Second-Price Common-Value Auctions under

Multidimensional Uncertainty1

Maria ¶Angeles de Frutos2

Universidad Carlos III

de Madrid

Lambros Pechlivanos3

Foundation for Economic & Industrial Research

Athens

February 2002

1Comments and suggestions by Mark Armstrong, Luis Corch¶on, Paul Klemperer, Jean-Jacques

La®ont, Jean Tirole, and Mark Walker, as well as participants at the 1st Game Theory World

Congress in Bilbao, at ESEM and EEA 2001 in Lausanne, at ASSET 2001 in Crete, and at seminars

in Madrid, Athens and Barcelona are gratefully acknowledged. The research started when the

second author was at IDEI, Toulouse. The ¯rst author's research was supported by the Spanish

Ministerio de Educaci¶on (grant DGICYT PB96-0118), and the second author's one by the European

Commission under the TMR program (grant ERBFMBICT961580). All remaining errors are ours.2Corresponding Author. Department of Economics, Universidad Carlos III de Madrid, Calle

Madrid 126, 28903 Getafe, Spain, E-mail: frutos@eco.uc3m.es3Foundation for Economic & Industrial Research, Tsami Karatassi 11, 117 42 Athens, Greece,

E-mail:lpech@iobe.gr

Abstract

The literature has demonstrated that second-price common-value auctions are sensitive to

the presence of asymmetries among bidders. Bikhchandani (1988) has shown that if it is

common knowledge that a bidder has a disadvantage compared to her opponent, that bidder

(almost surely) never wins the auction. This paper shows that this result does not carry

over when one allows for two-sided uncertainty. We show that even if the probabilities

that one of the bidders is an advantaged type while the other is a disadvantaged type

are arbitrarily large, in every equilibrium, the disadvantaged type bidder needs to win the

auction with strictly positive probability. We then solve for the equilibria in two cases, one

with two types and another with a continuum of types, and we show that they converge to

the symmetric equilibria of the corresponding symmetric auctions. This is also true when

considering convergence in terms of expected revenue performance. We thus reestablish a

lost linkage in the analysis of common-value and almost-common-value auctions.

Keywords: Common-value auctions, asymmetric bidders, spectrum auctions, mergers and

acquisitions, liquidity constraints.

JEL Classi¯cation Numbers: D44, D82, T96, G34.

1 Introduction

Auction design has been lately in vogue due to the on going licensing for the third gen-

eration mobile telephony services (UMTS) throughout Europe. In those auctions several

countries established rules to counter potentially deleterious e®ects created by the presence

of asymmetries among bidders. Asymmetries were considered to be prevalent because of the

coexistence in the auctions of incumbent and entrant ¯rms in the local telephony markets.1

Considerations about asymmetries among bidders come immediately into one's mind also

when, for example, the object being auctioned o® is a target ¯rm in a takeover contest, a

state-owned enterprise about to be privatized, or a bankrupt ¯rm under liquidation. In all

those cases, the market value of the assets is common to all bidders (i.e., they compete for

common-value objects), but at the same time, each of them may have an additional private

source of gains due to synergies.

Asymmetries among bidders can also be generated by the presence of liquidity constraints.

Firms that operate within imperfect capital markets face di®erent costs of raising the amount

of cash needed for their bids. Di®erences in retained earnings, in values of assets appropriate

for collateral, or, more generally, in access to external ¯nance may easily cause asymmetries

among bidders.

Auction theory, for reasons of technical convenience, had traditionally focused on sym-

metric environments. Nonetheless, attempts to extend the analysis to asymmetric environ-

ments have shown that most major insights derived in the symmetric case do not continue

to hold. Perhaps, the most celebrated, violent reversal of fortune has been delivered in the

analysis of single-object, second-price, common-value auctions. Bikhchandani (1988) has

demonstrated that if in such an auction it is common knowledge that one of the bidders

has a disadvantage, in a stochastic sense, compared to her opponent, this bidder (almost

surely) never wins the auction. Klemperer (1998) delivered the same result in a determinis-

tic asymmetric auction. As a consequence of these results, it is considered that second-price,

common-value auctions may have considerably less desirable properties in asymmetric en-

vironments compared to symmetric ones; such as, not attracting potential bidders to the

auction because they might feel intimidated by their opponents' prowess, or raising lower

revenues in expectation to the seller2. Actually, it is argued in Klemperer (2000) that these

1See Klemperer (2000).2Milgrom and Weber (1982) have shown that in symmetric common-value environments second-price

auctions, by reducing the winner's curse, generate larger expected revenues than ¯rst-price auctions do.

1

theoretical results have persuaded the designers of the spectrum auctions to establish rules

that attempt to \level the ¯eld" for the entrants in the telephone markets, who were consid-

ered to be weaker.

This paper is an attempt to further our understanding on the properties of asymmetric

second-price, common-value auctions. It departs from the existing literature in two ways.

First, by introducing two-sided uncertainty, i.e., it analyzes the case in which both bidders'

types are private information. This two-sided, stochastic departure from symmetry seems

more natural than the extensions already considered in the literature. For it is highly unlikely

that all additional bidders' characteristics, that cause asymmetries, are public information.

And, moreover, there is no reason to believe that it is more plausible that there is uncertainty

about the characteristics of a subset of bidders only, and not about the characteristics of all

of them. The second departure involves the explicit employment of discontinuous bidding

strategies. Bidding strategies are not required to be continuous and strictly increasing.

Our more general framework considerably alters the outcomes derived in the literature.

Even when the probabilities that one of the bidders is an advantaged type and the other a

disadvantaged type are arbitrarily large, in equilibrium, the disadvantaged type bidder has

to win the auction with strictly positive probability, even in the state of nature in which her

opponent is actually an advantaged type. This result is delivered in generality in a two-type

model (Proposition 1). Moreover, two applications are worked out explicitly; one in a two-

type and another in a continuum-of-types model. In these applications it is actually shown

that even if the stochastic comparative advantage a bidder enjoys is large, she is not able to

capitalize on it (Propositions 3 and 6). For example, it is shown, in the two-type application,

that when there is ² probability for a bidder to be an advantaged type, while the other bidder

is equally likely to be of an advantaged or a disadvantaged type, she wins the auction with

probability equal to one fourth! Note that this probability of winning is computed when

the state of nature is such that the ¯rst bidder is a disadvantaged type while her opponent

an advantaged type. Clearly these results should alter one's perception about the ability

of second-price auctions to attract bidders that could be intimidated by the presence of

opponents, who may even have a substantial \stochastic comparative advantage."

It is also found that as we consider a sequence of auctions converging to the symmetric

Bikhchandani (1988) and Klemperer (1998) argue that this result does not carry over when asymmetries

are introduced. Finally, in a takeover context, Bulow, Huang and Klemperer (1999) show that ¯rst-price

auctions are not as sensitive as second-price auctions to the introduction of asymmetries, when these are

deterministic.

2

one, the equilibria we construct converge to the unique symmetric equilibrium of the sym-

metric auction (Proposition 4 and 7). These results restore the \common sense" intuition

that a small advantage should not have an enormous impact on outcomes. We thus put

into a di®erent perspective the sharp discontinuity found in the literature as one moves from

common-values to almost-common-values.

Moreover, we show that the restriction to equilibria in continuous and strictly increasing

strategies imposed in the literature is a necessary condition to derive the chasm between

results in symmetric and asymmetric auctions (Proposition 5). This way, we attempt to

highlight the fact that this dichotomy was delivered due to speci¯c characteristics in the

extentions considered in the literature, and hence it can be misleading.

As a consequence of these results, it can also be shown that the expected revenues gen-

erated in these asymmetric environments are not necessarily much smaller than the ones

generated in \comparable" symmetric environments.3 Moreover we show that there does

not exist a monotonic relationship between the seller's expected revenues and the intensity

of asymmetry among bidders, parameterized by the degree of stochastic comparative ad-

vantage (Corollary 1). It is shown that as the degree of stochastic comparative advantage

becomes large enough, expected revenues actually become larger compared to the case where

both bidders are ex-ante symmetric, i.e., when no bidder enjoys the stochastic comparative

advantage.

Finally, we show that as the equilibria of the asymmetric auctions converge to the sym-

metric equilibria of the symmetric auctions, their expected revenues also converge to those

that are attainable in the symmetric auctions. Hence, one should put into a more rigor-

ous examination the widely held belief that second-price auctions should be avoided in the

presence of asymmetries.4

In short, by considering an encompassing and more natural framework to introduce asym-

metries among bidders, we reestablish a lost linkage in the analysis of pure-common-value

and almost-common-value auctions. We believe that this is of importance not only for its

aesthetic, theoretical appeal, but also because it brings back relevance to the intuitions

3An example is generated in which the di®erence between the expected revenues is less than 2%.4To the best of our knowledge, the literature has not performed direct comparative statics' with respect

to the relative performance of ¯rst-versus second-price auctions in almost-common-value environments. The

preference for ¯rst-price auctions comes from the fact that second-price auctions were considered to perform

very poorly. Here, we do not compare these two auctions either. Hence, we do not advocate the use of

second-price auctions over ¯rst-price ones.

3

gained by many theoretical results reached from the analysis of symmetric settings when

considering real case studies, where, no doubt, asymmetries abound.5

In recent years there is a renewed theoretical interest in the e®ects of the introduction

of asymmetries in various auctions.6 As already mentioned, the ¯rst paper to study the ef-

fects of asymmetries in second-price common-value auctions was Bickchandani (1988). Since

then, it was Bulow and Klemperer who in a sequence of papers popularized the idea that

the introduction of asymmetries greatly a®ects the outcome in a second-price auction in

di®erent contexts, and proposed ways to control for its e®ects. Bulow and Klemperer (2000)

discusses how the ratio of the number of objects being auctioned o® and of the number of

bidders considered to be advantaged can be manipulated to mitigate the unwanted e®ects

asymmetries create. Bulow, Huang and Klemperer (1999) analyzes the e®ects toeholds have

on takeover battles. Finally, Klemperer (1998) analyzes numerous real life examples, such as

the 1995 broadband spectrum auction in the US, in which he believes that the presence of

asymmetries shaped the outcomes. In a private-value framework, Maskin and Riley (2000)

perform e±ciency and revenue comparisons between ¯rst- and second-price auctions under

di®erent sorts of asymmetries in beliefs. Dasgupta and Maskin (2000) study the e±ciency

properties of various auctions, including one similar to ours (common values with multi-

dimensional uncertainty). Finally, Che and Gale (1998) also study the e®ect of ¯nancial

constraints on bidders' behavior, but on private-value auctions.

The paper is organized as follows. The next section sets up the model. Section 3 delivers

the general results in the two-type case. Moreover, it explicitly solves for a type-asymmetric

equilibrium (the only class of admissible candidate equilibria), and describes its convergence

and expected revenues properties. Section 4 constructs a type-symmetric equilibrium in a

continuum-of-types model. Finally, Section 5 concludes.

2 The Model

An auctioneer organizes an auction to sell an object. There are 2 risk-neutral bidders,

i 2 fI; IIg, competing to acquire it.5For an econometrician, such result actually implies that it does make sense to test implications coming

out of models on symmetric auctions; a meaningless exercise according to the perceived in the literature

fragility of the results in symmetric settings.6For a review of the literature, see Klemperer (1999).

4

We assume that the value of the object is common to both bidders, and equals to v 2[v; v ] ½ IR+. This value is unknown. Each bidder though receives a private signal xi, such

that v = v(xI ; xII ) = xI+xII.7 The signals are assumed to be independently and identically

distributed (i.i.d.) according to a cumulative distribution function F . The associated density

function, f, is assumed to be absolutely continuous and strictly positive over its closed and

bounded support [0; x] ´ X ½ IR+.8

Each bidder receives another private signal µi 2 [µA; µD ] ´ £ ½ IR++, i.i.d. across

bidders, distributed according to G, with absolutely continuous and strictly positive over its

support density function, g. This signal contains private information regarding bidder i's

marginal utility of income due to di®erential access to external sources of ¯nance.9 Finally,

there is no correlation between the two signals.

From now on, we will refer to µi as the type of bidder i, and to xi as the signal she has

received (i.e., a bidder who has observed the pair (x; µ) will be referred to hereafter as a

µ-type bidder with signal x).

Bidder i's utility is Ui = v ¡ µipi. Nonetheless, instead of understanding the auction asone of pure common-values in which bidders di®er in their marginal utility of income, it is

convenient to analyze it as an auction in which the bidders have identical marginal utilities of

income, but they value the object di®erently. In this case, bidder i's utility can be rewritten

as

Ui =v

µi¡ pi:

Clearly, a bidder who has drawn a low µ has a comparative advantage compared to one who

has observed a larger realization. Spanning through all possible realizations, the bidder with

µA is the most advantaged-type bidder, and the one with µD is the most disadvantaged-type

bidder.

The equilibrium notion we employ is stronger than the Bayesian-Nash equilibrium. In

addition, we restrict attention to strategies that satisfy individual rationality, i.e., bids have

7The normalization that the value of the object is the sum of the signals is only made for notational and

computational convenience. No results rely on it. An auction for such object is coined in the literature as

the \wallet game".8The lower bound of the support is chosen to be zero to relieve notational burden. The extent to which

this normalization a®ects the results will be explained as we proceed.9This is just one possible way to introduce asymmetries. Nonetheless, one can immediately see that the

employed canonical model can be used to analyze the e®ects of asymmetries created by any other cause

mentioned in the Introduction.

5

to be such that bi(xi; µi) ¸ xiµi.10

3 Being Disadvantaged Is Not Fatal

In this section, we consider that µ is distributed according to a two-point distribution £ ´fµA; µDg. A bidder who receives signal µA is called A-type, while if she receives µD, D-type.11

Bidder i is A-type with probability ¹i, where ¹i 2 (0; 1), and D-type with the complementaryprobability. The bidder with the larger ¹i is the one who has the stochastic comparative

advantage.

3.1 General Results

The most striking result in the literature on asymmetric, second-price, common-value auc-

tions is that a bidder who has a disadvantage compared to her opponent (almost surely)

never wins the auction. In a model with one-sided uncertainty about one bidder's type,

Bikhchandani (1988) shows that this result holds even if the probability that one bidder is

more advantaged than the other is vanishingly small. In a deterministic framework, Bulow

and Klemperer (2000) deliver the same result even when they allow the di®erences between

types to become negligible.

In this section, we show that this result is not robust to the introduction of two-sided

uncertainty about the type of the two bidders. Speci¯cally, we show that even if the prob-

abilities that one of the bidders is A-type and the other D-type are arbitrarily large, it is

possible that a D-type bidder wins the auction. It should be noted that she wins the auction

with strictly positive probability even in the state of nature in which her opponent is A-type.

This result holds regardless of whether the D-type bidder is the one considered to have the

stochastic comparative advantage or not. In other words, we prove a more general version

of the statement needed to show that the results shown in the literature do not carry over

when one allows for two-sided uncertainty. Most notably, it holds even when ¹I is ² close to

1 and ¹II is ² close to 0, and the actual state of nature is the most probable one, i.e., bidder

I is A-type and bidder II is D-type. The following proposition states formally the result.12

10This restriction is frequently made to rule out unreasonable equilibria, e.g., Bikhchandani (1988).11The notation i ¡ j bidder, where i = fI; IIg and j = fA; Dg, denotes in this section a j-type bidder i.12The normalization that the lower bound in the support of x is zero allows us to make the proposition

statement unconditionally. Otherwise, it would have been necessary to assume that¹x+xµD

>2xµA

. That is,

6

Proposition 1 8 (¹I; ¹II ), in every equilibrium in non-decreasing, pure strategies of this

auction, there exists a D-type bidder who must win over an A-type bidder with strictly positive

probability.

Proof: Assume not. In such case, in equilibrium, a D-type bidder wins with zero proba-

bility over an A-type bidder, and therefore an A-type bidder should assign zero probability to

the event that she ties with a D-type bidder. Concentrate ¯rst on type-symmetric strategies.

An A-type bidder must bid in equilibrium bA(x) = 2xµA; i.e., the strategies derived in Milgrom

(1981). In such equilibrium, for a D-type bidder to never win, she must bid bD(x) = 0.

Consider a D-type bidder with signal ¹x. The bid coming out of her individually rational

strategy is ¹xµD. By bidding it, she wins whenever 2x

µA< ¹x

µD. Hence, she wins against an

A-type bidder with signal x 2 [0; ¹xµA2µD), while making positive pro¯ts conditional on winning.

(The true value to her is ¹x+xµD, whereas she pays no more than ¹x

µD.) Since this deviation is

pro¯table it contradicts equilibrium strategies.

Consider now type-asymmetric strategies. The proof is split into two cases: (a) We ¯rst

show that there exists no equilibrium strategy in which one of the A-type bidders (say I-A)

always wins over a positive mass of II-A bidders with signals x 2 S µ X. We have to analyze

two situations: i) II-A bidder uses an increasing strategy. As D-type bidders always lose,

bII¡A(x) ¸ ¹xµD; 8x 2 S . Take y = ¹xµA

µD. Assume w.l.o.g. that y 2 S . Consider the I-A bidder

with signal ², where ² ! 0: She is making losses when winning over a II-A bidder with signal

x 2 [0; y). To see this, just notice that when winning the price is greater than the true value.Further, she does not make positive pro¯ts winning over a II-A bidder with signal larger

than y, as such bidder bids at least her individual rational bid. Thus, she is better o® by

deviating down to bII¡A(0). ii) II-A bidder's strategy has a °at interval, i.e., 9 [x1; x2] ½ S

such that bII¡A(x) = b;8 x 2 [x1; x2]. For such strategy to be individually rational, b ¸ x2µA.

But then the I-A bidder with signal ² is better o® by deviating down and not winning over

any II-A bidder with signal x 2 [x1; x2].

(b) We now show that there exists no equilibrium strategy in which I-A and II-A bidders

tie among themselves while D-type bidders always lose. Assume it does. Let II-A bidder use

an increasing strategy. Consider again a I-A bidder with signal ², ² ! 0. Assume that this

bidder ties with a II-A bidder with signal M . Bertrand competition among them implies

that the individually rational bid of the least advantaged type when receiving the highest possible signal is

larger than the individually rational signal of the most advantaged type when receiving the lowest possible

signal. This is nothing more than the weakest condition needed in an auction to ensure the presence of real

competition among bidders. This condition is trivially satis¯ed when x = 0.

7

that bI¡A(²) = ²+MµA

¸ ¹xµD. As ² ! 0, M

µA¸ ¹x

µDwhich implies that M ¸ µA

µD¹x > 0. Therefore,

M 9 0. Take a II-A bidder with signal m < M who ties with a I-A bidder with signal r.

In a tie, m+rµA

¸ ¹xµD. As m ! 0, r ¸ µA

µD¹x > ²; which contradicts increasing strategies. So

if an equilibrium exists, the bidding strategies must have °at intervals. Employing similar

arguments as in the previous case one can show that this is not possible. Q.E.D.

The intuition behind the result is the following: An A-type bidder cannot bid very

aggressively because she knows that with positive probability she may be playing against

a bidder who may also be an A-type. If both are aggressive, they end up regretting when

winning. Even if the probability of meeting an A-type is very small, the bidder's attention

is concentrated on this event. This reluctance, on the other hand, makes the D-type bidders

less cautious, and as a result the extreme situation analyzed in the literature is not obtained.

D-type bidders need to win with strictly positive probability in all equilibria of the auction.

The next proposition further characterizes the set of admissible equilibria in this auction.

Namely, it shows that there are no type-symmetric equilibria.

Proposition 2 There exist no type-symmetric equilibria in non-decreasing pure strategies.

Proof: We ¯rst show that type-symmetric equilibrium bidding functions must be single-

valued and continuous, i.e., there are no gaps. The proof is by contradiction. Assume

w.l.o.g. that an A-type bidder's equilibrium strategy has a gap at M, i.e., bA(M) = bH >

bA(M¡²) = bL. Two cases have to be considered: (a) Assume @ x such that bD(x) 2 (bL; bH).Conditional on winning, lim²!0 v(M ¡ ²; y) = v(M; y). Thus, either the bidder with signalM makes strictly positive pro¯ts, and hence the bidder with signal M ¡ ² has incentives toraise her bid; or she makes loses and hence has incentives to lower her bid. Note that if she

makes zero pro¯ts, the bidder with signal M ¡ ² makes pro¯ts and hence she has incentivesto lower her bid. (b) Assume 9 [xL; xH] ½ X such that bD(x) 2 (bL; bH); 8x 2 [xL; xH ]. Ina type-symmetric equilibrium, bA(x) ¸ bD(x);8x. Just notice that for any price a D-typebidder with signal x is willing to pay, an A-type bidder with the same signal makes strictly

more pro¯ts. This implies that xL > M ¡ ². Given that the bidder with signal M ¡ ²

loses over these D-type bidders whereas the one with signal M wins, she has incentives not

to raise her bid only if: E£bD(y)jy 2 [xL; xH ]

¤= M + E [y jy 2 [xL; xH ]] > 2M , where the

inequality holds because xL >M ¡ ². Hence, in equilibrium bA(M) > 2M . But then one ofthe A-type bidders, say bidder I, with signal M can make more pro¯ts by lowering her bid,

contradicting type-symmetric equilibrium. Hence, type-symmetric equilibria have to be in

continuous strategies.

8

Assume now, by way of contradiction, that there exists an equilibrium in type-symmetric

strategies. If this is the case, they have to be continuous. Further, an A-type bidder with

signal ¹x must win with probability 1. Since by Proposition 1, a D-type bidder cannot

always lose, and since strategies are shown to be continuous, there exists a signal Z < ¹x

such that bA(Z) = bD(¹x). Now, the unique equilibrium strategy for an A-type bidder with

signal greater than Z is bA(x) = 2xµA; 8x > Z. The continuity of the strategies implies that

bA(z) = bD(¹x) = 2ZµA. Now, consider an A-type bidder with signal Z ¡ ². If bA(Z ¡ ²) <

2ZµA, then by deviating and bidding 2Z

µA, she pays at most her bid, while the true value is

1µA

f(Z ¡ ²) + (¹iZ + (1 ¡ ¹i)¹x)g > 2ZµA. Since the deviation is pro¯table and strategies are

assumed non decreasing, it must be the case that bA(Z ¡ ²) = 2ZµA. By the same argument,

this must also be the bid of any A-type bidder with signal y 2 [z; Z], where z is such thatz +f¹i(E [xjx 2 [z; Z]]) + (1¡¹i)¹xg = 2Z. Given that z < Z, one of the A-type bidders canmake more pro¯ts by slightly increasing her bid. (The price she will pay is almost surely the

same but she will increase the probability of winning.) Therefore, the deviation is pro¯table;

a contradiction. This concludes the proof. Q.E.D.

Nonetheless, there exist type-asymmetric equilibria in such auction. The next subsection

fully characterizes a pure-strategy, type-asymmetric equilibrium, and discusses its properties

in an application with uniform distributions.

3.2 Type-Asymmetric Equilibria and Their Convergence Proper-

ties

3.2.1 Characterization of an Equilibrium

Consider the following example: xi's are uniformly drawn from [0; 1], µA = 1, and µD = 2.

The following proposition fully characterizes a type-asymmetric equilibrium in which a

D-type bidder has a strictly positive probability of winning over an A-type bidder.

Proposition 3 For all ¹I , and for each ¹II no larger than12, the following bidding strategies

constitute an equilibrium:

I) bI¡A(x) =

(x + s x 2 [0; Q]2x x 2 (Q; 1]

bI¡D(x) =

(x x 2 [0; s]x+ s x 2 (s; 1]

9

II) bII¡A(x) =

8>><>>:

s x 2 [0; s]2M x 2 (s;M ]2x x 2 (M; 1]

bII¡D(x) =

(x x 2 [0; s]s x 2 (s; 1]

where s =1¡¹II¡

p¹II¡¹2II

1¡2¹II , M = 1+s2 , and Q =

3+s4 .

Proof: We ¯rst show that there is no signal x 2 X, such that I-A bidder, by deviatingfrom the above strategies, can make more pro¯ts. If x = 0, by following the purported

equilibrium strategies, in the event of a tie, after updating her beliefs about the type of

the opponent she tied with, she makes zero pro¯ts: Just notice that, by the de¯nition of s,

¹IIs(s2¡s)+(1¡¹II)(1¡s)( 1+s2 ¡s) = 0. Moreover, there is no other bid that gives her larger

pro¯ts. If x 2 (0; Q), she makes positive pro¯ts by winning over a II-A bidder with signal in[0; s] and over any II-D bidder. By deviating up to win over a II-A bidder with signal in [s;M],

she only succeeds in reducing her pro¯ts. Just notice that x+M+s2 ¡2M < Q+M+s

2 ¡2M = 0.

When x = Q, she is indi®erent between bidding Q+ s and 2Q. The only payo®-di®erence

between these two strategies is that with the former she loses against a II-A bidder with

signal in [s;M ], whereas with the latter she wins over her. Since, conditional on winning,

expected pro¯ts are zero (just notice that Q+ M+s2 ¡ 2M = 3+s

4 + 1+3s4 ¡ (1 + s) = 0), the

two strategies are payo®-equivalent for the bidder. Clearly, if a I-A bidder with signal Q is

indi®erent between winning or not over a II-A bidder with signal in [s;M], a I-A bidder with

signal larger than Q does not regret winning. Hence, I-A bidder's strategy is best response

against bidder II's strategies.

We now show that there is no signal x 2 X, such that II-A bidder can pro¯tably deviatefrom her purported equilibrium strategy. If x 2 [0; s), by following the strategy, she loses

against any I-A bidder and a I-D bidder with signal in [s; 1], but she does not regret losing.

If she bids higher so as to win, she su®ers a loss equal to s ¡ x > 0. When x 2 [s;M ] shewins against a I-A bidder with signal in [0; Q), and against any I-D bidder. She does not

regret winning since, by doing so, she gains at least x¡ s > 0. For any signal x 2 [s;Q), sheloses against a I-A with signal in [Q; 1], but she does not regret losing. If she bids higher so

as to win, she loses at least x¡Q > 0. Finally, when her signal is larger than Q she followsthe strategy 2x. By doing so, she wins over a I-A bidder with signal smaller than hers and

makes pro¯ts upon outbidding them. By deviating up, to win over a I-A bidder with signal

larger than hers, she makes loses. As she also makes pro¯ts by winning over a I-D bidder,

there is no other strategy that gives her larger pro¯ts. Hence, no deviation is pro¯table.

10

Consider now D-type bidders. When they tie with their opponent, and according to the

equilibrium strategies they assign zero probabilities to the event that they have tied with

an A-type bidder, they play the unique symmetric equilibrium of an auction where: a) it

is common knowledge that both bidders have the same type, and b) the signal space is

truncated to contain only signals in which, by following the above speci¯ed strategies, they

would have never tied with an A-type bidder. This ensures that the purported equilibrium

strategies of D-type bidders with signal in [0; s) are indeed equilibrium strategies. We now

show that D-type bidders with larger signals do not want to deviate either. A I-D bidder

with signal s is indi®erent between bidding s and 2s. The only payo®-di®erence between

these two strategies is that with the former she ties against a II-A bidder with signal in

[0; s] and against a II-D bidder with signal in [s; 1], whereas with the latter she wins over

both types. Since, conditional upon winning, expected payo®s are zero (just notice that,

by the de¯nition of s, ¹IIs(3s4 ¡ s) + (1 ¡ ¹II )(1 ¡ s)(3s+14 ¡ s) = 0), the two strategies

are payo®-equivalent for this bidder. Since she is indi®erent between tying or losing against

those bidders, a I-D bidder with larger signal strictly prefers to win. Finally, notice that a

I-D bidder with signal in (s; 1] loses against a II-A bidder with signal larger than s, and she

does not regret it.

Now a II-D bidder with signal in [s; 1] loses against a I-D bidder with signal larger than s

and against any I-A bidder. We now show that a II-D bidder with signal 1 does not want to

deviate. She is willing to pay at most 1. Given that a I-D bidder with signal s bids 2s > 1,

she does not want to outbid her. Similarly, she does not regret losing against a I-A bidder

with signal 0, as the price she has to pay to win is s, while the true value conditional on

winning is 12< s. Clearly, if there is no pro¯table deviation for her, there cannot be any for

a II-D bidder with a smaller signal.

We have proved that the preceding strategies are best responses against each other.

Hence, they do constitute an equilibrium. Finally, notice that if ¹II is no larger than 1/2,

then s is no smaller than 1/2. This means that, when bidding s, a II-D bidder with signal

1 bids more than her individual rational bid, which is 1/2. Therefore, the strategies are

individually rational. Q.E.D.

It is worth highlighting an interesting property of the equilibrium: A D-type bidder has

a strictly positive probability of winning the auction over an A-type bidder. Just notice

that bidder I-D with signal x 2 [s; 1] beats bidder II-A with signal x 2 [0; s). Moreover,

the equilibrium is independent of the actual values of ¹I . So even if ¹I takes small values,

the probability with which bidder I-D wins the auction over bidder II-A can be pretty large.

11

This can be easily shown by checking the strategies in Figure 1.

s M Q 1

s

1+s

2

BII-D

BI-D

BII-A

BI-A

Figure 1: Bidding strategies

For example, by setting ¹II =12and ¹I = ², one can ¯nd that the probability bidder I-D

wins over bidder II-A is 14. Hence, we do not speak about an ² probability event. Even if the

stochastic comparative advantage bidder II enjoys is large, this bidder cannot capitalize on

this enhanced reputation and she ends up losing the auction with a large probability, even

in the state of nature in which she is A-type playing against a D-type opponent.

3.2.2 Convergence Properties

It is of great importance to understand the behavior of the equilibrium as one considers the

limit case in which the uncertainty about both or one of the bidders disappears. This can

be studied by considering di®erent scenarios. First, the two bidders may end up becoming

similar. In this case the auction converges to a symmetric one with no uncertainty. Alterna-

tively, the di®erence between the two bidders may become large. Finally, we will consider the

case in which the auction converges to an asymmetric auction with one-sided uncertainty.

Since the equilibrium strategies are independent of ¹I, one can perform these exercises by

just studying the strategies as ¹II approaches zero.

Let h¹IIni be a decreasing sequence of real numbers with ¹IIn > 0, and limn!1 ¹IIn =0. Let ¡¹I ;¹IIn be the auction game in which bidder I is A-type with probability ¹I and

12

bidder II with probability ¹IIn . For each n, and for every ¹I > 0, the set of strategiesnbI¡A¹IIn

; bII¡A¹IIn; bI¡D¹IIn

; bII¡D¹IIn

oin Proposition 3 constitute an equilibrium of the game ¡¹I;¹IIn ,

for sn = (1¡ ¹IIn ¡q¹IIn ¡ ¹2IIn)=(1 ¡ 2¹IIn ), Mn = (1 + sn)=2 and Qn = (3 + sn)=4.

We ¯rst show that as the auction converges to a symmetric one in which both bidders are

known to be D-type, the strategies in Proposition 3 converge to the symmetric equilibrium

strategies \µa la Milgrom", i.e., bI¡D(x) = bII¡D(x) = x for all x 2 [0; 1].

Proposition 4 If ¡¹I ;¹IIn ! ¡0;0, then bI¡D¹IInand bII¡D¹IIn

converge almost uniformly to Mil-

grom's symmetric equilibrium strategies.

Proof: As ¹IIn ! 0, bII¡D¹IInconverges pointwise boundedly to bII¡D(x) = x for every

x 2 [0; 1], whereas bI¡D¹IInconverges a.e. to bI¡D(x) = x: Notice that it converges pointwise

to the function f (x) = x whenever x 2 [0; 1) and to f(x) = 2 whenever x = 1. Hence, theresult follows from Egoro®'s theorem, as we have a sequence of measurable functions that

converge a.e. on a bounded set to a continuous function. (See Royden, chapter 3). Q.E.D.

This result should be contrasted to the one derived by Bikhchandani in his framework,

where it is shown that the type-symmetric equilibrium strategy does not belong to the set of

equilibrium strategies for a D-type bidder as one considers a sequence of auctions converging

to a symmetric auction.13 Moreover, it puts under a di®erent light the widely held belief

that the introduction of the slightest asymmetry alters completely the analysis of a second-

price, common-value auction. The symmetric auction is robust to small changes towards

asymmetry in the following sense: When considering a sequence of auctions converging to

the symmetric one, the type-asymmetric equilibrium we construct converges to the unique

symmetric equilibrium of the symmetric auction.

We now turn to the case in which the auction converges to the \Bikhchandani framework"

{ i.e., there is one-sided uncertainty. This can be done by taking ¡¹I ;¹IIn ! ¡0;¹IIn . In the

¯rst case, where ¹I ! 0, but ¹IIn 2 (0; 1=2), bidder I wins with strictly positive probabilityeven in the event that her opponent is A-type. This divergence from the result reached by

Bikhchandani is due to the fact that we have constructed an equilibrium with discontinuous

and/or °at strategies while the analysis in Bikhchandani is con¯ned to continuous and strictly

increasing strategies.14

13See Bikhchandani (1988), Proposition 3, p. 111.14The next section, by considering the continuum-of-types case, constructs an equilibrium with the same

qualitative properties as the one in Proposition 3, but in continuous and strictly increasing strategies. Hence,

the main results delivered in this section do not hinge on the fact that we allow for discontinuities.

13

Proposition 5 Consider an asymmetric, common-value, second-price auction with one-

sided uncertainty. There does not exist an equilibrium in which a D-type bidder wins over

an A-type opponent if and only if the equilibrium is in continuous and strictly increasing

strategies.

Proof: Su±ciency is proved in Bikhchandani (1988) (Proposition 1, p. 108). Necessity

is derived by the counterexample presented above. Q.E.D.

In contrast, were we considering the opposite situation where ¡¹I ;¹IIn ! ¡¹I;0, i.e., ¹IIn !0, and ¹I 2 (0; 1), where the limiting equilibrium employs strictly increasing strategies,

Bikhchandani's result is valid.15

Finally, consider the case in which the auction converges to a deterministic asymmetric

auction, as in Klemperer (1998). This is done by considering ¹I ! 1 and ¹II ! 0. In this

case the relevant strategies are those of A-type bidder I and D-type bidder II. It is clear

that now the D-type bidder will never win. Hence, the equilibrium outcome of the auction

converges almost uniformly to the one found in the literature.16

3.2.3 Expected Revenues

We now turn to the behavior of the auction in terms of expected revenues to the seller. Let

P i¡j(:) denote the ex-ante payment by bidder i when her type is j. The expected revenue

to the seller can be written as:

R(¹I ; ¹II) = ¹IPI¡A(¹II )+(1¡¹I )P I¡D(¹II)+¹IIP II¡A(¹I; ¹II )+(1¡¹II)P II¡D(¹I; ¹II );

where,

P I¡A(¹II) =

µs¡ 1

2s2

¶+ ¹II

µ49

192¡ 85

64s+

89

64s2+

35

192s3

¶;

P I¡D(¹II) =

µs¡ 3

2s2 +

2

3s3

¶+ ¹II

µ¡s+ 5

2s2 ¡ 5

3s3

¶;

P II¡A(¹I ; ¹II) =

µ1

2+1

2s¡ 2s2 + 3

2s3

¶+ ¹I

µ¡16+1

16s+

11

8s2 ¡ 85

48s3

¶;

P II¡D(¹I ; ¹II) = (1¡ ¹I)µ1

2s2 ¡ 1

3s3

¶;

15Note that as ¡¹I;¹IIn! ¡¹I;0, all bI¡A

¹IIn, bI¡D

¹IIn, and bII¡D

¹IInconverge almost uniformly to equilibrium

strategies ¶a la Bikhchandani. In particular, bI¡A¹IIn

(x) ! x + 1, bI¡D¹IIn

(x) ! x, and bII¡D¹IIn

(x) ! x.16Note that as ¡¹I;¹IIn

! ¡1;0, bI¡A¹IIn

, and bI I¡D¹IIn

converge uniformly to asymmetric equilibrium strategies

¶a la Milgrom. In particular, bI¡A¹IIn

(x) ! x + 1, and bII¡D¹IIn

(x) ! x.

14

and, recall, s=1¡¹II¡

q(¡¹2II+¹II)

1¡2¹II ¢Tedious but straightforward computations show that

R(¹I; ¹II ) =

µs¡ s2+ 1

3s3

¶+ ¹I

µ1

2s2 ¡ 1

3s3

¶+ ¹II

µ1

2¡ 1

2s+

1

6s3

¶

+ ¹I¹II

µ17

192¡ 17

64s+

49

64s2 ¡ 49

192s3

¶: (1)

It is easily veri¯ed that expected revenue is increasing in both ¹I and ¹II, so that the

expected revenue is bounded above by R(1; 0:5) = 0:568. Further, there are strategic revenue

substitutabilities between them, i.e., @2R(¹I ;¹II )@¹I@¹II

< 0. Note that when ¹I increases it is more

likely to have an I-A player in the auction. His expected payment is larger when facing a

II-A bidder than a II-D bidder, i.e., it is decreasing in ¹II .

As already mentioned, second-price auctions in almost-common-value environments have

acquired a quite negative reputation. The results we will present demonstrate that this

is due to the fact that the literature has focused on the deterministic or on the one-sided

uncertainty cases, and moreover because the analysis was con¯ned to continuous strategies.

To be able to appreciate the expected revenues' properties of the auction, we start by

computing the expected revenues as the auction converges either to a symmetric, or a de-

terministic asymmetric one. We compare them with the appropriate benchmark cases.

As ¹I ! ¹II ! 0, expected revenues can be easily computed; they converge to 13 . Just

notice that they are equal to the expected revenues coming out of the symmetric equilibrium

of the symmetric auction in which both bidders are known to be D-type. In other words, we

also get here, as expected, the same continuity result we got when analyzing the converging

properties of the equilibrium outcome per se.

As ¹I ! 1 and ¹II ! 0, expected revenues converge to 12 . When considering an asym-

metric auction, where it is common knowledge that there is an A-type and a D-type bidder

(as in Klemperer (1998)), expected revenues are 14.17

We now move to perform comparative statics' on the expected revenues with respect

to the degree of asymmetry between bidders, parametrized by the stochastic comparative

17There is actually a continuum of expected prices ranging from 0 to 12 . Nonetheless, the lower bound can

be reached only if the D-type bidder bids less than her individual rational bid. On the other hand, the upper

bound can be reached only if a D-type bidder with signal less than one-half bids \unnaturally" aggressively.

More aggressively than in the symmetric equilibrium had her opponent been of her own type.

15

advantage a bidder enjoys. This can be done by computing expected revenues as the di®er-

ence between ¹I and ¹II increases. To be able to make meaningful comparisons, it must be

the case that, as we change the comparative advantage, the \expected value of the object"

remains constant. We accomplish this by considering the case in which ¹I + ¹II = 1, i.e.,

by considering an example in which the expected value of the object is equal to 34.18 When

¹I = ¹II =12, no bidder has the comparative advantage, i.e., they are \stochastically sym-

metric" in an ex ante sense. At the other end, when ¹I = 1 and ¹II = 0 bidder I has the

maximum comparative advantage. Expected revenues can be readily computed by replacing

¹II with (1 ¡¹I) in (1). The following ¯gure graphs the expected revenues as a function of¹I .

0.492

0.494

0.496

0.498

0.5

0.5 0.6 0.7 0.8 0.9 1

Figure 2: Expected Revenues as a function of ¹I

Clearly, as it appears in Figure 2, there is a non-monotonic relationship between ex-

pected revenues and the degree of stochastic comparative advantage. Making bidders more

asymmetric in a stochastic sense may actually increase expected revenues.

Corollary 1 There exists a non-monotonic relationship between the seller's expected rev-

enues and the degree of stochastic comparative advantage.

Another interesting exercise is to compare the attainable expected revenues in two dif-

ferent auctions for which the expected value of the object being auctioned o® coincide: i.e.,

18The de¯nition of the \expected value of the object" is not straightforward. Given that each bidder may

value the ob ject di®erently, one could, in principle, condition the value of the object on the identity of the

winner. Note, though, that such approach is appropriate to compute the \expected value of the object to the

expected winner." But this is not the appropriate benchmark when one cares about maximizing expected

revenues before knowing the participating bidders' types. It makes more sense to compute the expected

value of the object to the group of bidders that can be attracted to the auction, or alternatively as it is done

in the paper, to the average bidder.

16

those presented in Figure 2, and those coming from an \expected value-equivalent" sym-

metric auction. The expected value-equivalent symmetric auction can be constructed by

considering an auction in which both bidders have the \¯ctitious type" µ = 43. It is then

found that in the symmetric equilibrium expected revenues are equal to 12. In Figure 2, it can

be seen that the seller's expected revenues are bounded below by 0:49091. That is, expected

revenues are not much di®erent compared to those attained in the comparable symmetric

auction.

4 Continuum-of-types case

We now characterize the equilibrium of the auction when x and µ are uniformly distributed

in their intervals. We show that in the continuous case there exists a type-symmetric equi-

librium in pure, continuous, and strictly increasing strategies.19

Proposition 6 The following strategies constitute the unique type-symmetric equilibrium in

continuous strategies:

b¤(x; µ) =x+ xE(b)

µ;

where

b = b¤(x; µ);

xE(b) =m(b) +M (b)

2;

and

m(b) ´ y 2 X such that b¤(y; µA) = b;

M(b) ´(y 2 X such that b¤(y; µD) = b if such y � ¹x exists

¹x if otherwise.

Proof: Any type-symmetric equilibrium strategies have to be decreasing in µ. Just,

notice that for any price (x; µ)-bidder is willing to pay for the object, (x; µ¡ ²)-bidder makesstrictly more pro¯ts when outbidding the same set of opponents. Similarly, it is easy to see

that they have to be increasing in x.

Further, in any type-symmetric equilibrium b(0; µ) = 0; 8 µ. Suppose the contrary: thereexists a bid function b(x; µ) such that (b; b) is an equilibrium and b(0; µ) = a > 0. When

19The two bidders' strategies are type-symmetric if bI (x; µ) = bI I(x; µ) = b(x;µ); 8x; µ.

17

bidder I observes (xI = a2; µI = µD) bidding 0 rather than the prescribed equilibrium bid

(i.e., a +¢a) will only matter in the event of a winning bid. The probability of such event

is positive. Now when bidding a+ ¢a and winning, she obtains an expected pro¯t which is

bounded above by 1µD( a2+ a4)¡ a: Notice that since strategies are decreasing in µ; this bidder

estimates the expected signal of her opponent as bounded above by a4(conditional on her

opponent signals' being below a2, they are uniformly distributed). As the upper bound on

her expected pro¯t is negative, she is hence strictly better o® by deviating down and bidding

0, contradicting equilibrium strategies.

Let (b0; b

0) be a type-symmetric equilibrium. Let us de¯ne its common bidding range

as bids below B; where B = b0(¹x; µA): Similarly, let bB = b

0(¹x; µD): Since (b

0; b

0) is a type-

symmetric equilibrium, for every b 2 [0; bB], there exist m(b) and M(b) in [0; ¹x], such thatb = b

0(m; µA) and b = b

0(M; µD). Similarly, for every b 2 ( bB;B] there exists m(b) 2 (0; ¹x]

and ~µ(b) 2 [µA; µD) such that b = b0(m; µA) and b = b0 (¹x; ~µ):

We now show that b¤(x; µ) constitute a type-symmetric equilibrium: Suppose that bidder

II adopts b¤ and consider the optimal response by bidder I. For every (x; µ)-type of bidder

I, she maximizes her expected pro¯t by bidding up to the point in which she is indi®erent

about whether she is selected as a winner when she is involved in a tie. Notice that such a

bid maximizes her probability of winning conditional on making non-negative pro¯ts when

winning. Now, using the properties of uniform distributions and the Bayesian inference of

the bidder, it is easy to see that the true value of the object when bidding b and facing a tie

is either x+(m(b)+M(b))=2µ if b � b¤(¹x; µD), or

x+(m(b)+¹x)=2µ if b > b¤(¹x; µD). As b¤(x; µ) is equal

to this expression, (b¤; b¤) constitute a type-symmetric equilibrium in continuous strategies.

Finally, the existence of a continuous function b¤ that satisfy all the above equations is

easily shown by construction. Consider an (x; µ)¡bidder such that b¤(x; µ) 2 [0; b¤(¹x; µD)]:Let b¤(x; µ) = b: The de¯nition of b¤ implies that the following system of equations must

hold:b¡ 1

µ(x+m(b)+M(b)

2 ) = 0

b¡ 1µA(m(b) + m(b)+M(b)

2 ) = 0

b¡ 1µD(M (b) + m(b)+M(b)

2 ) = 0

From the system it is clear that both m(b) andM (b) are linear function of b: Consequently for

each (x; µ) there is a unique b that solves the system. Further, the solution yields b¤(x; µ) =

4 x4µ¡µA¡µD which it is increasing in x, and decreasing in µ. A similar argument applies for all

(x; µ)-bidders such that b¤(x; µ) 2 (b¤(¹x; µD); b¤(¹x; µA)]. The argument completes the proof.Q.E.D.

18

In order to be able to perform comparative statics exercises, we set µA = 1, and ¹x = 1.

It is straightforward to calculate the equilibrium bid functions:

b¤(x; µ) =

8>><>>:

4x4µ¡µD¡1 if x 2

h0; 4µ¡µD¡13µD¡1

i;

3x+13µ¡1 if x 2

h4µ¡µD¡13µD¡1 ;1

i:

It is worth noting that the bid functions are piecewise linear in x. They have a steeper slope

in the ¯rst segment, and moreover this slope is decreasing in µ. The fact that the slope is

decreasing in µ shows that the more e±cient a bidder is (i.e., the smaller her µ is), the more

aggressively she bids.

As it was done in the two-type case, we investigate the behavior of the equilibrium as

the auction converges to a symmetric one. This is accomplished by performing comparative

statics as we let µD decrease. It can be easily checked out that the slope of the equilibrium

bidding function of a µD-type bidder is strictly increasing throughout the exercise, while the

one of a µA-type is weakly decreasing. In the limit, as µD ! µA, the equilibrium strate-

gies converge to the type-symmetric equilibrium strategies of the symmetric auction, i.e.,

limµ!1 b¤(x; µ) = b¤(x; 1) = 2x; 8 x.

Proposition 7 As µD ! 1, the type-symmetric equilibrium strategies of the asymmetric

auction converge pointwise to the symmetric equilibrium strategies of the symmetric auction.

Comparing Propositions 4 and 7, one can check that the results delivered in the two-type

case were not due to the fact that we do not restrict attention to continuous strategies.

This can be easily understood by Proposition 7: Although the equilibrium strategies are

continuous and strictly increasing, the properties of the equilibrium have nothing to do with

the results found in the literature. To the contrary, the equilibrium is qualitatively similar

to the one reached in the two-type case. In both cases, as we consider a sequence of auctions

converging to the symmetric one the equilibrium converges to the symmetric equilibrium of

the symmetric auction. Remember that according to Proposition 3 in Bickchandani (1988)

the symmetric equilibrum does not belong to the set of equilibria as one considers a sequence

of auctions converging to the symmetric auction.

19

5 Conclusions

The literature has demonstrated that second-price, common-value auctions are sensitive to

the presence of asymmetries among bidders. For example, Bikhchandani (1988) has shown

that if it is common knowledge that a bidder has a disadvantage compared to her opponent,

that bidder (almost surely) never wins the auction. We show that this result does not carry

over when one allows for two-sided uncertainty. Even if the probabilities that one of the

bidders is an advantaged type while the other is a disadvantaged type are arbitrarily large,

in every equilibrium, the disadvantaged type bidder needs to win the auction with strictly

positive probability. We then solve for the equilibria in two cases, one with two types

and another with a continuum of types, and we show that they converge to the symmetric

equilibria of the corresponding symmetric auctions. This is also true when considering

convergence in terms of expected revenue performance. In other words, we reestablish a

lost linkage in the analysis of common-value and almost-common-value auctions. We thus

highlight the fact that the presumed sensitivity to the presence of asymmetries was delivered

due to speci¯c characteristics in the extentions considered in the literature.

20

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21