Single Unit Auctions; Still Enigmatic

Michael Landsberger

Department of Economics

Haifa University, Haifa, Israel

e-mail:lmichael@econ.haifa.ac.il

Boris Tsirelson

School of Mathematical Sciences

Tel Aviv University, Tel Aviv, Israel

e-mail: tsirel@math.tau.ac.il

November, 1999

Abstract

We consider the well-known model proposed by Milgrom and We-

ber in 1982, where players are risk neutral, their valuations V1; : : : ; Vnare neither private nor common, they depend on privately observed

signals X1; : : : ;Xn and a common, unobservable, random variable S

(possibly, multidimensional). We prove that if

� bidders must entail participation cost, and

� signals are interdependent,

the existing results for single unit auctions are not valid anymore, in

many cases. In fact, it is not known how many equilibria (if any) exist

in such auctions, and how to de�ne (construct, select) the relevant

equilibrium. The reason is that in many cases if an equilibrium ex-

ists, it can not be supported by monotone (threshold) participation

strategies. The driving force of this result is the dual meaning of in-

terdependent signals. When signals are independent, a higher signalis good news. Under dependence, in a monotone equilibrium, a higher

signal is both, good and bad news.

1

Introduction

In addressing the case where bidding is costly, Milgrom&Weber, in their sem-inal [1982] paper, coined the term regular in reference to situations where par-ticipation strategy is a threshold x�, such that a potential participant choosesto participate in the auction (and incur participation cost) if, and only if, hissignalXk exceeds x

�: In their analysis of participation, MW assumed regular-ity, which indicates that they probably considered it to be typical; the term'regular' points in this direction. MW did not establish conditions (necessaryor su�cient) for regularity to take place, although this is an important issuesince regularity is necessary to obtain an overall monotone strategy, wherenon participation can be interpreted as not bothering how to bid, avoid thecost and submit 0 (or anything below the reserve price). In pursuing ouranalysis, we use the term threshold strategies instead of 'regularity'. We didso since the term 'threshold' has been widely used in economics, and maybe, therefore, more familiar to the readers.

We prove in this paper that when the cost of bidding are acknowledged,in a large class of environments, equilibria, if they exist, are not supportedby treshold strategies. Under such conditions, one have no di�erential equa-tions or other constructive approaches to equilibria. Game-theoretic argu-ments based on �xed point theorems are directed toward existence ratherthan uniqueness. In order to have any predictive power,1 an auction theorymust be based on a de�nite equilibrium (either unique or appropriately se-lected). For now such a basis is available under overall monotonicity only.Consequently, the existing literature provides no results for a large class ofenvironments.

We address this problem since we believe that in reality there are alwaysparticipation cost; sometimes, they may be large like in the case of procure-ments and privatization, and in other instances they may be small, �ndinginformation about the auction and processing it, or making sure what is thevalue of the object.2

The family of auctions which we consider is very large; in general, ourresults apply to all single unit auctions in which only the winner derivespositive utility from the auction. We do not stipulate anything about the

1By predictive power of a theory we mean the possibility to compute the complete nu-merical solution (strategies, probabilities, etc.) out of the complete numerical speci�cationof the model (distributions, utility functions, entry cost etc.)

2We are not the �rst to consider participation cost; see for example McAfee and McMil-lan (1987) Levin and Smith (1994), Stegeman (1996) and other papers listed in the Ref-erences. However, all these papers addressed the case where types were independent,whereas the crux of our problem is signal interdependence.

2

allocation rules. Our results cannot be invalidated by narrowing the scopeto, say, �rst price auctions, private valuations, normal correlation of signals,etc. (as long as participation cost and signal interdependence persist).3 Wealso assume that there is a large number of potential participants, n. This isa very innocuous assumption given that anybody who would derive positiveutility if he had the object is considered as a potential bidder. For example,many people would like to own a nice chateau in Provence, around Florence ormany other places around the globe; all those are potential bidders. Hencewe can assume that the number of potential bidders for such an object islarge. The fact that, e�ectively, only very few participate, is the outcomeof the game. In fact, we show that treshold strategies may not support anequilibrium even in the case when the number of potential bidders is 20 oreven 10. Hence, the assumption of a large number of potential bidders is notcrucial.

We derived conditions that are necessary for a threshold strategy to sup-port an equilibrium, and we proved that these conditions are violated inmany reasonable situations, such as the case where the n-dimensional distri-bution of signals is multivariate normal. Hence, if there exist an equilibrium,the set of participants may often not be an interval of types or signals, evenif the latter are one-dimensional; it may be that bidders with lower signalsparticipate and bidders with higher signals do not, which obviously impliesthat overall strategies are not monotone.

The spirit of the proof of this result runs as follows: Some integer-valuedrandom variables N are \tame" in the sense that P

�N > 0

�and EN are

reasonably related (say, for a Poisson distribution, 1� P�N > 0

�= e�EN ).

Some are \wild" in the sense that P�N > 0

�is small and nevertheless

EN is large. In equilibrium, the (random, endogeneous) number of e�ectivebidders is \tame", whereas, because of dependence, the number of signalsabove a threshold is \wild". These two properties can not co-exist, which iswhy threshold strategies may be incompatible with equilibrium.

We do not provide a proof of existence of an equilibrium. Since our

3The following elucidation could prevent a misunderstanding. Usually, positive state-ments are general statements, of the form \every object (say, auction) satisfying assump-tions : : : has the property : : : ". A general statement is stronger when assumptions aremore general (less restrictive). Negative statements are usually counterexamples, of theform \there is an object satisfying : : : that lacks the property : : : ". A counterexampleis stronger when assumptions are less general (more restrictive).Our main result is negative but general. It is negative in the sense that a desirable

property (threshold) is negated. Nevertheless it is general, of the form \every objectsatisfying assumptions : : : has the property : : : " (namely, no threshold for n largeenough). Proven under quite general assumptions, the result holds automatically undermore speci�c assumptions.

3

analysis aims at a proof of non existence of a certain type of equilibrium, itcan be conducted within a framework where we assume that an equilibriumexists. Should it turn out not to be true, our conclusion will hold trivially.

Equilibria in increasing strategies seem to be a conventional wisdom; or,to use Milgrom's expression, an 'educated guess' (see Milgrom (1981) p.928). The guess was educated because it turned out to be true in manyauction models.4 By this logic, equilibria with increasing strategies, are notconventional wisdom or an educated guess, when bidding cost are accountedfor.

On an intuitive level, we must be careful not to run into pitfalls that mayemerge when we try to build intuition based on models with independenttypes, or signals. In such environments, a higher signal is always good newsin the sense that it induces a higher expected utility. This intuition does notwork when types and signals are dependent; a higher signal may be associ-ated with a lower expected utility and nevertheless, equilibria supported byincreasing strategies are still common wisdom. All this breaks down whenbidding cost are accounted for.

To illustrate how a higher signal can be both good and bad news, wedistinguish between signals that are below and above the threshold. For aplayer whose signal is below the threshold, a higher signal is associated witha lower probability to be the only bidder and therefore, a lower probabilityof winning, which is bad news. For the other players, a higher signal meansalso a smaller probability to be the only bidder, but not a smaller probabilityof being the winner. What makes higher signals bad news for these players,is that they pay higher prices if they win. Other than that, for all players, ahigher signal has an element of good news since they enjoy the object morewhen they win. Consequently, it is the play between good and bad news thatmakes threshold strategies compatible, or incompatible, with equilibrium.

It is interesting to note that the dual meaning of signals is enhanced by thedegree of a�liation. Namely, the larger is a�liation, the stronger is the factorthat makes a higher signal bad news, and it becomes less likely to obtain athreshold equilibrium. This result may seem strange since a�liation wasintroduced in order to assure monotonicity (single crossing). It did the jobin the Milgrom Weber setting with costless bids, but it is couterproductivefor monotonicty in a model with bidding cost.

We treat as common knowledge all parameters of the model: the jointdistribution of signals and valuations, X1; : : : ; Xn; V1; : : : ; Vn, participationcost e, the reserve price r, allocation and payo� mechanisms, and the equi-

4See, Riley&Samuelson [1981], Milgrom [1981], Milgrom &Weber [1982], Riley&Maskin[1984], Matthews [1987] and many others.

4

librium played. However, we do not address it explicitly; after assuming thatequilibrium is played (for any reason), the common knowledge assumption isof no further use.

Explicit notations are given only to objects that are used explicitly. Inparticular, the joint distribution of X1; : : : ; Xn; V1; : : : ; Vn gets no notation.Neither do the allocation and payo� mechanisms, equilibrium, etc. We onlyneed some events, such as participation or win. The large and complicatedcollection of random variables, X1; : : : ; Xn; V1; : : : ; Vn, and indicators of theevents, are all interdependent, and the dependence is quite arbitrary, unlessexplicitly stated. The indicators, in particular, are tricky to handle sincethey implicitly represent equilibrium strategies in unspeci�ed auction games.However, using them makes it possible to consider very general environmentsand auction rules without having to address the complex issue of equilibriumstrategies, which would also require to restrict the family of auction ruleswhich we consider and take us away from the main goal of this paper.

All expectations are taken w.r.t. the single distribution (the joint distri-bution of \all that": signals, valuations and actions5). Conditioning, whenneeded, is shown explicitly. Thus, E (V1) is the (unconditional) expectationof V1, while E

�V1��X1

�is its conditional expectation, given X1 (as de�ned

in [15, Sect. 7.1]. Events of probability 0 may be neglected; strategies maybe treated as distributional strategies. Relations between random variables,such as X � Y , or events, such as A � B, are meant to hold with probability1. Due to symmetry, EV1 = EV2 = � � � = EVn , and we may write for short\EV1" rather than \EVk for any k". The indicator of an event A is denotedby 1A. Notations introduced within a lemma are treated as temporary, validwithin the lemma only; the same for propositions and theorems.

The plan of the paper is as follows. In Section 1 we derive necessaryconditions that must be satis�ed in equilibrium under any auction rule, par-ticipation strategy and signal dimensionality. In Section 2 we established thenecessary conditions that must be satis�ed if a threshold strategy is to becompatible with equilibrium. The proof that under certain type of signal de-pendence, threshold strategies are incompatible with equilibrium for large n,is presented in section 3. In Section 4 we illustrate numerically the necessaryconditions in the case of multivariate normal distribution. Finally, in Section5 we illustrate why a higher signal is both, good and bad news, and we showwhy it makes threshold strategies incompatible with equilibrium.

5Actions are not necessarily functions of signals, since strategies may be mixed.

5

1 Some properties of participation in auctions

We derive in this section a few properties of participation in auctions that areindependent of participation strategies and auction rules; all that is needed isequilibrium. As noted in the Introduction, we use the Milgrom-Weber setup.For completeness, however, and to avoid misunderstandings, we specify ex-plicitly a few basic assumptions that we use. Milgrom and Weber assumea�liation, valuations as functions of signals and an additional (randomizing)random variable (generally, multi-dimensional); we deviate from this setting.

1.1. Assumption. The auction is a game of n players. Initially, all playersare identical. Signals, X1; : : : ; Xn and valuations V1; : : : ; Vn 2 R are drawnaccordingly to a joint distribution, invariant under all permutations of then players (that is, of n pairs (X1; V1); : : : ; (Xn; Vn)). The one-dimensionaldistribution of valuations is integrable (that is, E jV1 j <1) and non-atomic.The private information available to the k-th player is his signal Xk.

The joint distribution of X1; : : : ; Xn; V1; : : : ; Vn need not have a density.Valuations are one-dimensional (each). Signals may be multidimensional.

1.2. Assumption. The action space (the same for each player) contains aspecial element \quit". If a player quits, his payo� is 0. If he participates(that is, does not quit), he incurs a cost e > 0, required to process the bid.6

Valuations are in monetary units, and players are risk neutral.

The action space is left unspeci�ed (except for the \quit" action anda \minimal bid" action introduced below); in principle, it may be multi-dimensional. The auction mechanism is also unspeci�ed, except the follow-ing.

1.3. Assumption. At most one participant wins. If the k-th player wins,his payo� does not exceed Vk�r. Otherwise his payo� is 0. There is anotherspecial action (\minimal bid") that ensures to the k-th player the payo� atleast Vk�r; whenever all other players quit. The (unconditional) expectationof V1 exceeds r + e. Payo�s stipulated in 1.2 and 1.3 combine additively.

Consequently, we consider symmetric single unit auctions that do not re-ward non-winning bidders. Other than that, allocation and payment mecha-nisms remain unspeci�ed. The \minimal bid" assumption may be weakenedas follows: for every " > 0 there is an action that ensures at least Vk � r � "whenever other players quit.

6We denote it by e following MW, though it could be confused with 2:71828 : : :

6

1.4. Assumption. A symmetric Nash equilibrium is considered.

Let ~V be the quantile function for valuations, that is, ~V : (0; 1)! R,

P�V1 � ~V (p)

�= p for all p 2 (0; 1) :

For every p 2 (0; 1) and every event A,

P�A�= p =)

Z p

0

~V (p) dp � E�V 1A

� �Z 1

1�p

~V (p) dp :(1.5)

That is too simple for being a theorem, and is well-known both in mathemat-ical analysis (see [3, (10.12.1)]) and in mathematical statistics (the proof ofthe famous Neyman-Pearson lemma is basically the same inequality appliedto a ratio of densities instead of ~V ; see for instance [1, Sect. 5.3]). For obviousreasons, we apply the inequality to (V � r)+ = max(V � r; 0) rather than V .De�ne numbers �n 2 (0;1), � 2 (0; 1), used throughout the paper, by

�n =1

e

Z 1

1� 1n

( ~V (p)� r)+ dp ;(1.6)

1

e

Z �

0

( ~V (p)� r) dp = 1 :(1.7)

Existence of such � 2 (0; 1) follows from 1.3, sinceR �

0( ~V (p)�r) dp = EV �r >

e. Such � is unique; though, the integral treated as a function of � canturn from decrease to increase, it surely is strictly increasing whenever itis positive (which follows from increase of its integrand). Note that �n; �are determined by the distribution of valuations (and r; e), irrespective ofproperties of signals.

Denote by pparticipation the probability of participation for player 1 (bysymmetry, it is the same for each player).

1.8. Proposition. pparticipation � �n.

Proof. Denote by A the event \player 1 participates", by B the event \player1 wins", and by � = (V1 � Y ) � 1B � e � 1A the pro�t of player 1, where Y isthe random payment for the object. Then A � B, and

� � (V1 � r) � 1B � e � 1A ;(1.9)

which follows from Assumptions 1.2 and 1.3. Namely, if the player quits, hegets 0; if he participates, he pays e and either wins and gets no more thanV1 � r, or looses and gets 0.

7

It follows from the de�nition of equilibrium that E� � 0; otherwise,player 1 could improve his strategy by always quitting.7 So,

0 � E� � E�(V1 � r) � 1B � e � 1A

�= E

�(V1 � r) � 1B

�� epparticipation :

However, P�B� � 1=n by symmetry (and 1.3). Using (1.5),

pparticipation � 1

eE�(V1 � r)1B

� � 1

e

Z 1

1�P(B)

�~V (p)� r

�dp �

� 1

e

Z 1

1�P(B)

�~V (p)� r

�+ dp � 1

e

Z 1

1� 1n

�~V (p)� r

�+ dp :

Proposition 1.8 conveys an obvious economic message that is absent fromthe existing auction literature that largely ignores the e�ects of participationcost; if there are many potential participants, the probability of winning issmall and therefore, so is the probability of participation. Indeed, �n ! 0for n!1 (which follows from integrability of V ). Even if the participationcost e is small, still pparticipation must be small provided, however, that theintegral in (1.6) is much smaller than e. If V is bounded then the integral� const =n (which means that the number of e�ective participants remainsbounded). Otherwise, the integral converges to 0 slower than 1=n. That is, ifthere are very large valuations at the very top of the distribution, the upperbound on the probability of participation is less restrictive. Namely, if thepossible gains are very large, then even if the probability of winning is small,potential bidders may participate; we refer to it as the lottery e�ect ; it canmitigate the e�ect of a large n when valuations are unbounded, see Lemma3.2 and Theorem 1.

The inequality E�(V1 � Y ) � 1B � e � 1A

� � 0; used in the proof, can berewritten as

enP�A� � nE

�(V1 � Y )1B

�:(1.10)

The left hand side of (1.10) is the expected cost incurred by all partici-pants. The correct interpretation of the right hand side is a little tricky, butafter refreshing elementary probability rules, it can be shown that it standsfor the expected gain of the winner. Hence, (1.10) may read as

no deadweight loss No matter how many potential participants there are,the expected total cost incurred by all the bidders does not exceed theexpected gain of the single winner.

7Moreover, E��

��X1

�� 0, but we do not need it.

8

This result provides an answer to claims that if participation cost must beincurred, possibly, by many bidders a dead weigh loss is incurred since thereis only a single winner. The result stated above indicates that regardless ofparticipation and auction rules, equilibrium considerations take care of thisconcern. Moreover, it also indicates that there won't be too many bidders,unless participation cost is very low.

The next Proposition complements the result established in Proposition1.8, showing that the probability that nobody participates in the auction cannot be too large unless valuations are very small.

Denote by pnobody the probability that all players quit. Recall � intro-duced in (1.7).

1.11. Proposition. pnobody � �.

Proof. Denote by A the event \player 1 quits", by B the event \players2; : : : ; n quit", and by � the pro�t of player 1. Due to 1.3 (the \minimalbid" assumption), player 1 is able to get at least the pro�t

�0 = (V1 � r) � 1B � e :(1.12)

Note that �0 stands for the part of the pro�t of player 1 generated by theevent that all the other players quit.

It follows from the de�nition of equilibrium that E (�0 �1A) � 0; otherwiseplayer 1 could improve his strategy by submitting the minimal bid insteadof quitting.8 So,

0 � E��0 � 1A

�= E

�(V1 � r)1B1A � e � 1A

�= E

�(V1 � r)1A\B

�� eP�A�:

However, P�A \ B �

= pnobody. Using (1.5),

e � e � P �A � � E�(V1 � r)1A\B

� �Z pnobody

0

( ~V (p)� r) dp ;

that is,

1

e

Z pnobody

0

( ~V (p)� r) dp � 1 =1

e

Z �

0

( ~V (p)� r) dp :

Propositions 1.8 and 1.11 jointly stipulate that regardless of participa-tion and auction rules and the number of potential participants, equilibrium

8Moreover, E��0 � 1A

��X1

�� 0, but we do not need it.

9

considerations ensure that the number of e�ective participants is typicallyneither very large nor very small.

An interesting question is which participation rules support an equilib-rium. In particular, we would like to know whether threshold strategies doso. The interest in such strategies stems from the fact that otherwise, we cannot use the existing literature not even to assure the existence of an equilib-rium. Such strategies seem also to be in spirit with economic conventionalwisdom, which is probably why Milgrom and Weber in their comprehensivepaper used the term regular in reference to such strategies, and restrictedtheir analysis to this case. See also Milgrom (1981), p. 928.

2 Treshold participation strategies

To address threshold strategies, we obviously assume that signals are one-dimensional, and therefore, are linearly ordered. It means that the stochasticdependence between signals and valuations should be monotone in some sense(otherwise, the linear order of signals would be irrelevant). As a specialcase, MW assumed a�liation which, one might think, gives a good chanceto monotonicity of equilibrium strategies in general, including participation.We show that this, reasonable hunch, is not true once participation cost isacknowledged; important distributions that exhibit 'strong' a�liation, suchas the multivariate normal with a large correlation coe�cient, make thresholdstrategies incompatible with equilibrium. In fact, we show that the higherthe correlation, the less likely it is that threshold strategies can support anequilibrium. So, it looks that if at all, a�liation may be interfering withthreshold equilibria.

Since we want to prove that, in many cases, threshold equilibria do notexist, assuming a�liation would be useless, which is why we did not assumeit.

2.1. Assumption. Signals X1; : : : ; Xn take on values in R.

2.2. De�nition. The equilibrium (referred to in 1.4) is called MW-regular,

or a threshold equilibrium if there exists a x� 2 R such that the k-th playerincurs participation cost and participates if Xk > x� and quits if Xk < x�.

A threshold participation strategy x� 2 R means that a k-th player par-ticipates if Xk > x� and quits if Xk < x�. Such participation strategies arenecessary for the overall strategies to be monotone. The question is whethersuch strategies are compatible with conditions that are necessary for an equi-librium, as stipulated in Propositions 1.8 and 1.11. To address this question,

10

we rewrite Propositions 1.8 and 1.11 for the case where the participationstrategy is a threshold.

2.3. Proposition. If the participation strategy is a threshold x�n, thenP�X1 > x�n

� � �n.

2.4. Proposition. If the participation strategy is a threshold x�n, thenP�max(X1; : : : ; Xn) � x�

� � �.

Introduce distribution functions used throughout the paper,

F1(x) = P�X1 � x

�;(2.5)

Fn(x) = P�max(X1; : : : ; Xn) � x

�:(2.6)

2.7. Corollary. For the participation strategy to be a threshold it is nec-essary that the following system of two inequalities (for a number x�n) isconsistent:

F1(x�

n) � 1� �n ;(2.8)

Fn(x�

n) � � :(2.9)

Corollary 2.7 is not intended to de�ne x�n. Rather, it states that twointervals, fx�n : F1(x

�

n) � 1� �ng and fx�n : Fn(x�n) � �g, must overlap.9 Weshow in the next section that under very reasonable signal structure, theydo not; namely, a threshold strategy is incompatible with equilibrium. Forindependent signals, see Note 2.12 below.

Note also that F1; Fn are determined by the joint distribution of signals,irrespective of valuations, while �n; � are determined by the one-dimensionaldistribution of valuations (and constants e; r), irrespective of signals. Cor-relations between valuations, or between signals and valuations, in uenceneither F1; Fn nor �n; �.

2.10. Note. Numbers �n; � satisfy the inequality

�n� � 1

n;

which follows from comparison of mean values:

1

�

Z �

0

( ~V (p)� r) dp � n

Z 1

1� 1n

( ~V (p)� r)+ dp :

9Illustrations given in Sect. 4 may help to understand the matter.

11

2.11. Note. If F1; Fn satisfy

maxx

Fn(x)�1� F1(x)

� � 1

n;

then the necessary condition is satis�ed (irrespective of ~V ; r; e). Indeed,Fn(xn) = � =) 1� F1(xn) � 1

n�� �n.

2.12. Note. If X1; : : : ; Xn are independent, then the necessary condition issatis�ed. Indeed,

maxx

Fn(x)�1� F1(x)

�= max

xF n1 (x)

�1� F1(x)

�=

= maxp2[0;1]

pn(1� p) =

�n

n+ 1

�n1

n+ 1<

1

n:

3 Large number of correlated signals

What makes threshold equilibria infeasible, are conditions de�ned in terms ofn and signals dependence. When signals are independent, threshold equilib-ria take place, and the necessary condition stated in Corollary 2.7 is satis�edfor all n. The latter still holds when dependence is weak enough, since thebound ( n

n+1)n 1

n+1guaranteed by 2.12 is strictly stronger than the bound 1

n

required by 2.11. Some \amount of dependence" is needed to violate 2.7.To proceed in this direction, we must have a notion of dependence that isindependent of n, since we let n!1 . Hence, we must �nd a formalizationof the idea of \the same dependence for a larger number of variables".

Fortunately, there is a simple and natural way to do it, at least for thesymmetric case (distributions invariant under permutations). A symmetricdependence is usually thought of as a manifestation of a common (maybeunobservable) random variable in uencing all X1; X2; : : : When the commonvariable, call it X0, is given, signals X1; X2; : : : become conditionally i.i.d.with a (one-dimensional) distribution GX0

drawn (via X0) from a family�Gx0

�x02R of distributions.

An example. Choose some � 2 (0; 1) and let G(x) = �(x=p�) and

Gx0(x) = ��(x� x0)=

p1� �

�. In other words, X0 � N(0; �) and condition-

ally (given that X0 = x0), Xk � N(x0; 1��) for k = 1; 2; : : : We get (uncon-ditionally) Xk � N(0; 1) and E (XkXl) = � for 1 � k < l <1; the joint dis-tribution of X1; : : : ; Xn is normal for every n; and F1(�) = �(�). For each x0,�1�Gx0(x)

�=�1�F1(x)

�! 0 for x! +1, since 1�F1(x) � const � 1xe�x

2=2

and 1�Gx0(x) � const � 1xe�x

2=2(1��) for x! +1.

12

A deeper mathematical discussion of the way introducing dependence,based on the theory of exchangeable sequences, is given in Appendix 1. Here,we restrict ourselves to the case mentioned in Assumption 3.1.

3.1. Assumption. The joint distribution function FX1;:::;Xn(x1; : : : ; xn) of

n signals X1; : : : ; Xn is of the form

FX1;:::;Xn(x1; : : : ; xn) =

ZGx0(x1) : : : Gx0(xn) dG(x0)

where G is a distribution function, and�Gx0(�)

�is a family of distribution

functions. When comparing models with di�erent n we assume that G and�Gx0(�)

�do not depend on n.

Monotonicity or continuity of Gx0(�) in x0 are not required (only measur-ability). Accordingly, it does not matter, whether x0 is one-dimensional ormulti-dimensional.

The following assumption on the conditional and unconditional distribu-tion of signals is crucial in establisihing our results. It contains a parameter� 2 [1;+1]. The higher the parameter, the stronger the restriction on sig-nal dependence (and the weaker the corresponding restriction on valuationsintegrability, see Theorem 3.6).

3.2. Assumption. For almost all x0 (w.r.t. G),�1�Gx0(x)

�1=�

1� F1(x)! 0 ;

here x ! +1 if supfx : F1(x) < 1g = +1, and x ! (xmax)�, if supfx :F1(x) < 1g = xmax <1.

The case � = 1 is treated as follows: 1=� = 0, and�1 � Gx0(x)

�1=� is

either 1 (when Gx0(x) < 1) or 0 (when Gx0(x) = 1). So, 3.2 for � =1 meansthat Gx0(x) = 1 but F1(x) < 1 for some x (that depends on x0). Usuallywe'll consider only the case x ! +1, leaving to the reader to replace it,when needed, with x! (xmax)�.

An example. Let X0 be uniform on (0; a), and Gx0 be uniform on (x0; x0+b); here a; b are positive parameters. Clearly, xmax = a+b, andGx0(x0+b) = 1but F1(x0+ b) < 1 whenever x0 2 (0; a). Here, 3.2 is satis�ed in its strongestcase, � =1.

Assumption 3.2 imposes a condition on the top of the right hand tail ofsignal distributions. The normal distribution of signals satis�es 3.2 if (and

only if) � <1

1� �. On the other hand, there are many cases that violate

13

3.2. For example, Gx0(x) = exp(�jx� x0j). Another example: all Gx0 withdensities uniformly bounded away from 0 and 1 on the same interval. Thefollowing result uses 3.2 in its weakest case, � = 1.

3.3. Lemma. There exist x1; x2; : : : such that

n�1� F1(xn)

�!1 ; Fn(xn)! 1

for n!1.10

Proof. Let a sequence a1; a2; : : : satisfy supn n�1�F1(an)

�<1, then n

�1�

Gx0(an)� ! 0 by 3.2, therefore Gn

x0(an) ! 1 (n ! 1) for G-almost allx0. By the bounded convergence theorem,

RGn

x0(an) dG(x0) ! 1, that is,

Fn(an)! 1.The rest of the proof is a standard trick of mathematical analysis. De�ne

am;n for 1 � m < n < 1 by 1 � F1(an;m) =mn, then limn Fn(an;m) = 1 for

each m, therefore we can choose n1 < n2 < : : : such that n � nm =)Fn(an;m) � 1� 1

mfor each m. De�ne xn = an;m for n 2 [nm; nm+1) (leaving

xn arbitrary for n < n1), then n�1 � F1(xn)

�= m and Fn(xn) � 1 � 1

mfor

n 2 [nm; nm+1).

The result stated in Lemma 3.3 violates the necessary condition for regu-larity stated in Corollary 2.7 if valuations are bounded. Indeed, Fn(xn)! 1implies that xn > x�n for large n, since by (2.9) Fn(x

�) � � and � < 1. By(1.6), n�n is bounded whereas, by Lemma 3.3, n

�1 � F1(xn)

� ! 1, whichimplies that �n < 1 � F1(xn) for all n large enough. For such n, Corollary2.7 implies x�n > xn. Consequently, a treshold strategy is impossible for largen.

3.4. Corollary. If the distribution of valuations have a compact support,then a treshold startegy is violated for all n large enough.

We show in Section 5 that the contradiction x�n < xn and x�n > xn isdue to the dual meaning of signals under a threshold strategy, where a highsignal is both, good and bad news.

Note that Assumption 3.2 imposes constraints on the upper tails of theconditional and unconditional distribution of signals. The fact that whatmatters are the upper tails is obvious since the treshold must be high inorder to assure that the number of bidders is not large. The condition itselfrepresents a certain measure of local dependence. For example, if signals areindependent, the conditional and unconditional distributions are the same,

10Here, as before, Fn(x) = P�max(X1; : : : ; Xn) � x

�.

14

and the ratio goes to 1. If all signals are the same, complete dependence, theratio goes to 0. This interpretation is illustrated in the next section wherewe consider the case of normal distribution of signals.

To accommodate the case where valuations are not bounded, we mustinvestigate how fast does n

�1� F1(xn)

�tend to in�nity. Indeed, if ~V (1�) =

+1 then n�n !1; in order to conclude that �n < 1� F1(xn) for large n,we need n

�1 � F1(xn)

�to grow faster than n�n. Lemma 3.3 deduces some

growth of n�1 � F1(xn)

�from the weakest form (� = 1) of 3.2. Stronger

restrictions (� > 1) will be handled in Theorem 3.6, admitting heavy tails ofvaluations. The following result holds under the weakest restriction (� = 1)on signals, at the expense of a strong restriction on the tail of valuations.The strong restriction is expressed in the form of integrability of a functionof V . The function cannot be speci�ed unless the rate of convergence

�1 �

Gx0(x)�=�1�F1(x)

�! 0 is given. In any case, the restriction does not forceV to be bounded.

3.5. Proposition. There exists a convex, strictly increasing function :[0;1) ! [0;1) such that the relation E

�(V1 � r)+

�< 1 implies that a

threshold strategy is violated for all n large enough.

Proof. Take xn according to Lemma 3.3. We can choose such that �n(1�

F1(xn))�grows as fast as we want; we choose such that

1

n �n(1� F1(xn))

�!1 :

Jensen inequality gives

�n

Z 1

1� 1n

( ~V (p)� r)+ dp

�� n

Z 1

1� 1n

�( ~V (p)� r)+

�dp :

The left-hand side is 1n (en�n) by (1.6), therefore

1

n (en�n) �

Z 1

0

�( ~V (p)� r)+

�dp = E

�(V1 � r)+

�:

So, the sequence 1n (en�n) is bounded, while 1

n �en(1 � F1(xn))

� ! 1;therefore �n < 1 � F1(xn) for all n large enough. On the other hand,Fn(xn) ! 1, therefore � < Fn(xn) for all n large enough. For such n,Corollary 2.7 implies both x�n > xn and x�n < xn, which makes a thresholdimpossible.

The most important case of 3.2, namely � 2 (1;1), is covered by the fol-lowing result. Note that if V has a density fV (�) such that fV (v) � const �v��

15

for v ! +1, then the condition E (V � r) + < 1 is satis�ed if and only if� > + 1. Illustrations given in Sect. 4 may help in understanding thematter.

3.6. Theorem. Let ; � 2 (1;1) satisfy1

+

1

�= 1, and E (V � r) + < 1.

Then, a threshold participation strategy is compatible with equilibrium onlyfor �nitely many n.

Proof. Jensen inequality gives

�n

Z 1

1� 1n

( ~V (p)� r)+ dp

�

� n

Z 1

1� 1n

�~V (p)� r

� + dp :

The left-hand side is (ne�n) , the right-hand side does not exceed nE (V �r) +,

therefore n �1� n � E�V�re

� + or, taking into account that = �

��1and

� 1 = 1��1

,

n��n � E� ; where E =

�E

�(V � r)+

e

� �1=

:(3.7)

On the other hand, for every " > 0,

P

�1�GX0

(x) � "�1� F1(x)

���! 1 for x!1 ;(3.8)

which follows from bounded convergence theorem applied to indicators of theevents 1� GX0

(xn) � "�1 � F1(xn)

��, xn ! 1. The left-hand side of (3.8)

may be estimated by Markov inequality:

P�1�GX0

(x) � "�1�F1(x)

���= P

�Gn

X0(x) � �

1�"(1�F1(x))��n� �

� EGnX0(x)�

1� "(1� F1(x))��n=

Fn(x)�1� "(1� F1(x))�

�n;

therefore by (3.8),

lim infn!1

Fn(xn)�1� "(1� F1(xn))�

�n� 1(3.9)

for every sequence (xn) such that xn !1. Choose xn such that

1� F1(xn) = n�1� (E + 1) ;(3.10)

16

then�1 � "(1 � F1(xn))

��n =

�1 � 1

n"(E + 1)�

�n ! exp

��"(E + 1)��for

n!1, and (3.9) gives

lim infn!1

Fn(xn) � exp��"(E + 1)�

�:

However, xn do not depend on ", the latter being arbitrary, so, lim inf Fn(xn) �1, that is,

Fn(xn)! 1 :(3.11)

Combining (3.10) and (3.7) we have

1� F1(xn) > �n :(3.12)

The necessary condition 2.7 implies xn < x�n by (3.12), and then Fn(xn) �Fn(x

�

n) � �; by (3.11), the latter can hold only for �nitely many n.

We turn to the case � = 1, = 1, not covered by Theorem 3.6. Dueto the strong restriction (� = 1) on signals, no restriction is imposed onvaluations (except for integrability stipulated in 1.1).

3.13. Proposition. Let 3.2 be satis�ed for � =1. Then a threshold existsonly for �nitely many n.

Proof. Introduce

F1(x) = limn!1

Fn(x) = limn!1

EGnX0(x) = P

�GX0

(x) = 1�;

then

F1(x)! 1 for x!1 ;(3.14)

which follows from 3.2 by the bounded (or monotone) convergence theoremapplied to indicators of the events GX0

(xn) = 1, xn !1.On the other hand, �n ! 0 (which follows from its de�nition and inte-

grability of V ). Therefore we can choose xn !1 such that 1�F1(xn) > �n.We have Fn(xn) � F1(xn)! 1 by (3.14). Now we �nish the proof similarlyto the proof of Theorem 3.6, since 3.11 and 3.12 hold.

4 Heavy tails and necessary conditions

This section contains some numerical examples that illustrate the necessarycondition 2.7 and results based on it (Theorem 3.6, and others). The distri-bution of valuations remains unspeci�ed; only its tail is restricted by various

17

integrability requirements. In contrast, the distribution of signals will becompletely speci�ed, allowing numerical computations.

Consider the normal distribution of signals: Xk � N(0; 1), E (XkXl) =� 2 [0; 1] for k < l. The distribution function Fn of max(X1; : : : ; Xn), plottedon Fig. 1 for some n and �, may be calculated as Fn(x) =

RGn

x0(x) dG(x0),

where G(x) = �(x=p�) and Gx0(x) = �

�(x� x0)=

p1� �

�. In other words,

Xk =p��0 +

p1� ��k ;(4.1)

where �0; �1; �2; : : : are i.i.d. N(0; 1) random variables. Thus,max(X1; : : : ; Xn) =

p��0 +

p1� �max(�1; : : : ; �n).

Note that the signal Xk in (4.1) is a linear combination of a commonand private value. The correlation coe�cient �; is not only a measure ofsignal dependence but also measures the relative importance of the commoncomponent. As such, � has a nice economic interpretation with an importantmessage. The relation between the common and private components statedin equation (4.1), isn't accidental, it admits a far-reaching generalization (seeAppendix); however, additivity is speci�c to the normal distribution.

Functions F1 and Fn are plotted on Fig. 1 for various n and �. Whensignals are independent (� = 0) and n grows, the distribution Fn ofmax(X1; : : : ; Xn) moves right but also concentrates.11 When signals are de-pendent (� > 0) and n grows, Fn still moves right, but does not concentratetoo much, since the variance of

p��0 is always a part of the variance of Fn.

0 2

� = 00 2

� = 0:10 2

� = 0:50 2

� = 0:9

Figure 1: Distribution functions Fn for n = 1; 2; 5; 10; 20; 50; 100.

According to Corollary 2.7 we are interested in the curve f�1 �F1(x); Fn(x)

�: x 2 Rg, plotted on Fig. 2. It is necessary for a threshold

participation that the point (1��n; �) is located above (that is, to the rightof) the curve. Typically, � is not close to 1, thus �n is forced to be nottoo small which, however, means a heavy tail of the distribution of valua-tions. If valuations are bounded then �n � const =n for large n. If valuations

11In fact, for large n it is concentrated nearp2 lnn.

18

� = 0 � = 0:1 � = 0:5 � = 0:9

Figure 2: Functional dependence between 1 � F1(x) (the horizontal axis) and

Fn(x) (the vertical axis).

0 5

� = 00 5

� = 0:10 5

� = 0:50 5

� = 0:9

Figure 3: Functional dependence between n�1� F1(x)

�(the horizontal axis) and

Fn(x) (the vertical axis).

0 5

� = 00 5

� = 0:10 5

� = 0:50 5

� = 0:9

Figure 4: Functional dependence between n2=3�1 � F1(x)

�(the horizontal axis)

and Fn(x) (the vertical axis).

0 5

� = 00 5

� = 0:10 5

� = 0:50 5

� = 0:9

Figure 5: Functional dependence between n1=3�1 � F1(x)

�(the horizontal axis)

and Fn(x) (the vertical axis).

19

have the second moment then �n � const =pn for large n. More generally,

if E (V � r) + < 1 then �n � const =n1=� where 1 + 1

�= 1. Thus we are

interested in curves f�n1=�(1 � F1(x)); Fn(x) : x 2 Rg, plotted on Figs 3, 4and 5 for � = 1, 1:5 and 3 respectively.

Fig. 3 is useful when V is bounded, P�V � vmax

�= 1, which implies

n�n � vmax=e. For a threshold equilibrium, the point (n�n; �) must be tothe right of the corresponding curve. Consider �rst the case where signalsare independent, � = 0, for which we know that there is no problem to havea threshold equilibrium. And indeed, regardless how large is n; there is alarge region where (n�n; �) can satisfy the constraints, being to the right ofall curves, see Fig. 3. Moreover, the region contains all points of the form(n�n; �), since they all satisfy n�n� � 1 by (2.10) (thus, � � e

vmax). We see

that independent signals never violate the necessary condition, as was notedin 2.12.

Once we introduce signal dependence, the feasibility region for a thresholdequilibrium is decreasing with n. Fig. 3 illustrates that the stronger is signaldependence, the more e�ective is n in restricting the possibilities to havea threshold equilibrium. Consequently, for highly correlated signals we see(Fig. 3, � = 0:9) that each point is admissible only for a �nite number of n.For instance, the center of the square (n�n = 2:5, � = 0:5) is admissible forn � 5 but inadmissible for n � 10. It means that for vmax=e = 2:5, � = 0:5and � = 0:9 a threshold equilibrium is possible at most when n < 10. For� = 0:5 the situation is qualitatively similar, but the bound for n is weaker(20 for the center). For � = 0:1 the picture is not so clear; the bound for n israther weak (much more than 100 for the center), but still �nite, which is notevident from the �gure but is ensured by the theory (Sect. 3). Every positive� excludes a threshold equilibrium for n large enough. However, when � issmall, the bound for n may be impractical.

When V is not bounded (vmax = 1), we need some moments. Assumethat

E =

�E

� (V � r)+e

� �1=

<1 ;

then n1=��n � E by (3.7), where 1 + 1

�= 1. The case = 3, � = 1:5 is shown

in Fig. 4. For highly correlated signals (� = 0:9) the situation is qualitativelysimilar to that for � = 1 (Fig. 3) but, naturally, the bound for n is weaker(50 for the center, E = 2:5, � = 0:5). For � = 0:5 the bound for n is ratherweak but still �nite, which is not evident from the �gure, but follows fromthe theory, since � < 1

1��. For � = 0:1 and � = 1:5, however (here � > 1

1��),

we see that the situation is dissimilar to � = 0:1 and � = 1; rather, it issimilar to � = 0 and � = 1. The same for � = 0:5 and � = 3, see Fig. 5.

20

A comparison between Figures 3{5 illustrates the e�ect of heavy tails;what we called, the lottery e�ect. The possibility of very large winningsmitigates the e�ects of both, dependence and large n, and makes a thresholdequilibrium more likely to happen. For example, when � 3 (� � 1:5)and � � 0:1, a large n does not seem to reduce the region of (n�n; �) thatmake it possible to have a threshold equilibrium; it takes a much strongerdependence to let n exert its e�ect. When tails are even heavier (Figure 5, = 1:5, � = 3), these e�ects are even more pronounced.

5 Good news bad news

We show in this section 'why' a threshold strategy may be incompatible withequilibrium, by considering some numerical examples. In contrast to Sect. 4,here we compute the very participation threshold (rather that a necessarycondition for its existence). To this e�ect, the joint distribution of signalsand valuations will be speci�ed. Before presenting the numerical results, wederive formulas underlying the computations.

Consider a second price auction, and assume that players know their val-uations: Vk = V(Xk) where V is a given strictly increasing function. At thebidding stage, e�ective participants bid their valuations. Consider a partic-ipation threshold strategy, x�; as a possible equilibrium. Then, denoting by�(x) the expected pro�t of a player whose signal is x, it must be that

x � x� =) �(x) � e ;

x � x� =) �(xi) � e ;(5.1)

which evidently implies e = �(x�). Let

Hx1(x) = P�max(X2; : : : ; Xn) � x

��X1 = x1�;

and assume that player 1 with signal X1 = x1; is considering to deviate fromthis strategy and participate regardless of his signal; what could he gain if hedid it. To evaluate his gain, we distinguish between the two cases: x1 � x�

and x1 � x�:If x1 � x�, his expected pro�t is

(5.2) �(x1) = (V(x1)� r)Hx1(x�) +

Z x1

x�

�V(x1)� V(x)� dHx1(x) =

=�V(x�)� r

�Hx1(x

�) +

Z x1

x�Hx1(x) dV(x) ;

21

which is basically the well-known integral of winning probability. Otherwise,

�(x1) =�V(x1)� r

�+Hx1(x

�):(5.3)

Using 3.1, Bayes formula for densities, and Corollary 2 from [15, Sect. 7.3]we get

Hx1(x) = E�P�max(X2; : : : ; Xn) � x

��X0; X1

� ��X1 = x1�=

= E�P�max(X2; : : : ; Xn) � x

��X0

� ��X1 = x1�= E

�Gn�1

X0(x)

��X1 = x1�=

=

ZGn�1

x0(x)

G0x0(x1)G0(x0)

F 01(x1)dx0 ;

provided that distributions G and Gx0 have densities G0 and G0x0.

Assume (once again) that signals are distributed normally N(0; 1) withcorrelation �. Then,

G0x0(x1)G0(x0)

F 01(x1)=

1

�0(x1)� 1p

1� ��0�x1 � x0p1� �

�� 1p

��0�x0p�

�=

=1p

2��(1� �)exp

�1

2x21 �

1

2(1� �)(x1 � x0)

2 � 1

2�x20

�=

=1p

2��(1� �)exp

�� 1

2�(1� �)(x0��x1)2

�=

1p�(1� �)

�0�x0 � �x1p�(1� �)

�;

therefore,

Hx1(x) =

ZGn�1

x0(x) d�

�x0 � �x1p�(1� �)

�= EGn�1

�x1+p

�(1��)�0(x)(5.4)

where �0 � N(0; 1).12

Inserting (5.4) into (5.2) we obtain

�(x1) =�V(x�)� r

�EGn�1

�x1+p

�(1��)�0(x�) +

Z x1

x�EGn�1

�x1+p

�(1��)�0(x) dV(x) ;

(5.5)

which may be computed as a one-dimensional integral plus a two-dimensionalintegral (though, one more level of integration is implicit in G). The formulaholds for x1 � x�. If x1 � x�;

�(x1) =�V(x1)� r

�+EG

n�1

�x1+p

�(1��)�0(x�):(5.6)

12The formula can also be obtained geometrically. Namely, the orthogonal projectionof the vector X0 =

p��0 onto the direction of the vector X1 =

p��0 +

p1� ��1 is �X1

(here �0; �1 are treated as orthogonal unit vectors), which means that the conditionaldistribution of X0 given X1 is N(�X1; �� �2).

22

Assume V(x) = x for x � r; that is, valuations are just equal to signals,at least above the reserve price r 2 (�1;+1). If r < 0, some valuations arenegative, which may seem strange, but is in fact harmless, since only V � rmatters. We may choose the origin on the axis of valuations (and bids) atwill; for convenience we choose the origin at the mean signal (and the unitas its mean square deviation). When V(x) < r, the exact value of V(x) doesnot matter, as far as each player knows his valuation (which was assumed).

r=0 x�=1

0:2

(a)

r x�x1

(b)r x1=x�

(c)r x� x1

(d)r x� x1

(e)r x� x1

(f)

Figure 6: A non-monotone pro�t function (above), and its values as integrals

(below).

Now we can compute the (expected) pro�t function �(�) numerically. Wedid it for n = 5; � = 0:9; r = 0; x� = 1, and plotted the result in Fig. 6(a),for x1 2 [0; 5]: The pro�t function is not monotone and x� is on the decliningsegment which, obviously, violates (5.1) and therefore, is at variance with athreshold equilibrium. To illustrate why it happens, we present graphicallyin Fig. 6(b{f) the � functions evaluated at di�erent signals; each �(x1) is anintegral over x 2 (r; x1): The integrals are shown in the �gure by the shadedareas (under the corresponding integrands).

Non participating signals; x 2 (r; x�): A puzzling property of signals in(r; x�) is that the probability of being the only bidder, and therefore,being the winner, is decreasing with the signal. This is just contrary towhat we get when signals are independent. This property is illustratedby the declining heights of the rectangular areas, which stand for theprobability of being the only bidder and therefore, winning the auction;see (5.2). This is the bad news associated with a higher signal. Thegood news is that since all winners pay the same price, r, a higher

23

signal conveys good news; a longer horizontal segment in Fig. 6(c) incomparison with Fig. 6(b). The simultaneous e�ects of these, good andbad, news results in the unimodal � function on (r; x�).

Participating signals; x 2 (x�;1). The reason for the continuing (beyondx�) decline of the � function is illustrated in Fig. 6(d). The area of therectangular area shrank a lot, which describes the declining probabilityof getting x� � r. A player with a signal higher than x� can count onwinning the auction also when he is not the only bidder, see the curvi-linear area in Fig. 6(d). However, as the picture shows, these winningsare costly since instead of paying r, the wining bidder is paying pricesthat are increasingly higher than x�. Note that the upper boundaries ofthe shaded areas indicate the probability distributions of the price thewinner pays for the object, see the second equality in (5.2). The jumpfrom paying r to paying prices increasingly higher than x�, is respon-sible for the continuing decline immediately after x�. After a while,however, the e�ect of the low buying price has almost disappeared, seeFig. 6(e) and (f), and the good news of a higher signal get the upperhand and the � function starts rising. Remember that since � < 1,x1 � �x1 is increasing.

It is interesting to note that deviating from a threshold strategy is prof-itable for most of participating signals. The drawing in Fig. 6(a) covers morethan 99 percent of the signals and yet, the largest expected pro�t for partici-pating signals is smaller than �(x�) = e. More than half of non participatingsignals will gain if they deviate.

An interesting question is what will our example produce if we consideredother values of x�. A partial answer to this question is given in Fig. 7 wherewe considered thresholds higher and lower than x� = 1.

We may consider other values of x� and corresponding curves, shown onFig. 7.

�(x�)=0:020

x�=0:1

�(x�)=0:041

x�=0:2

�(x�)=0:12

x�=0:5

�(x�)=0:27

x�=1

�(x�)=0:74

x�=2x�

e=�(x�)

0:27

0:058

Figure 7: Low values of x� are equilibrium thresholds, and give increasing pro�t

functions; high values do not.

We see that a small value of x� (say, 0:1 or 0:2) determines an increasingpro�t function. Such x� satis�es (5.1) and describes an equilibrium with a

24

participation threshold for the corresponding e = �(x�). However, a largevalue of x� (say, 0:5, 1, or 2) determines a non-monotone pro�t function thatviolates (5.1) and cannot describe an equilibrium. A more subtle situationappears near the critical point, see Fig. 8. The value x� = 0:25 is \small" in

�(x�)=0:0527

x�=0:25

0:2 0:450:050

0:056

�(x�)=0:0550

x�=0:26

0:4 0:60:0568

0:0569

�(x�)=0:0580

x�=0:273

0:25 0:800:057

0:059

Figure 8: Near the critical point: non-monotonicity emerges outside of x�; ulti-

mately it violates equilibrium, but not immediately.

the sense used above. However, the value 0:26 determines a non-monotonepro�t function and nevertheless satis�es (5.1), since the minimal value of thefunction still exceeds its value at x�. The value 0:273 is \large", it violates(5.1). The critical value is x�crit = 0:272 � � � � 0:27, and the correspondingecrit = �(x�crit) is � 0:58. Each e 2 (0; ecrit) leads (or at least, can lead) to athreshold-participation equilibrium, and the corresponding threshold x� maybe seen on Fig. 7 (the leftmost graph). However, no one e 2 (ecrit;1) does.

A Exchangeability

The formula of 3.1,

FX1;:::;Xn(x1; : : : ; xn) =

ZGx0(x1) : : : Gx0(xn) dG(x0) ;(A.1)

de�nes a family�FX1;:::;Xn

�n=1;2;::: of multidimensional distribution functions.

The family is consistent,

FX1;:::;Xn(x1; : : : ; xn) = lim

xn+1!1FX1;:::;Xn;Xn+1

(x1; : : : ; xn; xn+1)

(see [15, (6.4.8)]), and exchangeable13 in the sense that each FX1;:::;Xnis sym-

metric, that is, invariant under every permutation of arguments (see [15,Exercise 6.3.10]). Consistency and exchangeability hold for every distribu-tion G (on R, or Rn , or a more general space) and every family

�Gx0

�of

distributions on R; it is only needed that Gx0(x) is a Borel function of x0 foreach x.

13Or \interchangeable".

25

Interestingly, the converse it true. That is, every consistent and exchange-able family

�FX1;:::;Xn

�n=1;2;::: can be represented in the form (A.1). It is a

version of de Finetti theorem, see [15, 7.3] or [2, 4.6, p. 269]. In otherwords, every exchangeable probability measure on the space R1 (of all in�-nite sequences (x1; x2; : : : )) is a mixture of exchangeable product measures(that describe i.i.d. random variables). Reconstructing G and

�Gx0

�out of�

FX1;:::;Xn

�is not evident and may be visualized as follows.

First, if random variables X1; X2; : : : are i.i.d., then the empirical distri-

bution function F !n , de�ned by F !

n (x) =1n

Pnk=1 1(�1;x](Xk), converges (for

n!1) almost surely to the distribution function F ofX1 (Glivenko-Cantellitheorem, see [15, 8.2, Th. 2]).

Second, if random variables X1; X2; : : : are conditionally i.i.d. given an-other random variable X0, then the empirical distribution function F !

n con-verges to the conditional distribution function GX0

of X1 given X0 (which isa conditional version of Glivenko-Cantelli theorem).

Third, if random variables X1; X2; : : : are exchangeable then the empir-ical distribution function F !

n converges almost surely to a limit F !1. The

latter is a random variable, and we may use as X0 just that random variable!Accordingly, GX0

is nothing but X0. The only problem is that F !1

takes onvalues in the in�nite-dimensional space of all distributions (on R). Some-times (the exchangeable normal distribution is an example), all F !

1belong

to a parametric family, which makes the reconstructed X0 e�ectively �nite-dimensional. In general it is in�nite-dimensional, which does not invalidatearguments of Sect. 3 (insensitive to the dimension of x0).

14

Note also that de Finetti theorem holds only for in�nite sequences; see[15, Exercises 6.3.10, 7.3.7]. In fact, every in�nite exchangeable sequenceX1; X2; : : : with EX2

1 < 1 satis�es Cov(X1; X2) � 0. However, a �nite ex-changeable sequence X1; : : : ; Xn can satisfy Cov(X1; X2) = � 1

n�1Var(X1) <

0. Especially, that is the case for n exchangeable normal random variablessuch that X1 + � � �+Xn = 0.

Further information on exchangeability can be found in [4].

References

[1] G.P. Beaumont, Intermediate mathematical statistics, Chapman andHall, London 1980.

14In fact, the considered in�nite-dimensional Borel space is standard, hence, Borel-isomorphic to R. Therefore, the reconstructed in�nite-dimensional X0 can be convertedinto a one-dimensional random variable which, however, is rather unnatural.

26

[2] R. Durrett, Probability: theory and examples, (2nd ed) Duxbury Press,Belmont 1996.

[3] G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, (2nd ed.) Cam-bridge Univ. Press, 1959.

[4] G. Koch and F. Spizzichino (eds), Exchangeability in probability and

statistics, (International conference, Rome 1981.) North-Holland 1982.

[5] D. Levin, and J. Smith, "Equilibrium in Auctions with Entry", Ameri-

can Economic Review, 84 (1994), 585-599.

[6] P. McAfee,and J. McMillan, "Auctions with Entry", Economics Letters,23 (1987), 343-7.

[7] E. Maskin, J. Riley, "Optimal Auctions With Risk Averse Buyers"Econometrica, 52 (1984), 1473-1518.

[8] S. Matthews, "Comparing Auctions for Risk Averse Buyers: A Buyer'sPoint of View" Econometrica, 55 (1987), 633-646.

[9] S. Matthews, "Information Acquisition In Discriminatory Auctions"Bayesian Models in Economic Theory, ed. M. Boyer and R. Kihlstrom,Elsevier Science Publishers B.V. (1984).

[10] P.R. Milgrom, R.J. Weber, \A theory of auctions and competitive bid-ding\, Econometrica 1982, 50:5, 1089{1122.

[11] P.R. Milgrom, "Rational Expectations, Information Acquisition, andCompetitive Bidding" Econometrica 49:4 (1981), 921-943.

[12] Samuelson, W. "Competitive Bidding with Entry Costs", Economics

Letters, 17 (1985), 53-57.

[13] Stegeman, M. "Participation Costs and E�cient Auctions", Journal ofEconomic Theory, 71 (1996), 228-259.

[14] Tan, G. "Entry and R&D in Procurement Contracting", Journal of Eco-nomic Theory, 58 (1992), 41-60.

[15] Yuan Shih Chow, H. Teicher, Probability theory: independence, inter-

changeability, martingales, Springer 1978.

27