Asymmetric English auctions

Journal of Economic Theory 112 (2003) 261–288

Asymmetric English auctions

Vijay Krishna�

Department of Economics, Penn State University, University Park, PA 16802, USA

Received 28 February 2001; final version received 7 August 2002

Abstract

This paper studies equilibria of the n-bidder single-object English, or open ascending

price, auction in a setting with interdependent values and asymmetric bidders. Maskin (in:

H. Siebert (Ed.), Privatization, Institut fur Weltwirtschaften der Universitat Kiel, Kiel, 1992,

pp. 115–136.) showed that if the values satisfy a ‘‘single crossing’’ condition, then the two-

bidder English auction has an efficient equilibrium. In this paper, two extensions of the single

crossing condition, the ‘‘average crossing’’ condition and the ‘‘cyclical crossing’’ condition, are

introduced. The main result is that under either of these conditions the n-bidder English

auction has an efficient equilibrium.

r 2003 Elsevier Science (USA). All rights reserved.

JEL classification: D44; D82

Keywords: Auctions; Efficiency

1. Introduction

The English, or open ascending price, auction is the oldest and perhaps the mostprevalent auction mechanism. Indeed, the word auction itself is derived from theLatin augere, which means ‘‘to increase.’’ Beginning with the pioneering work ofVickrey [18], the English auction has been the subject of extensive study in theanalysis of auctions as games of incomplete information. In the private values modelintroduced by him, Vickrey [18] observed that the English auction always allocatesthe object efficiently whereas, when bidders are asymmetric, the first-price sealed-bidauction may not.

ARTICLE IN PRESS

�Fax: +1-814-863-4775.

E-mail address: [email protected].

0022-0531/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved.

doi:10.1016/S0022-0531(03)00074-7

The English auction is also attractive from the perspective of the seller. In asymmetric model with interdependent values and affiliated signals, Milgrom andWeber [16] established the important result that the English auction fetches a higherexpected selling price than both the sealed-bid second-price auction and the sealed-bid first-price auction. Of course, in such a symmetric setting all common auctionforms allocate efficiently under fairly weak conditions.This paper follows the work of Maskin [13] in considering the efficiency properties

of the (single-object) English auction in a model where values are interdependent butbidders may be asymmetric. Maskin [13] pointed out that in this setting thepossibility of allocating efficiently requires that the valuations satisfy a ‘‘singlecrossing’’ condition: every bidder’s signal has a greater influence on his own valuethan on any other bidder’s value. If this condition is not satisfied there may be no

efficient mechanism. Maskin [13] also showed that with two bidders the singlecrossing condition is sufficient to guarantee that the English auction has an efficientequilibrium. The single crossing condition by itself does not suffice once there arethree or more bidders.In this paper, I introduce two different extensions of the single crossing

condition that are individually sufficient to guarantee that the general n-bidderEnglish auction has an efficient equilibrium. The first, called the average crossing

condition, is just a single crossing condition between a particular bidder’s valueand the average of all bidders’ values with respect to some other bidder’s signal.The second, called the cyclical crossing condition, requires that the influencesof different signals on different bidders’ values can be cyclically ordered. Bothimply the single crossing condition and both reduce to it when there are only twobidders.The conditions are flexible enough to accommodate the polar cases of pure private

values and pure common values. Moreover, if the valuations are additively separable

into a pure private component and a pure common component, then both theaverage and cyclical crossing conditions are satisfied. Thus, for instance, Wilson’s[19] ‘‘log-normal model’’ satisfies the conditions introduced here.There has been a renewed interest in the design of efficient mechanisms

with interdependent values. Following Maskin [13], Dasgupta and Maskin [6]have constructed an ingenious mechanism that has an efficient equilibrium wheneverthe single crossing condition is satisfied. Their mechanism asks bidders to submitbids that are contingent on the realized values of other bidders. Thus, bidders submitfunctions mapping others’ values into bids and the seller chooses a vector ofbids that are consistent with the submitted functions—that is, the bids are a fixedpoint of the submitted functions. As a result, the workings of the mechanism aresomewhat complicated. Perry and Reny [17] have proposed an alternative efficientmechanism which is simpler and more intuitive. Their mechanism asks bidders toengage in pairwise competition by submitting a vector of bids directed at thedifferent bidders. The form of the associated efficient direct mechanism has, ofcourse, been known since the work of Cremer and McLean [5]. It is a generalizationof the well-known Vickrey auction for private values to the case of interdependentvalues.

ARTICLE IN PRESSVijay Krishna / Journal of Economic Theory 112 (2003) 261–288262

The concerns of this paper are different in that instead of constructing alternativeefficient mechanisms, I ask whether there are circumstances under which a well-known and time-tried mechanism, the English auction, allocates efficiently.

1.1. Other related literature

Wilson [19] studies English auctions in a class of environments in which bidders’values have both private and common components and each is distributed lognormally but not necessarily symmetrically. He then shows that there exists anefficient equilibrium with log-linear strategies that can be explicitly computed as afunction of the parameters. The fact that this model is tractable for bothcomputation and estimation makes it particularly important for empirical andexperimental work. It will be shown below that Wilson’s [19] model is included in theclass of problems studied in this paper. It satisfies both the average crossing and thecyclical crossing conditions.Lopomo [12] shows that the English auction is revenue maximizing in the class of

‘‘posterior implementable’’ rules. Posterior implementability is related to, butslightly weaker than, the notion of ex post implementability. Lopomo’s [12] result isderived in the context of the Milgrom and Weber [16] symmetric model and pertainsto the equilibrium identified there.It is convenient to model the English auction as one in which the price rises and

bidders only indicate whether they are active at that price or not. This is the modelmost commonly adopted and the one studied in this paper. Avery [3] has considereda more realistic model in which rather than indicating whether they are active or not,bidders actually call out bids. He exhibits that in this situation there are equilibria inwhich successive bids involve ‘‘jumps,’’ and indeed these may be revenue superior toother equilibria. Avery [3] also works in the context of the Milgrom and Weber [16]symmetric model.An open ascending price auction for multiple objects has been proposed by

Ausubel [2]. For the case of multi-dimensional private values, the outcomes of thismechanism are the same as those of Vickrey’s [18] multiple object auction and hence,are also efficient. Maskin [13] and Dasgupta and Maskin [6] show, however, thatwhen values are interdependent and signals are multi-dimensional, no mechanismcan allocate efficiently. Jehiel and Moldovanu [9] have generalized and refined theseresults. Basically, the single crossing condition cannot be satisfied when signals aremulti-dimensional.Kirchkamp and Moldovanu [10] have tested the efficiency properties of the

English auction experimentally. In an asymmetric setting with three bidders, theycompare the English auction to the second-price sealed-bid auction and find that theEnglish auction is indeed superior on grounds of efficiency. The valuations used intheir experiment satisfy the cyclical crossing condition.In varying guises, the single crossing condition is pervasive in the economics of

information. It was explicitly used in the context of auctions by Cremer and McLean[5] to construct mechanisms that extract all the surplus from buyers and are also, byimplication, efficient.

ARTICLE IN PRESSVijay Krishna / Journal of Economic Theory 112 (2003) 261–288 263

Bidder asymmetries cause great technical difficulties in first-price sealed-bidauctions. Maskin and Riley [14] have compared first- and second-price auctions in aprivate value setting with two bidders. They provide conditions under which theexpected selling prices in the two auctions can be ranked. Athey [1], Jackson andSwinkels [8], and Lebrun [11], provide general results on the existence of equilibriumin asymmetric first-price sealed-bid auctions.

2. Preliminaries

There is a single object to be auctioned and a set N ¼ f1; 2;y; ng of bidders. Eachbidder i receives a real-valued signal, SiA½0; 1� prior to the auction that affects thevalue of the object.1 As usual, I will denote by s ¼ ðs1; s2;y; snÞ; the vector of signalsof all the bidders and by sA ¼ ðsjÞjAA the vector of signals of the bidders in the set

ACN: Thus, I will write s ¼ ðsA; sN\AÞ and when A ¼ fig; s ¼ ðsi; siÞ:If the realized signals are s; the value of the object to bidder i is

Vi ¼ viðs1; s2;y; snÞ;

where vi is a twice continuously differentiable function. The function vi; hereafterreferred to as i’s valuation, contains all the information regarding the value of theobject to i that is available collectively to the bidders. Since other bidders’ signalsmay affect the value of a given bidder, the values are interdependent. The valuationssatisfy, for all i;

við0; 0;y; 0Þ ¼ 0: ð1Þ

It will be supposed throughout that every bidder’s information has a positiveinfluence on his own value, that is, for all i;

@vi

@si

ðsÞ40 ð2Þ

and also has a positive influence on the sum of all bidders’ values, that is, for all i;

Xn

k¼1

@vk

@si

ðsÞ40: ð3Þ

The signals S1;S2;y;Sn are distributed according to a joint density function f

with full support and f is assumed to be strictly positive on the interior of its support.Other assumptions on f will be introduced as and when needed.Our formulation follows that of Maskin [13] and can be thought of as a reduced

form of the general model of Milgrom and Weber [16], extended to allow for bidderasymmetries.

ARTICLE IN PRESS

1The assumption that all signals come from ½0; 1� is made only for convenience and does not affect anyof what follows.

Vijay Krishna / Journal of Economic Theory 112 (2003) 261–288264

2.1. English auctions

The English auction is an open ascending price auction. Specifically, the price ofthe object rises and bidders indicate whether they are willing to buy the object at thatprice or not. A bidder that is willing to buy at the current price is said to be an active

bidder. At a price of 0 all bidders are active and as the price rises bidders can chooseto drop out of the auction. The decision to drop out is both public and irrevocable.Thus, if bidder i drops out at a price p; both the identity of the bidder dropping outand the price p at which he dropped out are commonly known to the bidders.Furthermore, once bidder i drops out he cannot then ‘‘re-enter’’ the auction at ahigher price.It may be useful to think of each bidder pressing a button to signify that he is

active and this is indicated by a numbered light that is observed by all bidders. Oncea bidder drops out, his light goes off and the bidder cannot re-enter the auction. Inthis form, the auction specified above is sometimes referred to as the ‘‘buttonauction.’’In an English auction, a strategy for bidder i determines the price at which he

would drop out given his private information and given the history of who droppedout at what price. Formally, a bidding strategy for bidder i is a collection of

functions bAi : ½0; 1� R

N\Aþ -Rþ; where iAADN and #A41: The function bA

i

determines the threshold price bAi ðsi; pN\AÞ such that bidder i will drop out at any

price pXbAi ðsi; pN\AÞ when the set of active bidders, including i; is A; his own signal is

si; and the bidders in N\A have dropped out at prices pN\A ¼ ðpjÞjAN\A: I will require

that bAi ðsi; pN\AÞ4maxfpj: jAN\Ag:

The strategy bi calls on bidder i to behave as follows. If the set of active bidders,

including bidder i; is A; then bidder i compares bAi ðsi; pN\AÞ against the current

price p: As long as pobAi ðsi; pN\AÞ; bidder i remains active. He exits as soon as

p ¼ bAi ðsi; pN\AÞ: If he finds himself still active at a price p4bA

i ðsi; pN\AÞ; say as theresult of an error, he exits immediately.A Bayes–Nash equilibrium is a collection of strategies such that no bidder can

improve his expected payoff by following some other strategy.An ex post equilibrium of the English auction is a Bayes–Nash equilibrium with

the property that given all the signals ðs1; s2;y; snÞ and given the strategies of theother bidders no bidder can gain by behaving differently. In other words, an ex postequilibrium is a Bayes–Nash equilibrium that induces a Nash equilibrium for everyrealization of the signals.This paper is concerned with the existence of an efficient ex post equilibrium—that

is, one in which for every realization of the signals, the object is obtained by thebidder with the highest value. It known that the possibility of allocating the objectefficiently hinges on the single crossing condition (see [13]).2

ARTICLE IN PRESS

2Cremer and McLean [5] used a similar condition in their discrete signal model.

Vijay Krishna / Journal of Economic Theory 112 (2003) 261–288 265

3. Single crossing condition

Let

DvðsÞ ¼ @vi

@sj

ðsÞ� �

be the matrix whose ði; jÞth element is the partial derivative of i’s value vi with respectto j’s signal sj; that is, it is a measure of how j’s signal influences i’s value. We will

refer to Dv as the influence matrix and in order to economize on notation, in what

follows, I will write v0ijðsÞ @vi

@sjðsÞ:

Define

EijðvÞ ¼ s : viðsÞ ¼ vjðsÞ ¼ maxkAN

vkðsÞ� �

ð4Þ

to be the set of signal vectors such that bidder i’s value is the same as bidder j’s value

and both values are maximal. Recall that because of (1), 0AEijðvÞ:The version of the single crossing condition used in this paper is that introduced

by Dasgupta and Maskin [6].

Definition 1. The valuations v satisfy the (pairwise) single crossing condition if for allj and iaj;

v0jjðsÞ4v0ijðsÞ ð5Þ

at every sAEijðvÞ:

The pairwise single crossing condition requires that a given bidder’s information hasa greater influence on his own value than it does on some other bidder’s value. In otherwords, every diagonal element v0jj of the influence matrix Dv is greater than any off-

diagonal element v0ij in the same column. In the weak form stated above, it is required

that this holds only at points s where the values vi and vj are the same and maximal.

Dasgupta and Maskin [6] have shown that if the (pairwise) single crossing conditionis violated, then it may be that no incentive compatible mechanism can allocateefficiently. Moreover, as Cremer and McLean [5] and Dasgupta and Maskin [6] haveshown, this condition is also sufficient to guarantee that there is an efficient mechanism.Perry and Reny [17] also use this condition to construct efficient mechanisms.Maskin [13] showed that the single crossing condition is sufficient to guarantee the

existence of an efficient equilibrium in an English auction with only two bidders.

3.1. Two-bidder auctions

Theorem 1 (Maskin). Suppose that the valuations v satisfy the single crossing

condition. Then there exists an ex post equilibrium of the two-bidder English auction

that is efficient.


The equilibrium of the two-bidder English auction identified by Maskin [13]consists of a pair of increasing and continuous bidding functions bi : ½0; 1�-Rþ thatdetermine the prices b1ðs1Þ p1 and b2ðs2Þ p2 at which bidders 1 and 2,respectively, plan to drop out.The equilibrium bidding functions are determined as follows. Suppose that there

exists a pair of continuous and non-decreasing functions si :Rþ-½0; 1� satisfying:for all p such that siðpÞo1; si is strictly increasing;

3 and

v1ðs1ðpÞ; s2ðpÞÞ ¼ p;

v2ðs1ðpÞ; s2ðpÞÞ ¼ p: ð6Þ

Define biðsiÞ ¼ minfp : siðpÞXsig as the inverse of si: Then the bi constitute an expost equilibrium.Suppose that there exists a solution ðs1; s2Þ to (6) with the required properties and

suppose that b1ðs1Þ ¼ p14p2 ¼ b2ðs2Þ: Then s1ðp2Þo1 and so from (6),v1ðs1ðp2Þ; s2ðp2ÞÞ ¼ p2

and since s1 ¼ s1ðp1Þ4s1ðp2Þ and s2ðp2Þ ¼ s2;

v1ðs1; s2Þ4p2

because v01140: This implies that the winning bidder makes an ex post profit when hewins and since he cannot affect the price he pays, he cannot do better.It is also the case that

v2ðs1ðp1Þ; s2ðp1ÞÞ ¼ p1

and since s2ðp1Þ4s2ðp2Þ ¼ s2 and s1ðp1Þ ¼ s1;

v2ðs1; s2Þop1

because v02240: This implies that the losing bidder has no incentive to raise his bidsince if he were to do so and win the auction, it would be at a price that is so high asto make it not worthwhile.In addition, for any p4p2;

v2ðs1ðpÞ; s2ðpÞÞ ¼ p

and since v02240 and s2ðpÞ4s2ðp2Þ ¼ s2;

v2ðs1ðpÞ; s2Þop:

This means that if bidder 2 is still active at any p4p2; say, as a result of an error, hecannot benefit by remaining active since the value is less than the current price p andthus he should exit immediately.Thus, if there is an appropriate solution to (6), there exists an ex post equilibrium.4

ARTICLE IN PRESS

3This means that if p and p0 are such that pop0 and siðpÞo1 then siðpÞosiðp0Þ:4Milgrom [15] showed that with pure common values there is a continuum of such equilibria. The

reason is that if v1 ¼ v2; the system (6) is degenerate and so has a continuum of solutions.


The equilibrium constructed above is efficient because from (6),

v1ðs1ðp2Þ; s2ðp2ÞÞ ¼ v2ðs1ðp2Þ; s2ðp2ÞÞand again since s1 ¼ s1ðp1Þ4s1ðp2Þ and s2ðp2Þ ¼ s2;

v1ðs1; s2Þ4v2ðs1; s2Þbecause v0114v021 by the single crossing condition.Note that (6) asks a bidder, say 1; to stay in until a price b1ðs1Þ such that if bidder

2 were to drop out at b1ðs1Þ; and his signal s2 ¼ s2ðb1ðs1ÞÞ were inferred, bidder 1would just ‘‘break-even’’ since

v1ðs1; s2ðb1ðs1ÞÞÞ ¼ b1ðs1Þ:Eqs. (6) will thus be referred to as the break-even conditions.In two-bidder auctions, the single crossing condition guarantees that there is a pair

of continuous and non-decreasing functions ðs1; s2Þ satisfying the break-evenconditions (6), and as argued above, in that case, ðb1; b2Þ ¼ ðs11 ; s12 Þ constitutes anefficient equilibrium.5

3.2. Three or more bidders

The single crossing condition by itself is not sufficient to guarantee the existence ofan efficient equilibrium when there are more than two bidders. Consider thefollowing example.6

Example 1. Suppose there are three bidders with valuations:

v1ðs1; s2; s3Þ ¼ s1 þ 2s2s3 þ aðs2 þ s3Þ;

v2ðs1; s2; s3Þ ¼ 12

s1 þ s2;

v3ðs1; s2; s3Þ ¼ s3;

where a is a parameter satisfying 0oao 118:

To verify that the single crossing condition is satisfied, first consider changes inbidder 2’s signal, s2: Now v012 ¼ 2s3 þ a and v022 ¼ 1: But notice that if v1 ¼ v2; then

ðv022 v012Þs2 ¼ð1 2s3 aÞs2¼ 12

s1 þ as3

where the second equality follows from the fact that v1 ¼ v2: If either s140 or s340;then v0224v012: On the other hand, if both s1 ¼ 0 and s3 ¼ 0; then v022 ¼ 14a ¼ v012:Thus, whenever v1 ¼ v2; it is the case that v0224v012:

ARTICLE IN PRESS

5Chung and Ely [4] have shown that in a two-bidder English auction, if the single crossing condition is

satisfied, then the efficient equilibrium identified above can be obtained by iterative removal of weakly

dominated strategies.6This example is adapted from Perry and Reny [17] and I owe the idea to Phil Reny.


Likewise, whenever v1 ¼ v3; v0334v013: All other comparisons are straightforward.Suppose, by way of contradiction, that there is an efficient equilibrium in the English

auction and let b denote the strategies when all bidders are active. If s2 and s3 are both

greater than 12; then for all s1; v1 is greater than both v2 and v3: Efficiency requires,

therefore, that when all bidders are active, bidder 1 is never the first to drop out.But now consider signals s1; s2 and s3 such that bidders 2 and 3 have the same

value and bidder 1 has a lower value. (For example, if s1 ¼ 18; s2 ¼ 1

4; and s3 ¼ 5

16;

then v2 ¼ 516¼ v3 whereas v1 ¼ 9

32þ a 9

16o 516since ao 1

18:) Clearly, b2ðs2Þ ¼ b3ðs3Þ is

impossible since then bidder 1 would win the object—he is never the first to dropout—and that is inefficient. Suppose that b2ðs2Þob3ðs3Þ; so that bidder 2 drops outfirst. For small e40; if bidder 1’s signal is s1 þ e; then bidder 2 has the highest valueand it is inefficient for bidder 2 to drop out. On the other hand, supposeb2ðs2Þ4b3ðs3Þ; so that bidder 3 drops out first. Now if bidder 1’s signal is s1 e; thenbidder 3 has the highest value and it is inefficient for bidder 3 to drop out. This is acontradiction, so there cannot be an efficient equilibrium.

4. Main results

The single crossing condition is a bilateral condition; it is separately applied topairs of bidders and it has been shown that by itself the condition is not sufficient toguarantee that the English auction has an efficient equilibrium once there are threeor more bidders.I now introduce two conditions on the matrix Dv that are multilateral extensions

of the pairwise single crossing condition. Each condition is separately sufficient toguarantee that there is an efficient equilibrium in the English auction. Both involve astrengthening of the single crossing condition.

4.1. Average crossing condition

Define

%vðsÞ ¼1

n

Xn

i¼1viðsÞ

to be the average of the values of all the bidders. Denote by %v0jðsÞ ¼ @ %v@sjðsÞ the

derivative of the average value %v with respect to sj: The average crossing condition is

just a single crossing condition between a bidder’s value vi and the average value %v

with respect to signals sj; jai:7 It is required to hold only on the set of signals

EðvÞ ¼[jai

EijðvÞ;

where at least two bidders’ values are the same and maximal (recall (4)).

ARTICLE IN PRESS

7 ‘‘Average crossing’’ is short for the more descriptive but cumbersome term, ‘‘single crossing with

respect to the average.’’


Definition 2. The valuations v satisfy the average crossing condition if for all j andiaj;

%v0jðsÞ4v0ijðsÞ

at every sAEðvÞ:

The average crossing condition requires that the influence of any bidder’s signal onsome other bidder’s value is smaller than its influence on the average of all bidders’values. In other words, every off-diagonal element v0ij of the matrix Dv is smaller than

the average of the elements in that column.The first result of this paper is

Theorem 2. Suppose that the valuations v satisfy the average crossing condition. Then

there exists an ex post equilibrium of the English auction that is efficient.

The proof of Theorem 2 is provided in the next section.

4.2. Cyclical crossing condition

Definition 3. The valuations v satisfy the cyclical crossing condition if for all j;

v0jj4v0jþ1;jXv0jþ2;jX?Xv0j1;j ð7Þ

holds at every sAEðvÞ; where j þ k ð j þ kÞmod n:

The cyclical crossing condition requires that each column of the influence matrixDv can be ordered ‘‘in the same way’’ with the proviso that in each column, thediagonal element is (strictly) the largest. Cyclical crossing implies that for any two‘‘adjacent’’ bidders i and i þ 1; every signal sj; with the exception of siþ1; has agreater influence on vi than it does on viþ1: The exception is necessitated, of course,by the single crossing condition.Notice that the cyclical crossing condition requires only that there be some fixed

ordering of the bidders for which (7) holds. For instance, it may be that in a three-bidder situation, (7) holds only after bidders 1 and 2 interchange names.As an example, when n ¼ 3; cyclical crossing requires that Dv have the following

structure (perhaps after a relabelling of the indices):

v011 v012 v013k k

v021 v022 v023k k

v031 v032 v033U U

26666666664

37777777775;


where the arrow k denotes4 (orX) and the cyclical arrowU denotes4 (orX) thefirst element in the column.The second result of this paper is

Theorem 3. Suppose that the valuations v satisfy the cyclical crossing condition and

DvðsÞX0 for all sAEðvÞ: Then there exists an ex post equilibrium of the English auction

that is efficient.

The proof of Theorem 3 is also provided in the next section.

4.3. Discussion of the conditions

Both conditions imply the single crossing condition and when n ¼ 2; both reduceto it.When n ¼ 3; an alternative graphical representation is a useful aid in under-

standing the nature of the various crossing conditions. Consider the matrix Dv andlet v0j be the jth column of Dv: Suppose that each column of Dv is non-negative and

lies in the unit simplex D (alternatively, consider a separate rescaling of each columnof Dv so that it lies in the unit simplex). The three conditions can then beconveniently depicted as in Fig. 1. The single crossing condition requires that eachv0jASj ¼ fzAD: zj4zi; iajg: The average crossing condition requires that eachv0jAAj ¼ fzAD: 1

34zi; iajg and the cyclical crossing condition that each

v0jACj ¼ fzAD: zj4zjþ14zjþ2g:8

As is apparent from Fig. 1, the average crossing condition neither implies, nor isimplied by, the cyclical crossing condition.

4.3.1. Flexibility

Both conditions are flexible enough to accommodate both pure private values, thatis, for all i; viðsÞ ¼ uiðsiÞ; and, as a limiting case, pure common values, that is, for

ARTICLE IN PRESS

S3 S1

S2

e3 e1

e2

············

··············

·· · ·············

Single

····························

···························

······

······

······

······

···

A3 A1

A2

e3 e1

e2

Average

·················································

············

············

·

············

············

············

·

C3

C1

C2

e3 e1

e2

Cyclical

Fig. 1. Crossing conditions.

8The cyclical crossing condition is also satisfied if each v0jASj\Cj : This is just the result of a relabelling of

bidder indices.


all i; viðsÞ ¼ wðsÞ:9 (In Fig. 1, these correspond to the extreme points and center ofthe simplex, respectively.)More generally, if the valuations are additively separable into a private value and a

common value component, that is, for all i; viðsÞ ¼ uiðsiÞ þ wðsÞ; where u0i40; then

both the average and cyclical crossing conditions are satisfied. To see this, note firstthat with separability, for all j and iaj;

%v0jðsÞ ¼

1

nu0

jðsjÞ þ w0jðsÞ

4w0jðsÞ

¼ v0ijðsÞ

and so average crossing is satisfied. Also, for all j; and iaj;

v0jjðsÞ ¼ u0jðsjÞ þ w0

jðsÞ4w0jðsÞ ¼ v0ijðsÞ

so that cyclical crossing is satisfied. The additively separable formulation iscommonly employed in empirical work.

4.3.2. Heritability

Both conditions are inherited by the valuations of subsets of the bidders.Specifically, suppose that the valuations v satisfy the average (cyclical) crossingcondition. Then for all ACN and for all signals sN\A; the valuations ðvið�; sN\AÞÞiAA

also satisfy the average (cyclical) crossing condition.To see that the average crossing condition is inherited in this sense, suppose that

the ith row and column of Dv are deleted. This means that in each of the remainingcolumns the entry that was deleted was below the average for that column. This canonly cause the average of the remaining n 1 entries to increase and so the averagecrossing condition continues to apply to the remaining matrix Dvi: The argumentcan now be iterated. The fact that the cyclical crossing condition is inherited isstraightforward.This feature of the conditions is attractive because it means that once the

conditions have been verified for the set of all bidders, they no longer need to beseparately verified at different stages in the English auction when only some subset ofthe bidders is still active.

4.3.3. Ordinality

Both conditions can be weakened as follows. In a manner analogous to (4), definefor all ADN

EAðvÞ ¼ s : for all iAA; viðsÞ ¼ maxkAN

vkðsÞ� �

ð8Þ

ARTICLE IN PRESS

9The strict inequality in the definition of single crossing rules out the case of pure common values.


to be the set of signals such that the values of all bidders in A are maximal. Clearly,

for all A; 0AEA: Also, for every ACN; define

%vAðsÞ ¼1

#A

XiAA

viðsÞ

to be the average value of the bidders in A and define %v0Aj ¼ @ %vA

@sjto be the derivative of

%vA with respect to sj:

The weak average crossing condition is the following.

Definition 4. The valuations v satisfy the weak average crossing condition if for allADN for all i; jAA; iaj;

%v0AjðsÞ4v0ijðsÞ

at every sAEAðvÞ:

The only difference between this and Definition 2 is that when sAEAðvÞ theaverage is computed only over bidders in A and only derivatives with respect to thesignals of bidders in A are deemed relevant. Thus, the average crossing conditionimplies the weak average crossing condition. The conclusion of Theorem 2 obtainsunder this weaker condition.The weaker condition has an important desirable property that is not shared by

the stronger version in Definition 2.Suppose f :Rþ-Rþ is an increasing and differentiable function such that fð0Þ ¼

0: If the valuations v satisfy weak average crossing, and for all i; wið�Þ ¼ fðvið�ÞÞthen the valuations w also satisfy weak average crossing. To see this, first note that vi

is maximal at s if and only if wi is maximal at s: This means that for all

ADN; EAðvÞ ¼ EAðwÞ: Then, for sAEAðvÞ and i; jAA; iaj;

w0ijðsÞ ¼ f0ðviðsÞÞv0ijðsÞ ¼ f0ð%vAðsÞÞv0ijðsÞ:

Now for all iAA and for all sAEAðwÞ; viðsÞ ¼ %vAðsÞ and so

%w0AjðsÞ ¼

1

#A

XiAA

w0ijðsÞ

¼f0ð%vAðsÞÞ %v0AjðsÞ

4f0ðviðsÞÞv0ijðsÞ

¼w0ijðsÞ:

Thus, the weak average crossing condition is ordinal in this sense. The originaldefinition is not.If the valuations are multiplicatively separable, that is, viðsÞ ¼ uiðsiÞwðsÞ; then by

applying the transformation fðtÞ ¼ logðtÞ this is reduced to an additively separableformulation (assuming, of course, that both uiðsÞX1 and wðsÞX1). The previousremark then implies that the multiplicatively separable formulation also satisfies


weak average crossing condition. Thus, the valuations resulting fromWilson’s (1998)log-normal model satisfy the weak average crossing condition.An analogous weak cyclical crossing condition that is also ordinal can be

formulated. Basically, it requires that for every ADN; the cycling condition holdonly for bidders in A and only with respect to signals of bidders in A:The average crossing condition can also be weakened so that it concerns weighted

averages rather than simple averages. The conclusion of Theorem 2 obtains underthis weaker condition.The weighted (weak) average crossing condition also neither implies nor is implied

by the (weak) cyclical crossing condition.

5. Proofs of Theorems 2 and 3

Recall that a bidding strategy for i is a collection of functions bAi : ½0; 1�

RN\Aþ -Rþ; where iAADN and #A41: The function bA

i determines the threshold

price bAi ðsi; pN\AÞ such that bidder i will drop out at any price pXbA

i ðsi; pN\AÞ whenthe set of active bidders, including i; is A; his own signal is si; and the bidders in N\A

have dropped out at prices pN\A ¼ ðpjÞjAN\A:

Let b ¼ ððbAi ÞiAAÞADN be the collection of all bidders’ strategies.

If there is an equilibrium b such that for all ADN; the bAi are increasing functions

of si then if bidder i drops out at some price pi ¼ bAi ðsiÞ all remaining bidders jai

deduce that Si ¼ si: In that case, with a slight abuse of notation bAi ðsi; pN\AÞ will be

written as bAi ðsi; sN\AÞ:

Let GðA; sN\AÞ denote the ‘‘sub-auction’’ in which the set of active bidders is ACN

and the signals of the bidders who have dropped out are sN\A:

Lemma 1. Suppose that for all ADN; for all sN\A; there exists a unique set of

continuous and non-decreasing functions sj :Rþ-½0; 1�; jAA such that for all p

vjðrAðpÞ; sN\AÞpp: ð9Þ

and for any i if p is such that siðpÞo1; then si is strictly increasing and

viðrAðpÞ; sN\AÞ ¼ p: ð10Þ

Let bAi : ½0; 1� ½0; 1�N\A-Rþ be the inverse of si; that is,

bAi ðsi; sN\AÞ ¼ minfp: siðpÞXsig:

Then b is an ex post equilibrium of the English auction.10

ARTICLE IN PRESS

10To ease the notational burden I have suppressed the dependence of si on the set of active bidders A

and the signals of the bidders who have already dropped out sN\A: Thus, when there is no danger of

confusion, I write siðpÞ instead of sAi ðp; sN\AÞ:


Proof. Consider bidder 1, say, and suppose that all other bidders ia1 are followingthe strategies bA

i as specified above. Suppose that bidder 1 gets the signal s1 but

deviates and decides to drop out at some price other than bA1 ðs1Þ: It will be argued

that no such deviation is profitable.First, suppose that bidder 1 gets the signal s1 and wins the object by following the

strategy b1 as prescribed above. Bidder 1 cannot affect the price he pays for theobject and so the only way that a deviation could be profitable is if winning leads to aloss for bidder 1 and the deviation causes him to drop out. Suppose that he wins theobject when the set of active bidders is some ADN and without loss of generality,suppose that A ¼ f1; 2;y; ag: This can only happen if all bidders 2; 3;y; a drop outat the same price, say p�:11 Since all equilibrium strategies in every sub-auction areincreasing, all bidders in A can infer the signals sN\A of the inactive bidders from the

prices at which they dropped out. The price that bidder 1 pays is p� ¼ bA2 ðs2Þ ¼

bA3 ðs3Þ ¼ ? ¼ bA

a ðsaÞ at which the other bidders in A drop out. Since

bA1 ðs1Þ4p�; s1ðp�Þo1 and the break-even conditions (10) imply that

v1ðrAðp�Þ; sN\AÞ ¼ p�:

Now for i ¼ 2; 3;y; a; si ¼ siðp�Þ and since bA1 ðs1Þ4p�; s14s1ðp�Þ: Now since

v01140; this implies that

v1ðs1; sA\1; sN\AÞ4p�

showing that in equilibrium 1 makes an ex post profit whenever he wins with a bid of

bA1 ðs1Þ: Thus, any deviation that causes him to drop out is not profitable.Second, suppose that the strategy b1 calls on bidder 1 to drop out at some price p�

1

but bidder 1 deviates and remains active longer than bA1 ðs1Þ ¼ p�

1 in some sub-auction

GðA; sN\AÞ: This makes a difference only if he stays active until all other bidders havedropped out and he actually wins the object. So suppose this is the case and suppose,without loss of generality, that the bidders in A ¼ f1; 2;y; ag drop out in the ordera; ða 1Þ;y; 2 at prices pappa1p?pp2; respectively, so that bidder 1 wins theobject at a price of p2: We will argue that this cannot be profitable for 1.

Let ðs jþ1i Þ jþ1

i¼1 denote the inverse bidding strategies being played when the set ofactive bidders is f1; 2;y; j þ 1g:When bidder j þ 1AA drops out at price pjþ1; then(10) imply that for all i ¼ 1; 2;yj; j þ 1;

viðs jþ11 ðpjþ1Þ; s jþ1

2 ðpjþ1Þ;y; s jþ1jþ1 ðpjþ1Þ; sjþ2;y; snÞppjþ1 ð11Þ

with an equality for any bidder i with s jþ1i ðpjþ1Þo1: Now since bidder j þ 1 drops

out at pjþ1; sjþ1

jþ1 ðpjþ1Þ ¼ sjþ1 and the break-even conditions when the set of active

bidders is f1; 2;y; jg imply that for all i ¼ 1; 2;y; j;

viðs j1 ðpjþ1Þ; s j

2 ðpjþ1Þ;y; s jj ðpjþ1Þ; sjþ1; sjþ2;y; snÞppjþ1

ARTICLE IN PRESS

11Of course, the probability that more than one bidder drops out at the same price is zero and so with

probability 1, a ¼ 2:


with an equality for any bidder i with s ji ðpjþ1Þo1: Since the break-even conditions

have a unique solution, for all joa and i ¼ 1; 2;y; j;

s ji ðpjþ1Þ ¼ s jþ1

i ðpjþ1Þ ð12Þ

and since pjXpjþ1 and s ji is non-decreasing this implies that for all joa and

i ¼ 1; 2;y; j;

s ji ðpjÞXs jþ1

i ðpjþ1Þ: ð13ÞSimilarly, at the last stage, when 2 drops out at price p2 it must be that for i ¼ 1; 2;

viðs21ðp2Þ; s2; s3;y; snÞpp2

with an equality if s2i ðp2Þo1: Now applying (13) repeatedly when i ¼ 1 resultsin s21ðp2ÞXs31ðp3ÞX?Xsa

1ðpaÞ: But pa4p�1 and so sa

1ðpaÞ4sa1ðp�

1Þ ¼ s1: Thus,

s21ðp2Þ4s1 and since v01140;

v1ðs1; s2;y; snÞop2

and by staying in and winning the object at a price p2; bidder 1 makes a loss. Thus,

bidder 1 cannot benefit by remaining active longer than bA1 ðs1Þ:

In addition, for any p4p�1 and any set of active bidders f1; 2;y; jg; the break-even

conditions imply that

v1ðs j1 ðpÞ; s

j2 ðpÞ;y; s j

j ðpÞ; sjþ1; sjþ2;y; snÞpp

and since v1140 and s j1 ðpÞXs jþ1

1 ðpjþ1ÞXsa1ðpaÞ4sa

1ðp�1Þ ¼ s1 we have that

v1ðs1; s j2 ðpÞ;y; s j

j ðpÞ; sjþ1; sjþ2;y; snÞop:

This means that if bidder 1 is still active at any p4p�1 he cannot benefit by remaining

active since the value is less than the current price p and thus he should exitimmediately.Finally, note that the strategies specified in the statement are well defined in the

sense that

bAi ðsi; pN\AÞ4maxfpj : jAN\Ag:

To see this, again suppose, as above, that the bidders drop out in the order n; n 1;y; 1: Consider the price pjþ1 at which bidder j þ 1 drops out. As above, it must bethat (12) holds. But since bidders 1; 2;y; j did not drop out at pjþ1; it must be thatfor all i ¼ 1; 2;y; j;

si4s jþ1i ðpjþ1Þ

and from (12) we obtain that

si4s ji ðpjþ1Þ;

which is the same as saying that

b ji ðsi; pjþ1; pjþ2;y; pnÞ4pjþ1:

This shows that indeed the strategies in the statement are well defined.


Since all the arguments above were of an ex post nature, it has been shown that b

is an ex post equilibrium. &

Lemma 2. Suppose that the valuations v satisfy the average crossing condition. Then

for all ADN; for all sN\A there exists a unique set of continuous and non-decreasing

functions sj :Rþ-½0; 1�; jAA such that for all p,

vjðrAðpÞ; sN\AÞpp


viðrAðpÞ; sN\AÞ ¼ p: ð14Þ

Proof. First, consider A ¼ N: Then the break-even conditions (14) may becompactly written as

vðrðpÞÞ ¼ pe; ð15Þwhere eARn is a vector of 1’s. Recall from (1) that vð0Þ ¼ 0 and so when p ¼ 0; it ispossible to set rð0Þ ¼ 0:Differentiating the above equation with respect to p results in

DvðrðpÞÞr0ðpÞ ¼ e;

where r0ðpÞ ¼ ðs0iðpÞÞni¼1: In other words, a differentiable and increasing solution r to

(15) exists if and only if there is an increasing solution to the differential equationsystem

DvðrÞr0 ¼ e;

rð0Þ ¼ 0: ð16ÞLemma A.1 in Appendix A implies that if the valuations v satisfy (3) and the

average crossing condition, then Dv1 exists and so by the Cauchy–Peano theorem,

there exists a unique solution to this system for all ppminiAN s1i ð1Þ: Lemma A.1 inAppendix A also implies that r0ðpÞc0 for any ppmin s1i ð1Þ:The solution can then be extended to all p as follows. Suppose, without loss of

generality, that arg mini s1i ð1Þ ¼ 1 and let p1 be the price at which this occurs. In

other words, the solution to the differential equation system above is such that s1reaches 1 first and this occurs at a price of p1: Let s1i ¼ siðp1Þ denote the signal valueof bidder i at the price p1 as solved by the system (16). We then set s1ðpÞ ¼ 1 for allpXp1:For bidders other than j; consider the new system

Dv1ð1; r1ðpÞ; Þr01ðpÞ ¼ e;

r1ðp1Þ ¼ s11;

which is analogous to (16) except that bidder 1 is excluded by setting s1ðpÞ ¼ 1: Asbefore, the average crossing condition guarantees that this new system also has anincreasing solution for all p4p1 and we extend the solution until the price reaches a


level such that bidder 2’s solution s2ðp2Þ ¼ 1; say. Now we set s2ðpÞ ¼ 1 for all pXp2and solve a third system that excludes both 1 and 2. Proceeding in this way, weobtain a solution r for all p: Notice that the r obtained in this fashion satisfy (14) forall pop1 and the si are increasing in this region.

12

It remains to argue that the solution so obtained satisfies viðrðpÞÞpp for all pXp1:To see this suppose that p is such that for all bidders l in some subsetMDN; slðpÞ ¼ 1 whereas for bidders leM; slðpÞo1 and viðrðpÞÞ ¼ p: Considera bidder iAM and notice that the average crossing condition implies that

d

dpviðrðpÞÞ ¼

XlAM

v0ilðrðpÞÞs0lðpÞ þXleM

v0ilðrðpÞÞs0lðpÞ

pXlAM

%v0lðrðpÞÞs0lðpÞ þXleM

%v0lðrðpÞÞs0lðpÞ

¼ d

dp%vðrðpÞÞ

since for all lAM; s0lðpÞ ¼ 0 and for all leM; s0lðpÞ40 and v0ilðrðpÞÞo%v0lðrðpÞÞ: Thisimplies that for all lAM; vlðrðpÞÞp%vðrðpÞÞ and since for all leM; vlðrðpÞÞ ¼ p wehave that for all lAM; vlðrðpÞÞpp:The same argument applies in a sub-auction GðA; sN\AÞ once the initial conditions

are chosen with some care. As an example, consider the sub-auction where one of thebidders, say n; with signal sn has dropped out. Let A ¼ N\fng and consider the sub-auction GðA; snÞ: From the solution to the game GðNÞ as above, this must have beenat a price pn such that sN

n ðpnÞ ¼ sn: For all iAA; let %si ¼ sNi ðpnÞ; where sN

i are the

inverse bidding strategies in GðNÞ: Then in the sub-auction GðA; snÞ; a solution of thesystem

DvAðrAÞr0A ¼ e;

rAðpnÞ ¼ %sA

determines the inverse bidding strategies. This has a solution and the averagecrossing condition guarantees that the matrix DvAðrAÞ satisfies the conditions ofLemma A.1 in Appendix A and so r0

Ac0: The solution may then be extended for allpositive p:Proceeding recursively in this way results in strategies satisfying (15) in all sub-

auctions. &

Lemma 3. Suppose that the valuations v satisfy the cyclical crossing condition and

DvðsÞX0; for all sAEðvÞ: Then for all ADN; for all sN\A there exists a unique set of

continuous and non-decreasing functions sj :Rþ-½0; 1�; jAA such that for all p

vjðrAðpÞ; sN\AÞpp

ARTICLE IN PRESS

12 If two or more of the si ’s, say s1 and s2; reach 1 simultaneously, the construction is similar. The newsystem excludes both.



viðrAðpÞ; sN\AÞ ¼ p:

Proof. The proof is the same as that of Lemma 2 except that now the cyclicalcrossing condition and DvðsÞX0; for all sAEðvÞ; guarantee that the conditions ofLemma A.2 in Appendix A are satisfied. So r0

Ac0 in every sub-auction. Once again

the solution is extended to all p in the same manner as in Lemma 2. In other words,once sj reaches 1 it remains fixed at that level whereas the other si’s are obtained as

solutions to the appropriate differential equation system.It only remains to argue that the solution so obtained satisfies viðrðpÞÞpp for all p:

To see this suppose that p is such that for all bidders l in some subsetMDN; slðpÞ ¼ 1 whereas for bidders leM; slðpÞo1 and viðrðpÞÞ ¼ p: Consider abidder iAM and let jeM be such that i4j and there is no j0 in M that satisfiesi4j04j in M (here 4 denotes the cyclical order so that if i ¼ 1 then it may be thatj ¼ n). The cyclical crossing condition implies that

d

dpviðrðpÞÞ ¼

XlAM

v0ilðrðpÞÞs0lðpÞ þXleM

v0ilðrðpÞÞs0lðpÞ

pXlAM

v0jlðrðpÞÞs0lðpÞ þXleM

v0jlðrðpÞÞs0lðpÞ

¼ d

dpvjðrðpÞÞ

since for all lAM; s0lðpÞ ¼ 0 and for all leM; v0ilðrðpÞÞpv0jlðrðpÞÞ: This shows thatviðrðpÞÞpvjðrðpÞÞ ¼ p:

As in Lemma 2, the argument can then be replicated in all sub-auctions. &

Lemmas 1 and 2 together imply that under the average crossing condition thereexists an ex post equilibrium satisfying the break-even conditions (9) and (10).Similarly, Lemmas 1 and 3 together imply that under the cyclical crossing conditionthere also exists an ex post equilibrium satisfying the break-even conditions. Tocomplete the proofs of Theorems 2 and 3, I now show that in each case, theseequilibria are efficient.

Lemma 4. Suppose that the valuations v satisfy the average crossing condition and b is

an equilibrium of the English auction such that bAi are continuous and increasing

functions whose inverses satisfy the break-even conditions (9) and (10). Then b is

efficient.

Proof. Consider the case when all bidders are active. To economize on notation, let

bNi bi and si ¼ b1i : Suppose that the signals are s1; s2;y; sn and that biðsiÞ ¼ pi:Without loss of generality, suppose that p1Xp2X?Xpn14pn so that bidder n is the


first to drop out.13 Now (10) implies that for all i;

viðs1ðpnÞ; s2ðpnÞ;y; snðpnÞÞ ¼ pn;

that is, all the values at rðpnÞ are the same. Thus, they all equal the average value andso, in particular,

vnðs1ðpnÞ; s2ðpnÞ;y; snðpnÞÞ ¼ %vðs1ðpnÞ; s2ðpnÞ;y; snðpnÞÞ:

Since snðpnÞ ¼ sn and for all ian; siðpiÞ4siðpnÞ; the average crossing conditionimplies that

vnðs1; s2;y; snÞo%vðs1; s2;y; snÞ:

Since the ex post value of bidder n is less than the average ex post value of all thebidders, it must be that

vnðs1; s2;y; snÞomaxi

viðs1; s2;y; snÞ:

Thus, the person who is the first to drop out does not have the highest value.The same argument can be made in every sub-auction GðA; sN\AÞ and so at no

stage does the bidder with the highest value drop out. Thus, the equilibrium isefficient. &

Lemma 5. Suppose that the valuations v satisfy the cyclical crossing condition and b is

an equilibrium of the English auction such that bAi are continuous and increasing

functions whose inverses satisfy the break-even conditions (9) and (10). Then b is

efficient.

Proof. Again consider the case when all bidders are active. To economize on

notation, let bNi bi and si ¼ b1i : Suppose that the signals are s1; s2;y; sn and that

biðsiÞ ¼ pi: Without loss of generality, suppose that p1Xp2X?Xpn14pn so thatbidder n is the first to drop out. Now (10) implies that for all i;

viðs1ðpnÞ; s2ðpnÞ;y; snðpnÞÞ ¼ pn;

that is, all the values at rðpnÞ are the same. Thus,vnðs1ðpnÞ; s2ðpnÞ;y; snðpnÞÞ ¼ vn1ðs1ðpnÞ; s2ðpnÞ;y; snðpnÞÞ:

Now note that cyclical crossing implies that for all sAEðvÞ; for all jan

v0njðsÞpv0n1;jðsÞ

and

v0n;n1ðsÞov0n1;n1ðsÞ:

Since snðpnÞ ¼ sn and for all ian; siðpiÞ4siðpnÞ;vnðs1; s2;y; snÞovn1ðs1; s2;y; snÞ

ARTICLE IN PRESS

13Again, in the interests of simplicity it is assumed that multiple bidders do not drop out at the same

price.


and so the ex post value of bidder n is less than the ex post value of bidder n 1:Thus, the person who is the first to drop out does not have the highest value.The same argument can be made in every sub-auction GðA; sN\AÞ and so at no

stage does the bidder with the highest value drop out. Thus, the equilibrium isefficient. &

6. Some remarks on the main results

The proofs of both Lemmas 4 and 5 show only that the bidder who drops out isnot the one with the highest value. They do not assert that the bidders drop out ‘‘inorder’’ with the person with the lowest ex post value dropping out first, followed bythe person with the next lowest value, etc. Bidders do drop out according toincreasing ex post values in a completely symmetric model, provided the singlecrossing condition is satisfied, but not necessarily when bidders are asymmetric. TheEnglish auction is remarkable in that it allocates efficiently even when the bidders donot drop out in order. The average crossing and cyclical crossing conditions allowsuch interesting behavior as the following example shows.

Example 2.

v1ðs1; s2; s3Þ ¼ s1 þ 13

s3;

v2ðs1; s2; s3Þ ¼ 13

s1 þ s2;

v3ðs1; s2; s3Þ ¼ 13

s2 þ s3:

Both the average and cyclical crossing conditions are satisfied.When all bidders are active, the strategies prescribed by the efficient equilibrium

are bNi ðsiÞ ¼ 4

3si:

Suppose that s ¼ ðs1; s2; s3Þ ¼ ðe2; e; 1 eÞ; where eo12: Then bidder 1 is the first to

drop out. But for small enough e; the ex post values are such that v2ðsÞov1ðsÞov3ðsÞ:Thus, while the equilibrium is efficient, bidders do not necessarily drop out in orderof increasing ex post values. &

Lemma 1 shows that as long as there are increasing functions ðsAi ÞiAA that satisfy

the break-even conditions (9) and (10), their inverses constitute an equilibrium of theEnglish auction. It is also possible to show that every such equilibrium is efficient.The average crossing and cyclical crossing conditions guarantee that there is anincreasing solution to the break-even conditions. More general sufficient conditionscan be provided, say, based on Theorem 4 in the appendix, but these are clumsy tostate and unintuitive. More importantly, one may wonder whether an efficientequilibrium of the English auction must always satisfy the break-even conditions. Inother words, are the break-even conditions necessary? The answer is no, as thefollowing example shows.


Example 3.

v1ðs1; s2; s3Þ ¼ s1 þ 23

s2 þ 23

s3;

v2ðs1; s2; s3Þ ¼ s2;

v3ðs1; s2; s3Þ ¼ s3:

It is easy to verify that there does not exist an increasing solution to the break-evenconditions.But the following constitutes an efficient equilibrium. Regardless of the situation,

bidders 2 and 3 drop out at p2 ¼ s2 and p3 ¼ s3; respectively. Bidder 1 stays in aslong as bidders 2 and 3 are active. If bidder ja1 drops out a price pj then bidder 1

drops out at 3s1 þ 2pj: It is easy to see that this is an equilibrium.

This is efficient because if bidder 2; say, drops out, then it must be that v2ov3and so 2 does not have the highest value. In the ‘‘sub-auction,’’ with only bidders 1and 3 remaining, the person with the higher value will win. If both 2 and 3 wereto drop out at some price p; bidder 1 would be happy to win the object at that

price since his value would then be at least 43

p and it would also be efficient for him

to win.

7. Conclusion

An analogue of the English auction for the case when there are many identicalobjects for sale has been introduced by Ausubel [2]. In the case of private values, thismechanism allocates efficiently. The question of whether there are conditions underwhich this mechanism has an efficient equilibrium in asymmetric situations in whichbidders’ values are interdependent and signals are one-dimensional, remains to beexplored.Both the conditions introduced in this paper have the virtue of being relatively

simple extensions of the single crossing condition of Maskin [13] and Dasgupta andMaskin [6]. Both reduce to the single crossing condition when there are only twobidders. Neither condition is necessary for the existence of an ex post equilibriumthat is efficient. Nor is their union. Whether there exist some simply stated andintuitive conditions that are both necessary and sufficient remains an open problem.

Acknowledgments

I am very grateful to Sergiu Hart, Sergei Izmalkov, Motty Perry, Phil Reny,a referee and an associate editor for many helpful suggestions. I also thankthe participants at the 2000 Stony Brook Workshop on Auctions for theircomments.


Appendix A

In this appendix it is shown that the average crossing and cyclical crossing

conditions imply that for all sAEðvÞ; the systemDvðsÞx ¼ e

has a unique solution xc0; where eARn denotes the vector of 1’s. When s ¼ rðpÞ;this implies that r0ðpÞc0:

A.1. Matrices with dominant averages

Definition A.1. An n n matrix A satisfies the dominant average condition if in everycolumn the off-diagonal terms are less than the average of the column:

8iaj; aijo1

n

Xn

k¼1akj ðA:1Þ

and the average of each column is positive:

8j; 0o1

n

Xn

k¼1akj : ðA:2Þ

Observe that if A satisfies the dominant average condition and Ai is obtained by

deleting the ith row and ith column of A then Ai also satisfies the condition. This isbecause if from any column an entry that is less than the average is deleted, then theaverage of the remaining entries increases.

Let eiARn denote the ith unit vector and let e ¼Pn

i¼1 ei denote the vector of 1’s.

Although the same symbols will be used for different n; the sizes of these vectors willbe apparent from the context.

Lemma A.1. Suppose A is an n n matrix that satisfies the dominant average

condition. Then there exists an xc0 such that

Ax ¼ e: ðA:3Þ

Furthermore, detAa0 and so (A.3) has a unique solution.

Proof. We first show that there is a strictly positive solution to (A.3). The proof is byinduction on n:

Step 1: For n ¼ 1; the fact that there is a strictly positive solution is immediate.Now suppose that the result holds for all matrices of size n 1:Let A be an n n matrix. Define Ai to be the ðn 1Þ ðn 1Þ matrix obtained

from deleting the ith row and the ith column of A: From the induction hypothesis,

for each i ¼ 1; 2;y; n; there exists an xic0 such that

Aixi ¼ e


which is the same as: for all kaiXjai

akjxij ¼ 1: ðA:4Þ

Let Xjai

aijxij ¼ ci: ðA:5Þ

Step 2: Adding the n 1 equations (A.4) with (A.5) results inXjai

Xn

k¼1akj

!xi

j ¼ðn 1Þ þ ci

4 0

which is positive because of (A.2) and the fact that xic0: But now (A.1) implies that

ci Xjai

aijxij

oXjai

1

n 1Xkai

akj

!xi

j

¼Xkai

1

n 1

� � Xjai

akjxij

!

¼ 1

using (A.4). Thus, ðn 1Þ þ ci40 and cio1:Step 3: Since ðn 1Þ þ ci40 and cio1; for all i; 1

1ci41

nand so

Xn

i¼1

1

1 ci

41: ðA:6Þ

Now let yiARnþ be the vector obtained by appending to xiARn1

þþ ; 0 in the ith

coordinate. Then for all i:

Ayi ¼

1

^

1

ci

1

^

1

2666666666664

3777777777775¼

1

^

1

1

1

^

1

2666666666664

3777777777775 ð1 ciÞ

0

^

0

1

0

^

0

2666666666664

3777777777775;

which can be compactly rewritten as

Ayi ¼ e ð1 ciÞei:


Dividing through by the positive quantity ð1 ciÞ results in

A1

1 ci

yi

� �¼ 1

1 ci

e ei:

Adding n such equation systems, one for each i yields

AXn

i¼1

1

1 ci

yi

!¼

Xn

i¼1

1

1 ci

!e e

or equivalently

AXn

i¼1

1

Kð1 ciÞyi ¼ e;

where K ¼ ½ðPn

i¼111ci

Þ 1�40 from (A.6). Since each yiX0 with only the ith

component equal to zero, and ð1 ciÞ40 we have that

x ¼Xn

i¼1

1

Kð1 ciÞyic0

is a solution to system (A.3).Thus, there is a strictly positive solution to (A.3).Step 4: We now verify that the solution is unique by arguing that detAa0; and

hence x ¼ A1e: Again, the proof is by induction on n:For n ¼ 1; it is immediate that the solution is unique. Now suppose that for all

matrices of size n 1 there is a unique solution to the system. Let A be of size n andlet xc0 be such that Ax ¼ e:If A is singular then there exists a column, say the kth, which is a linear

combination of the other n 1 columns, that is, for all jak there exists a zj such that

8i; aik ¼Xjak

aijzj ðA:7Þ

and since akk40; not all the zj can be zero.

Of course, (A.3) is equivalent to

8i;Xn

j¼1aijxj ¼ 1

and substituting from (A.7) yields

8i;Xjak

aijðzjxk þ xjÞ ¼ 1: ðA:8Þ

As before, let Ak be the ðn 1Þ ðn 1Þ matrix obtained from A by eliminatingthe kth row and the kth column of A: From the induction hypothesis, there exists a

unique yc0 such that Aky ¼ e which is equivalent to

8iak;Xjak

aijyj ¼ 1: ðA:9Þ


Since the solution is unique, comparing (A.9) and the equations in (A.8) for iak;implies that

8jak; ðzjxk þ xjÞ ¼ yj:

This means that the kth equation in (A.8) can be rewritten asXjak

akjyj ¼ 1: ðA:10Þ

Step 5: Now adding the n 1 equations in (A.9) and dividing by n 1 results inXjak

1

n 1Xiak

aij

!yj ¼ 1: ðA:11Þ

But (A.1) implies that

8j; akjo1

n 1Xiak

aij ðA:12Þ

and since yj40; (A.12), implies thatXjak

akjyjo1;

contradicting (A.10). Thus A is not singular and Ax ¼ e has a unique solution. &

The dominant average condition may be weakened as follows. An n n matrix Asatisfies the dominant weighted average condition if there exist positive weightsl1; l2;y; ln with

Pi li ¼ 1 such that

8iaj; aijoXn

k¼1lkakj

and

8j; 0oXn

k¼1lkakj:

The conclusion of Lemma A.1 follows under this weaker condition.Suppose A is a matrix that satisfies the dominant diagonal condition and for all

iaj; aijp0: Then A satisfies the dominant weighted average condition.

A.2. Cyclically ordered matrices

Definition A.2. An n n matrix A is cyclically ordered if

8j; ajj4ajþ1;jXajþ2;jX?Xaj1;j ; ðA:13Þwhere j þ k ð j þ kÞmod n:

Observe that if A is cyclically ordered and Ai is obtained by deleting the ith row

and ith column of A then Ai is also cyclically ordered.


Lemma A.2. Suppose AX0 is an n n matrix that is cyclically ordered. Then there

exists an xc0 such that

Ax ¼ e: ðA:14ÞFurthermore, detAa0 and so (A.14) has a unique solution.

Proof. The proof of Lemma A.2 is very similar to the proof of Lemma A.1. In fact,Steps 1, 3 and 4 are identical and in the interests of space these are not repeated here.The remaining steps are replaced by Steps 20 and 50; respectively.

Step 20: In this case, the fact that AX0 (buta0) and xic0; implies that ci40 and

so ðn 1Þ þ ci40: Now (A.13) implies that for all jai; ai1;jXaij and

ai1;i14ai;i1: Since xij40;

ci ¼Xjai

aijxijoXjai

ai1;jxij ¼ 1

using (A.4) and identifying the index 0 with n; if necessary. Thus, ðn 1Þ þ ci40 andcio1:

Step 50: Now (A.13) implies that for all jak; ak1;jXakj and ak1;k14ak;k1:Since, yj40;X

jak

akjyjoXjak

ak1;jyj ¼ 1 ðA:15Þ

contradicting (A.10). Thus, A is not singular and Ax ¼ e has a unique solution. &

A.3. A theorem of the alternative

Necessary and sufficient conditions for the existence of a strictly positive solutionto the linear system Ax ¼ e can, of course, be derived by considering the relevantdual system (see [7]). The resulting ‘‘theorem of the alternative’’ is

Theorem A.1. Exactly one of the following alternatives holds. Either there exists an

xc0 such that

Ax ¼ e

or there exists a y such that

yT ½A : e�X0 but a0:

References

[1] S. Athey, Single crossing properties and the existence of pure strategy equilibria in games of

incomplete information, Econometrica 69 (2001) 861–889.

[2] L. Ausubel, An efficient ascending-bid auction for multiple objects, Working Paper 97-06,

Department of Economics, University of Maryland, June 1997.


[3] C. Avery, Strategic jump bidding in English auctions, Rev. Econom. Stud. 65 (1998) 185–210.

[4] K.-S. Chung and J. Ely, Efficient and dominance solvable auctions with interdependent valuations,

Mimeo, Department of Economics, Northwestern University, May 2001.

[5] J. Cremer, R. McLean, Full extraction of the surplus in Bayesian and dominant strategy auctions,

Econometrica 56 (1988) 1247–1257.

[6] P. Dasgupta, E. Maskin, Efficient auctions, Quart. J. Econom. 115 (2000) 341–388.

[7] D. Gale, The Theory of Linear Economic Models, The RAND Corporation, 1960.

[8] M. Jackson, J. Swinkels, Existence of equilibrium in single and double private value auctions, Mimeo,

California Institute of Technology, November 2001.

[9] P. Jehiel, B. Moldovanu, Efficient design with interdependent values, Econometrica 69 (2001)

1237–1259.

[10] O. Kirchkamp, B. Moldovanu, An experimental analysis of auctions with interdependent valuations,

Games Econom. Behav. (2003), to appear.

[11] B. Lebrun, First price auctions in the asymmetric N bidder case, Internat. Econom. Rev. 40 (1999)

125–142.

[12] G. Lopomo, Optimality and robustness of the English auction, Games Econom. Behav. 36 (2001)

219–240.

[13] E. Maskin, Auctions and privatization, in: H. Siebert (Ed.), Privatization, Institut fur Weltwirtschaf-

ten der Universitat Kiel, Kiel, 1992, pp. 115–136.

[14] E. Maskin, J. Riley, Asymmetric auctions, Rev. Econom. Stud. 67 (2000) 413–438.

[15] P. Milgrom, A convergence theorem for competitive bidding with differential information,

Econometrica 47 (1979) 679–688.

[16] P. Milgrom, R. Weber, A theory of auctions and competitive bidding, Econometrica 50 (1982)

1089–1122.

[17] M. Perry, P. Reny, An efficient auction, Econometrica 70 (2002) 1199–1212.

[18] W. Vickrey, Counterspeculation, auctions and competitive sealed tenders, J. Finance 16 (1961) 8–37.

[19] R. Wilson, Sequential equilibria of asymmetric ascending auctions: the case of log-normal

distributions, Econom. Theory 12 (1998) 433–440.


Documents

Asymmetric English auctions