This paper analyzes a two-stage sealed-bid auction that is frequently employed in privatization, takeover, and merger and acquisition contests. This auction format yields the same expected revenue as the open ascending (English) auction, yet is less susceptible to preemptive bidding and collusion.Auctions

Elmar G. Wolfstetter

Motty Perry

Shmuel Zamir

English

Research Papers in Economics

CESifo Working Paper Series
February 2000
CESifo
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________________________
* Financial support was received by the Center for Rationality, Hebrew University in Jerusalem, the
Deutsche Forschungsgemeinschaft, SFB 373 (“Quantifikation und Simulation Ökonomischer
Prozesse”), Humboldt-Universität zu Berlin, the Koret Foundation and the EU-TMR Research Network
ENDEAR (FMRX-CT98-0238). We are grateful to Filippo Caffarelli for bringing to our attention the ENI
auction procedure. We are also indebted to Todd Kaplan for very useful discussions.
A SEALED-BID AUCTION THAT
MATCHES THE ENGLISH AUCTION
Motty Perry
Elmar Wolfstetter
Shmuel Zamir*
Working Paper No. 249CESifo Working Paper No. 249
February 2000
A SEALED-BID AUCTION THAT MATCHES
THE ENGLISH AUCTION
Abstract
This paper analyzes a two-stage sealed-bid auction that is frequently
employed in privatization, takeover, and merger and acquisition contests.
This auction format yields the same expected revenue as the open
ascending (English) auction, yet is less susceptible to preemptive bidding
and collusion.
Keywords: Auctions
JEL Classification: D44
Motty Perry
Hebrew University of Jerusalem
Department of Economics
Mount Scopus
Jerusalem 91905
Israel
e-mail: motty@sunset.huji.ac.il
Elmar Wolfstetter
Humboldt University Berlin
Department of Economics
Spandauer Str. 1
10178 Berlin
Germany
e-mail: wolf@wiwi.hu-berlin.de
Shmuel Zamir
Hebrew University of Jerusalem
Department of Statistics
Mount Scopus
Jerusalem 91905
Israel
e-mail: zamir@math.huji.ac.il1 Introduction
An important issue in the theory of auction design is whether one should
employ an open, ascending bid (English) or a sealedbid auction. The
open format is generally advised on the ground that it furnishes bid-
ders with valuable information. This reduces the winner's curse, which
contributes to more aggressive bidding, to the benet of the seller.
In their seminal contribution, Milgrom and Weber (1982) showed that
the open-bid ascending auction yields higher revenues than the sealed-
bid (second or rst price) auctions. However, the exchange of informa-
tion in the course of an English auction also has a drawback: It may
invite preemptive or jump bidding,1 and it may be misused by bidders
to communicate and enforce collusion.
Casual observation of real world auctions reveals that in many cases
the seller employs a dierent twostage sealedbid auction that does not
t the usual distinction of auction formats. For example, in Italy, the for-
merly state owned industrial conglomerate ENI was privatized using such
a two-stage procedure. In the rst round bidders submitted sealedbids
and reorganization plans. Then, the auctioneer screened out the low-
est bidders and nalized the sale in the second round, in a sealedbid
auction, with the proviso that bids could not be lowered.2 A similar two
round sealedbid format is frequently observed in takeover and merger
and acquisition contests. Indeed, if one talks to investment bankers, they
describe this twostage format as the standard procedure, and they jus-
tify it with the high cost of bidding and the fact that preemptive bidding
would be a serious problem in an open, ascending auction.
Motivated by the above examples the purpose of this note is to ratio-
nalize the idea of two round auctions and to provide some useful insights
regarding its properties. We study two-stage auctions in the context of
the Milgrom and Weber aliated values model. In the rst stage, all
1A preemptive bid, also called a jump bid, is a high initial bid, with the intention
of encouraging others to quit the auction. For explanations of jump bidding see
Fishman (1988) and Avery (1998).
2For a detailed account of the breakingup of ENI (Italian Oil and Energy Corpora-
tion) see Caarelli (1997). A peculiar feature of the ENI auction was that all bids (not
just the losing bids) were made known. Therefore, the secondround auction was
potentially an auction in which the ranking of valuations was common knowledge.
Such auctions are analyzed in Landsberger, Rubinstein, Wolfstetter and Zamir (1998).
1agents simultaneously submit sealed bids. Only the two highest bidders
continue to the second stage. All bids that fail to pass to the second stage
are publicly revealed. In the second stage, the two remaining bidders play
a sealedbid second price auction where each bidder is bounded by his
or her own rst round bid. We show that this design which, compared to
the English auction, reduces the opportunity of bidders to send signals
back and forth via their bids, yields an expected revenue as high as the
ascending open-bid (English) auction.
2 The Model
Consider the wellknown symmetric aliated values model by Milgrom
and Weber (1982), which includes the symmetric private values and the
symmetric common value model as special cases.
One indivisible unit of a certain good is auctioned to n  2 risk neu-
tral bidders. Prior to the auction, each bidder receives a signal or value
estimate Xi of the object for sale. The vector of signals X  Xii2N is
drawn from a continuous and symmetric joint distribution F with sup-
port 0;1n. Bidders' valuations are an identical function of their own
signal and the set of rivals' signals X−i :
 
Xj

ji
VXi;X −i:
V is nonnegative, continuous, and strictly increasing in each of its vari-
ables and symmetric in

Xj
	
ji.
The auction rules have two rounds, as follows:
Round 1 Eachbidder, afterobservinghisprivatesignal, submitsaclosed
sealed bid. The two highest bidders are allowed to continue; for all
others the game is over. Ties are resolved by an equal probability
rule.
Round 2 The auctioneer publicly announces the bids rejected in round 1
and runs a secondprice auction among the two remaining bidders.
Bidders must bid at least their own bid from round 1 and no one
is allowed to withdraw from bidding. Again, ties are broken by an
equal probability rule.
2We denote by bx the 1st round bid function, dened on 0;1,
and by x;B the 2nd round bid, where x is the bidder's signal and B
the set of rejected 1st round bids that were announced after the 1st
round.
The joint probability distribution F is symmetric, and the random
variables X1;X 2;:::;X n are aliated, as dened in Milgrom and Weber
(1982). For technical convenience we assume that the joint distribution
of signals has a density f satisfying 0 <c<f x < c<1 on 0;1n.
We denote the highest, the second highest etc. signal received by rival
bidders by the order statistics Y1, Y2;:::;Y n−1 respectively, and let Z be
the vector of order statistics of the lowest n − 2 rival bidders' signals,
Z Y2;::;Yn−1:
Finally, we restrict the analysis to symmetric equilibria with (strictly)
increasing bid functions. In this case, the signals of the rejected bidders
can be deduced from the rejected bids B. In view of the symmetry of V in
the coordinates of X−i, the dependence of all relevant functions on B is
only through Y1 and the vector of order statistics Z. Therefore, by slight
abuse of notation, we shall keep the same names for the functions and
write x;z;x;Z;Vx;y1;z;Vx;Y1;Z, etc.
3 Equilibrium
In this section we show that all Nash equilibria yield the same equilibrium
outcome as the open ascending (English) auction. Among these equilibria
there is a unique strict equilibrium.
Let b be a strictly increasing function on 0;1 and consider the fol-
lowing brestricted auction which is equivalent to the original auction
except that bidders are restricted to play the strategy b in the rst round.
Proposition 1 (SecondRound Bidding)
x;z : maxfbx;Vx;x;zg (1)
is the unique symmetric equilibrium of the brestricted auction (in the class
of pure monotone bid strategies).
Proof In the absence of the constraint on 2nd round bidding this is a
straightforward adaptation of Milgrom and Weber (1982), Sect. 5. Recall
3that only two bidders are allowed to participate in the second round,
and note that in this case a bidder who plays this strategy would never
benet from bidding dierently even if he knew the rival's signal. In the
presence of the constraint it may happen that a rstround winner with
a signal x could benet from lowering his bid so that he loses the auction
when his rival is constrained by his 1st round bid. However, as one can
see easily, this can happen only if the winning bidder cannot lower his
bid due to his own constraint (note that the 1-st round winning bidder's
constraint exceeds that of his rival, due to the monotonicity of b). 
In view of Proposition 1 we refer to b as a strategy with the under-
standing that 2nd round bidding is according to (1).
Lemma 1 Consider a bidder with a signal X  x:Suppose rival bidders bid
according to the same strict monotone increasing bid function b.I f~ bx
is a (strictly) better response than bx; then ~ b  x <b  x  .
Proof Suppose instead of bidding bx, a bidder with signal x bids
~ b  x >b  x while all others are bidding according to b. This bid can
make a dierence only if he thus wins the auction, but would have lost it
while bidding bx. Therefore, it must be true that x<Y 1. Thus, when
winning the auction, the bidder pays VY1;Y 1;z which is greater than
his own valuation Vx;Y1;z. Hence, if everyone else bids according to
strategy b, it never pays to bid higher than bx. 
Lemma 2 Any strict monotone increasing b such that bx  Vx;x;0 is
a symmetric Nash equilibrium.
Proof Firstnotethatwhenallbiddersbidaccordingtobx  Vx;x;0,
a bidder with signal x who is allowed to participate in the 2nd round
never regrets bidding bx in the 1st round, even if he knew rivals' sig-
nals, since Vx;x;z  Vx;x;0  bx. By Lemma 1 any protable
deviation ~ bx from bx must satisfy ~ b  x <b  x  . Such a deviation
makes a dierence only in states where x>Y 1 and ~ b  x <b  Y 2. In such
a state, by bidding ~ bx he does not qualify for the 2-nd round and thus
foregoes a net prot of Vx;Y1;z− VY1;Y 1;z > 0, which he would
have as a winner by playing bx. 
4Lemma 3 Assume b is strict monotone increasing with bx  Vx;x;0;
8x. If all rival players bid according to b then ~ b 2 bx;Vx;x;0 is a
best reply while any ~ b  x <b  x is not a best reply.
Proof The rst part is true because 8z;Vx;x;0  Vx;x;z; hence,
bidding Vx;x;0 does not constrain 2nd round bidding. On the other
hand, when bidding ~ b  x <b  x  , while all rivals bid according to b, there
is a positive probability that the signal x is the highest, but the bidder
will not get into the 2nd round and hence foregoes a positive prot. 
Corollary 1 bx : Vx;x;0 is a symmetric strict equilibrium and
any equilibrium b Þ b is not strict.
Proof We already know that b is an equilibrium. By Lemma 3, no
deviation b  x <b x is a best reply to b. It remains to be shown that
no deviation b  x >b x is a best reply to b. In fact, by Lemma 1,
bx is not a strictly better response than bx. To see that it cannot be
a best reply to b, observe that for suciently small z, there is y1 >x
s.t. b  x >V  y 1;y 1;z > Vx;y 1;z. In such an event (which is of
positive probability), the player will win the object paying Vy1;y 1;z,
which is more than what it is worth to him, Vx;y1;z. Thus, bx is a
strictly worse reply than bx, completing the proof that b is a strict
equilibrium. Any equilibrium b Þ b is not strict, since by Lemma 3, b
is also a best reply to b. 
Remark 1 While formally speaking trembling hand perfectness is not
dened in our game (since it has a continuum of moves), there is a sense
in which the equilibrium b is the only equilibrium that satised the
spirit of trembling hand perfection. In order to see this, consider an
equilibrium b Þ b. When in the perturbed game all players use a uni-
formly perturbed strategy (tremble) around b, then a best reply is bx
which is strictly better than bx (since with positive probability x>y 1
and b  x <b  y 2<b x in which case bx is strictly better than
bx.) Hence, as the perturbation tends to 0, the limiting best reply is
bx.
Theorem 1 (FirstRound Bidding) ThereisnoNashequilibriumbx ß
bx and bx : Vx;x;0 is the only (trembling hand) perfect equi-
librium.
5Proof Suppose there exists another function bx ß bx that gener-
ates an equilibrium prole. We then show that, unless bx  bx, the
bid function b is not a best reply assuming all others play b. This proves
that there is no equilibrium above b from which the last assertion fol-
lows by Remark 1.
To see the heuristics of the main argument, let x0 > 0 and bx0 
bx0  x0 where x0>0. If the bidder with the signal x0 bids in
the rst round bx0 −" rather than bx0 he increases his payo in the
event Y2 <x 0 − "<Y 1 <x 0 in the order of x0", and decreases it in
the event x0 −"<Y 2 <Y 1 <x 0 in the order of "2. Thus, if x0>0, for
suciently small " the payo increases.
The full details of the proof are spelled out in the Appendix. 
Recall that in the equilibrium of the English auction, once only two
bidders are active, they play as in a secondprice auction after the n − 2
lowest signals have become known (see Milgrom and Weber (1982)). In
equilibrium this is precisely the situation in the secondround of our
game. Therefore, we conclude:
Theorem 2 (Revenue Equivalence) The strict and perfect equilibrium
b (as well as each other nonstrict equilibrium) is revenue equivalent to
the corresponding English auction.
4 Conclusion
We have analyzed a simple twostage sealedbid auction and showed that
it has a unique symmetric strict equilibrium that is payo equivalent
to the symmetric equilibrium of the English auction. We now briey
summarize and discuss the merits of this auction rule.
It is useful to compare it to two revenue-equivalent mechanisms: 1)
theopenascending(English)auction, and2)theassociateddirectincentive
compatible mechanism.3 Of course, the latter exists, by the revelation
principle, and it can be implemented in one stage and as a closedseal
bid. We evaluate these mechanisms according to simplicity of rules, ease
3Other standard auctions like rstprice and Dutch auctions are ignored, since they
are less protable for the seller in any case (see Milgrom and Weber (1982)).
6of implementation, revenue, and susceptibility to collusion and jump
bidding.
The rules of the open ascending (English) auction are fairly simple,
but implementation is complex and costly, and it is vulnerable to both
collusion and jump bidding. The associated direct incentive compati-
ble mechanism is easy to implement, but the rules are complicated and
perhaps too dicult for buyers to understand. The proposed twostage
auction combines the best of 1) and 2). Indeed, the rules of the two
stage auction are at least as simple as those of the English auction, and
their implementation is as simple as that of the direct incentive compat-
ible mechanism. In addition, it is less susceptible to collusion and jump
bidding.
In view of these results it is not surprising that the twostage sealed
bid auction format is commonly employed in privatization, takeover, and
merger and acquisition contests.
Appendix
Proof of Theorem 1 Assume bx : bxx with x  0;8x;
is an increasing function that generates a symmetric equilibrium prole.
We show that bx is not a best reply to bx (used by all other players)
unless x  0.
Suppose that for some x0 > 0 one has x0>0. Since b is strictly
monotone it has at most countably many discontinuities. Therefore, we
can assume that there is such an x0 where b is continuous.
Consider a bidder with signal x0 and assume all other bidders play the
strategy prole generated by b. Denote by x0;x 0 the payo of that
bidder if he follows the same strategy and by x0;x 0 − " his payo if
he deviates to bx0 −" (followed by the  continuation). We shall show
that if x0>0, then for suciently small " one has x0;x 0 − " >
x0;x 0.
Note rst that if x0 is not the highest signal, bidding bx0 −" yields
the same payo as bidding bx0 because that bidder loses the auction
in both cases. Similarly, if Y2 <Y 1 <x 0−"<x 0, then bidding bx0 also
yields the same payo as bx0−" because in both cases the bidder wins
the auction and pays the bid of the player with signal Y1. Therefore, we
7only need to evaluate the change in payo in the following two events
A1 :f Y1;Y 2 j Y2 <x 0 − "<Y 1 <x 0g
A2 :f Y1;Y 2 j x0 − "<Y 2 <Y 1 <x 0g:
In the event A1 the bidder enters the 2-nd round both with bx0 and
with bx0 − ". However, bidding bx0 − " makes him less constrained
in the 2-nd round bidding, and hence yields a (conditional) payo at least
as large as that from bidding bx0. To compute the increment in payo
in this event dene
Zx0 :fZ j EfY1jY1<x0g Vx0;Y 1;Z
−maxfbY1;VY1;Y 1;Zg < −
x0
4
g
In words: Zx0 is the set of the n−2 lowest signals at which the bidder
expects a loss of at least x0=4 if he wins the auction. In this event he
will not revise his 1-st round bid bx0, resp. bx0 − ". If in addition
x0 − "<Y 1 <x 0 the bidder loses the auction and thus avoids the loss if
his 1-st round bid is bx0 − " rather than bx0.
Therefore, the increment in payo in event A1, denoted by 1,i sa t
least
1 
x0
4
PrfZx0 \f x0 − "<Y 1 <x 0gg

x0
4
PrfZx0 j x0 − "<Y 1 <x 0gPrfx0 − "<Y 1 <x 0g

x0
4
c"PrfbY1 − Vx0;Y 1;Z>
x0
4
j x0 − "<Y 1 <x 0g
To evaluate the probability in the last expression use the uniform conti-
nuity of  and of V and choose ">0 such that
x0 − "<y 1 <x 0 =)
8
<
:
 Vx0;y 1;z− Vy1;y 1;z
  
x0
8 8z
and y1 
x0
2 :
Choose >0 such that 8z;jzj <and8y1
jVx0;y 1;z− Vy1;y 1;0j
x0
8
:
8(Note that  does not depend on ".) Then 8z;jzj < and
x0 − "<y 1 <x 0 we have
by1 − Vx0;y 1;zVy1;y 1;0  y1 − Vx0;y 1;z

x0
2
−j Vx0;y 1;z− Vy1;y 1;0j

x0
2
−j Vx0;y 1;z− Vy1;y 1;zj
−j Vy1;y 1;z− Vy1;y 1;0j

x0
4
:
It follows that
PrfbY1 − Vx0;Y 1;Z>
x0
4
j x0 − "<Y 1 <x 0g
PrfjZjg;
and hence
1 c
n−2c"
x0
4
cx0";
for some constant c>0.
To assess the eect of the deviation from bx0 to bx0 −" in event
A2 let L : maxx jVx;y;zj. Then, clearly, the payo increment in this
event, denoted by 2, satises
2 − LPrfA2g− L c"
2:
We conclude that
x0;x 0 − "− x0;x 0  cx0" − L c"
2;
which is positive for " suciently small, unless x0  0. This com-
pletes the proof. 
9References
Avery, C. (1998). Strategic jump bidding in English auctions. Review of
Economic Studies, 65:185210.
Caarelli, F. V. (1997). Auction mechanisms for privatization: The ENI
group multiple round auction procedure. Working paper, CORIPE,
Torino.
Fishman, M. J. (1988). A theory of preemptive takeover bidding. Rand
Journal of Economics, 19:88101.
Landsberger, M., Rubinstein, J., Wolfstetter, E., and Zamir, S. (1998). First
price auctions when the ranking of valuations is common knowl-
edge. Discussion paper, Center for Rationality, Hebrew University,
Jerusalem.
Milgrom, P. and Weber, R. J. (1982). A theory of auctions and competitive
bidding. Econometrica, 50:10891122.
10

A Sealed-Bid Auction that Matches the English Auction

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A Sealed-Bid Auction that Matches the English Auction

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