small. As a result, the condition in Proposition 5 is violated.

The condition in Proposition 5 is not sufficient for a symmetric revenue-maximizing entry and its unique implementation. However, it can be strengthened by a monotone hazard rate property to guarantee the optimality of symmetric entry in terms of revenue maximization. (29) The following proposition presents this result.

PROPOSITION 6. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the hazard rate G(x)/g(x) weakly increases then (i) the revenue-maximizing entry must be symmetric; (ii) the revenue-maximizing entry is uniquely implemented through the Proposition 4(ii) auction.

Proof See Appendix.

The intuition behind Proposition 6 is the following. Suppose the revenue-maximizing entry [C.sup.e[dagger]] is asymmetric across bidders [i.sub.1] and [i.sub.2] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The corresponding revenue-maximizing auction is specified by Proposition 3. Without loss of generality, we assume [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are not on the boundaries. Then, the revenue-maximizing entry fees for these two bidders are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] respectively. When G(x)/g(x) weakly increases, threshold type of bidder [i.sub.1] faces a higher entry fee on top of the higher information acquisition costs. Based on similar arguments that follow Lemma 4, we have when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is relatively small compared to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the difference between the expected payoffs of threshold types [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] must be negative. However, this conflicts with the fact that the expected payoffs of these two threshold types must be zero. (30)

The conditions of Propositions 5 and 6 completely resolve the entry coordination problem. When these conditions hold, the desirable entries are symmetric and they are uniquely implemented by the proposed auctions. Both Propositions 5 and 6 require sufficient dispersion in information acquisition costs. Note that this is also necessary as illustrated by the analysis of Section IIIC.

B. Second-Price Auctions with a General Reserve Price and No Entry Fee

In previous sections, we have allowed ex ante entry fees for auction design and identified sufficient conditions for unique implementation of the desirable entry. Although the ex ante efficient auction does not really require ex ante entry fee, generally, the revenue-maximizing auction needs to employ it to extract the expected surplus of entrants. It remains an issue whether similar sufficient condition can be identified for the unique implementation of revenue-maximizing entry when ex ante entry fee is not allowed.

It is well expected that when no ex ante entry fee is allowed, the seller can employ reserve price to enhance revenue. However, for any second-price auction with a reserve price, generally, there exist multiple entry equilibria. For example, for second-price auction [A.sub.0] with a reserve price equal to the seller's valuation, the Section IIIC example has shown this point. This multiple entry equilibria issue clearly complicates the search for the revenue-maximizing reserve price, as different entry equilibria deliver different expected revenues. What makes it worse is that for a given reserve price, it is technically difficult to pin down all entry equilibria associated. In this section, we will show that the "sufficient dispersion" condition identified in Section IVA still applies to any second-price auction with a reserve price but no ex ante entry fee. Thus, this condition significantly simplifies the search for the revenue-maximizing reserve price.

Define [Y.sub.n](r), n [greater than or equal to] 0 as the expectation of the highest value among reserve price r and the valuations of n([greater than or equal to] 0) bidders. Note that we have [Y.sub.n]([v.sub.0]) = [W.sub.n]. Similar to Lemma 3, the following lemma provides useful properties of series [Y.sub.n](r), n [greater than or equal to] 0. The proof is omitted.

LEMMA 5. Both [Y.sub.n + 1](r) - [Y.sub.n](r) and ([Y.sub.n + 1] (r) - [Y.sub.n](r)) - ([Y.sub.n + 2](r) - [Y.sub.n + 1](r)) decrease with n([greater than or equal to] 0) and r [member of] [v.sub.0], [bar.v]]

The results of Lemma 5 are rather intuitive. From Lemma 2, [Y.sub.n](r) - [Y.sub.n - 1](r) is the expected payoff of a representative bidder with zero information acquisition cost when there are altogether n bidders in a second-price auction with reserve price r and no ex ante entry fee. (31) Lemma 5 says that this expected payoff decreases with the number of bidders n and reserve price r and the rate of decrease also decreases with n and reserve price r. Lemma 5 leads to the following results.

PROPOSITION 7. If G(c) - G(c')/c-c' [less than or equal to] 1/([W.sub.1] - [W.sub.0])-([W.sub.2]-[W.sub.1]), [for all] c, c' then there exists a unique entry equilibrium for a second-price auction with any reserve price r [member of] [[v.sub.0], [bar.v]] and no ex ante entry fee. The unique entry equilibrium is symmetric.

Proof. See Appendix.

Proposition 7 implies that when the condition G(c) - G(c')/c-c' [less than or equal to] 1/([W.sub.1] - [W.sub.0])-([W.sub.2]-[W.sub.1]), [for all] c, c' holds, the search for the revenue-maximizing reserve price is greatly simplified. Under this condition, the unique entry is characterized by a uniform entry threshold [c.sup.e] given by:

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [C.sup.n.sub.N] = n!(N - n)!/N!. (32) The expected revenue generated from this entry is: (33)

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The search for the revenue-maximizing reserve price is reduced to a program of maximizing Equation (15) subject to constraint (14). For this argument, we see that the sufficient condition identified in Proposition 7 greatly facilitates revenue-maximizing auction design without ex ante entry fee.

V. CONCLUDING REMARKS

Auction design with endogenous entry is complicated by an issue of entry coordination due to multiple entry equilibria problem. This article studies auction design with an emphasis on entry coordination in a setting where bidders' information-acquisition costs are their private information. We find that sufficient dispersion in the information acquisition cost resolves the entry coordination issue.

Unlike the case of fixed costs, bidders enjoy information rents, when information acquisition costs are their private information. Due to these information rent, revenue-maximizing entry diverges from the ex ante efficient one. The ex ante efficient entry is always implemented through a second-price auction with no entry fee and a reserve price equal to the seller's valuation, while the revenue-maximizing auction involves personalized ex ante entry fees for bidders. We find that the optimal ex ante entry fees for bidders are given by the hazard rates of the information acquisition cost distribution, evaluated at the optimal entry thresholds of the bidders. Our analysis is carried out in a private value framework. These results are not extendable to a common value setting due to the implied divergence between the social benefit and investor's individual benefit from his investment in information acquisition.

While both the ex ante efficient entry and revenue-maximizing entry can be asymmetric rather than symmetric, we find that sufficient dispersion in the information acquisition costs guarantees the optimality of symmetric entry. Specifically, we show that when the information acquisition cost is more disperse than a particular uniform distribution by the Bickel-Lehman dispersive order, the efficient entry must be symmetric across bidders. If further the hazard rate of the information acquisition cost distribution is increasing then the revenue-maximizing entry must also be symmetric. These conditions also guarantee that these optimal entries are uniquely implemented by the proposed auctions. These results mean that sufficient dispersion in information acquisition costs can coordinate bidders and implement uniquely the desirable entry.

Given that the sufficient dispersion condition holds, we further find that any second-price auction with a general reserve price and no entry fee must induce a unique entry equilibrium, which is symmetric. This result facilitates the search for the revenue-maximizing reserve price in a second-price auction framework when ex ante entry fee is not allowed.

This article focuses on simultaneous entry while it is well recognized in the literature that allowing sequential entry improves auction design when information acquisition costs exist. Ye (2007) and Cremer, Spiegel, and Zheng (2009) among others have explored this direction while assuming information acquisition costs are fixed. It is in our future research agenda to investigate the impact of private information acquisition costs on auction design while allowing sequential entry.

ABBREVIATION

IPV: Independent Private Value

doi: 10.1111/j.1465-7295.2009.00216.x

APPENDIX

Proof of Lemma 1

Let us consider any simultaneous entry equilibrium [epsilon] implemented by an auction rule. If all bidders other than i adopt the equilibrium entry strategy in e, the bidder i's equilibrium entry strategy in e must be his best entry strategy. Given all bidders other than i adopt the equilibrium entry strategy in e, there must exist an entry threshold [c.sup.e.sub.i] [member of] [[c.bar], [bar.c]] such that bidder i's best entry strategy is described by property (ii) in Lemma 1. This is true because the expected payoff of bidder i from participating in any given auction decreases strictly with his information acquisition cost, given all bidders other than i adopt the equilibrium entry strategy in [epsilon].

Proof of Proposition 1

Let us first consider auction [A.sub.0]. Suppose all bidders other than i participate in auction [A.sub.0] according to thresholds [C.sup.e.sub.-i], = ([c.sup.e.sub.1], ..., [c.sup.e.sub.-i], [c.sup.e.sub.i+1], ..., [c.sup.e.sub.N]). Denote bidder i's expected payoff by [[pi].sub.i]([c.sup.e.sub.i]; [C.sup.e.sub.-i]) if he participates in [A.sub.0] while his information acquisition cost is [c.sup.e.sub.i]. Set a Time 2 ex ante entry fee (or subsidy) for bidder i as [E.sub.i] = [[pi].sub.i]([c.sup.e.sub.i]; [C.sup.e.sub.-i]), [for all]i [member of] N. Clearly, for a second-price auction with ex ante entry fee (or subsidy) [E.sub.i] for bidder i and a reserve price equal to the seller's valuation, bidder i's expected payoff is [c.sup.e.sub.i] - [c.sub.i] if he participates and his information acquisition cost is [c.sub.i]. Hence, the above auction with ex ante entry fee [E.sub.i] for bidder i implements entry thresholds [C.sup.e]. Note that for any auction implementing participation thresholds [C.sup.e], the total expected information acquisition costs are the same. Thus, the auction designed above achieves the highest attainable expected total surplus among the class of auctions implementing [C.sup.e], as the auction always awards the item to the participant (including the seller) with the highest valuation.

Moreover, for any auction implementing entry thresholds [C.sup.e], the expected payoff of bidder i with information acquisition cost [c.sub.i] [less than or equal to] [c.sup.e.sub.i] cannot be smaller than [c.sup.e.sub.i] - [c.sub.i]. This is due to the fact that a type [c.sub.i] of bidder i can always adopt the same bidding strategy of a type [c.sup.e.sub.i], and by doing so he gets at least a payoff of [c.sup.e.sub.i] - [c.sub.i]. Recall that in a second-price auction with ex ante entry fee [E.sub.i] for bidder i and a reserve price equal to the seller's valuation, bidder i's expected surplus is exactly [c.sup.e.sub.i] - [c.sub.i] if his information acquisition cost is [c.sub.i]. As a result, this auction achieves the highest attainable seller's expected revenue among all auctions implementing any given entry threshold vector [C.sup.e].

Proof of Lemma 3

We use [H.sub.n](x) to denote the cumulative distribution function of the highest valuation of the seller and n([greater than or equal to] 0) symmetric bidders. Then, [H.sub.n](x) = [F.sup.n](x) on its support [[v.sub.0], [bar.v]], [for all]n [greater than or equal to] 1. [H.sub.n](x) may have a mass point at [v.sub.0]. It follows [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This leads to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, we have both [W.sub.n + t] - [W.sub.n] and ([W.sub.n + 1] - [W.sub.n]) - ([W.sub.n + 2] [W.sub.n + 1]) decrease with n([greater than or equal to] 0).

Proof of Lemma 4

We prove the proposition using contradiction. Note that a symmetric entry equilibrium always exists for this auction. Suppose that there is another asymmetric entry equilibrium [C.sup.e]. Then, we can find [i.sub.1], [i.sub.2] [member of] N such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in [C.sup.e]. We use Pr(n), n = 0, 1, ..., N - 2 to denote the probabilities of which there are n bidders from the other N - 2 bidders participating in the auction. According to Lemma 2, [W.sub.n + 1] - [W.sub.n] is the expected payoff of a bidder from participating in auction [A.sub.0], if his information acquisition cost is zero and there are n other participants.

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be corner solutions, we must have that:

(A1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(A2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where E is the ex ante entry fee. These two conditions lead to:

(A3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From Lemma 3, we have

[[summation].sup.N-2.sub.n-0] Pr(n)[([W.sub.n+1] - [W.sub.n]) - ([W.sub.n+2] - [W.sub.n+1])] < ([W.sub.1] - [W.sub.0]) -([W.sub.2] - [W.sub.1]). As G(c) - G(c')/c-c [less than or equal to] 1/([W.sub.1] - [W.sub.0]) - ([W.sub.2] - [W.sub.1], [for all] c, c', we thus have the left-hand side of Equation (A3) must be smaller than [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This contradicts Equation (A3). Therefore, no asymmetric entry equilibrium exists.

Proof of Proposition 6

We prove the proposition using contradiction. Suppose that the revenue-maximizing entry [C.sup.e] is asymmetric. Then, we can find [i.sub.1], [i.sub.2] [member of] N such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in [C.sub.e]. We use Pr(n), n = 0, 1, ..., N - 2 to denote the probabilities of which there are n bidders from the other N - 2 bidders participating in the auction. According to Lemma 2, [W.sub.n + 1] - [W.sub.n] is the expected payoff of a bidder from participating in auction [A.sub.0], if his information acquisition cost is zero and there are n other participants.

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be corner solutions, Proposition 3 gives:

(A4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(A5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

These two conditions lead to:

(A6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From Lemma 3, we have [[summation].sup.N-2.sub.n=0]Pr(n)[([W.sub.n+1] - [W.sub.n]) - ([W.sub.n+2] - [W.sub.n+1])] < ([W.sub.1] - [W.sub.0]) - ([W.sub.2] - [W.sub.1]). As G(c) - G(c')/c - c' [less than or equal to] 1/([W.sub.1] - [W.sub.0]) - ([W.sub.2] - [W.sub.1]), [for all]c, c' we thus have the left-hand side of Equation (A6) must be smaller than [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This contradicts Equation (A6). Therefore, the revenue-maximizing entry must be symmetric.

According to Proposition 4(ii), this symmetric revenue-maximizing entry is implemented through the Proposition 4(ii) auction. From Lemma 4, the Proposition 4(ii) auction has a unique entry equilibrium. We thus have the result in Proposition 6(ii).

Proof of Proposition 7

The proof is similar to that of Lemma 4 while setting entry fee E = 0 and replacing [W.sub.n], by [Y.sub.n](r). Applying Lemma 2 while treating r as the seller's valuation, we have [Y.sub.n + 1] (r) - [Y.sub.n](r) is the expected payoff of a bidder from participating in the second-price auction with reserve price r and no entry fee, if his information acquisition cost is zero and there are n other participants. The properties in Lemma 5 are applied.

REFERENCES

Bag, P. K. "Optimal Auction Design and R&D." European Economic Review, 41, 1997, 1655-74.

Bergemann, D., and J. Valimaki. "Information Acquisition and Efficient Mechanism Design." Econometrica, 70, 2002, 1007-33.

--. "Information in Mechanism Design." Cowles Foundation Discussion Paper No. 1532, 2005.

Bickel, P. J., and E. L. Lehman. "Descriptive Statistics for Nonparametric Models, III, Dispersion." Annals" of Statistics, 4, 1976, 1139-58.

Cremer, J., Y. Spiegel, and C. Zheng. "Auctions with Costly Information Acquisition." Economic Theory, 38 (1) 2009, 41-72.

Dasgupta, S. "Competition for Procurement Contracts and Underinvestment." International Economic Review, 31, 1990, 841-65.

Engelbrecht-Wiggans, R. "On Optimal Reserve Prices in Auctions." Management Science, 33, 1987, 763-70.

--. "Optimal Auction Revisited.'" Games and Economic Behavior, 5, 1993, 227-39.

Harstad, R. "Alternative Common-Value Auction Procedures: Revenue Comparisons with Free Entry." Journal of Political Economy, 98, 1990, 421-29.

Levin, D., and J. L. Smith. "Equilibrium in Auctions with Entry." American Economic Review, 84, 1994, 585-99.

Lu, J. "Auction Design with Opportunity Cost." Economic Theory, 38 (1) 2009, 73-103.

McAfee, R. P., and J. McMillan. "Auctions with Entry." Economics Letters, 23, 1987, 343-47.

McAfee, R. P., D. Quan, and D. Vincent. "How to Set Minimum Acceptable Bids, with an Application to Real Estate Auctions." Journal of Industrial Economics, L(4), 2002, 391-416.

Milgrom, P. "Rational Expectations, Information Acquisition, and Competitive Bidding." Econometrica, 49, 1981, 921-43.

Myerson, R. B. "Optimal Auction Design." Mathematics of Operation Research, 6, 1981, 58-73.

Persico, N. "Information Acquisition in Auctions." Econometrica, 68, 2000, 135-48.

Piccione, M., and Tan, G. "A Simple Model of Expert and Non-Expert Bidding in First-Price Auctions." Journal of Economic Theory, 70, 1996, 501-15.

Samuelson, W. F. "Competitive Bidding with Entry Costs." Economics Letters, 17, 1985, 53-57.

Stegeman, M. "Participation Costs and Efficient Auctions." Journal of Economic Theory, 71, 1996, 228-59.

Tan, G. "Entry and R&D in Procurement Contracting" Journal of Eeonomic Theory, 58, 1992, 41-60.

Ye, L. "Optimal Auctions with Endogenous Entry" Contributions to Theoretical Economics, 4, 2004, Article 8.

--. "Indicative Bidding and a Theory of Two-Stage Auctions." Games and Economic Behavior, 2007, 58, 181-207.

(1.) In the case of the U.S. timber auctions, bidders engage in prebid activities such as acquiring and analyzing information. Other examples include construction procurements, and so forth.

(2.) Here, the performance of an auction may refer to efficiency or revenue maximization.

(3.) When the marginal entry cost is sufficiently high and does not depend on entry probability, it is always incentive compatible in the second-price auction of Levin and Smith (1994) that a subset of bidders enter with a high probability (e.g., 1) and the others enter with a lower probability (e.g., 0). The constant marginal cost plays an essential role. On one hand, it prevents the less active bidders from increasing their participation since their marginal cost is not low: on the other hand, it does not provide enough incentive for the more active bidder to enter less since their marginal entry cost is not high.

(4.) Private information acquisition cost also features the Outer Continental Shelf wildcat auctions as pointed out by Piccione and Tan (1996).

(5.) Please refer to Bergemann and Valimaki (2005) for a thorough review of the literature. Information acquisition costs refer to the costs for bidders to discover their valuations. Many other studies focus on entry costs that are incurred by bidders who know their valuations. These studies include Samuelson (1985), Stegeman (1996), and Lu (2009) among others. In this later setting, the optimal reserve generally is not equal to the seller's value. This leads to a divergence between the revenue and the total surplus.

(6.) Following the majority of literature on auctions with entry, we focus on simultaneous entry in this article.

(7.) The entry fees (subsidies) make the entry threshold types indifferent between participating and not participating. Note that these entry fees (subsidies) do not affect the expected total surplus as they are transfers between the seller and the bidders.

(8.) This auction may not uniquely induce the given entry pattern. At this stage, the entry coordination problem is temporarily ignored.

(9.) When ex ante entry fees are not allowed then a more sophisticated reserve is in demand.

(10.) It is clear that the threshold type's contribution to the total efficiency must equal the corresponding hazard rate for an inner revenue-maximizing threshold.

(11.) Note that the Levin and Smith (1994) setting is a degenerate case of our setup where the range of the distribution is zero. Intuitively, if the range the private information cost is sufficiently small then quite likely multiple entry equilibria exist in our setup as in Levin and Smith (1994).

(12.) The setting with fixed acquisition cost is a degenerate case with zero dispersion. It cannot satisfy the sufficient dispersion condition we identified.

(13.) Dasgupta (1990) finds that if the principal is unable to precommit to a mechanism then an underinvestment outcome results.

(14.) This assumption is widely adopted in the literature, such as McAfee and McMillan (1987), Engelbrecht-Wiggans (1987, 1993), Levin and Smith (1994), and McAfee, Quan, and Vincent (2002). However, this assumption precludes the possibility that a bidder may simply submit a bid equal to the unconditional expected value. When [Ev.sub.i] < [v.sub.0], the assumption is innocent as the buyers who do not know their values prefer not to bid in the auctions that will be presented.

(15.) If the seller sets a Time 2 entry fee then a bidder must pay the entry fee to the seller to enter. Hereafter, an entry fee refers to a Time 2 entry fee.

(16.) Every participant may or may not observe the other participants. The auctions designed later work in both cases.

(17.) The proposed auctions may not induce only the desirable entry. Section IV addresses this issue.

(18.) It is well recognized in the literature that allowing sequential entry improves auction design when information acquisition costs exist. Ye (2007) and Cremer, Spiegel, and Zheng (2009) among others have explored this direction while assuming information acquisition costs are fixed. It is in our future research agenda to investigate toe impact of private information acquisition costs on auction design while allowing sequential entry.

(19.) This simplification is reasonable because if [c.sup.e.sub.i] > [c.bar] then bidder i with cost [c.sup.e.sub.i] at least weakly prefers participation and if [c.sup.e.sub.i] = [c.bar] then bidder i with cost [c.sup.e.sub.i] at least weakly prefers nonparticipation. Moreover, this simplification only further specifies the participation of the threshold type [c.sup.e.sub.i] of bidder i. The expected total surplus and seller's expected revenue are not affected.

(20.) Note that in the setting of Myerson (1981) that does not involve endogenous entry, ex ante entry fees are impossible as buyers are endowed with private information.

(21.) These two expressions can be easily extended to the case with asymmetric bidders. Similar analysis then follows.

(22.) Please refer to their arguments on page 593.

(23.) The case where ex ante entry fee is not allowed will be discussed in Section IVB within a framework of second-price auctions with a general reserve price.

(24.) I thank Jeff Ely and an anonymous referee for pointing out this issue.

(25.) Consider a case with two buyers. Given one adopts a lower entry threshold, the equilibrium entry threshold of the other has to be higher if the entry fee for him is sufficiently close lo zero.

(26.) Section IV will provide conditions for optimality of symmetric entry.

(27.) I am grateful to an anonymous referee for his insight on this point.

(28.) Both threshold types enjoy zero-expected payoff when [c.sup.e.sub.i1] and [c.sup.e.sub.i2] are in ([c.bar], [bar.c]). Similar arguments apply when entry thresholds are allowed to be at the boundaries.

(29.) Monotone hazard rate property is well adopted in the literature. It is satisfied by a wide class of distributions. For example, the uniform distributions mentioned above satisfy this property.

(30.) If entry thresholds [c.sup.e[dagger].sub.i1] and [c.sup.e[dagger].sub.i2] are allowed to be at the boundaries, similar arguments can be made.

(31.) We can simply treat r as the seller's valuation and apply Lemma 2.

(32.) Without loss of generality, we assumed that the entry threshold is an inner solution.

(33.) The proof is available from the author upon request. Readers can also refer to Proposition 1 of Engelbrecht-Wiggans (1993).

JINGFENG LU *

* I am very grateful to Preston McAfee and two anonymous referees for insightful comments and suggestions, which greatly improved the quality of this article. I thank Murali Agastya, Parimal Bag, Indranil Chakraborty, Jeff Ely, Daniel Friedman, Atsushi Kajii, Chenghu Ma, Steven Morris, and Ruqu Wang for helpful discussions, comments, and suggestions. Previous versions have been presented at the 17th International Game Theory Conference at Stony Brook, the 2006 Hong Kong Economic Association Meeting and the 2007 NUS economic theory conference. Ali errors are mine. Financial support from National University of Singapore (R-122-000-106-112) is gratefully acknowledged.

Lu: Assistant Professor, Department of Economics, National University of Singapore, Singapore 117570. Tel (65) 6516-6026, Fax (65) 6775-2646, E-mail [email protected]