Auction Design with Opportunity Cost - CiteSeerX

Auction Design with Opportunity Cost∗

Jingfeng Lu Department of Economics, University of Southern California, CA, USA January 2004

∗ Address: Department of Economics, University of Southern California, Los Angeles, CA 90089-0253. Tel : (213) 740-2114. Fax : (213) 740-8543. Email: [email protected].

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Abstract This paper considers the optimal auction design when potential bidders have the same known positive opportunity cost of bidding. I show that a modified Vickrey auction with a uniform reserve price and a uniform participation subsidy is the optimal symmetric-shutdown mechanism under the usual regularity conditions. When this mechanism is adopted, I find that the seller’s expected revenue decreases as the number of the potential bidders increases, if the bidders’ private values are heavily distributed near the highest value. Thus, in those cases the overall optimal auction mechanism must be discriminatory, in the sense of implementing asymmetric shutdown across bidders. JEL classifications: D44; D82 Keyword: Auction design; Endogenous participation; Opportunity cost

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1

Introduction

Opportunity costs are ubiquitous in an economic world. For example, it is costly and timeconsuming for potential bidders to travel to the auction site and stay for the duration of the auction. The bidders, rather, can invest those resources in other opportunities. Opportunity cost leads to a different auction design problem in which the bidders have a positive instead of a zero reservation utility adopted by Myerson [10]. To induce participation, an auction mechanism must give the participants at least this positive reservation utility. Myerson [10] characterizes the optimal auction mechanism with the bidders’ reservation utility set at zero. When the bidders are symmetric, the optimal auction is a modified Vickrey auction with a reserve price. In this mechanism there is no shutdown of lower private value types and the reserve price is independent of the number of the potential bidders. Here and hereafter, “shutdown” refers to the outcome that the bidder prefers not to participate in the auction given an auction mechanism. These properties turn out to be the feature of the optimal auction mechanism under the assumption of zero reservation utility for bidders. In my paper, due to the positive reservation utility of the bidders, the optimal mechanism potentially involves shutdown of the bidders with lower private values. Since participation can be uncertain, the procedure used in Myerson [10] to derive the optimal mechanism needs to be modified to accommodate for the shutdown of the bidders with lower private values. It will be shown that the problem cannot be solved in a zero reservation utility setting by simply renormalizing the bidders’ values by their reservation utility. Jehiel, Moldovanu and Staccheti [4] and Brocas [1] consider the optimal auction design 3

when bidders have a negative contingent opportunity cost. Jehiel, Moldovanu and Staccheti [4] examine the case in which the auctioneer has the ability to commit giving the object to a third party if no participant appears. They show that such a credible threat will induce all potential bidders to participate, and therefore a second-price auction with an entry fee is the optimal auction. Brocas [1] considers the case in which the auctioneer has no ability to commit giving the object to a third party if no participant appears. Brocas [1] shows that an optimal auction will not induce all potential bidders to participate. In other words, some lower types are shut down at the optimum. There is a connection between the auction designs with opportunity cost and participation cost. In fact, both opportunity cost and participation cost lead to a mathematically equivalent auction design problem. There have been some papers considering the effect of endogenous entry due to the participation cost on the bidding behavior of bidders and the auction design. These papers can be divided into two groups based on whether the bidders observe their private values before or after their participation decisions. The first group is formed by McAfee and McMillan [7], Engelbrecht-Wiggans [2, 3], Tan [13] and Levin and Smith [5] who study the endogenous entry when the bidders learn their valuations for the item for sale after paying the entry costs. The second group is formed by Samuelson [11], Stegeman [12] and Menezes and Monteiro [9] who consider endogenous entry when the potential bidders learn their private values before their entry decisions. This setting is mathematically equivalent to the one considered in my paper. Menezes and Monteiro [9] provide a detailed survey on this literature. Here I only highlight the results related to my study. Samuelson [11] shows that excluding some buyers ex ante may

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improve the efficiency of the first-price sealed-bid auction. Stegeman [12] also concentrates on analyzing the efficient mechanism. He shows that the efficient equilibrium may be asymmetric even if the bidders are symmetric. Menezes and Monteiro [9] establish the equilibrium purestrategy entry decisions and bidding strategies for both first and second price sealed-bid auctions without reserve price or entry fees.1 At those equilibria only the bidders with private values higher than a critical value participate. Menezes and Monteiro [9] also study the optimal auction with participation cost that maximizes the sellers’ expected revenue. Their conclusion is that the optimal auction is a modified Vickrey auction with an optimal reserve price and an entry subsidy equal to the participation cost. I will show in this paper that the mechanism proposed by Menezes and Monteiro [9] is actually a “restricted” optimum among the “symmetric-shutdown” class. This paper first provides concrete proof to show the symmetric-shutdown optimal auction in a more general setting than that of Menezes and Monteiro [9]. Specifically, my setting allows the auctioneer’s value to be different from zero. However, I reconsider this problem due to the following additional reasons. First, the mechanism in Menezes and Monteiro [9] requires all potential bidders to reveal their true types irrelevant of their participation decisions, and the allocation rules are based on the signals from all potential bidders. Thus they need to divide the auction into two stages and adopt a strong assumption that there is no participation cost at the first stage of the auction when the bidders are required to submit their bids. This is somewhat 1

Tan and Yilankaya [14] study asymmetric equilibrium entry across bidders in a second-price auction setting

when entry costs exist.

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restrictive in the sense that these mechanisms rule out the possibility that bid submitting may be costly in itself. This drawback is due to the fact that Menezes and Monteiro [9] rely on the traditional revelation principle, which essentially requires a prior stage at which agents send free messages to a prior coordinator. When one consider endogenous entry due to opportunity cost, the opportunity cost occurs if and only if the bidders submit bids to participate in the auction in light of the essence of the opportunity cost. In order to catch the essence of the shutdown policy of the seller and the endogenous entry of the potential bidders, based on a semirevelation principle in Stegeman [12, Lemma 1] my mechanism requires that only the participating bidders reveal their types, and the potential bidders who do not participate in the auction are not required to submit signals. Second, in the Menezes and Monteiro [9] mechanism the winning probability of a participant does not depend on the participation of the other bidders. This restriction does not catch the essence of shutdown policy, and thus their proof based on this restriction is not convincing. Third, this paper addresses explicitly whether allowing stochastic participation can lead to a better mechanism in the two potential bidders case. Based on the optimal symmetric-shutdown mechanism obtained, I studied its properties. Menezes and Monteiro [9] show that in both first-price and second-price sealed-bid auctions without reserve price or entry fees, the increase in the number of potential bidders may decrease the seller’s expected revenue. This paper will demonstrate that the same result holds even when the seller adopts the optimal symmetric-shutdown auction mechanism, i.e., a modified Vickrey auction with an optimal reserve price and an entry subsidy equal to the participation cost, if the bidders’ private values are distributed near the highest value. The following example shows that

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the seller prefers 1 potential bidder rather than 2 potential bidders when the optimal symmetricshutdown mechanisms are adopted. Suppose that the seller’s value of the object is 0, potential bidders’ opportunity cost is 0.5, and the potential bidders’ values have a cumulative distribution function of F (t) = t10 on [0, 1]. In the 1 potential bidder case, a modified Vickrey auction with both a reserve price of 0.85 and a participation subsidy of 0.5 is the optimal symmetric-shutdown mechanism. This mechanism provides the seller an expected revenue of 0.28. In the 2 potential bidders case, a modified Vickrey auction with a reserve price of 0.95 and a participation subsidy of 0.5 is the optimal symmetric-shutdown mechanism. The latter mechanism provides the seller an expected revenue of 0.21, which is smaller than 0.28. The implication of the above result is two-fold. On one hand, the result means that if the auctioneer cannot discriminate the bidders, then he wants to limit the number of the potential bidders in many situations. On the other hand, the result also indicates that the overall optimal auction mechanism must be discriminatory in many cases, in the sense of implementing asymmetric shutdown across symmetric potential bidders. Thus asymmetry among the bidders is not a necessary condition for obtaining a discriminatory optimal mechanism. The optimality of the discriminatory mechanism from the view of the auctioneer is parallel to that in Stegeman [12] in terms of the efficiency. One policy implication of the above result is that the seller does not necessarily engage in limiting the number of the potential bidders. What the seller needs to do is to discriminate the ex ante symmetric bidders by shutting down the potential bidders asymmetrically which can be implemented through adopting different reserve prices across bidders. This point is illustrated by an example in the following example. There are 2 potential

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bidders with opportunity cost of 0.2, and the potential bidders’ values have a cumulative distribution function of F (t) =

t−0.6 0.4

on [0.6, 1.0]. The seller’s value of the object is 0. A second

price auction with a uniform reserve price of 0.76 and a participation subsidy of U0 = 0.2 is the optimal symmetric-shutdown mechanism for the two potential bidders case, which gives the auctioneer an expected revenue of 0.427. When the seller shuts down one potential bidder, he gets the optimal expected revenue of 0.40 by setting a reserve price of 0.60 for the remaining bidder. However, reserve prices 0.66 and 0.86 respectively for bidder 1 and bidder 2 provide the auctioneer the best expected revenue of 0.431 when considering a second price auction with a participation subsidy of U0 = 0.2. Note that these reserve prices shut down bidder 1 and bidder 2 respectively at 0.66 and 0.86. This example shows that at the optimum the auctioneer may want to shut down bidders asymmetrically, even if his expected revenue increases wrt. the number of potential bidders when the optimal symmetric-shutdown mechanisms are adopted. In section 2, I set up the model and derive the optimal symmetric-shutdown auction mechanism with positive reservation utility for bidders. In section 3, I show the properties of the derived optimal symmetric-shutdown auction mechanism and discuss the implications for the optimal number of potential bidders as well as the optimal auction design. The properties of the optimal symmetric-shutdown auction are illustrated by numerical examples.

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The Optimal Symmetric-Shutdown Auction

Opportunity cost leads to a different auction design problem, in which the bidders have a positive instead of a zero reservation utility. In this section I solve for the optimal symmetric-shutdown 8

auction mechanism when the potential bidders have a strictly positive reservation utility. Following Stegeman [12], I define participating in the auction by submitting a bid. Due to the positive reservation utility, the optimal mechanism potentially involves shutdown of a portion of the bidders with lower private values. An auction mechanism implementing shutdown policy essentially cannot require the bidders who are shut down to submit bids as they do not participate in the auction. The bidders who do not participate reveal that they are the lower types by not submitting signals. In this paper, I consider the mechanisms based on the signals submitted by the participating bidders. I assume that the bidders not participating have no chance to win the object and their payments to the auctioneer are set equal to zero. This assumption is consistent with the no passive reassignment (NPR) assumption adopted by Stegeman [12]. I solve the optimal symmetric-shutdown auction mechanism problem by the following three steps. First, I set up the model with symmetric bidders. Second, I solve the optimal mechanism for any given uniform cutoff point tc . As I study the symmetric-shutdown mechanism, I adopt the same cutoff point tc for every bidder who is assumed to be symmetric ex ante. Third, I determine the optimal cutoff point t∗c . The optimal symmetric-shutdown mechanism corresponding to the optimal cutoff point is then the overall optimal symmetric-shutdown mechanism.

2.1

The Model

There is one auctioneer who wants to sell one indivisible object to N potential bidders through an auction, N is assumed to be common knowledge. The seller’s private value for the object

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is t0 which is public information. The ith bidder’s private value of the object is ti which is his private information. These values ti , i = 1, 2, ...N are i.i.d. distributed on interval [t, t] following cumulative distribution function F (·) with density function f (·). It is assumed that F (·) satisfies the regularity condition in Myerson [10], i.e., function J(t) = t − (1 − F (t))/f (t) increases wrt. t on interval [t, t]. The density f (·) is assumed to be common knowledge. All the potential bidders have a strictly positive reservation utility U0 which is also common knowledge. Every bidder observes his private value before his participation decision. The auctioneer and the bidders are assumed to be risk neutral. Specifically, the timing of the auction is the following. Time 0: The number of potential bidders N , the reservation utility of the bidders U0 and the auctioneer’s private value t0 are revealed by nature as common knowledge. Every bidder i observes his private value ti , i = 1, ..., N. Time 1: The auctioneer announces the rule of the auction. Time 2: The bidders simultaneously and confidentially make the participation decisions and announce their bids if they decide to participate. Time 3: The payoffs of the auctioneer and all the participating bidders are realized according to the announced rule at time 1. Stegeman [12, Lemma 1] provides a “semirevelation” principle for the case allowing no participation, which justifies that I need only to look at the truthful direct revelation mechanism, which requires the bidder reveals his true type if and only if he participates. Based on this semirevelation principle, there is no loss of generality to consider only truthful direct revelation

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mechanisms for the optimal mechanism. At time 1, the seller publicly announces a truthful direct revelation auction mechanism allowing no participation with the following form: if i1 th, i2 th, ..., ik th bidders participate, then for each ik0 ∈ {i1 , i2 , ..., ik }, the probability that he wins the object is pik0 {i1 ,i2 ,...,ik } (ti1 , ti2 , ..., tik ), and the his payment is xik0 {i1 ,i2 ,...,ik } (ti1 , ti2 , ..., tik ) upon the signals submitted by the participants (ti1 , ti2 , ..., tik ), with 1 ≤ k ≤ N and 1 ≤ i1 < i2 < ... < ik ≤ N . Hereafter, I use the following notations. (ti1 , ti2 , ..., tik ) by tIk ,

Q

i∈Ik

f (ti ) by f (tIk ),

Denote {1, ..., N } by N , {i1 , i2 , ..., ik } by Ik , Q

i∈Ik

dti by dtIk , pik0 {i1 ,i2 ,...,ik } (ti1 , ti2 , ..., tik ) by

pik0 Ik (tIk ), and xik0 {i1 ,i2 ,...,ik } (ti1 , ti2 , ..., tik ) by xik0 Ik (tIk ). This mechanism is denoted by (p, x). I say (p, x) is a feasible auction mechanism which achieves shutdown at tc ∈ [t, t] if (p, x) satisfies:2 (1) The bidders with private values lower than tc do not participate, i.e., if they participate they get utility equal to or lower than their reservation utility. Thus these bidders do not submit signals; (2) The bidders with private values equal to or higher than tc participate and reveal their true types. (3) For ∀k ∈ N , ∀Ik ⊂ N , pIikk0 (tIk ) ≥ 0, ∀ik0 ∈ Ik , with

Pk

Ik k 0 =1 pik0 (tIk )

≤ 1.

For any feasible truthful direct revelation mechanism (p, x) implementing shutdown policy 2

I first focus on the cutoff shutdown mechanism described below. I will show later in Proposition 3 that this

is not restrictive for deriving the optimal mechanism implementing symmetric participation at least in the two potential bidders case.

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at given tc ∈ [t, t], the seller’s expected revenue is given by: N

R0 (p, x) = t0 F (tc ) +

N X

(

F (tc )

)  Z t k k X X Ik Ik t0 (1 − ... pik0 (tIk ))+ xik0 (tIk ) f (tIk )dtIk . (1) ∀Ik ⊂N | tc {z tc} k0 =1 k0 =1

N −k

k=1

X Z

t

k

For bidder i with private value ti , if he submits signal t0i , then his interim expected utility is given by: 0

Ui (p, x, ti , ti )

= (ti pi

{i}

0

(ti ) − xi

{i}

0

(ti ))F (tc )

N −1

+

N −1 X

(

k=1

Z |

t tc

... {z k

Z

X

F (tc )N −k−1

∀Ik ⊂N \{i}

t

{ti pi

Ik ∪{i}

(tIk , t0i )

− xi

Ik ∪{i}

(tIk , t0i )}f (tIk )dtIk

)

,

(2)

tc

}

where pi Ik ∪{i} (tIk , t0i ) and xi Ik ∪{i} (tIk , t0i ) are respectively bidder i’s winning probability and payment if he submits signal t0i and any other participant ik ∈ Ik submits signal tik .

The seller’s optimization problem is to find the optimal feasible truthful direct revelation mechanism implementing shutdown policy at given tc ∈ [t, t], i.e., max R0 (p, x)

(3)

Subject to: (i) Ui (p, x, ti , ti ) ≥ U0 ; ∀i, ∀ti ∈ [tc , t],

(4)

(ii) Ui (p, x, ti , ti ) ≥ Ui (p, x, ti , ti 0 ); ∀i, ∀ti ∈ [tc , t], ti 0 ∈ [tc , t],

(5)

(iii) Ui (p, x, ti , ti ) ≥ Ui (p, x, ti , ti 0 ); ∀i, ∀ti ∈ [tc , t], ti 0 ∈ [t, tc ),

(6)

(iv) Ui (p, x, ti , ti 0 ) ≤ U0 ; ∀i, ∀ti < tc , ∀ti 0 ∈ [t, t],

(7)

(v) pIikk0 (tIk ) ≥ 0, ∀ik0 ∈ Ik ,

k X

pIikk0 (tIk ) ≤ 1, ∀k ∈ N , ∀Ik ⊂ N .

(8)

k0 =1

Compared to Myerson [10], (7) is a different constraint which guarantees that low-type bidders do not participate; (5) and (6) guarantee that high-type bidders reveal their true types; 12

while (4) guarantees that higher-type bidders do participate. A mechanism (p, x) is a feasible truthful direct revelation mechanism implementing shutdown at tc , if and only if (4), (5), (6), (7) and (8) are satisfied.

2.2

The Optimal Auction Implementing Symmetric-Shutdown at tc

In this part I derive the optimal feasible truthful direct revelation mechanism implementing shutdown at tc . For any direct revelation mechanism (p, x), I define Qi (p, ti ) = pi

{i}

(ti )F (tc )

N −1

+

N −1 n X

F (tc )

X

N −k−1

Z

t

∀Ik ⊂N \{i} | tc

k=1

... {z k

Z

t

o pi Ik ∪{i} (tIk , ti )f (tIk )dtIk . (9)

tc

}

If (p, x) is a feasible truthful direct revelation mechanism implementing shutdown at tc , then Qi (p, ti ) is the conditional expected probability that bidder i wins the object if his private value is ti ≥ tc . The following Lemma which is parallel to Myerson [10, Lemma 2], gives the necessary and sufficient conditions for a direct revelation mechanism to be a feasible truthful direct revelation mechanism implementing shutdown at tc . Lemma 1: Direct revelation mechanism (p, x) is a feasible truthful direct revelation mechanism implementing shutdown at tc , if and only if ∀i ∈ N the following conditions and (6), (7) and (8) hold:

Qi (p, ti ) ≥ Qi (p, si ), ∀tc ≤ si ≤ ti ≤ t, ∀i ∈ N , Ui (p, x, ti , ti ) = Ui (p, x, tc , tc ) +

Z

ti

Qi (p, si )dsi , ∀ti ∈ [tc , t], tc

13

(10) (11)

Ui (p, x, tc , tc ) ≥ U0 .

(12)

Proof of Lemma 1: see Appendix A. Based on Lemma 1, I can replace (4) and (5) by (10), (11) and (12) in the seller’s optimization problem. Applying Lemma 1, the expected revenue of the auctioneer from a feasible truthful direct revelation mechanism (p, x) implementing shutdown at tc ∈ [t, t] is given in the following Lemma. Lemma 2: For a feasible truthful direct revelation mechanism (p, x) implementing shutdown at tc ∈ [t, t], the auctioneer’s expected revenue can be written as N

R0 (p, x) = t0 F (tc ) − (1 − F (tc ))

N X

Ui (p, x, tc , tc ) +

i=1

+

Z |

t tc

... {z k

Z

t

[

k X

N X

(

F (tc )N −k

k=1

Ik

pik0 (tIk )(tik0

tc k0 =1 }

X  ∀Ik ⊂N

1 − F (tik0 ) )]f (tIk )dtIk − t0 − f (tik0 )

(1 − F (tc ))k t0

)

.

(13)

Proof of Lemma 2: Appendix A. Hereafter I consider a relaxed problem by ignoring constraints (6) and (7). I then verify that these ignored constraints are actually satisfied by the optimal solution of the relaxed problem. Thus the solution of the relaxed problem is then the solution of the original problem. Note that x only appears in the second term of the objective function (13) and in constraints (11) and (12) as I temporarily ignore constraints (6) and (7). From (11), (2) and (9), I have Ui (p, x, tc , tc ) = Ui (p, x, ti , ti ) − Z h = ti pi {i} (ti ) − +

N −1 X k=1

(

Z

ti

Qi (p, si )dsi tc

ti

i pi {i} (si )dsi − xi {i} (ti ) F (tc )N −1

tc

F (tc )N −k−1

X

Z

t

∀Ik ⊂N \{i} | tc

14

... {z k

Z th tc

}

ti pi Ik ∪{i} (tIk , ti )

−

Z

ti

pi

Ik ∪{i}

(tIk , si )dsi − xi

Ik ∪{i}

) i (tIk , ti ) f (tIk )dtIk .

(14)

tc

For ∀i ∈ N , k ∈ N \ {N }, ∀Ik ⊂ N \ {i}, I define xi {i} (ti ) = ti pi {i} (ti ) −

Z

ti

pi {i} (si )dsi − U0 , tc

xi Ik ∪{i} (tIk , ti ) = ti pi Ik ∪{i} (tIk , ti ) −

Z

ti

pi Ik ∪{i} (tIk , si )dsi − U0 .

(15)

tc

Therefore from (14) and (15), I have Ui (p, x, tc , tc ) = U0

N −1 X

CkN −1 F (tc )

(N −1)−k

k

(1 − F (tc )) = U0 [F (tc ) + (1 − F (tc ))]

N −1

= U0 . (16)

k=0

Thus (15) defines the optimal solution for x given p. From (13), it is obvious that the optimal p should satisfy the following conditions. Each set of {pIikk0 (tIk )}

k k 0 =1

corresponding to Ik ⊂ N

should maximize Z |

t tc

... {z k

Z

t

[

k X

tc k0 =1 }

pik0 Ik (tIk )(tik0 − t0 −

Under the regularity condition J(t) = t −

1−F (t) f (t)

1 − F (tik0 ) )]f (tIk )dtIk . f (tik0 )

(17)

increases wrt. t, it is easy to see from (17)

that the optimal solution for p is given by the following formula. For each k ∈ N , and each Ik ⊂ N , I define Ik

pik0 (tIk ) =

    1 if

tik0 > ziIkk0 (t−ik0 )

   0 if

tik0 ≤

(18)

ziIkk0 (t−ik0 ),

where ziIkk0 (t−ik0 ) = max{J −1 (t0 ),

max il ∈Ik \{ik0 }

ti` }, 1 ≤ k 0 ≤ k.

(19)

From (18) and (19), Z

tik0 tc

pik0 Ik (ti1 , ti2 , ..., sik0 , ...tik )dsik0 =

    ti 0 − max{tc , ziIk0 (t−i 0 )} if k k k

tik0 > ziIkk0 (t−ik0 )

if

ziIkk0 (t−ik0 ).

   0

15

tik0 ≤

(20)

The optimal solution for x in (15) is then simplified as follows.

xik0 Ik (tIk ) =

    max{tc , ziIk0 (t−i 0 )} − U0 k k    −U0

if

tik0 > ziIkk0 (t−ik0 )

if

ziIkk0 (t−ik0 ).

tik0 ≤

(21)

From (18), (19) and (21), it is easy to check that constraints (6) and (7) are satisfied from the following arguments. First, let us check that constraint (7) is satisfied. Suppose that bidder i’s private value ti is less than tc and he participates with signal t0i ∈ [t, t]. If tc < J −1 (t0 ) and t0i ∈ [t, J −1 (t0 )), then his utility from participation is U0 . If tc < J −1 (t0 ) and t0i ∈ [J −1 (t0 ), t], then his utility is less than U0 , because he has a positive possibility to win and pay at least J −1 (t0 ) − U0 , which is higher than ti − U0 . If tc ≥ J −1 (t0 ) and t0i ∈ [t, J −1 (t0 )), then his utility from participation is U0 . If tc ≥ J −1 (t0 ) and t0i ∈ [J −1 (t0 ), t], then his utility is less than U0 , because he has a positive possibility to win and pay at least tc − U0 , which is higher than ti − U0 . Thus constraint (7) holds. Second, let us check that constraint (6) is satisfied. Suppose that bidder i’s private value ti belongs to [tc , t] and he participates with signal t0i ∈ [t, tc ). If tc < J −1 (t0 ), then he has no chance to win, so his utility is U0 . If tc ≥ J −1 (t0 ), reporting t0i < tc ≤ ti only reduces his chance to win a positive surplus equal to the difference between his private value and the price he pays upon winning. Thus constraint (6) is satisfied. Therefore (18), (19) and (21) give the optimal solution of the original problem with constraints (6) and (7). When I derive the optimal mechanism implementing symmetric-shutdown at tc , I only restrict

16

that the tc is same across all bidders, the winning probability functions and payment functions are not required to be symmetric wrt. the arguments. Thus the above result implies that there must exist a symmetric optimal mechanism implementing symmetric-shutdown policy.

2.3

The Optimal Symmetric Cutoff Point

The optimal cutoff point must not be less than J −1 (t0 ), because otherwise the optimal symmetricshutdown mechanism means that the seller incurs costs from subsidizing some participants who never bid high enough to win. Thus tc < J −1 (t0 ) is not the interesting case and I can focus on the tc ∈ [J −1 (t0 ), t] case. When tc ∈ [J −1 (t0 ), t], the seller’s expected revenue is given by the following equation under the optimal auction mechanism implementing symmetric-shutdown policy at tc . N

∗

R0 (tc ) = t0 F (tc ) − N U0 (1 − F (tc )) +

Z

t

[t(1) −

tc N

= t0 F (tc ) − N U0 (1 − F (tc )) +

Z

t

1 − F (t(1) ) (1) (1) (1) ]f (t )dt f (t(1) )

t(2) f (2) (t(2) )dt(2)

tc

+N tc (1 − F (tc ))F (tc )N −1 ,

(22)

where t(1) , t(2) are respectively the first and second order statistics of t = (t1 , t2 , ..., tN ), and f (1) (t(1) ), f (2) (t(2) ) are respectively the densities of t(1) , t(2) . Since f (2) (t(2) ) = N (N − 1)(1 − F (t(2) ))F (t(2) )

N −2

f (t(2) )),3 I have dR0 ∗ (tc ) U0 = N f (tc )F (tc )N −1 (t0 + − J(tc )). dtc F (tc )N −1

3

F (2) (t) = F (t)N + N F (t)N−1 (1 − F (t)).

17

(23)

From (23), I have the following proposition 1 which gives the optimal symmetric-shutdown point and the overall optimal symmetric-shutdown auction mechanism. Proposition 1: The optimal cutoff point tc ∗ is defined by (24). The optimal symmetricshutdown auction mechanism is defined by (18), (19) and (21) with tc = tc ∗ . t0 +

U0 F (tc ∗ )N −1

= J(tc ∗ ) = tc ∗ −

1 − F (tc ∗ ) . f (tc ∗ )

(24)

Note that tc ∗ ≥ J −1 (t0 ), the above results lead to the following proposition which is on the implementation of the optimal symmetric-shutdown mechanism. Proposition 2: Equations (18), (19) and (21) indicate that a modified Vickrey auction with a reserve price of tc ∗ defined by (24) and a participation subsidy of U0 , is the optimal mechanism implementing symmetric-shutdown. The highest bidder among the participants wins the object and pays the second highest bid less his reservation utility, or the value of the reserve price less his reservation utility if the winner is the only participant, all the losing participants get a participation subsidy equal to their reservation utility. In the above mechanism, the highest bidder always wins the auction and he has to pay at least the “minimum price” tc ∗ . As a result of this “minimum price”, the bidders with their types lower than tc ∗ is strictly shut down by the optimal symmetric-shutdown mechanism given in Proposition 2, because they may have to pay at lease tc ∗ − U0 if they participate. From Proposition 2, the optimal symmetric shutdown point tc ∗ is the optimal reserve price in a modified Vickrey auction with a participation subsidy of U0 . Based on this insight, the following argument provides the intuition for obtaining (24) which determines the optimal sym18

metric shutdown point tc ∗ . Consider a modified Vickrey auction with a participation subsidy of U0 and a reserve price of tc . We ask what is the optimal reserve price t∗c . Consider an increase in reserve price tc , say 4tc > 0. The auctioneer’s saving in participation subsidy from this change in tc is N (F (tc + 4tc ) − F (tc ))U0 ; the auctioneer’s gain from this higher reserve price is approximately N (1 − F (tc + 4tc ))F N −1 (tc ); and the auctioneer’s loss due to less participation is (F N (tc + 4tc ) − F N (tc ))(tc − t0 ). At the optimal t∗c , the gain should be equal to the loss. This condition gives exactly the result in Proposition 1. An interesting issue is what is the relation between the auction mechanism in Proposition 2 and the optimal auction when bidders’ opportunity cost is zero and their values are lower by U0 . Direct computations using (24) show that the optimal reserve price in Proposition 2 is strictly higher than the sum of U0 and the optimal reserve price obtained in a setting in which opportunity cost is zero and the bidders’ values are lower by U0 . This means when the bidders have positive opportunity cost U0 , the renormalization of the model’s components including the bidders’ values and opportunity cost is restrictive for deriving the optimal mechanism for the original setting. In other words, the optimal solution of the original problem with U0 > 0 cannot be obtained through solving the transformed problem obtained by a renormalization. The following example illustrates this point. Suppose in the original problem, t0 = 0, U0 = 1 and bidders’ values follow a uniform distribution on [1, 2]. Then Proposition 2 gives that the optimal reserve price is tc ∗ = 1 + (1/2)1/2 . The expected revenue for the auctioneer from the corresponding optimal symmetric-shutdown mechanism is R∗ = 0.27613. For the transformed ˜0 = 0 and bidders’ values follow a uniform distribution on [0, 1], problem in which t0 = 0, U

19

either Myerson [10] or Proposition 2 above gives the optimal reserve price to be 1/2. Adding U0 to this reserve price gives 1.5, which is lower than tc ∗ = 1 + (1/2)1/2 . A second price auction adopting a reserve price of 1.5 and a participation subsidy of U0 = 0.5 gives the auctioneer an expected revenue of 0.1667 which is lower than 0.27613.

2.4

Stochastic Participation Issue

The following proposition justifies why I only need to consider the cutoff shutdown for the optimal symmetric-shutdown mechanism in the two potential bidders case. Proposition 3: In terms of the auctioneer’s expected revenue, any mechanism implementing stochastic participation of bidders is dominated by a cutoff shutdown truthful direct revelation mechanism in the two potential bidders case. Proof of Proposition 3: see Appendix C. Appendix B shows a revelation principle allowing stochastic participation of bidders. Based on the revelation principal, I only need to show that any truthful direct revelation mechanism implementing stochastic participation of bidders is dominated by a cutoff shutdown truthful direct revelation mechanism. The intuition why this is true is the following. If bidder i with value ti participates with a probability ∈ (0, 1) at the equilibrium of a truthful direct revelation mechanism, then as Lemma C.2 illustrates, the equilibrium conditions require that he wins the object with zero probability when he participates. However, in order to get him to participate with a positive probability, the auctioneer’s expected subsidy to this bidder is the reservation utility U0 when he participates. Thus if the auctioneer shuts down this bidder, he will save

20

this subsidy but has nothing to lose. This is the basic intuition why the cutoff point shutdown mechanism dominates the stochastic shutdown mechanism. According to Proposition 3, in the two potential bidders case there is no loss of generality to consider the cutoff shutdown truthful direct revelation mechanisms for the optimal auction mechanism, as long as the shutdown points are allowed to be different across bidders. Although the same intuition applies to the N -bidder case, how to generalize the proof of Proposition 3 to the N -bidder case remains an open question.

3

Properties of the Optimal Symmetric-Shutdown Mechanism and Their Implications

This section presents the properties of the optimal symmetric-shutdown mechanism and discusses their implications. I will first present the properties of the optimal symmetric cutoff point t∗c : (i) If t0 + U0 < t, there is a unique optimal cutoff value tc ∗ ∈ (t, t) since the left-hand-side of (24) decreases and right-hand-side of (24) increases in tc ∗ . (ii) The optimal tc ∗ is an increasing function in the number of the potential bidders N . As a matter of fact, if it is not increasing then the lefthand-side of (24) is strictly decreasing in N , while the right-hand-side does not depend on N . (iii) As N approaches to ∞, the optimal tc ∗ approaches t. If there is an upper bound for tc ∗ which is smaller than t, then the left-hand-side approaches to ∞ as N approaches to ∞, while the right-hand-side is lower than t. (iv) The optimal tc ∗ is increasing in U0 if t0 + U0 < t. If

21

U0 = 0, then tc ∗ = J −1 (t0 ) and I have the same optimal mechanism in Myerson [10]. (v) If U0 > 0, the reserve price in the optimal symmetric-shutdown mechanism given by Proposition 2 is always the optimal cutoff point defined as max{tc ∗ , J −1 (t0 )} = tc ∗ since tc ∗ > J −1 (t0 ). If U0 = 0, I go back to the case of zero opportunity cost as considered in Myerson [10]. The optimal symmetric-shutdown mechanism given in section 2 is equivalent to that given in Myerson [10]. Proposition 4 below shows the optimal expected revenue increases wrt. the number of potential bidders N for this optimal mechanism. Proposition 4: If U0 = 0, t0 ∈ [0,1], and bidder’s private values follow any distribution on [0, 1] satisfying J 0 (t) > 0, then R∗ = R0 ∗ (tc ∗ ) increases in the number of the potential bidders N . Moreover, tc ∗ does not depend on N , and equals to the reserve price in Myerson [10]. Proof of Proposition 4: see Appendix A. Proposition 5 shows that if the optimal symmetric-shutdown mechanism is adopted, the auctioneer wants to have as many as possible potential bidders even if U0 > 0, as long as the bidders’ values do not concentrate near the upper bound. Proposition 5: For any U0 ≥ 0, t0 ≥ 0, U0 + t0 < 1, if F (t) = tk with support [0, 1], then R* increases wrt. the number of the potential bidders N if 1 < k ≤ 2. Proof of Proposition 5: see Appendix A. Notice J −1 (t) > 0 if k ≥ 1, so Propositions 1 and 2 apply. Proposition 6 shows, however, that the auctioneer adopting the optimal symmetric-shutdown mechanism wants to limit the number of potential bidders, if U0 > 0 and the bidders’ values do concentrate near the upper bound.

22

Proposition 6: For ∀ U0 > 0, t0 ≥ 0, U0 + t0 < 1, F (t) = tk on [0,1], I have approaches to ∞, if k >

2 1−t0 (1

−

dR∗ dN

< 0 as N

log(U0 /(1−t0 )) U0 /(1−t0 ) ).

Proof of Proposition 6: see Appendix A. Note that

2 1−t0 (1

−

log(U0 /(1−t0 )) U0 /(1−t0 ) )

≥

2 1−t0

≥ 2.

Proposition 6 tells us that if the optimal symmetric-shutdown mechanism is adopted, the seller’s expected revenue can decrease as the number of potential bidders increases if the bidders’ opportunity cost is positive and the bidders’ private values are distributed near the highest value. Thus in these cases, it is in the seller’s benefit to limit the number of potential bidders if the optimal symmetric-shutdown mechanisms are adopted. The following example illustrates this point. Suppose that the seller’s value of the object is 0, potential bidders’ opportunity cost is 0.5, and the potential bidders’ values have a cumulative distribution function of F (t) = t10 on [0, 1]. In the 1 potential bidder case, a modified Vickrey auction with both a reserve price of 0.85 and a participation subsidy of 0.5 is the optimal symmetric-shutdown mechanism. This mechanism provides the seller an expected revenue of 0.28. In the 2 potential bidders case, a modified Vickrey auction with a reserve price of 0.95 and a participation subsidy of 0.5 is the optimal symmetric-shutdown mechanism. The latter mechanism provides the seller an expected revenue of 0.21, which is smaller than 0.28. Proposition 4 shows that U0 > 0 is also a necessary condition for obtaining the above result; while proposition 5 shows that having the bidders’ private values distributed close enough to the highest value is also a necessary condition. Figure 1 shows the seller’s expected revenue R∗ as a function of the number of potential bidders N when t0 = 0, U0 = 0.3 and k takes different values of 2.0, 5.0 or 10.0. We can see

23

that the seller’s expected revenue R∗ is a strictly increasing function of the number of potential bidders N if k = 2.0, while the seller’s expected revenue R∗ is strictly decreasing as the number of potential bidders N gets big if k = 5.0 or k = 10.0. Figure 1 confirms that having the bidders’ private values distributed close enough to the highest value is a necessary condition for obtaining the result in Proposition 6. Figure 1 also shows that the optimal number of potential bidders decreases wrt. the power k of the private value distribution.

Figure 1 here

Figure 2 shows that the seller’s expected revenue as a function of the number of potential bidders when t0 = 0, k = 10.0 and U0 takes different values of 0.0, 0.1, 0.2 or 0.3. Figure 2 here

We can see from Figure 2 that if U0 = 0 then the seller’s expected revenue is always a strictly increasing function of the number of potential bidders; while if U0 > 0, the seller’s expected revenue is always a strictly decreasing function of the number of potential bidders as N gets big. Figure 2 confirms that U0 > 0 is a necessary condition for obtaining the result in Proposition 6. Figure 2 also shows that the optimal number of potential bidders decreases wrt. the opportunity cost U0 . In particular, when k = 10.0, U0 = 0.1, t0 = 0, R∗ reaches a maximum at 0.7016 when N = 3 and approaches to 0.66974 when N approaches to ∞. When k = 10.0, U0 = 0.3, t0 = 0, R∗ reaches a maximum at 0.4485 when N = 1 and approaches to 0.3388 when N approaches to ∞. 24

The latter example above means that if there are 2 potential bidders, considering the optimal auction mechanism among the “symmetric-shutdown” class is restrictive, i.e., the overall optimal auction mechanism in that setting must implement asymmetric shutdown. Generally, Proposition 6 implies that as the number of the potential bidders increases, the overall optimal auction mechanism must implement asymmetric shutdown across symmetric bidders in plenty of situations if U0 > 0. In other words, the seller should discriminate the ex ante symmetric bidders by shutting down the potential bidders asymmetrically. This discrimination can be implemented through adopting different reserve prices across bidders. Note that in a second price auction with a participation subsidy of U0 , bidder i is shut down if ti is less than the reserve price set to him. In particular, setting the reserve price for a potential bidder at the maximum private value is equivalent to shutting down completely this bidder. Consider a setting in which N = 2, t0 = 0, U0 = 0.2, F (t) =

t−0.6 0.4

on [0.6, 1.0]. A second price auction with a uniform

reserve price of 0.76 and a participation subsidy of U0 = 0.2 is the optimal symmetric-shutdown mechanism for the two potential bidders case, which gives the auctioneer an expected revenue of 0.427. When the seller shuts down one potential bidder, he gets the optimal expected revenue of 0.40 by setting a reserve price of 0.60 for the remaining bidder. However, reserve prices 0.66 and 0.86 respectively for bidder 1 and bidder 2 provide the auctioneer the best expected revenue of 0.431 when considering a second price auction with a participation subsidy of U0 = 0.2. Note that these reserve prices shut down bidder 1 and bidder 2 respectively at 0.66 and 0.86. This example shows that at the optimum the auctioneer may want to shut down bidders asymmetrically, even if his expected revenue increases wrt. the number of potential bidders when the

25

optimal symmetric-shutdown mechanisms are adopted. McAfee and McMillan [8] show that the optimal procurement is in general discriminatory if the bidders’ cost distributions are different. Proposition 6 means that if the bidders’ opportunity cost is positive, discriminatory optimal mechanism can arise even with symmetric bidders. If the auctioneer cannot discriminate bidders, this result indicates that he may want to limit the number of the potential bidders. These results are put into the following corollaries. Corollary 1: If the bidders’ opportunity cost is positive, then the optimal auction is discriminatory in terms of inducing asymmetric participation, if the bidders’ private values are distributed close enough to the highest value. Corollary 2: If the bidders’ opportunity cost is positive and the auctioneer cannot discriminate bidders, then the auctioneer wants to limit the number of the potential bidders if the bidders’ private values are distributed close enough to the highest value. Moreover when t0 = 0, it is easy to see that both the limit value of R∗ and R∗ itself decrease in U0 . The following figure 3 illustrates that the seller’s expected revenue is a decreasing function of the bidders’ opportunity cost U0 when t0 = 0, N = 2 and k takes different values of 2.0, 5.0 or 10.0. It also shows that the seller’s expected revenue is an increasing function of the power k of the private value distribution. Figure 3 Propositions 7 and 8 give distribution free properties of the optimal symmetric-shutdown mechanism. Proposition 7 studies the property of the auctioneer’s expected revenue at limit as N approaches to ∞. Proposition 8 gives the expected number of participation as N approaches to 26

∞. Interestingly, both the auctioneer’s expected revenue at limit and the expected participation at limit are distribution free. Proposition 7: ∀ U0 ≥ 0, t0 ≥ 0 and U0 + t0 < 1, R∗ approaches to (1 − U0 ) + U0 log U0 − U0 log(1 − t0 ) as N approaches to ∞. Moreover, the limit value of R∗ does not depend on the private value distribution function F (t) on [0, 1]. Proof of Proposition 7: see Appendix A. Figure 1 confirms the result of Proposition 7. When U0 = 0.3 and t0 = 0, the optimal expected revenue of the seller R∗ approaches to 0.3388 as N → ∞. Proposition 8: ∀ U0 ≥ 0, t0 ≥ 0 and U0 + t0 < 1. If the optimal symmetric-shutdown U0 mechanism is adopted, the expected number of participants is − log( 1−t ) as N approaches to 0

∞, which does not depend on the private value distribution function F (t) on [0, 1]. Proof of Proposition 8: see Appendix A. Proposition 8 says that the lower of the bidders’ reservation utility U0 and the auctioneer’s valuation t0 , the higher the expected participation.

4

Conclusion

The contribution of this paper is two-fold. First, in this paper I relax the assumption of zero reservation utility for bidders as postulated in Myerson [10], and provide concrete proof to derive the optimal symmetric-shutdown auction mechanism maximizing seller’s expected revenue. A modified Vickrey auction with a reserve price and a participation subsidy is shown to be the optimal symmetric-shutdown mechanism under the usual regularity condition on the bidder’s 27

private value distribution. Because of the positive reservation utility of the bidders, at the optimum the seller shuts down bidders with low private values, and subsidizes the participants by their reservation utility. The participation subsidy encourages bidders’ participation while the reserve price selects participants with higher private values, as this optimal symmetricshutdown mechanism precludes bidders with private values lower than the reserve price from participating. It is shown that the problem cannot be solved in a zero reservation utility setting by simply renormalizing the bidders’ values by their reservation utility. Moreover in the case of two potential bidders, a constructive proof is given to show that an auction mechanism implementing stochastic participation of the bidders is never optimal. Second, the properties of the optimal uniform reserve price and the optimal symmetricshutdown mechanism have been examined. The optimal reserve price is an increasing function of the bidders’ reservation utility, seller’s private value and the number of the potential bidders. In particular, the optimal uniform reserve price is strictly higher than the reserve price in Myerson [10] and it approaches to the upper boundary of the bidders’ private values as the number of potential bidders approaches to ∞, as long as the bidders’ reservation utility is strictly positive. More interestingly, I find that when the optimal symmetric-shutdown mechanism is adopted the seller’s expected revenue can decrease as the number of the potential bidders increases, if the bidders’ private values are distributed near the highest value. The economic intuition is that one additional potential bidder may contribute less than he costs as the auctioneer needs to subsidize the participants. This result has two implications. On one hand, if the auctioneer cannot discriminate bidders, then he will want to limit the number of potential bidders in many

28

situations. On the other hand, the overall optimal auction mechanism must be discriminatory in many cases, in the sense of implementing asymmetric shutdown across bidders. This optimality of the discriminatory mechanism in terms of the auctioneer’s expected revenue is parallel to that in Stegeman [12] in terms of efficiency. The above results imply that asymmetry among the bidders is not a necessary condition for obtaining a discriminatory optimal mechanism. A natural extention is to study the unrestricted optimal mechanism when there is positive opportunity cost for symmetric bidders even in the case of two potential bidders, as numerical examples given in section 3 shows that the optimal auction mechanism must implements asymmetric shutdown across bidders. From Proposition 3, there is no loss of generality to consider the mechanisms implementing cutoff-shutdown for the overall optimal mechanism as long as asymmetric shutdown points across bidders are allowed. The procedure developed in section 2.1 can be slightly modified and applied to derive the optimal shutdown mechanism implementing the asymmetric cutoff points across bidders. The result can then be used to determine the optimal shutdown points. Moreover, the optimal mechanism when opportunity cost is private information of the bidder should be considered. Following this direction, Lu [6] considers the auction design when bidders have two-dimensional continuous private signals, namely their private value and their opportunity cost.

Acknowledgments I am grateful to Jean-Jacques Laffont for his advice. I thank Isabelle Brocas, Juan Carrillo, Jinwoo Kim, Vijay Krishna, Preston McAfee, Isabelle Perrigne, Guofu Tan and Quang Vuong 29

for helpful comments. All errors are mine.

30

Appendix A Proof of Lemma 1: It is easy to verify from (2) that Ui (p, x, ti , ti 0 ) = Ui (p, x, ti 0 , ti 0 ) + (ti − ti 0 )Qi (p, ti 0 ), ∀ti , ti 0 ∈ [tc , t], ∀i ∈ N .

(A.1)

From (5) and (A.1), I have Ui (p, x, ti , ti ) ≥ Ui (p, x, ti 0 , ti 0 ) + (ti − ti 0 )Qi (p, ti 0 ), ∀ti , ti 0 ∈ [tc , t], ∀i ∈ N .

(A.2)

Thus (5) is equivalent to (A.2). Using (A.2) twice, I have for ∀ ti , ti 0 ∈ [tc , t], ∀ i ∈ N , (ti − ti 0 )Qi (p, ti 0 ) ≤ Ui (p, x, ti , ti ) − Ui (p, x, ti 0 , ti 0 ) ≤ (ti − ti 0 )Qi (p, ti ).

(A.3)

(A.3) implies (10). From (A.3), I have for ∀ si , si + δ ∈ [tc , t], ∀ i ∈ N , Qi (p, si )δ ≤ Ui (p, x, si + δ, si + δ) − Ui (p, x, ti , ti ) ≤ Qi (p, si + δ)δ.

(A.4)

Since Qi (p, si ) is increasing in si , (A.4) implies Ui (p, x, si , si ) = Qi (p, si ), ∀i ∈ N , ∀ si ∈ [tc , t], dsi

(A.5)

where Qi (p, si ) is Riemann integrable, so Z

ti

Qi (p, si )dsi = Ui (p, x, ti , ti ) − Ui (p, x, tc , tc ).

(A.6)

tc

(A.6) implies (11), and (12) is directly from (4). Thus (10), (11) and (12) are derived from (4) and (5). Now I have to show (4) and (5) from (7), (8), (10), (11) and (12). ∀tc ≤ si ≤ ti ≤ t, ∀i ∈ N , (10) and (11) imply Ui (p, x, ti , ti )

= Ui (p, x, si , si ) + ≥ Ui (p, x, si , si ) +

Z

ti

Qi (p, ri )dri

si Z ti

Qi (p, si )dri

si

= Ui (p, x, si , si ) + (ti − si )Qi (p, si ).

31

Similarly, ∀tc ≤ ti ≤ si ≤ t, ∀i ∈ N , (10) and (11) imply Ui (p, x, ti , ti )

Z

= Ui (p, x, si , si ) +

si

Qi (p, ri )dri

ti Z ti

≥ Ui (p, x, si , si ) +

Qi (p, si )dri

si

= Ui (p, x, si , si ) + (ti − si )Qi (p, si ). Thus I have (A.2), i.e., (5) is shown. Eq. (4) is directly derived from (8), (11) and (12). 2 Proof of Lemma 2: From (2), Rt

Ui (p, x, ti , ti )dti Z t( (ti pi {i} (ti ) − xi {i} (ti ))F (tc )N −1

tc

=

tc

+

N −1 n X

F (tc )

=

t

∀Ik ⊂N \{i} | tc

k=1

Z

Z

X

N −k−1

... {z

Z

t

[ti pi

Ik ∪{i}

(tIk , ti ) − xi

Ik ∪{i}

(tIk , ti )]f (tIk )dtIk

o

)

f (ti )dti

tc

}

k

t

(ti pi {i} (ti ) − xi {i} (ti ))f (ti )dti F (tc )N −1 tc N −1n X

+

F (tc )

Z

Z

t o ... [ti pi Ik ∪{i} (tIk , ti )−xi Ik ∪{i} (tIk , ti )]f (tIk )f (ti )dtIk dti . (A.7) ∀Ik ⊂N \{i} | tc {z tc}

N −k−1

k=1

X

t

k+1

From (A.7), I have PN

i=1

=

Z

t

Ui (p, x, ti , ti )f (ti )dti tc

N n X

F (tc )

N −k

k=1

X Z ∀Ik ⊂N

|

t tc

... {z k

Z

t

k X

o [tik0 pik0 Ik (tIk ) − xik0 Ik (tIk )]f (tIk )dtIk .

(A.8)

tc k0 =1 }

From (1) and (A.8), N

R0 (p, x) = t0 F (tc ) +

N X k=1

Z |

t tc

... {z k

Z

t

[

k X

tc k0 =1 }

Ik

(

F (tc )

N −k

X nZ |

∀Ik ⊂N

pik0 (tIk )(tik0 − t0 )]f (tIk )dtIk +

t tc

Z |

32

... {z

Z

t

t0 f (tIk )dtIk + tc

}

k

t tc

... {z k

Z

t

[

k X

tc k0 =1 }

xIikk0 (tIk )

−

pIikk0 (tIk )tik0 ]f (tIk )dtIk

o

)

= t0 F (tc )N −

N Z X i=1

+

N X

F (tc )

N −k

k=1

t

Ui (p, x, ti , ti )f (ti )dti tc

(

X

k

t0 (1 − F (tc )) +

Z |

∀Ik ⊂N

t tc

... {z

Z

t

[

k X

tc k0 =1 }

k

pIikk0 (tIk )(tik0

)

− t0 )]f (tIk )dtIk .

(A.9)

From (11), I have Z

t

Ui (p, x, ti , ti )f (ti )dti tc

=

Z

t

[Ui (p, x, tc , tc ) +

Z

ti

Qi (p, si )dsi ]f (ti )dti tc

tc

= (1 − F (tc ))Ui (p, x, tc , tc ) + = (1 − F (tc ))Ui (p, x, tc , tc ) + = (1 − F (tc ))Ui (p, x, tc , tc ) +

Z Z Z

t

[ tc t

[ tc

Z Z

ti

Qi (p, si )dsi ]f (ti )dti tc t

f (ti )dti ]Qi (p, si )dsi si

t

[1 − F (si )]Qi (p, si )dsi .

(A.10)

tc

From (9), I have Z

t

[1 − F (si )]Qi (p, si )dsi ( Z

tc

t

=

[1 − F (si )] pi {i} (si )F (tc )N −1

tc

+

N −1 n X

F (tc )

X

N −k−1

t

∀Ik ⊂N \{i} | tc

k=1

Z

Z

... {z

Z

k

t

pi

Ik ∪{i}

(tIk , si )f (tIk )dtIk

o

)

dsi

tc

}

t

1 − F (si ) {i} ]pi (si )f (si )dsi f (si ) tc ( ) Z t Z t N −1 X X 1 − F (si ) Ik ∪{i} N −k−1 f (tIk )f (si )dtIk dsi . (A.11) F (tc ) + ... pi (tIk , si ) f (si ) k=1 ∀Ik ⊂N \{i} | tc {z tc} = F (tc )N −1

[

k+1

From (A.10) and (A.11), I have N Z X i=1

t

Ui (p, x, ti , ti )f (ti )dti tc

= (1 − F (tc ))

N X

Ui (p, x, tc , tc )

i=1

33

+

N X

(

F (tc )

X Z

N −k

k=1

∀Ik ⊂N

|

t tc

... {z

Z

k

t tc

}

k X k0 =1

pik0

Ik

1 − F (tik0 ) (tIk ) f (tik0 )

!

)

f (tIk )dtIk .

(A.12)

From (A.9) and (A.12), I have the desired result. 2 Proof of Proposition 4: (22) is equivalent to R∗

N

= R∗ (t∗c ) = t0 F (tc ∗ ) − N U0 (1 − F (tc ∗ )) + t − tc ∗ F (2) (tc ∗ ) Z t N −1 − F (2) (t(2) )dt(2) + N tc ∗ (1 − F (tc ∗ ))F (tc ∗ ) ,

(A.13)

tc ∗

where F (2) (t) = F (t)N + N F (t)N −1 (1 − F (t)). Thus N

R∗ = t0 F (tc ∗ ) − N U0 (1 − F (tc ∗ )) + t − tc ∗ F N (tc ∗ ) −

Z

t

F (2) (t(2) )dt(2) .

(A.14)

tc ∗

Applying the Envelope Theorem, I have dR∗ dN

= t0 F (tc ∗ )N log(F (tc ∗ )) − U0 (1 − F (tc ∗ )) − tc ∗ F (tc ∗ )N log(F (tc ∗ )) Z t F (t)N −1 {[ log F (t) + (1 − F (t)) ] + (N − 1)(1 − F (t)) log F (t)}dt. −

(A.15)

tc ∗

Since both [ log F (t) + (1 − F (t)) ] and (N − 1)(1 − F (t)) log F (t) are negative on [0, 1], it is easy to see that

dR∗ dN

> 0 for ∀t0 ≥ 0, since (24) implies that tc ∗ ≥ t0 holds. 2

Proof of Proposition 5: From (A.15), dR∗ N N = t0 F (tc ∗ ) log(F (tc ∗ )) − U0 (1 − F (tc ∗ )) − tc ∗ F (tc ∗ ) log(F (tc ∗ )) dN Z t − F (t)N −1 {[ log F (t) + (1 − F (t))] + (N − 1)(1 − F (t)) log F (t)}dt tc ∗ N

= t0 F (tc ∗ ) log(F (tc ∗ )) − U0 (1 − F (tc ∗ )) + tc ∗ F (tc ∗ ) −tc ∗ {F (tc ∗ ) −

Z

N −1

N −1

(1 − F (tc ∗ ))

[F (tc ∗ ) log(F (tc ∗ ) + (1 − F (tc ∗ ))]}

t

F (t) tc

N −1

{[ log F (t) + (1 − F (t))] + (N − 1)(1 − F (t)) log F (t)}dt

∗

34

N

= t0 F (tc ∗ ) log(F (tc ∗ )) + t0 F (tc ∗ )

N −1

(1 − F (tc ∗ ))

+(1 − F (tc ∗ ))[−U0 + (tc ∗ − t0 ∗ )F (tc ∗ )

N −1

]

N −1

−tc ∗ {F (tc ∗ ) [F (tc ∗ ) log(F (tc ∗ ) + (1 − F (tc ∗ ))]} Z t N −1 − F (t) {[ log F (t) + (1 − F (t))] + [(N − 1)(1 − F (t)) log F (t)]}dt tc ∗

= t0 F (tc ∗ )

N −1

[F (tc ∗ ) log(F (tc ∗ )) + (1 − F (tc ∗ ))]

+(1 − F (tc ∗ ))

1 − F (tc ∗ ) F (tc ∗ )N −1 f (tc ∗ )

−tc ∗ {F (tc ∗ )N −1 [F (tc ∗ ) log(F (tc ∗ ) + (1 − F (tc ∗ ))]} Z t N −1 F (t) {[ log F (t) + (1 − F (t))] + [(N − 1)(1 − F (t)) log F (t)]}dt. −

(A.16)

tc ∗

Note that [F (tc ∗ ) log(F (tc ∗ )) + (1 − F (tc ∗ ))] > 0. I already know that both [ log F (t) + (1 − 2

F (t))] and (N − 1)(1 − F (t)) log F (t) are negative on [0, 1]. For k ∈ [1, 2], I have (1 − F (tc ∗ )) − f (tc ∗ )tc ∗ [F (tc ∗ ) log(F (tc ∗ ) + (1 − F (tc ∗ )] = (1 − F (tc ∗ ))2 − kF (tc ∗ )[F (tc ∗ ) log(F (tc ∗ ) + (1 − F (tc ∗ )] > 0. Thus the proposition holds for k ∈ [1, 2]. Actually as long as 2F (t) ≥ tf (t) is satisfied, the proposition holds following the same reasoning. 2 Proof of Proposition 6: From (A.16), dR∗ = t0 F (tc ∗ )N −1 [F (tc ∗ ) log(F (tc ∗ )) + (1 − F (tc ∗ ))] dN 1 − F (tc ∗ ) N −1 F (tc ∗ ) +(1 − F (tc ∗ )) f (tc ∗ ) N −1

−tc ∗ {F (tc ∗ ) [F (tc ∗ ) log(F (tc ∗ ) + (1 − F (tc ∗ ))]} Z t N −1 − F (t) {[ log F (t) + (1 − F (t))] + [(N − 1)(1 − F (t)) log F (t)]}dt tc ∗

= (t0 − tc ∗ )F (tc ∗ ) +(1 − F (tc ∗ )) −

Z

N −1

[F (tc ∗ ) log(F (tc ∗ )) + (1 − F (tc ∗ ))]

1 − F (tc ∗ ) N −1 F (tc ∗ ) f (tc ∗ )

t

tc

F (t)

N −1

{[ log F (t) + (1 − F (t))] + [(N − 1)(1 − F (t)) log F (t)]}dt

∗

35

U0 1 − F (tc ∗ ) N −1 + )F (tc ∗ ) [F (tc ∗ ) log(F (tc ∗ )) + (1 − F (tc ∗ ))] N −1 f (tc ∗ ) F (tc ∗ ) 1 − F (tc ∗ ) F (tc ∗ )N −1 +(1 − F (tc ∗ )) f (tc ∗ ) Z t N −1 − F (t) {[ log F (t) + (1 − F (t))] + [(N − 1)(1 − F (t)) log F (t)]}dt

= −(

tc ∗

1 − F (tc ∗ ) N F (tc ∗ ) log(F (tc ∗ )) − U0 [F (tc ∗ ) log(F (tc ∗ )) + (1 − F (tc ∗ ))] f (tc ∗ ) Z t F (t)N −1 {[ log F (t) + (1 − F (t))] + [(N − 1)(1 − F (t)) log F (t)]}dt. −

=−

(A.17)

tc ∗

Let h = 1 − F (tc ∗ ), using tc ∗ = (1 − h)

1/k

, f (tc ∗ ) = k(1 − h)

k−1 k

and F N −1 (tc ∗ ) =

−U0 1−F (t ∗ ) t0 −tc ∗ + f (t ∗c)

,I

c

have 1 − F (tc ∗ ) N F (tc ∗ ) log(F (tc ∗ )) f (tc ∗ ) 1 1 = − (1 − h)1/k hU0 log(1 − h) 1/k k (1 − h) − t0 − −

=

h(1−h)1/k k(1−h)

U0 h2 + o(h2 ), as N → ∞ ( i.e., F (tc ∗ ) → 1), (1 − t0 )k

(A.18)

and −U0 (F (tc ∗ ) log(F (tc ∗ )) + (1 − F (tc ∗ ))) =−

U0 2 h + o(h2 ), as N → ∞ ( i.e., F (tc ∗ ) → 1). 2

(A.19)

I next show that N (1 − F (tc ∗ )) = − log

 U  0 + o(1), 1 − t0

(A.20)

as N → ∞ ( i.e., F (tc ∗ ) → 1). Since F N −1 (tc ∗ ) =

U0 ∗

tc − t0 −

1−F (tc ∗ ) f (tc ∗ )

,

(A.21)

I have  1 − F (tc ∗ )  (N − 1) log F (tc ∗ ) = log U0 − log tc ∗ − t0 − , f (tc ∗ )

36

(A.22)

and thus (N − 1) log F (tc ∗ ) = log

U0 + o(1). 1 − t0

(A.23)

(A.23) implies that 1 − F (tc ∗ ) = O



1  . N −1

(A.24)

From (A.22), I have  1 − F (tc ∗ ) i log U0 − log tc ∗ − t0 − f (tc ∗ ) i h U0 2 N −1 (1 − F˜ ) − log + o(1) = N (1 − F (tc ∗ ) − (N − 1)(1 − F (tc ∗ ) − 2 1 − t0 N −1 U0 2 − (A.25) (1 − F˜ ) + o(1), = − log 1 − t0 2

N (1 − F (tc ∗ ))

= [N (1 − F (tc ∗ ) + (N − 1) log F (tc ∗ )] −

h

where F˜ ∈ [F (tc ∗ ), 1]. (A.24) and (A.25) imply that
U0 + o(1). N 1 − F (tc ∗ ) = − log 1 − t0

(A.26)

Thus I have −

Z

t

F (t)

N −1

{[ log F (t) + (1 − F (t))] + [(N − 1)(1 − F (t)) log F (t)]}dt

tc ∗

≤ −{[log(F (tc ∗ )) + (1 − F (tc ∗ ))] + N (1 − F (tc ∗ )) log(F (tc ∗ ))}(1 − F (tc ∗ )1/k ) = −{[ log(1 − h) + h] + (− log( =

U0 ) + o(1)) log(1 − h)}(1 − (1 − h)1/k ) 1 − t0

U0 −1 log( )h2 + o(h2 ), as N → ∞ ( i.e., F (tc ∗ ) → 1). k 1 − t0

Form (A.17), (A.18), (A.19) and (A.27), when k >

2 1−t0 (1

−

log(U0 /(1−t0 )) dR∗ (U0 /(1−t0 )) ), dN

N → ∞. 2 Proof of Proposition 7: From (A.14) and (A.26), N

R∗ = t0 F (tc ∗ ) − N U0 (1 − F (tc ∗ )) + t − tc ∗ F N (tc ∗ ) −

Z

t

F (2) (t(2) )dt(2) tc ∗

37

(A.27) < 0 holds when

N

= (t0 − tc ∗ )F (tc ∗ ) − N U0 (1 − F (tc ∗ )) + t −

Z

t

F (2) (t(2) )dt(2) tc ∗

= −(

U0 1 − F (tc ∗ ) N + )F (tc ∗ ) − N U0 (1 − F (tc ∗ )) + t − ∗ f (tc ) F (tc ∗ )N −1

Z

t

F (2) (t(2) )dt(2) . (A.28) tc ∗

Thus R∗ → (1 − U0 ) + U0 log U0 − U0 log(1 − t0 ) as t = 1. Note that (A.14) and (A.26) hold for any distribution for bidders’ private values as long as the corresponding function J(·) is increasing. Thus the above result holds for any distribution satisfying the regularity condition J 0 (·) > 0. 2 Proof of Proposition 8: (A.26) gives the desired result directly since the left hand side of (A.26) is the expected participation when there are N potential bidders. Note that (A.26) holds for any distribution of the bidders’ private values as long as the corresponding function J(·) is increasing. Thus the above result holds for any distribution satisfying the regularity condition J 0 (·) > 0. 2

38

Appendix B Appendix B presents the proof for a revelation principle allowing stochastic participation of bidders in a setting with an opportunity cost for potential bidders which does not depend on the message space. In a general auction game, each bidder has a set of message space Mi , and there is a set of outcome functions of the following form: if i1 th, i2 th, ..., ik th bidders participate ( participation is defined as submitting a signal), for each ik0 ∈ Ik = {i1 , i2 , ..., ik }, the probability that he gets the object is ˆIikk0 (mi1 , ..., mik ) upon the submitted signals (mi1 , ..., mik ). pˆIikk0 (mi1 , ..., mik ), and his payment is x ˆ), Where mik0 ∈ Mik0 , 1 ≤ k ≤ N and 1 ≤ i1 < i2 < ... < ik ≤ N . This mechanism is denoted by (M, pˆ, x where M =

QN

i=1

Mi .

An auction mechanism is any such auction game together with a description of the strategic plans that the bidders are expected to adopt in playing the game. Formally, a strategic plan (pp (·), m(·)) = (i)

(i)

(pp (ti ), mi (ti ))N ×1 is a mapping from T N to [0, 1]N × M, where pp (ti ) is bidder i’s participation probability and mi (ti ) is bidder i’s strategy if he participates, T is the type space of the bidders. (i)

Denote the reservation utility of bidder i by U0 , denote (t1 , ..., tN ) by t, and (t1 , ..., ti−1 , ti+1 , ..., tN ) (i)

(i)

by t−i . For generality, I do not restrict U0 s, pp (·)s and mi (·)s to be same across all the bidders. 0(i)

p, x ˆ, ti , pp (·), m(·); pp , m0 i ) to denote the expected utility of bidder i with value ti if he Use Ui (ˆ 0(i)

and signals m0 i when participating. I have ( Z Y {i} {i} 0(i) 0 0(i) Ui (ˆ (ti pˆi (m0i ) − x p, x ˆ, ti , pp (·), m(·); pp , m i ) = pp ˆi (m0i )) (1 − p(j) p (tj ))f (t−i )dt−i

participates with probability of pp

T−i j6=i

+

Z

−1 n NX

T−i



X

k=1 Ik ⊂N \{i}

h Y

k0 ) (t p(i ik0 ) × p

ik0 ∈Ik

I ∪{i} ti pˆi k (mi1 (ti1 ), ..., m0i , ..., mik (tik ))

Y

(1 − p(j) p (tj )) ×

j ∈I / k ∪{i}

−

I ∪{i} x ˆi k (mi1 (ti1 ), ..., m0i , ..., mik (tik ))

io

f (t−i )dt−i

)

(i)

+(1 − p0(i) p )U0 . Definition 1: The strategic plan (pp (·), m(·)) is a Bayesian Nash equilibrium of the auction game

39

(M, pˆ, x ˆ) if it satisfies the following incentive conditions: for any bidder i ∈ N with ti ∈ [t, t], 0 0(i) 0 Ui (ˆ p, x ˆ, ti , pp (·), m(·); p(i) p, x ˆ, ti , pp (·), m(·); p0(i) p (ti ), mi (ti )) ≥ Ui (ˆ p , m i ), ∀ pp ∈ [0, 1], ∀ m i ∈ Mi .

Definition 2: An auction mechanism (M, pˆ, x ˆ; pp (·), m(·)) is feasible if and only if (i) For ∀k ∈ N , ∀Ik ⊂ N , pˆIikk0 (mi1 , mi2 , ..., mik ) ≥ 0;

Pk

k0 =1

pˆIikk0 (mi1 , mi2 , ..., mik ) ≤ 1.

(ii) (pp (·), m(·)) is a Bayesian Nash equilibrium of the auction game (M, pˆ, xˆ). For a feasible auction mechanism (M, pˆ, x ˆ; pp (·), m(·)), the auctioneer’s expected revenue is: R0 (ˆ p, xˆ, ti ; pp (·), m(·)) Z ( h N X  Y k i Y X X Ik (j) k0 ) (t = t0 1 − p(i ) · (1 − p (t )) p ˆ (m (t ), ..., m (t )) i j i i i i 1 1 p p k k ik0 k0 T

+

N X

k=1 Ik ⊂N

X  Y

k=1 Ik ⊂N

ik0 ∈Ik

ik0 ∈Ik

k0 ) (t p(i ik0 ) p

·

k0 =1

j ∈I / k

Y

(1 −

p(j) p (tj ))

k X

xˆIikk0 (mi1 (ti1 ), ..., mik (tik ))

) 

f (t)dt

k0 =1

j ∈I / k

Definition 3: An auction mechanism (M, pˆ, x ˆ; pp (·), m(·)) is a direct revelation mechanism if the message spaces Mi = T, ∀i ∈ N . Definition 4: A direct revelation mechanism is truthful if reporting type truthfully when participating is a Bayesian Nash equilibrium, i.e., mi (ti ) = ti , ∀i ∈ N . Revelation principle with stochastic participation: Given any feasible auction mechanism, there exists an equivalent feasible direct revelation mechanism which gives the auctioneer and all bidders the same expected utilities as those from the given mechanism. Proof: Suppose that the given mechanism is (M, pˆ, x ˆ; pp (·), m(·)). Define a direct revelation mechanism (T , p, x; pp (·), t(·)) in the following way, where T = T N , ti (ti ) = ti , ∀i ∈ N . If i1 th, i2 th, ..., ik th bidders participate, then for each ik0 ∈ Ik , the probability that he gets the object is pik0 Ik (t0i1 , ..., t0ik ) = pˆIikk0 (mi1 (t0i1 ), ..., mik (t0ik )), and his payment is xik0 Ik (t0i1 , ..., t0ik ) = x ˆIikk0 (mi1 (t0i1 ), ..., mik (t0ik )) upon the submitted signals (t0i1 , ..., t0ik ). Here tik0 ∈ T , 1 ≤ k ≤ N and 1 ≤ i1 < i2 < ... < ik ≤ N . First, Let us verify that (pp (·), t(·)) is a Bayesian Nash equilibrium of auction game (T , p, x). Use

40

0(i)

Ui (p, x, ti , pp (·), t(·); pp , t0 i ) to denote the expected utility of bidder i with value ti if he participates with 0(i)

and signals t0 i when participating. I have

probability of pp

0 Ui (p, x, ti , pp (·), t(·); p0(i) p , t i) (

=

p0(i) p

·

{i} (ti pi (t0i )

−

{i} xi (t0i ))

Z

Y (1 − p(j) p (tj ))f (t−i )dt−i T−i j6=i

+

Z

−1 n NX

T−i

h Y

X

ik0 ∈Ik

k=1 Ik ⊂N \{i}

=

(

{i}

{i}

(1 − p(j) p (tj )) ×

j ∈I / k ∪{i}

io  I ∪{i} I ∪{i} (tIk , t0i ) − xi k (tIk , t0i ) f (t−i )dt−i ti pi k p0(i) p

Y

k0 ) (t p(i ik0 ) × p

· (ti pˆi (mi (t0i )) − x ˆi (mi (t0i )))

Z

)

(i)

+ (1 − p0(i) p )U0

Y (1 − p(j) p (tj ))f (t−i )dt−i T−i j6=i

+

Z

−1 n NX

T−i

X

k=1 Ik ⊂N \{i}

h Y

k0 ) (t p(i ik0 ) × p

ik0 ∈Ik

Y

(1 − p(j) p (tj )) ×

j ∈I / k ∪{i}

 io I ∪{i} I ∪{i} ti pˆi k (mi1 (ti1 ), ..., mi (t0i ), ..., mik (tik )) − x ˆi k (mi1 (ti1 ), ..., mi (t0i ), ..., mik (tik )) f (t−i )dt−i

)

(i)

+(1 − p0(i) p )U0

0 = Ui (ˆ p, x ˆ, ti , pp (·), m(·); p0(i) p, x ˆ, ti , pp (·), m(·); p(i) p , mi (ti )) ≤ Ui (ˆ p (ti ), mi (ti )) ( Z Y {i} {i} = pip (ti ) · (ti pˆi (mi (ti )) − x ˆi (mi (ti ))) (1 − p(j) p (tj ))f (t−i )dt−i T−i j6=i

+

Z T−i

−1 n NX

X

k=1 Ik ⊂N \{i}

h Y

k0 ) (t p(i ik0 ) × p

ik0 ∈Ik

Y

(1 − p(j) p (tj )) ×

j ∈I / k ∪{i}

 io I ∪{i} I ∪{i} ti pˆi k (mi1 (ti1 ), ..., mi (ti ), ..., mik (tik )) − x ˆi k (mi1 (ti1 ), ..., mi (ti ), ..., mik (tik )) f (t−i )dt−i

)

(i)

+(1 − pip (ti ))U0

= Ui (p, x, ti , pp (·), t(·); pip (ti ), ti (ti )). Second, by construction (T , p, x; pp (·), t(·)) is truth reporting, and the auctioneer’s and bidders’ expected revenues from the new mechanism are the same as those from the original given mechanism (M, pˆ, x ˆ; pp (·), m(·)). 2

41

Appendix C Proof of Proposition 3: Based on the revelation principal in Appendix B, I only need to show that any truthful direct revelation mechanism implementing stochastic participation of bidders is dominated by a cutoff shutdown truthful direct revelation mechanism. The basic idea of the proof is that for any truthful direct revelation mechanism implementing stochastic participation, I can construct a cutoff shutdown truthful direct revelation mechanism, which dominates the original one in terms of the auctioneer’s expected revenue. Due to complex notation, I only give the proof for the two potential bidders case. Denote the original direct revelation mechanism by {p, x} = {(p(1) (t1 ), p(2) (t2 ), p(1) (t1 , t2 ), p(2) (t1 , t2 )), (x(1) (t1 ), x(2) (t2 ), x(1) (t1 , t2 ), x(2) (t1 , t2 ))}, where the p(i) functions are the probabilities for bidder i to win the object if he participates and the x(i) functions are the bidder i’s payments if he participates, and ti is signal submitted by bidder i when he (1)

(2)

(i)

participates. Denote the equilibrium participation by pp (·) = (pp (t1 ), pp (t2 )), where pp (·) function is (i)

the probability of bidder i’s participation. Use U0 (i)

Appendix B, for generality I allow that U0

denote bidder i’s reservation utility level. Like in

(i)

and pp (·) to be different across bidders and {p, x} are not

restricted to be symmetric across bidders. 0(i)

Following the notation in Appendix B, use Ui (p, x, ti , pp (·), t(·); pp , t0 i ) to denote the expected utility 0(i)

of bidder i with value ti if he participates with probability of pp

and signals t0 i when participating.

(i)

Then Ui (p, x, ti , pp (·), t(·); pp (ti ), ti ) is the expected utility of bidder i with value ti if he participates (i)

with probability of pp (ti ) and signals ti when participating. Then we have (i)

(i)

(i)

(i)

Ui (p, x, ti , pp (·), t(·); pp (ti ), ti ) = pp (ti )Ui (p, x, ti , pp (·), t(·); 1, ti ) + (1 − pp (ti ))U0 . Define tci = inf {Ui (p,x,ti ,pp (·),t(·);p(i) (ti ),ti )>U (i) } (ti ). I have the following Lemma. p

0

(i)

(i)

Lemma C.1: For any i ∈ {1, 2}, ∀ti > tci , I must have Ui (p, x, ti , pp (·), t(·); pp (ti ), ti ) > U0 , which im(i)

plies that Ui (p, x, ti , pp (·), t(·); 1, ti ) > U0

(i)

and pp (ti ) = 1; for ti < tci , I must have Ui (p, x, ti , pp (·), t(·);

42

(i)

(i)

(i)

pp (ti ), ti ) = U0 , which implies Ui (p, x, ti , pp (·), t(·); 1, ti ) ≤ U0 ; moreover Ui (p, x, tci , pp (·), t(·); 1, tci ) = (i)

(i)

(i)

U0 , which implies Ui (p, x, tci , pp (·), t(·); pp (tci ), tci ) = U0 . Proof of Lemma C.1: see Appendix D. (i)

(i)

Lemma C.2: For any ti < tci with pp (ti ) > 0, I must have Ui (p, x, ti , pp (·), t(·); 1, ti ) = U0 , moreover bidder i with value ti has no chance to win the object, and the auctioneer’s expected subsidy to this bidder (i)

is U0

if he participates in order to get him to participate with a positive probability.

Proof of Lemma C.2: The intuition behind this result is the following. If this is not the case, (i) then for t˜i ∈ (ti , tci ], Ui (p, x, t˜i , pp (·), t(·); 1, ti ) > Ui (p, x, ti , pp (·), t(·); 1, ti ) = U0 , which conflicts with (i)

(i)

Ui (p, x, t˜i , pp (·), t(·); pp (t˜i ), t˜i ) = U0 . Thus no matter bidder i participates or not, he has no chance to win when ti < tci . The following is a detailed proof for the above mentioned results. (1)

Consider bidder 1, ∀ t1 < tc1 and pp (t1 ) > 0. When bidder 2 participates with the equilibrium (2)

probability pp (t2 ) and reveals his true type when participating, I must have U1 (p, x, t1 , pp (·), t(·); 1, t1 ) Z t (1) = t1 [p(1) (t1 , t2 )p(2) (t1 )(1 − p(2) p (t2 ) + p p (t2 )] t

 (1) (1) −[p(2) (t1 , t2 ) + (1 − p(2) (t1 )] f (t2 )dt2 p (t2 )x p (t2 ))x

(1)

= U0 .

(C.1)

For ∀ t˜1 ∈ (t1 , tc1 ), I have U1 (p, x, t˜1 , pp (·), t(·); 1, t1 ) Z t (1) = (t1 )(1 − p(2) t˜1 [p(1) (t1 , t2 )p(2) p (t2 ) + p p (t2 )] t

 (1) (1) −[p(2) (t1 , t2 ) + (1 − p(2) (t1 )] f (t2 )dt2 p (t2 )x p (t2 ))x Z t  (1) ˜ p(1) (t1 , t2 )p(2) = U1 (p, x, t1 , t1 ) + (t1 − t1 ) (t1 )(1 − p(2) p (t2 ) + p p (t2 ) f (t2 )dt2 t

Z t  (1) (1) ˜ = U0 + (t1 − t1 ) (t1 )(1 − p(2) p(1) (t1 , t2 )p(2) p (t2 ) + p p (t2 ) f (t2 )dt2 . t

43

(C.2)

(1)

Since I must have U1 (p, x, t˜1 , pp (·), t(·); 1, t1 ) ≤ U1 (p, x, t˜1 , pp (·), t(·); 1, t˜1 ) ≤ U0 , thus Z t  (1) (2) p(1) (t1 , t2 )p(2) (t ) + p (t )(1 − p (t ) f (t2 )dt2 = 0, ∀t1 < tc1 , p(1) 2 1 2 p p p (t1 ) > 0.

(C.3)

t

(C.3) implies that Z

t2 c

  (1) c (1) p(1) (t1 , t2 )p(2) (t1 )(1 − p(2) p (t2 ) + p p (t2 ) f (t2 )dt2 = 0, ∀t1 < t1 , pp (t1 ) > 0,

t

Z

t t2 c

  (1) (2) p(1) (t1 , t2 )p(2) (t ) + p (t )(1 − p (t ) f (t2 )dt2 = 0, ∀t1 < tc1 , p(1) 2 1 2 p p p (t1 ) > 0.

Similarly, I have Z t  (2) (1) p(2) (t1 , t2 )p(1) (t ) + p (t )(1 − p (t ) f (t1 )dt1 = 0, ∀t2 < tc2 , p(2) 1 2 1 p p p (t2 ) > 0.

(C.4)

t

(C.4) implies that Z

t1 c

  (2) c (2) p(2) (t1 , t2 )p(1) (t2 )(1 − p(1) p (t1 ) + p p (t1 ) f (t1 )dt1 = 0, ∀t2 < t2 , pp (t2 ) > 0,

t

Z

t t1 c

  (2) (1) p(2) (t1 , t2 )p(1) (t ) + p (t )(1 − p (t ) f (t1 )dt1 = 0, ∀t2 < tc2 , p(2) 1 2 1 p p p (t2 ) > 0.

(C.1) and (C.3) give that Z

t

(1)

(C.5)

(2)

(C.6)

(1) (1) −[p(2) (t1 , t2 ) + (1 − p(2) (t1 )]f (t2 )dt2 = U0 , ∀t1 < tc1 , p(1) p (t2 )x p (t2 ))x p (t1 ) > 0. t

Similarly, I have Z

t

(2) (2) −[p(1) (t1 , t2 ) + (1 − p(1) (t2 )]f (t1 )dt1 = U0 , ∀t2 < tc2 , p(2) p (t1 )x p (t1 ))x p (t2 ) > 0. t

(C.3) and (C.4) indicate that at the equilibrium if bidder i with ti < tci participates with a positive probability, then he has no chance to win when he participates. (C.5) and (C.6) indicate that in order to get him to participate with a positive probability, the auctioneer’s expected subsidy to this bidder is (i)

U0 if he participates. 2

44

Thus based on Lemma C.2, if the auctioneer shuts down this bidder, he will save this subsidy but has nothing to lose. This is the basic reason why the cutoff shutdown policy dominates the stochastic shutdown policy. Next I will construct a new cutoff shutdown mechanism from the original one but providing a higher expected revenue to the auctioneer. The auctioneer’s expected revenue from the original mechanism is:

R0

Z tZ tn  (2) (1) (2) (1) t0 1 − p(1) = (t1 ) − p(1) (t1 , t2 ) + p(2) (t1 , t2 )) p (t1 )(1 − pp (t2 ))p p (t1 )pp (t2 )(p t

t

  (2) (2) (2) (1) (2) (2) −(1 − p(1) (t2 ) + p(1) (t1 ) + (1 − p(1) (t2 ) p (t1 ))pp (t2 )p p (t1 )(1 − pp (t2 ))x p (t1 ))pp (t2 )x (2) (1) +p(1) (t1 , t2 ) + x(2) (t1 , t2 )) p (t1 )pp (t2 )(x (1)

(2)

(3)

o

f (t1 )f (t2 )dt1 dt2

(4)

= R0 + R0 + R0 + R0 , (1)

(2)

(3)

(4)

where R0 , R0 , R0 , R0 (1) R0

=

Z

tc1 t

Z

tc2

(C.7)

are defined as below.

n  (2) (1) (2) (1) t0 1 − p(1) (t1 ) − p(1) (t1 , t2 ) + p(2) (t1 , t2 )) p (t1 )(1 − pp (t2 ))p p (t1 )pp (t2 )(p

t

  (2) (2) (2) (1) (2) (2) −(1 − p(1) (t2 ) + p(1) (t1 ) + (1 − p(1) (t2 ) p (t1 ))pp (t2 )p p (t1 )(1 − pp (t2 ))x p (t1 ))pp (t2 )x (2) (1) +p(1) (t1 , t2 ) + x(2) (t1 , t2 )) p (t1 )pp (t2 )(x Z tc1 Z tc2 = t0 f (t1 )f (t2 )dt1 dt2 t

−

Z

−

Z

+

Z

+

Z

t tc1

t0 p(1) p (t1 )

Z

t0 p(2) p (t2 )

Z

t tc2 t tc1

p(1) p (t1 )

Z

p(2) p (t2 )

Z

t tc2 t

o

f (t1 )f (t2 )dt1 dt2

tc2

   (1) (2) p(1) (t1 , t2 )p(2) f (t1 )dt1 (t ) + p (t )(1 − p (t ) f (t )dt 2 1 2 2 2 p p

tc1

   (2) p(2) (t1 , t2 )p(1) (t2 )(1 − p(1) p (t1 ) + p p (t1 ) f (t1 )dt1 f (t2 )dt2

t

t tc2



  (1) x(1) (t1 , t2 )p(2) (t1 )(1 − p(2) p (t2 ) + x p (t2 ) f (t2 )dt2 f (t1 )dt1

tc1



  (2) x(2) (t1 , t2 )p(1) (t2 )(1 − p(1) p (t1 ) + x p (t1 ) f (t1 )dt1 f (t2 )dt2

t

t

= A1 + B1 + C1 + D1 + E1 ,

(C.8)

45

where

A1

Z

=

t

B1

Z

tc1

=−

Z

=−

Z

tc2

t0 f (t1 )f (t2 )dt1 dt2 , t

tc1

t0 p(1) p (t1 )

Z

t0 p(2) p (t2 )

Z

t

C1 D1 E1

=

Z

=

Z

tc2

t tc1

p(1) p (t1 )

Z

p(2) p (t2 )

Z

t tc2 t

(2) R0

=

Z

tc1

tc2



  (1) (t1 )(1 − p(2) p(1) (t1 , t2 )p(2) p (t2 ) + p p (t2 ) f (t2 )dt2 f (t1 )dt1 ,

tc1



  (2) (t2 )(1 − p(1) p(2) (t1 , t2 )p(1) p (t1 ) + p p (t1 ) f (t1 )dt1 f (t2 )dt2 ,

t

t tc2

   (1) x(1) (t1 , t2 )p(2) (t1 )(1 − p(2) p (t2 ) + x p (t2 ) f (t2 )dt2 f (t1 )dt1 ,

tc1

   (2) (1) x(2) (t1 , t2 )p(1) (t ) + x (t )(1 − p (t ) f (t )dt 1 2 1 1 1 f (t2 )dt2 . p p

t

t

Z tn  (2) (1) (2) (1) (t1 ) − p(1) (t1 , t2 ) + p(2) (t1 , t2 )) t0 1 − p(1) p (t1 )(1 − pp (t2 ))p p (t1 )pp (t2 )(p tc2

t

  (2) (2) (2) (1) (2) (2) −(1 − p(1) (t2 ) + p(1) (t1 ) + (1 − p(1) (t2 ) p (t1 ))pp (t2 )p p (t1 )(1 − pp (t2 ))x p (t1 ))pp (t2 )x (2) (1) +p(1) (t1 , t2 ) + x(2) (t1 , t2 )) p (t1 )pp (t2 )(x Z tc1 Z t = t0 f (t1 )f (t2 )dt1 dt2 tc2

t

−

Z

−

Z

+

Z

+

Z

tc1

t0 p(1) p (t1 )

Z t  tc2

t t tc2

t0 p(2) p (t2 )

Z



f (t1 )f (t2 )dt1 dt2

  (1) (2) p(1) (t1 , t2 )p(2) f (t1 )dt1 (t ) + p (t )(1 − p (t ) f (t )dt 2 1 2 2 2 p p   (2) (1) (t ) + p (t )(1 − p (t ) f (t )dt p(2) (t1 , t2 )p(1) 1 2 1 1 1 f (t2 )dt2 p p

t

tc1

p(1) p (t1 )

Z t tc2

t t tc2

tc1

o

p(2) p (t2 )

Z

tc1



  (1) x(1) (t1 , t2 )p(2) (t1 )(1 − p(2) p (t2 ) + x p (t2 ) f (t2 )dt2 f (t1 )dt1   (2) x(2) (t1 , t2 )p(1) (t2 )(1 − p(1) p (t1 ) + x p (t1 ) f (t1 )dt1 f (t2 )dt2

t

= A2 + B2 + C2 + D2 + E2 ,

(C.9)

(2)

with pp (t2 ) = 1, where A2

=

Z

tc1 t

B2

=−

Z t

Z

t

t0 f (t1 )f (t2 )dt1 dt2 , tc2

tc1

t0 p(1) p (t1 )

Z t tc2

  (1) p(1) (t1 , t2 )p(2) (t1 )(1 − p(2) p (t2 ) + p p (t2 ) f (t2 )dt2 f (t1 )dt1 ,

46

C2

=−

D2

=

Z

=

Z

Z

t

tc2 tc1

t0 p(2) p (t2 )

(3)

R0

=

Z tZ tc1

tc2



  (2) (1) p(2) (t1 , t2 )p(1) f (t2 )dt2 , (t ) + p (t )(1 − p (t ) f (t )dt 1 2 1 1 1 p p

t

t

p(2) p (t2 )

tc2

tc1

 Z t  (1) x(1) (t1 , t2 )p(2) p(1) (t ) (t1 )(1 − p(2) 1 p p (t2 ) + x p (t2 ) f (t2 )dt2 f (t1 )dt1 , tc2

t

E2

Z

Z

tc1

   (2) x(2) (t1 , t2 )p(1) (t2 )(1 − p(1) p (t1 ) + x p (t1 ) f (t1 )dt1 f (t2 )dt2 .

t

n  (2) (1) (2) (1) t0 1 − p(1) (t1 ) − p(1) (t1 , t2 ) + p(2) (t1 , t2 )) p (t1 )(1 − pp (t2 ))p p (t1 )pp (t2 )(p

t

  (2) (2) (2) (1) (2) (2) −(1 − p(1) (t ))p (t )p (t ) + p(1) (t1 ) + (1 − p(1) (t2 ) 1 2 2 p p p (t1 )(1 − pp (t2 ))x p (t1 ))pp (t2 )x (2) (1) (t1 , t2 ) + x(2) (t1 , t2 )) +p(1) p (t1 )pp (t2 )(x Z t Z tc2 = t0 f (t1 )f (t2 )dt1 dt2 tc1

o

f (t1 )f (t2 )dt1 dt2

t

−

Z

−

Z

+

Z

+

Z

t tc1

t0 p(1) p (t1 )

Z



  (1) p(1) (t1 , t2 )p(2) (t1 )(1 − p(2) p (t2 ) + p p (t2 ) f (t2 )dt2 f (t1 )dt1

Z t 

  (2) (1) p(2) (t1 , t2 )p(1) f (t2 )dt2 (t ) + p (t )(1 − p (t ) f (t )dt 1 2 1 1 1 p p

tc2 t

tc2

t0 p(2) p (t2 )

tc1

t t tc1

p(1) p (t1 )

Z



  (1) x(1) (t1 , t2 )p(2) (t1 )(1 − p(2) p (t2 ) + x p (t2 ) f (t2 )dt2 f (t1 )dt1

Z t

  (2) (t2 )(1 − p(1) x(2) (t1 , t2 )p(1) p (t1 ) + x p (t1 ) f (t1 )dt1 f (t2 )dt2

tc2 t

tc2

p(2) p (t2 )

t

tc1

= A3 + B3 + C3 + D3 + E3 ,

(C.10)

(1)

with pp (t1 ) = 1, where A3

=

Z tZ tc1

B3

tc2

t0 f (t1 )f (t2 )dt1 dt2 , t

=−

Z

=−

Z

t

tc1

C3

t0 p(1) p (t1 )

tc2

Z



  (1) (2) p(1) (t1 , t2 )p(2) f (t1 )dt1 , (t ) + p (t )(1 − p (t ) f (t )dt 2 1 2 2 2 p p

Z t

  (2) (1) p(2) (t1 , t2 )p(1) (t ) + p (t )(1 − p (t ) f (t )dt 1 2 1 1 1 f (t2 )dt2 , p p

t

t0 p(2) p (t2 )

t

D3 E3

=

Z

=

Z

t tc1

p(1) p (t1 )

tc2 t

tc2

Z

tc2

tc1

   (1) x(1) (t1 , t2 )p(2) (t1 )(1 − p(2) p (t2 ) + x p (t2 ) f (t2 )dt2 f (t1 )dt1 ,

t

 Z t  (2) (2) x(2) (t1 , t2 )p(1) pp (t2 ) (t2 )(1 − p(1) p (t1 ) + x p (t1 ) f (t1 )dt1 f (t2 )dt2 . tc1

47

(4)

R0

=

Z tZ tn  (2) (1) (2) (1) t0 1 − p(1) (t1 ) − p(1) (t1 , t2 ) + p(2) (t1 , t2 )) p (t1 )(1 − pp (t2 ))p p (t1 )pp (t2 )(p tc1

tc2

  (2) (2) (2) (1) (2) (2) −(1 − p(1) (t2 ) + p(1) (t1 ) + (1 − p(1) (t2 ) p (t1 ))pp (t2 )p p (t1 )(1 − pp (t2 ))x p (t1 ))pp (t2 )x o (2) (1) +p(1) (t1 , t2 ) + x(2) (t1 , t2 )) f (t1 )f (t2 )dt1 dt2 p (t1 )pp (t2 )(x Z tZ t  t0 (1 − p(1) (t1 , t2 ) + p(2) (t1 , t2 )) + (x(1) (t1 , t2 ) + x(2) (t1 , t2 )) f (t1 )f (t2 )dt1 dt2 = tc1

(C.11)

tc2

(1)

(2)

as pp (t1 ) = 1, pp (t2 ) = 1. (C.3) and (C.4) imply that I have B1 = 0, C1 = 0, B2 = 0, C3 = 0, thus from (C.7), (C.8), (C.9), (C.10) and (C.11), I have R0

(1)

(2)

(3)

(4)

= R0 + R0 + R0 + R0 (4)

= R0 + (A1 + A2 + A3 ) + (D1 + D2 ) + (E1 + E3 ) + (C2 + E2 ) + (B3 + D3 ).

(C.12)

Let us define a new direct revelation mechanism by {˜ p, x ˜} = {(˜ p(1) (t1 ), p˜(2) (t2 ), p˜(1) (t1 , t2 ), p˜(2) (t1 , t2 )), ˜(1) (t1 , t2 ), x˜(2) (t1 , t2 ))} as below, where the p˜i functions are the bidder i’s probabilities (˜ x(1) (t1 ), x˜(2) (t2 ), x to win if he participates and the x ˜i functions are bidder i’s payments if he participates.

(1)

p˜ (t1 ) =

x ˜(1) (t1 ) =

(2)

p˜ (t2 ) =

x ˜(2) (t2 ) =

   

1 F (tc2 )

R tc2  t

 (2) (2) p(1) (t1 , t2 )pp (t2 ) + p(1) (t1 )(1 − pp (t2 ) f (t2 )dt2

   0    

1 F (tc2 )

R tc2  t

 (2) (2) x(1) (t1 , t2 )pp (t2 ) + x(1) (t1 )(1 − pp (t2 ) f (t2 )dt2

   0    

1 F (tc1 )

R tc1  t

 (1) (1) p(2) (t1 , t2 )pp (t1 ) + p(2) (t2 )(1 − pp (t1 ) f (t1 )dt1

   0    

1 F (tc1 )

R tc1  t

 (1) (1) x(2) (t1 , t2 )pp (t1 ) + x(2) (t2 )(1 − pp (t1 ) f (t1 )dt1

   0

48

if

t1 ≥ tc1 ,

if

t1 < tc1 .

if

t1 ≥ tc1 ,

if

t1
tci , I have Ui (p, x, ti , pp (·), t(·); 1, ti ) ≥ Ui (p, x, ti , pp (·), t(·); 1, tˆi ) ≥ (i)

(i)

Ui (p, x, tˆi , pp (·), t(·); 1, tˆi ) > U0 . The above result implies pp (ti ) = 1. (i)

(i)

When ti < tci , from the definition of tci , I must have Ui (p, x, ti , pp (·), t(·); pp (ti ), ti ) = U0 as alterna(i)

(i)

tive participation probability of 0 guarantees that Ui (p, x, ti , pp (·), t(·); pp (ti ), ti ) = U0 (i)

cannot be less

(i)

than U0 . Thus I have Ui (p, x, ti , pp (·), t(·); 1, ti ) ≤ U0 . (i)

I will show Ui (p, x, tci , pp (·), t(·); 1, tci ) = U0 by the following two steps. First, as ti − tci → 0+ , I have Ui (p, x, ti , pp (·), t(·); 1, ti ) − Ui (p, x, tci , pp (·), t(·); 1, ti ) → 0, which implies (i)

that I must have Ui (p, x, tci , pp (·), t(·); 1, tci ) ≥ U0 . Suppose this is not the case, as Ui (p, x, tci , pp (·), t(·); 1, (i)

ti ) ≤ Ui (p, x, tci , pp (·), t(·); 1, tci ) and Ui (p, x, ti , pp (·), t(·); 1, ti ) > U0 , Ui (p, x, ti , pp (·), t(·); 1, ti )−Ui (p, x, tci , pp (·), t(·); 1, ti ) → 0 cannot hold. Second, as ti − tci → 0− , I have Ui (p, x, ti , pp (·), t(·); 1, tci ) − Ui (p, x, tci , pp (·), t(·); 1, tci ) → 0, which (i)

implies that I must have Ui (p, x, tci , pp (·), t(·); 1, tci ) ≤ U0 . Suppose this is not the case, as Ui (p, x, ti , pp (·), (i)

t(·); 1, ti ) ≥ Ui (p, x, ti , pp (·), t(·); 1, tci ) and Ui (p, x, ti , pp (·), t(·); 1, ti ) ≤ U0 , Ui (p, x, ti , pp (·), t(·); 1, ti ) − Ui (p, x, tci , pp (·), t(·); 1, ti ) → 0 cannot hold. (i)

Aggregating the above results leads to Ui (p, x, tci , pp (·), t(·); 1, tci ) = U0 , which implies Ui (p, x, tci , pp (·), (i)

(i)

t(·); pp (tci ), tci ) = U0 . 2

53

References [1] I. Brocas, Endogenous entry in auctions with negative externalities, Theory and Decision, 54 (2003), 125-149 [2] R. Engelbrecht-Wiggans, On optimal reserve prices in auctions, Management Science, 33 (1987), 763-770. [3] R. Engelbrecht-Wiggans, Optimal auction revisited, Games and Economic Behavior, 5 (1993), 227-239. [4] P. Jehiel, B. Moldovanu and E. Staccheti, How (not) to sell nuclear weapons, Am. Econom. Rev., 86 (1996), 814-929. [5] D. Levin and J.L. Smith, Equilibrium in auctions with entry, Am. Econom. Rev., 84 (1994), 585-599. [6] J. Lu, Optimal auction design with two-dimension private signals, typescript, University of Southern California, March, 2003, available at http://www-rcf.usc.edu/∼jlu. [7] R. P. McAfee and J. McMillan, Auctions with entry, Economics Letters, 23 (1987), 343-347. 43 (1987), 1-19. [8] R. P. McAfee and J. McMillan, Government procurement and international trade, J. of International Economics, 26 (1989), 291-308. [9] F. M. Menezes and P. K. Monteiro, Auction with endogenous participation, Rev. of Econom. Design, 5 (2000), 71-89. [10] R. B. Myerson, Optimal auction design, Mathematics of Operation Research, 6 (1981), 58-73. [11] W.F. Samuelson, Competitive bidding with entry costs, Economics Letters, 17 (1985), 53-57. [12] M. Stegeman, Participation costs and efficient auctions, J. of Econom. Theory, 71 (1996): 228-259. [13] G. Tan, Entry and R&D in procurement contracting, J. of Econom. Theory, 58 (1992), 41-60. [14] Tan G. and Yilankaya O., Equilibria in second price auctions with participation costs, Working Paper, 2003

54

Seller's Expected Revenue 0.5 0.45

R*

0.4 0.35 0.3

k=2

0.25

k=5

0.2

k=10

0.15 0.1 0.05 0 1

3

5

7

9

11

13

15

N

Figure 1: Uo = 0.3, to = 0.0

55

17

19

Seller's Expected Revenue 1.2 1

R*

0.8

Uo=0.0

Uo=0.1

0.6

Uo=0.2

Uo=0.3

0.4 0.2 0 1

3

5

7

9

11

13

15

N

Figure 2: k = 10.0, to = 0.0

56

17

19

Seller's Expected Revenue 1 0.9

R*

0.8 0.7 0.6

k=2

0.5

k=5

0.4

k=10

0.3 0.2 0.1 0 0

0.1

0.2 N

Figure 3: N = 2, to = 0.0

57

0.3