Cross-Owned Firms Competing in AuctionsOttorino Chillemi 1

Department of Economics, University of Padua, Italy

Abstract

The paper studies the e¤ect of ownership links among bidders in auctions. Firstly,it is shown that in …rst-price, in second-price, and in all-pay auctions, ownership linksdamage both the seller and society; the bidders too may be impaired by the seller’sstrategic reaction. Secondly, the optimal selling procedure is characterized: in sharpcontrast with standard auctions, both the seller and society gain from ownership links.In the last part of the paper the analysis is extended to the case of strategic entry.

Running Title: Cross-Owned Firms Competing in Auctions. Keywords J .E .L:D44, D62.

Address for correspondence: Ottorino Chillemi , Via del Santo 33, 35123 PadovaItaly.

(tel: +39 049 8274224; fax: +39 0498274211; e-mail: chillemi@decon.unipd.it)

1 I am grateful to Michele Piccione for early discussions on the theme, and toGiovanni Colombo, Gianni De Fraja and Benedetto Gui for useful advice. I alsothank the audiences at the 2000 Conference on Economic Design (Istanbul), at theIRC University of Minnesota, at the University of Bologna, and at the 2003 LGS3Conference (Siena). The …nancial support of the Italian Ministry for Universities isgratefully acknowledged.

1

1 IntroductionDoes cross holding among competing …rms hurt society ? While a …rm’s takeoverof a competitor raises obvious concern about adverse e¤ects on competition,several economists have expressed worry also about partial ownership, and evenpassive investment when neither anticompetitive intention nor tacit collusionexists.2 For instance, Reynolds and Snapp (1986) show that ownership linkslessen competition in non-collusive oligopolistic markets and, as a consequence,bring about both an increase in the equilibrium price and a reduction in the out-put level (see also Bresnahan and Salop, 1986). American antitrust authorities,on the other hand, have challenged ownership arrangements, under the ShermanAct if there was evidence of anticompetitive intent, and under the Clayton Actif they conveyed control or accounted for a large share of industry output. Inestablishing their tolerant approach, American agencies have probably consid-ered both the pros and cons of lessening competition – in particular the positivee¤ects on R&D and technology transfers (for more on this, see Reynolds andSnapp, 1986). However, thorough examination of sentences reveals that theyseem to ignore the importance of a …rm’s strategic motivation for passive in-vestments, namely that a commitment to a less competitive behavior, which isimplicit in the investment decision, might induce rivals to behave less competi-tively themselves; hence the claim that antitrust agencies should be less toleranton passive ownership links between competitors (Gilo, 2000).

This paper focuses on passive ownership links among bidders in auctions.In the literature, property linkages between the seller and a bidder have beenconsidered in Bulow et al. (1999), where it is shown that bidding becomesmore aggressive. However, to the best of my knowledge, ownership links amongbidders have not been examined. The legal practice for auctions and for publicprocurement in the U.S. and the E.U. makes no speci…c provision about passiveownership. One exception is the Italian law for public procurement, whichforbids the simultaneous participation in the same auction of both a controlledcorporation and its controlling one; double participation is also forbidden, inthat a member of a consortium that participates in the auction is not allowedto participate individually. The rationale may be to prevent collusive behavior(De Fraja, 2002). All in all, passive ownership links among bidders does notseem to have attracted the attention of economists or legislators.

The impact of cross holding on the outcome of an auction is not obvious.Two opposite intuitions come to mind. The …rst one is that by decreasing itsbid, a …rm may induce rivals to do the same, with a positive impact on thewinner’s surplus, an adverse impact on the seller’s revenue, and perhaps anadverse impact on social surplus if the seller’s best response is to increase hisreservation price. Another intuition, however, points out that bidding …rms dohave an interest in the mere fact that the object is sold, because losing …rms willshare in the winner’s surplus. A clever seller could …nd a way to take advantage

2 Passive investments are those in which investors do not seek to gain in‡uence over thecompetitor’s activities or to access the competitor’s sensitive information (see Gilo, 2000, foran extensive discussion of passive investments).

2

of this by extracting some of the surplus from losing bidders; in this manner,perhaps he could gain, rather than lose from the presence of ownership links.

In this paper, we show that the optimal auction in the presence of ownershiplinks among bidders exploits the second intuition. The seller extracts part ofeach …rm’s (direct and indirect) surplus by committing not to run the auctionunless each potential bidder pays for the auction to be held.3 For concreteness,we shall refer to the following set up. The seller of an indivisible object facesN ¸ 2 buyers and M ¸ N holding companies. Each holding company controlsat most one buyer and owns minority shares in the remaining ones; each buyer’sdecisions are made in the sole interest of the controlling holding; the seller hasno proprietary links with any buyers. This set up captures important aspects ofproprietary links in the real world. For instance, in Italy, the most common formof company control is through pyramidal groups headed by a holding companyat the top (Brunello et al. 2001); pyramidal groups are also very frequentin France and Belgium (see Becht and Roell, 1999, who compare ContinentalEurope with the U.K. and the U.S.).4 Holding companies are very importantalso in East Asia: in 1980, in Japan 65 of the 100 largest …rms belonged tothe 16 largest holding companies, which controlled 26% of the capital of all non…nancial companies (Hamilton et al., 1990).

The paper is organized as follows. Section 2 deals with one-stage auctions;since entry decisions and bids are simultaneous, we can safely assume that eachpotential bidder participates in the auction. Firstly, the equilibrium of the …rst-price sealed-bid auction is characterized; then, the analysis is extended to thesecond-price and the all-pay auctions; …nally, the revenue-maximizing sellingprocedure is studied. Section 3 deals with two-stage auctions in which bids aremade after each player has observed the entry decisions of the others; in this case…rms face a trade-o¤ between entering and staying away. Section 4 summarizesthe results and concludes.

2 The one-stage modelThe following is common knowledge. A risk-neutral, revenue maximizing sellerhas a single, indivisible object to sell. His valuation of the object is zero. Thereare N risk-neutral …rms indexed by i 2 I = f1; ::;Ng. Each …rm’s possible valu-ation of the object, v; is continuously distributed and drawn independently froma common distribution, F , which is strictly increasing and at least twice contin-uously di¤erentiable on the interval T = [0; v+]; with F (0) = 0 and F (v+) = 1.

3 The situation is reminiscent of Jehiel et al.(1996), where the players that would be dam-aged by the sale of an object pay the seller to renounce selling it. However, both in thatpaper and a later one by the same authors (Jehiel et al., 1999), which extends the analysisto auctions with negative externalities, it is assumed that external e¤ects are of a given size;instead, in our framework, they are endogenously determined.

4 Interestingly, the control premium for quoted stocks is very high in Italy: Zingales (1994)estimates that the private bene…t of control is worth more than 60 percent of the value ofnonvoting equity. See also Bianco and Casavola, 1999, who give empirical evidence on thelimited protection of minority shareholders in Italy.

3

The probability density for v on T is f = F0. The following standard regularity

condition holds: f(v)1¡F (v) is strictly increasing with v. Each …rm maximizes its

total expected payo¤, which is a weighted sum of the surpluses of all …rms:the non-negative weight …rm i gives to the surplus of …rm k is denoted by ®ik;

k = 1; :::; N . It is assumed that ®ii >P

k 6=i ®ik; i; k 2 I; andP

k 6=i®ik

®ii= ¸ for

all i; so bidders’ symmetry obtains, as will be clear in a moment. We will call¸ 2 [0; 1) the concern parameter. Finally, we assume that the bidders do notcollude.

The model naturally extends the benchmark auction model (see e.g. McAfeeand McMillan, 1987) to the case of ownership links. The structure of weightsmirrors the property structure we discussed in the introduction. For example,let N = 2, M = 3 and suppose that the holding in control of …rm 1 owns a 40%share in this …rm and a 20% share in …rm 2, the holding in control of …rm 2owns a 60% share in this …rm, and a 30% share in …rm 1, and the third holdingowns the remaining shares, namely 30% of …rm 1 and 20% of …rm 2. Then itis ®11 = :4; ®12 = :2; ®21 = :3; ®22 = :6; and therefore ¸ = 1=2: 5

Having described our general setting, we are ready to examine some speci…cselling procedures.

2.1 The …rst-price sealed-bid auctionIn this auction, the seller openly announces a reserve price bs 2 [0; v+]; next,bidder i submits a sealed bid bi 2 [0; vi] i 2 I; …nally, if the highest bid is greaterthan the reserve price, the highest bidder is awarded the object at a price equalto its o¤er — otherwise the seller keeps the object. The losers pay nothing.

Nash equilibria of this auction could be characterized by standard meth-ods; however, it is illuminating to resort to the revelation mechanism approach.In this approach, …rst the seller announces pi = pi(vd) : TN ! R+; andxi = xi(vd) : TN ! R+; i 2 I, which are, respectively, bidder i’s probabil-ity of getting the object and its payment to the seller when vd is the vectorof reported types, that is the vector of reported valuations. Then, the bidderssimultaneously and con…dentially report their valuations to the seller. Finally,payments are made and the good is assigned through a lottery with the an-nounced probabilities. Thanks to the revelation principle we can restrict our at-tention to the truth-telling equilibria. Let f(v) =

Qj f(vj); dv =

Qj dvj ;v¡i =

(v1; v2; ::; vi¡1;vi+1; :::; vn) 2 T¡i;¡vd

i ;v¡i¢

= (v1;v2; :::; vi¡1; vdi ; vi+1; ::; vn); f(v¡i) =Q

j 6=i f(vj); and dv¡i =Q

j 6=i dvj : Bidder i’s total payo¤, when it is of type vi

and reports vdi while the other bidders are telling the truth, is Ui

¡vi;vd

i¢

=5 A similar structure applies to the case in which …rms have direct ownership links. For

instance, let N = 2 and suppose each …rm owns 60% of its equity and 40% of the other …rm.Then, letting Spi and Ui denote …rm i’s expected own surplus and overall surplus respectively,we have: U1 = 6

10Sp1 + 4

10U2 and U2 = 610Sp2 + 4

10U1; hence U1 = ®11Sp1 + ®12S

p2 implies

®11 = 6084 ; ®12 = 24

84 ; and analogously U2 = ®22Sp2+®21S

p1 implies ®22 = 60

84 ; ®21 = 2484 ; …nally

¸ = 410 :

4

RT¡i

Pk ®ik

£vkpk

¡vd

i ;v¡i¢

¡ xk(vdi ;v¡i)

¤f(v¡i)dv¡i. The seller’s expected

revenue when all bidders tell the truth is Us =RTN

PNk=1 xk(v)f (v) dv:

In searching for an optimal solution, we only consider policy functions thatare piecewise continuously di¤erentiable. Let Qi

¡vd

i¢

=RT¡i

pi¡vd

i ;v¡i¢f(v¡i)dv¡i

denote the probability that bidder i will get the object when it reports vdi while

the other bidders are telling the truth. Noticing that Ui¡vi; vd

i¢

= Ui¡vd

i ; vdi¢+

®ii¡vi ¡ vd

i¢Qi

¡vd

i¢

allows the incentive compatibility constraint IC(vi; vdi ) to

be written Ui (vi; vi) ¸ Ui¡vd

i ; vdi¢+ ®ii

¡vi ¡ vd

i¢Qi

¡vd

i¢. Lemma 2 in Myerson

(1981) ensures that in any feasible mechanism IC(vi; vdi ) can be replaced by the

following conditions: for all vi 2 T; i 2 I½a) Ui (vi; vi) = Ui (0; 0) + ®ii

R vi

0 Qi (si) dsib) Qi increasing (1):

Recalling the de…nition of Qi¡vd

i¢

allows us to writeRT¡i

¡Pk ®ik [vkpk (vi;v¡i) ¡ xk(vi;v¡i)] ¡ ®ii

R vi

0 pi (si;v¡i) dsi¢f (v¡i) dv¡i

= Ui (0; 0), vi 2 T; i 2 I (2).The rules of the …rst-price sealed-bid auction translate into the following

additional restrictions on the direct revelation mechanism. For v 2 TN ; i 2 I

R1 : pi (v) =½

1; if Bi (vi; bs) > Max(bs; Bj (vj ; bs)all j 6=i)0 otherwise

R2 : xi(v) =½

Bi (vi; bs) ; if pi (v) = 10; otherwise

where Bi (vi; bs) is bidder i’s optimal bid when it is of type vi and the reserveprice is bs:Now bidder i’s action set is the interval [0; vi] of possible reports, the seller’saction set is made up of the interval of possible reserve prices and the space offunctions Bi = Bi(vi; bs) : [0; vi]£T ! [0; vi] ; i 2 I: Thanks to our assumptions,we can focus on the class of bidding functions that are monotonically strictlyincreasing with the bidder’s type (later we will con…rm that this class of solutionsis not empty). The principal solves:

MaxfBigi2I ; bs Us

s.t. (1); R1; R2;PN

k=1 pk(v) · 1; pk (v) ¸ 0; v 2 TN ; k 2 I.Using R1 and R2, we can write (1), for vi ¸ bs and any given bs; as

®ii

h(vi ¡ Bi(vi))

Qj 6=i F

¡B¡1

j (Bi(vi))¢

¡R vi

bs

Qj 6=i F

¡B¡1

j (Bi(z))¢dz

i+

Pk 6=i ®ik

R v+

vi(vk ¡ Bk(vk))

Qj 6=i;j 6=k F

¡B¡1

j (Bi(vk))¢f (vk) dvk =

Pk 6=i ®ik

R v+

bs(vk ¡ Bk(vk))

Qj 6=i;j 6=k F

¡B¡1

j (Bi(vk))¢f (vk) dvk; i 2 I (3).

(Here I am simplifying notation, omitting the reserve price when confusion doesnot arise.) The left side of (3) is Ui(vi; vi) ¡ ®ii

R vi

0 Qi (si) dsi; the right side isUi(0; 0):

We will show that a symmetric solution exists and is unique in the classof symmetric solutions. Let Bi(vi) = B(vi); @B

@vi> 0, vi 2 (bs; v+) ; i 2 I.

Straightforward manipulations of (3) yieldB(vi) = vi ¡ 1

F (vi)N¡1 £

5

³R vi

bsFN¡1 (z) dz + ¸

R vi

bs(z ¡ B(z))F

N¡2

(z)f (z) dz´

; bs · vi · v+ (4):We can obtain a closed form for the bidding function as follow. Di¤erentiat-

ing (4) with respect to vi we get B0(vi) = (N ¡1¡¸) (vi ¡ B(vi)) f (vi) =F (vi).

This linear di¤erential equation, together with the condition B(bs) = bs; yieldsthe bidding function

B(vi) = vi¡ 1FN¡1¡¸(vi)

R vi

bsFN¡1¡¸(v)dv; bs · vi · v+ (5).

B has the properties we assumed above: it is continuously di¤erentiable andstrictly increasing with the bidder’s valuation.6 Notice also that B can be seen asa standard bidding function, in which the distribution of types, F , has replacedwith the distribution G± = F 1¡±, where ± = ¸=(N ¡ 1). G± is such that G±0

…rst order stochastically dominates G±00 whenever ±0 < ±00. This means that, forgiven bs; the bid of a …rm of a given valuation is decreasing with the intensityof ownership links:

Finally, the equilibrium bidding function Be is such that Be (vi) = B(vi; b¤s);

where b¤s maximizes Us: Now we are ready to prove the following.

Proposition 1. In the equilibrium of the …rst-price sealed-bid auction:a) If the object is sold, it is assigned to the bidder with the highest valuation.b) The optimal reserve price is higher than the seller’s valuation, hence the

selling procedure is not ex post e¢cient.c) Both the seller’s revenue and the social surplus are strictly decreasing

with the concern parameter ¸.Proof: See Appendix.Remark. In equilibrium, the ex ante expected surplus to a bidder may be

either decreasing or increasing with the concern parameter ¸ (see appendix).The intuition behind these results is as follows. Assume ¸ = 0 initially,

and then let ¸ increase by a small amount. Suppose that both the seller’sreserve price and bids remain …xed at the values which are optimal at ¸ = 0.A bidder’s expected marginal gain from increasing its bid, which previouslywas zero, now becomes negative, since winning the auction implies losing the“external” surplus deriving from ownership links. Therefore, bids decrease. Theonly instrument the seller has to oppose this e¤ect is to increase the reserve price,which, however, raises the probability of not selling the object and causes thesocial surplus to decrease. This line of reasoning can be repeated for any initialvalue of ¸. The seller’s revenue decreases because the e¤ect of bid reduction is…rst order while the increase in reserve price is only second order. Instead, bidderi’s ex ante expected surplus,

R 1b¤

s[vi ¡ Be(vi)]FN¡1(vi)f(vi)dvi; is a¤ected by

two …rst order e¤ects of opposite sign.Example 1. There are two bidders, and valuations are uniformly distrib-

uted in the unit interval. Under these assumptions, the bidding function isB(vi) = vi ¡ 1

vi1¡¸

R vi

bsv1¡¸dv; bs · vi · v+; i = 1; 2: The …rst order con-

dition for maximization of the seller’s objective function, 2R 1bs

B(vi)vidvi, is6 For 0 · v < bs, B(v) is not determined, but this is irrelevant as, by assumption, B(v) ·

v for all v:

6

¡b1¡¸s + 2b2s + b2

s¸ = 0: Therefore, the optimal reserve price is b¤s =

³1

¸+2

´ 11+¸

:Finally, substituting b¤

s for bs and calculating the integral yield the equilibriumbidding function Be(vi) = 1

2¡¸

³vi(1 ¡ ¸) + v¡(1¡¸)

i (2 + ¸)¡ 2¡¸1+¸

´: It is imme-

diately seen that b¤s is increasing with ¸; while the seller’s expected revenue and

a bidder’s ex ante expected surplus are decreasing with ¸:

2.2 Other selling procedures: the second-price sealed-bidand the all-pay auctions

Are the above results particular to the …rst-price sealed-bid auction or are theycommon to other auctions ? In this section we shall verify that Proposition 1also holds for the second-price and the all-pay auctions.

The second-price sealed-bid auctionIn this auction the winner, if any, is the highest bidder and it pays either the

second highest bid or the seller’s reserve price, whichever is higher. Proceedingin the same manner as in Section 2.1 we can write (2) for bs · vi · v+; asviF (vi)N¡1¡bsF (bs)N¡1¡

R vi

bsF (u)N¡1du¡P

k 6=iR vi

bsB(vk)F (vk)N¡2f(vk)dvk+

Pk 6=i

®ik®ii

R v+

vi

¡vkF (vk)N¡2 ¡ B(vi)F (vi)N¡2

¢f(vk)dvk¡

Pk 6=i

®ik®ii

R v+

vi(1 ¡ F (u))B(u)(N ¡ 2)F (u)max(0;N¡3) f(u)du =

Pk 6=i

®ik®ii

R v+

bs

¡vkF (vk)N¡2 ¡ bsF (bs)N¡2

¢f(vk)dvk¡

Pk 6=i

®ik®ii

R v+

bs(1 ¡ F (u))B(u)(N ¡ 2)F (u)max(0;N¡3) f(u)du:

(6):B(vi) = 1

¸F (vi)N¡2(1¡F (vi))£

[(N ¡ 1)R vi

bs(v ¡ B(v))F (v)N¡2f (v) dv ¡ ¸

R vi

bsvF (v)N¡2f(v)dv+

¸bsF (bs)N¡2 (1 ¡ F (bs))+¸R vi

bs(1 ¡ F (u))B(u)(N¡2)F (u)max(0;N¡3) f(u)du]:

We can obtain a closed form for the bidding function by di¤erentiating (6)with respect to vi: It results B0(vi) = (N¡1)(1¡¸)

¸ (vi ¡ B(vi))f(vi)

1¡F (vi): Integrat-

ing yieldsB(vi) = (1 ¡ F (vi))

(N¡1)(1¡¸)¸ ( (1¡¸)(N¡1)

¸ G(vi) + C1) (7);

where G(vi) is an antiderivative of vif(vi)1¡F (vi)

(1 ¡ F (vi))¡ (N¡1)(1¡¸)

¸ , and C1 is suchthat B(bs; bs) = bs:

In the appendix it is shown that in equilibrium the seller’s expected revenueand the reserve price are, respectively, strictly decreasing and strictly increasingwith ¸.

The intuition behind this result is the following. Assume ¸ = 0 initially,and then let ¸ increase by a small amount. Suppose bids remain …xed at thevalues which are optimal at ¸ = 0. Firm i perceives that it gains from marginallydecreasing its bid, since such a deviation is without e¤ect on its marginal surplus,

7

which is zero, but increases the expected surplus of the other …rms, which paysless when they win.7

Example 2. Let N = 2; F (v) = v; v+ = 1: Then B(vi; bs) = vi¸¡vi+¸2¸¡1 ¡

¸(1¡bs)2¸¡1

¸ (1¡vi)¡ ¸¡1

¸

2¸¡1 for ¸ 6= 12 ; and B(vi; bs) = (1¡ vi) [ln (1 ¡ vi) ¡ ln (1 ¡ bs)]+

vi; for ¸ = 1=2. b¤s is a maximizer of Us = 2

R 1bs

B(v2; bs)(1 ¡ F (v2)) f(v2)dv2

+2(1¡bs)b2s: It results b¤

s = 1¡¸+p

5¸2+6¸+12(¸+2) for all ¸: It is immediately seen that

b¤s is increasing with ¸; while both the seller’s expected revenue and a bidder’s

ex ante expected surplus are decreasing with ¸:The all-pay auctionThis auction only di¤ers from the …rst-price auction in that each bidder, and

not the winner only, pays what it o¤ers. Proceeding as in Section 2.1 we canwrite (2), for bs · vi · v+; i 2 I; as

viFN¡1(vi) ¡ B(vi) ¡R vi

v± FN¡1(v)dv+Pk 6=i

®ik

®ii

³R v+

vivkFN¡2 (vk) f (vk) dvk ¡

R v+

v± B(vk)f (vk) dvk

´=P

k 6=i®ik

®ii

³R v+

v± vkFN¡2 (vk) f (vk) dvk ¡R v+

v± B(vk)f (vk) dvk

´; where v± is

the valuation that renders bidder i indi¤erent between o¤ering the seller’s re-serve price and zero, and therefore v±FN¡1 (v±) = bs: From this expression weget

B(vi) =

8><>:

viFN¡1(vi) ¡R vi

v± FN¡1(v)dv ¡ ¸R vi

v± vFN¡2 (v) f (v) dv;if v± · vi · v+

0; if vi <v±

(8):

Simple calculations con…rm that the seller’s expected revenue in equilibriumis strictly decreasing with ¸; while the optimal reserve price is strictly increasingwith ¸:

Example 3. Let N = 2; F (v) = v; v+ = 1: Then B(v; bs) = 12v2(1 ¡ ¸) +

12(v±)2(1 + ¸): Furthermore, the optimal value of bs is b¤

s = (1+¸)2

(2+¸)2 ; and hence

B(v) = 12v2(1 ¡ ¸) + 1

2(1+¸)2

(2+¸)2 (1 + ¸):Notice that in deriving the bidding functions we did not impose any condition

on Ui(0; 0) – of course it is non negative, since no bidder’s own expected surpluscan be negative in an equilibrium of a standard auction. Therefore, one cannotexpect that the revenue equivalence theorem holds in our framework – recall thata necessary condition for the theorem is that in equilibrium Ui(0; 0) be the samefor any auction procedure. Indeed, it is easily seen that, under the assumptionsof the examples, for any value of ¸; the …rst-price auction is more pro…table thanthe second-price auction, which on its turn is better than the all-pay auction (see

7 Suppose N = 2. When …rm 2 bids its true valuation while …rm 1 is of valuation v1 and bids

b1; …rm 1’s expected payo¤ is: U1(b1) =R b10

(v1 ¡ v2) f(v2)dv2+ ¸R v+b1

(v2 ¡ b1) f(v2)dv2:Now, it is dU1

db1= (v1 ¡ b1) f(b1) ¡ ¸ [1¡ F (b1)] : When b1 = v1 this derivative is strictly

negative for v1 < v+.

8

Figure 1). It is also noteworthy that, in our examples, the …rst-price and the all-pay auctions have the same probability of e¢cient trade, while the second-priceis the most ine¢cient of the three. These comparative results can be generalizedto some extent. Using the fact, noticed in Section 2.1., that, for given reserveprice, the …rst-price auction with ownership links is revenue equivalent to a…rst-price auction without ownership links and with a transformed distributionof valuations, it is possible to see that the …rst- price auction is more pro…tableto the seller than the all-pay auction, whatever the distribution of valuationsis. Instead, how the second-price auction compares with the …rst-price and theall-pay auctions in the general case is an open question. We summarize theresults of this section as follows.

Proposition 2. a) Proposition 1 also holds for both the second-price and theall-pay auctions. b) Such auctions are not revenue equivalent.

Since standard auctions are not equivalent, the question arises of what is theoptimal selling procedure.

<< Insert Figure 1 here >>

2.3 The revenue-maximizing selling procedureThe formal characterization of the best procedure is not di¢cult. The seller’sproblem is Maxfpi;xigi2I

Us; s.t : truth-telling constraint (1),PN

k=1 pk(v) · 1;pi (v) ¸ 0; v 2 TN ; Ui (vi; vi) ¸ 0; vi 2 T; i 2 I; which has the same format asMyerson’s (1981).

Notice that the individual rationality constraint requires the total payo¤U (vi; vi) to be non-negative for all vi: this is a crucial di¤erence with respect tostandard auctions, in which, as we know, it is required that a bidder’s expectedown surplus be non-negative.

The following proposition contains the main results.Proposition 3. The optimal selling procedure is such that:a) If the object is sold, it is assigned to the bidder with the highest valuation.b) The payo¤ to the bidder with the lowest possible valuation is zero.c) The procedure is not ex post e¢cient.d) Both the seller’s revenue and social surplus are strictly increasing with

the concern parameter ¸.Proof. By simple manipulations, we can put Us in a form that parallels equation(4.9) in Myerson (1981): Us =

RTn

Pk2I (vkpk (v) ¡ [vkpk (v) ¡ xk (v)]) f (v) dv.

Thanks to symmetry, from (2) we get Ui (0; 0) =R

Tn

Pk2I aik (vkpk (v) ¡ xk(v))£

f(v)dv ¡ ®iiR v+0

R vi0 Qi (si) dsif (vi) dvi; for all vi 2 T; i 2 I. From this it

follows, after an integration, ®ii (1 + ¸)RT n [vkpk (v) ¡ xk(v)] f(v)dv =

®iiR v+0 Qi (vi) [1 ¡ F (vi)]dvi + Ui (0; 0) ; i 2 I.

Finally, recalling the de…nition of Qi (vi) allows the seller’s expected revenue tobe written

Us = ¡Pi2I

Ui(0;0)®ii(1+¸) +

RTN

Pi2I [vi ¡ 1

1+¸1¡F (vi)

f(vi)]pi (v) f (v) dv (9).

From (9) we see that it is in the seller’s interest to make Ui (0; 0) equal to zero forall i; and to set pi (vi;v¡i) equal to one (zero) whenever c (vi) = vi¡ 1

1+¸1¡F (vi)

f(vi)

9

is positive (non-positive). Under our assumptions, c(vi) is a monotone strictlyincreasing function of vi: For any vector v¡i consider z (v¡i) = inffs : c (s) ¸ 0and c (s) ¸ c (vj) for all j 6= ig: Then, in equilibrium

pi (si;v¡i)=½

1 if si ¸ z (v¡i)0 if si < z (v¡i)

, and

R vi

0 pi (si;v¡i) dsi =½

vi ¡ z (v¡i) if vi ¸ z (v¡i)0 if vi < z (v¡i)

:

pi (v) i 2 I , v 2 TN is piecewise continuously di¤erentiable, as assumedabove; it depends positively upon ¸ via the condition c (si) ¸ 0. Since the values¤ that solves c (si) = 0 is a strictly decreasing function of ¸, the social surplusis strictly increasing with ¸. Furthermore, s¤ is greater than zero, so tradeine¢ciency occurs with positive probability. Finally, from (9) it is immediatelyseen that the seller’s expected revenue is increasing with ¸.

The characterization of the bidders’ payments is as follows. Recall thatUi (0; 0) =

RT¡i

©Pk2I ®ik [vkpk (vi;v¡i) ¡ xk(vi;v¡i)] ¡ ®ii

R vi

0 pi (si;v¡i) dsiª

£f (v¡i) dv¡i, vi 2 T; i 2 I: To have Ui(0; 0) = 0, we set the expression in bracesat zero for all (vi;v¡i). Therefore, …x state v = (vi;v¡i) ; and suppose a winningbidder exists. Let w 2 I denote the winner. Then in equilibrium, for each statev in which a winner exists, xj(v); j = 1; 2; :::; N; solve the system of N linearequations:

Pk ®wkxk(vw;v¡w) = ®wwz (v¡w) ;

Pk ®jkxk(vw;v¡w) = ®jwvw;

j 2 I n fwg. Finally, in those states in which the object is not assigned, allpayments are zero.

The intuition behind the above results is as follows. Thanks to the individualparticipation constraint — non-negative total payo¤ – the seller can ask eachlosing bidder to pay an amount exactly equal to its ex post external surplus.Therefore, for encouraging truth-telling the seller lowers the winner’s paymentwith respect to the case of no ownership links – note that the weight of a unitpayment in the winner’s payo¤ is 1 if it pays but only ¸ if a losing bidder pays.In addition, the seller has less need to ine¢ciently renounce selling the object inorder to encourage truth-telling — this explains why social surplus is increasingwith ¸:

Example 4. Let N = 2; F (v) = v; v+ = 1: The optimal payment arexw(vw; v3¡w) = max(v3¡w;s¤)¡¸2vw

1¡¸2 ; and x3¡w (vw; v3¡w) = ¸(vw¡max(v3¡w;s¤)¡¸2vw1¡¸2 );

where w denotes the winning bidder:8 Moreover, from c (s¤) = s¤ ¡ 1¡s¤

1+¸ = 0we get s¤ = 1

¸+2 ; the optimal reserve price in the …rst-price auction is b¤s =

³1

¸+2

´ 11+¸

; and therefore, for ¸ > 0; it is b¤s > s¤; which means that ine¢ciency

is lower in the optimal selling procedure.8 It is xw(vw; v3¡w) = max(v3¡w; s¤) ¡ ¸x3¡w (vw; v3¡w) ; and x3¡w (vw; v3¡w) = ¸

(vw ¡ xw(vw; v3¡w)) : Substituting the expression of xw(vw; v3¡w) in that of x3¡w (vw; v3¡w)yields the payments in the body text.

10

3 Extensions. Strategic entry in two-stage auc-tion games

Until now we have assumed that all the …rms take part in the auction. This isan innocuous assumption in a static setting, each …rm will participate, possiblybidding zero, because entry does not entail any (opportunity) cost. However, letthere be two stages in the game, an entry stage and a bidding one; and supposeit is common knowledge that in the second stage, at the moment of bidding,the number of active bidders is known to all players. Then, when there areownership links, entry may reveal individual valuation, and hence a …rm might…nd it pro…table not to enter and let its payo¤ be determined by its shares inother …rms. Let’s reconsider the …rst-price auction in this new setting.

3.1 The …rst-price sealed-bid auctionWe assume that entries are simultaneous and that the number of active biddersis known to all players at the moment of bidding. In addition, we let aik = aih forall h; and all k; i = 1; ::; N; so symmetry is preserved, no matter which biddersstay away. (Recall that ®ik denotes the non-negative weight …rm i assigns tothe expected surplus of …rm k.) Finally we assume that the seller’s reserve pricecan be made contingent on the number of active bidders.

Firstly, we show that the seller’s revenue is not less than when entry is notstrategic (the case we dealt with in Section 2.1). Let bm, m = 1; 2; ::; N; denotethe reserve price when m bidders are active, and b¤

s denote the optimal reserveprice in the one-stage game. Now suppose bm = v+ for m < N , and bN = b¤

s;then all the …rms will participate, for no one will be better o¤ staying away.This proves our assertion.

Secondly, we show that the seller’s revenue is decreasing with the concernparameter ¸; just like under non-strategic entry. Two cases are to be consid-ered. The …rst case, in which bidders learn their own valuations after entry,is easy. The above strategy — selling only if N bidders enter — is optimalfor the seller, since a bidder of a given valuation bids more aggressively whenthere are more bidders. Therefore, it is con…rmed that the seller’s revenue inequilibrium is decreasing with the concern parameter ¸. The second case, inwhich bidders learn their own valuations before entry, is more complex, becauseof the possibility of bidder self-selection according to valuation. For simplicity,we examine the case of two bidders only. When deciding whether to participateor not, bidder i supposes that its rival adopts the monotone increasing (notstrictly) participation function ´ = ´ (v) : [0; v+] ! [0; 1] – later we shall verifythat this actually happens in equilibrium. Let Vm(vi) denote bidder i’s surpluswhen it wins, it is of type vi and the number of active bidders is m = 1; 2. Thenbidder i solves the following program:Max ´i

´iV2(vi)R vi

0 ´(v)f(v)dv + ´i¸R v+vi

´(v)V2(v)f(v)dv+´iV1(vi)

R v+0 (1 ¡ ´(v)) f(v)dv + (1 ¡ ´i)¸

R v+0 ´(v)V1(v)f(v)dv;

11

where the maximand is proportional to its total payo¤ and ´i is its probabilityof entry.

We posit that each bidder’s payo¤ is increasing with its valuation — later weshall verify that this really occurs in equilibrium. Since bidders are symmetric,in equilibrium all of them use the same participation function. Now notice thatbidder i’s optimal entry strategy is deterministic since the objective function islinear in ´i an increasing in vi; hence in equilibrium ´(vi) = 0 for vi less than acut-o¤ value, which we call c, and ´(vi) = 1 otherwise. Therefore, for vi ¸ c

V2(vi) [F (vi) ¡ F (c)] + V1(vi)F (c) + ¸R v+

viV2(v)f(v)dv¡

¸R v+c V1(v)f(v)dv ¸ 0; (10).

The …rst three addends in the sum in (10) add up to type vi’s (normalized)payo¤ in the case it enters, the last term is its (normalized) payo¤ in case itstays away. The bidding function when only one bidder is active equals theseller’s reserve price, b1; for vi ¸ b1; and equals zero otherwise; so V1(vi) =max(0; vi ¡b1). Furthermore, letting B denote the equilibrium bidding functionwhen two bidders are active, we have V2(vi) = vi ¡ B(vi). (Of course, bidder iwins if B(vi) > max(b2;B(v3¡i)).) Proceeding as in Section 2 yields the biddingfunction B(vi) = vi ¡ 1

[F (vi)¡F (c)]1¡¸

R vi

b2(F (v) ¡ F (c))1¡¸ dv; b2 · vi · v+:9

Notice that bidder i’s payo¤ is increasing with its valuation and strictly sofor vi ¸ c, as was assumed.10 Notice also that B(vi) is increasing with c; whichmeans that, for given b2, bidder self-selection induces more aggressive bidding– as was expected.

The seller chooses the values of the reserve prices and c that maximize Us =2R v+b2

B(v) [F (v) ¡ F (c)] f(v)dv+ 2 [1 ¡ F (c)]F (c)b1; under constraint (10):In the appendix we prove that Us is strictly decreasing with the concern

parameter ¸: Therefore we can state the following.Proposition 4. Consider a …rst-price sealed-bid auction, in which entry

decisions are simultaneously and strategically made by cross-owned …rms in anon-cooperative manner. Then:

a) The seller’s revenue is strictly decreasing with the concern parameter ¸:b) When the reserve price is contingent on the number of active bidders, the

seller’s revenue in equilibrium is not less than when entry is non-strategic.It is noteworthy that, if the reserve price is not contingent on the number

of active bidders, the seller’s revenue may fall with respect to the case of non-strategic entry. To see this, reconsider example 1 in Section 2.1. The biddingfunction is B (v) = (1¡¸)v+c

2¡¸ , c · v · v+; where c > b > 0; and b denotes the9 We can assume b2 ¸ c; without loss of generality. Noticing that

U (b2; b2) = ¸R v+b2

(v ¡B(v)) f(v)dv, we get B(vi) = vi ¡ 1F (vi)¡F (c)£R vi

b2fF (z)¡ F (c)¡ ¸(z ¡B(z))f (z)g dz; b2 · vi · v+. Therefore, it is

B0(vi) = (1¡¸)[vi¡B(vi)]f(vi)F (vi)¡F (c) : Integrating yields the bidding function in the body text.

10 Di¤erentiating the left side of (10) we get [1 ¡ B0(vi)] [F (vi)¡ F (c)]+

(1¡ ¸) [vi ¡ B(vi)] f(vi) + F (c): Substituting B0(vi) = (1¡¸)[vi¡B(vi)]f(vi)F (vi)¡F (c) in the above

expression yields F (vi)¡ F (c) ¡(1¡ ¸) [vi ¡B(vi)] f(vi) + (1¡ ¸) [vi ¡ B(vi)] f(vi) + F (c);which is equal to F (vi):

12

seller’s reserve price.11 The equilibrium value of c is determined by the equation(c ¡ b)c ¡ ¸

R 1c (B(v) ¡ b) dv = 0 (11),

Solving (11) yields the equilibrium value of c, c¤ = c(b), which satis…es require-ments for a cut-o¤ value.12 The seller chooses the reserve price in order tomaximize 2

R 1c¤ B(v)(v ¡ c¤)dv+ 2(1 ¡ c¤)c¤b: Figure 1 depicts the equilibrium

for ¸ = :05: The seller’s revenue is lower than in Section 2.1 — the values are0:412 55; and 0:414 92; respectively. Also social surplus is lower, since the equi-librium value of b is greater than b¤

s — the respective values are 0:506 24 and0:504 77. Numeric simulations suggest that these qualitative results hold truefor all values of ¸:

<< Insert Figure 2 here >>

3.2 On the optimal selling procedureWe only observe that all the …rms will certainly participate if the seller an-nounces that the object will not be sold if a …rm stays away. Therefore, thefollowing holds true.

Proposition 5. Consider the optimal selling procedure for the case in whichentry decisions are simultaneously and strategically made by cross-owned …rmsin a non-cooperative manner. Then, the seller’s revenue is not less than whenentry is non-strategic.

4 Summary and conclusionsIn this paper we have examined the e¤ect of passive ownership links amongbidders in auctions. The analysis has focused on the case of symmetric bidderswith no collusion. Initially, it has been assumed that all potential bidders par-ticipate in the auction, provided a bidder’s total expected payo¤ is non-negative.Firstly, we proved that: a) ownership links weaken the willingness to bid highand thus hurt the seller in …rst-price, in second-price, and in all-pay auctions; b)such auctions are not revenue equivalent; c) social surplus strictly decreases asan e¤ect of the seller’s reaction, and also the bidders’ own surplus may decrease.Next, we characterized the revenue-maximizing selling procedure: in sharp con-trast with standard auctions, the result is that both the seller’s revenue andsocial surplus are strictly increasing with the intensity of ownership links. Thereason for this surprising result is that standard auctions impose a more restric-tive individual participation constraint than the optimal selling procedure: inthe former a bidder’s own surplus must be non-negative, in the latter only thetotal payo¤ must be non-negative. Therefore, in the optimal auction, the selleris able to extract surplus from the losing bidders and has less need to destroypotential surplus for encouraging truth-telling.

11 c = 0 is unfeasible, since this requiresR v+b

([v ¡ B(v)] dv ¡R v+b

(v ¡ b)dv ¸ 0; which isimpossible.

12 It is c¤ = 1¸( 3¡¸

2¡¸ )+2

µb+ ¸

2¡¸ + b¸+q¸3(1¡b)2¡¸(3b2¡6b+2)¡2b2

¡2+¸

¶

13

In the second part of the paper, we dispense with the assumption that allpotential bidders participate, and explore the consequences of bidder self selec-tion in a two-stage game, in which the potential bidders …rst decide to enterthe auction or stay away, and then the auction is held, all players knowing thenumber of active bidders. With reference to the …rst-price sealed-bid auction,it is con…rmed that the seller’s revenue is strictly decreasing with the intensityof ownership links. As for the optimal selling procedure, it results that theseller’s revenue is strictly increasing with the intensity of ownership links, pro-vided he can commit to the strategy of selling if and only if all potential biddersparticipate.

A possible extension of the paper would consider asymmetric bidders, char-acterized by ownership links of di¤erent intensity. Although a formal analysis isleft for future research, intuitively it is clear that ownership links decrease bidsin this case as well: the bidders with strong links bid less aggressively for thereasons explained in the paper, those with weak links do the same because theyhave less need to bid high in order to beat their opponents. As regards the op-timal selling procedure, it is clear that the seller can extract surplus selectivelyfrom each bidder, so the seller certainly gains under asymmetric ownership linksas well.

14

References

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Bianco, M., and Casavola, P. (1999). “Italian Corporate Governance: E¤ectson Financial Structure and Firm Performance.” European Economic Review, 43,1057-1069.

Bresnahan T. F., and Salop S. C. (1986). “Quantifying the CompetitiveE¤ects of Production Joint Ventures.” International Journal of Industrial Or-ganization, 4(1), 155-175.

Brunello, G., Graziano, C., and Parigi, B. (2001). “Executive Compensationand Firm Performance in Italy.” International Journal of Industrial Organiza-tion, 19, 133-161.

Bulow, J. I., Huang, M., and Klemperer, P. D. (1999). “Toeholds andTakeovers.” Journal of Political Economy, 107, 427-454.

De Fraja, G. (2002). “The Legal Framework for Public Procurement inItaly.” In Baldassarri, M., Galli, G., and Piga G. (Eds.), La competitività inItalia, Milan: Il Sole 24 Ore.

Gilo, D., (2000). “The Anticompetitive E¤ect of Passive Investment.” Michi-gan Law Review, 99(1), 1-47.

Hamilton, G., Zeile, W., and Kim, W. J. (1990). “The Network Structureof the East Asian Economy.”. In Clegg, S., Redding, G. (Eds), Capitalism inContrasting Cultures. Berlin: de Gruyter.

Hartman Philip, Ordinary Di¤erential Equations. Wiley &Sons, New York1964.

Jehiel, P., Moldovanu, B., and Stacchetti, E. (1996). “How (not) to SellNuclear Weapons.” American Economic Review, 86 (4), 814-29.

Jehiel, P., Moldovanu, B., and Stacchetti, E. (1999). “MultidimensionalMechanism Design for Auctions with Externalities.” Journal of Economic The-ory, 85, 258-293.

McAfee, R. P., and McMillan, J. (1987). “Auctions and Bidding.” Journalof Economic Literature, 25 (2), 699-738.

Myerson, R. B. (1981). “Optimal Auction Design.” Mathematics of Opera-tions Research, 6(1), 58-73.

Reynolds, R. J., and Snapp, B. R. (1986). “The Competitive E¤ects ofPartial Equity Interests and Joint Ventures.” International Journal of IndustrialOrganization, 4(1), 141-153.

Zingales, L. (1994). “The Value of the Voting Right: A Study of the MilanStock Exchange Experience.” Review of Financial Studies, 7(1), 125-48.

15

AppendixProof of Proposition 1 (Section 2.1.)a) It is a trivial consequence of the fact that B is strictly increasing with v.b) b¤

s maximizes the seller’s expected revenue Us = NR v+

bsB (v)FN¡1(v)f(v)dv:

Thus it solves the following equation:hR v+

bsBbs (v)FN¡1(v)f(v)dv ¡ bsFN¡1(bs)f(bs)

i(bs ¡ 0) = 0, where bs = 0

if the term in square brackets is negative. If it results that, for any given ¸;limbs!0+

@Us@bs

is strictly positive, then bs = 0 can never be optimal. It is easilyshown that Bbs (v) is always positive and strictly positive for v > bs; thus thedesired result is proven.

c) As to the seller’s revenue, it is in equilibrium dUsd¸ jbs=b¤

s= @Us

@¸ jbs=b¤s=

NR v+

b¤s

@B@¸ (v)FN¡1(v)f(v)dv: The desired result is proven if we show that, for

any given ¸; @B@¸ is negative for all v; and strictly negative for v > b¤

s. We canuse the following lemma(Gronwall’s): Let I ½ R be an interval and x0 2 I: Letu; v : I ! R be two continuous functions on I , u be non-negative and c ¸ 0:If v(x) · c+ j

R xx0

u(t)v(t)dt j for all x 2 I; then v(x) · c exp(jR xx0

u(t)dt j):(For a proof see Hartman (1964), Ch. 3). Now, for any v ¸ b¤

s; we get from

(4): B¸(v) = ¡R v

b¤s(z¡B(z))FN¡2(z)f(z)dz

F (v) + ¸

R v

b¤s

B¸(z)FN¡2(z)f(z)dz

F (v) and therefore

B¸(v) ·R vb¤

s

¸FN¡2(z)f(z)F (b¤) B¸(z)dz: Since B¸ and ¸f(:)

F (b¤s) are continuous; we can

apply Gronwall’s lemma and conclude from the non-negativity of f that B¸(v)is strictly negative for v > b¤

s and is zero for v = b¤s :

As to the joint surplus in equilibrium, its derivative with respect to ¸ isopposite in sign to db¤

sd¸ = ¡

³@2Us@bs@¸ =@2Us

@2bs

´jbs=b¤

s: As Us is locally strictly

concave, by assumption, it is sign³

db¤s

d¸

´= sign

³@2Us@bs@¸

´jbs=b¤

s=

sign(R v+

bsBbs¸ (v) (F (v))N¡1 f(v)dv) jbs=b¤

s. For any v ¸ bs; we get from (5)

Bbs¸ (v) = @[F (bs)=F (v)]N¡1¡¸

@¸ > 0: Therefore we conclude that db¤s

d¸ > 0.Remark to Proposition 1Following is an example in which, contrary to example 1 in Section 2.1.,

SB is always increasing with ¸. There are two bidders, and v is distrib-uted in the interval [0; 1] according to the function F (v) = v8. It results

b¤s = (9 + 8¸)

1¡8¡8¸ ;Be(v) = 8v(1¡¸)+v¡8(1¡¸)(9+8¸)

8¸¡98(1+¸)

9¡8¸ ; and …nally SB =

17(9+8¸)8¸¡9

8(1+¸) ¡8(1¡¸)17(8¸¡9)(1+¸) + +(9+8¸)

¡ 178(1+¸)

17(1+¸) ; which is strictly increasing with ¸.Proof of Proposition 2 (Section 2.2.)A) We have to prove that proposition 1 holds true for the second-price sealed-

bid auction.Firstly, we show that B¸(vi; bs) < 0 for bs · vi < v+. This is easily seen via

Gronwall lemma. Indeed, we can write B(vi) = bs+R vi

bs(v ¡ B(v)) f(v)

1¡(F (v))(1¡¸)(N¡1)

¸ dv;

and hence B¸(vi) = ¡R vi

bsB¸(v) f(v)

1¡(F (v))(1¡¸)(N¡1)

¸ dv¡R vi

bs(v ¡ B(v)) f(v)

1¡(F (v))(N¡1)

¸2 dv:

16

Now, for any vi 2 (bs; v+) B¸(vi) <jR vi

bsB¸(vi) (1¡¸)(N¡1)

¸f(v)

1¡(F (v))dvi j, whichimplies that B¸(vi) is strictly negative Since dUs

d¸ jbs=b¤s= @Us

@¸ jbs=b¤s

is equal insign to B¸(vi), we conclude that the seller’s revenue is decreasing with ¸: Sec-ondly, for any bs; B¸bs(vi) = (N¡1)

¸2

R vi

bsBb(v) f(v)

1¡F (v)dv¡ (1¡¸)(N¡1)¸

R vi

bsB¸bs(v) f(v)

1¡F (v)dv:

Applying again Gronwall lemma yields B¸bs(vi) > 0: Since in equilibrium db¤s

d¸

is equal in sign to B¸bs(vi), we conclude that db¤s

d¸ > 0:The seller’s expected revenue in equilibrium is strictly decreasing

with ¸ (Section 3.1.)Notice …rst that c = 0; b1 = v+; b2 = b¤

s is a feasible solution to the seller’sproblem. In fact, when c = 0; (10) becomes, for all vi; [vi ¡ B(vi)]F (vi)I[b2;v+](vi)

+¸R v+

max(b2;vi)[v ¡ B(v)] f(v)dv ¡¸

R v+

b1(v ¡ b1)f(v)dv ¸ 0; where I[b2;v+](b1)

equals 1 for b1 2 [b; v+] ; and equals zero otherwise. This further impliesR v+

b2

[v ¡ B(v)] f(v)dv ¸R v+

b1(v ¡ b1)f(v)dv; which is satis…ed for b1 = v+; b2 = b¤

s:Next, notice that, if an optimal solution with c > 0 exists, then it must beb1 < c < b2: To see this, suppose b2 is given, and take b1 to be such that (10)is satis…ed as an equality with vi = c and c = 0 — this requires b1 > b2. Thenlet b1 decrease: (10) can be satis…ed for vi = c only if c > b1, which impliesb1 < b2. Therefore, it is

(c ¡ b1)F (c) ¡¸R b2

b1(v ¡ b1)f(v)dv ¡¸

R v+

b2(B(v) ¡ b1) f(v)dv ¸ 0;

with equality if c > 0 (1a):Thanks to this result, we can replace (10) with (1a) in the seller’s problem.

Finally, notice that the partial derivative, with respect to ¸; of the left side of(1a) is negative, while that of Us is strictly negative; therefore, by the envelopetheorem, Us in equilibrium is strictly decreasing with ¸. This proof generalizesto the case of N bidders — we simply repeat the main argument.

17

0.3

0.32

0.34

0.36

0.38

0.4

0.2 0.4 0.6 0.8

Figure 1. The seller’s expected revenue as a function of ¸: N =2; F (v) = v; v 2 [0; 1] :The dashed curve depicts the …rst-price sealed-bid auction, the solid curve depicts the second-price sealed bid auction,and the dot curve depicts the all pay auction.

18

0

0.2

0.4

0.6

0.8

1

c

0.2 0.4 0.6 0.8 1b

Figure 2. The equilibrium in the …rst-price sealed-bid auction with strategic entryand one reserve price. N = 2; F (v) = v; v 2 [0; 1] ; ¸ = 5

100 : The U-shaped curve isan isoquant of the objective function, the dotted curve is the participation constraint(see eq. 10).

19