AUCTIONS

An Introduction

y

Elmar Wolfstetter

April

Humb oldtUniversitat zu Berlin

Institut f Wirtschaftstheorie I

Wirtschaftswissenschaftliche Fakultat

Spandauerstr

Berlin

Germany

email wolfwiwihub erlinde

Abstract

This is a fairly detailed review of theory It b egins with

basic results on private value with particular emphasis on the

generality and limitations of the of a large class of

distinct auction rules The basic framework is then gradually mo died

to admit for example risk aversion a minimum price entry fees and

other xed costs of multiunit auctions and bidder collusion

There follows an intro duction to the theory of optimal auctions and

to common value auctions and the asso ciated winners curse problem

It closes with a sample of applications of in economics

such as the regulation of natural monop olies the theory of oligop oly

and the government securities market

Diese Arb eit ist im Sonderforschungsb ereich Quantikation und Simulation

Okonomischer Prozesse Humb oldtUniversitat zu Berlin entstanden und wurde auf seine

Veranlassung unter Verwendung der ihm von der Deutschen Forschungsgemeinschaft zur

Verfugung gestellten Mittel gedruckt

y

Comments byFriedel Bolle UweDulleck Peter Kuhbier Michael Landsb erger

Wolfgang Leininger Georg Merdian and in particular by Dieter Nautz are gratefully

acknowledged

Contents

Intro duction

Private value auctions

Some basic results on Dutch and English auctions

Revenue equivalence theorem

The case of uniformly distributed valuations

Generalization

Robustness

Removing risk neutrality

Removing symmetry

Intro ducing a minimum price or participation fee

Intro ducing numb ers uncertainty

Endogenous quantity

Multiunit auctions

Removing indep endence correlated b eliefs

Auction rings

Optimal auctions

Application of the revelation principle

Illustration two bidders and twovaluations

Discussion endogenizing bidder participation

Common value auctions and the winners curse

Further applications

Auctions and oligop oly

Natural gas and electric p ower auctions

Treasury bill auctions

Men prize the thing ungainedmore than it is

Shakesp eare Troilus and Cressida

Intro duction

Supp ose you inherit a unique and valuable go o d say a painting by the late

van Gogh or a copy of the edition of Adam Smiths Wealth of Nations

For some reason you are in desp erate need for money and decide to sell You

have a clear idea of your reservation price Of course you hop e to earn more

than that You wonder isnt there some way to reach the highest p ossible

price Howshouldyou go ab out it

If you knew the p otential buyers and their valuations of the ob ject for sale

your pricing problem would have a simple solution You would only need to

call a nonnegotiable price equal to the highest valuation and wait for the right

customer to come and claim the item Its as simple as that

Information problem The trouble is that you the seller usually haveonly

incomplete information ab out buyers valuations Therefore you have to gure

out a pricing scheme that p erforms well even under incomplete information

Your problem b egins to b e interesting

Basic assumptions Supp ose there are N p otential buyers eachof

whom knows the ob ject for sale well enough to decide howmuch he is willing

V V Since you the seller to pay Denote buyers valuations by V

N

do not know V you must think of it as a random variable distributed by some

joint probability distribution function CDF F v v v v

N

N

F v v PrV v V v Similarlypotential buyers

N N N

know their own valuation but not that of others Therefore buyersalsoview

valuations as a random variable except their own

Cournot monop oly approach Of course you could stick to the takeit

orleaveit pricing rule even as you are sub ject to incomplete information

ab out buyers valuations You would then call a nonnegotiable price p at or

ab oveyour reservation price r and hop e that some customer has the right

valuation and is willing to buyYour pricing problem would then b e reduced

toavariation of the standard Cournot monop oly problem The only unusual

part would b e that the tradeo b etween price and quantity is replaced byone

between price and the probabilityofsale

Alluding to the usual characterization of Cournot monop olyyou can even

draw a demand curve with the nonnegotiable price p on the price axis

and the probability that at least one bidders valuation exceeds the listed price

p

p Gp GpF pp

on the quantity axisas in Figurebelow And you would then pickthat

price p that maximizes the exp ected value of your gain from trade pp r

Equivalentlyyou can state the decision problem in quantity co ordinates

as is customary in the analysis of Cournot monop oly where quantity is here

represented by the probabilityofsale q Gp

maxP q r q P q G q

q

Computing the rstorder condition the optimal price p can then b e char

een marginal revenue acterized in the familiar form as a relationship b etw

MR and marginal cost MC

Gp

r MC MR p

G p

p

v

Gp

marginal revenue p

G p

p

r

q Gp demand

v

q

Figure Optimal takeitorleaveit price

As always this condition applies only if the maximization problem is wellb ehaved

Wellb ehavedness requires that revenue is continuous and marginal revenue strict monotone

increasing in p

Example Suppose valuations V are independent and uniformly distributed

i

over the interval and the reservation priceis r Then for al l

N N

p GpF p p Hence the optimal takeitorleaveit priceis

s

N

p N

N

The resulting expectedpricepaid price asked times probability of success is

N

pN p N Gp N p N

N

Both p N and pN are strict monotone increasing in N starting from



p p and approaching as N

Takeitorleaveit pricing is one waytogo But generally you can do

b etter by setting up an auction This brings us to the analysis of auctions

and the design of optimal auction rules Interestingly the Cournot monop oly

price p will continue to play a role as optimal minimum price in the optimal

auction setting as you will learn in the section on optimal auctions

Auctions what where and why An auction is a bidding mechanism

describ ed by a set of auction rules that sp ecify how the winner is determined

and howmuch he has to pay In addition auction rules may restrict partici

pation and feasible bids and imp ose certain rules of b ehavior

Auctions are widely used in many transactions not just in the sale of

art and wine Every week the Treasury auctionso billions of dollars of

bills notes and Treasury b onds Governments and private corp orations solicit

deliveryprice oers of pro ducts ranging from oce supplies to tires or con

struction jobs The Department of the Interior sells the rights to drill oil and

other natural resources on federally owned prop erties And private rms sell

pro ducts ranging from fresh owers sh and tobacco to diamonds and real

estate Altogether auctions account for an enormous volume of transactions

Essentially auctions are used for three reasons

sp eed of sale

to reveal information ab out buyers valuations

to prevent dishonest dealing b etween the sellers agent and the buyer



For a simple pro of of monotonicitywork with lnp whichisobviously a monotone

transformation ofp

The latter agency role of auctions is particularly imp ortantifgovernment agen

cies are involved in buying or selling As a particularly extreme example just

think of the current privatization programs in Eastern Europ e Clearly if the

agencies involved such as the Treuhandanstalt that is in charge of privatizing

the state run corp orations of former communist East Germanywere free to

negotiate the terms of sale the lucky winner probably would b e the one who

made the largest brib e or p olitical contribution However if assets are put up

for auction cheating the taxpayer is much more dicult and costly and hence

less likely to succeed

Popular auctions There are many dierent auction rules Actually the

word itself is something of a misnomer Auctio means increase but not all

auctions involve calling out higher and higher bids

One distinguishes b etween oral and written auctions In oral auctions

bidders hear each others bids and can makecounteroers each bidder knows

his rivals In a written or closedseal bid bidders submit their bids simultane

ously without revealing them to others often bidders do not even knowhow

many rival bidders participate

The b est known and most frequently used auction is the ascending price or

English auction followed by the rstprice closedseal bid or



and the secondprice closedseal bid also known as

Oral sealBid

ascending price secondprice

English

descending price rstprice

Dutch

Table The most p opular auctions

In the ascending price or the auctioneer seeks increasing

bids until all but the highest bidders are eliminated If the last bid is at or

ab ove the reserve price the item is awarded or kno cked down to the remaining

bidders If a tie bid o ccurs the item is awarded for example byachance

rule In one variation used in Japan the price is p osted on a screen and raised

continuouslyAny bidder who wants to b e active at the current price pushes

a button Once the button is released the bidder has withdrawn altogether



Manyvariations of these basic auctions are reviewed in CassadyR Auctions

and Auctioneering University of California Press

Instead of letting the price rise the descending price or Dutch auction

follows a descending pattern The auctioneer b egins by asking a certain price

and gradually lowers it until some bidders shout Mine to claim the item

Frequently the auctioneer uses a mechanical device called the Dutch clo ck

This clo ck is started and the asking price ticks down until someone calls out

The clo ck then stops and the buyer pays the indicated price Again provisions



are made to deal with tie bids

Finally in a written auction bidders are invited to submit a closedseal

bid with the understanding that the item is awarded to the highest bidder

Under the rstprice rule the winner actually pays as much as his own bid

whereas under the secondprice rule the price is equal to the second highest

bid

Of course this terminology is not always used consistently in the aca

demic and in the nancial literature For example in the nancial commu

nitya multipleunit singleprice auction is termed a Dutch auction and a

multipleunit closedseal bid auction is termed an English auction except by

the English who call it an American auction Whereas in the academic litera

ture the lab els English and Dutchwould b e exactly reversed In order to avoid

confusion keep in mind that here we adhere to the academic usage of these

terms and use English as equivalent to ascending and Dutch as equivalentto

descending price

Early history of auctions Probably the earliest rep ort on auctions is found

in Hero dotus in his account of the bidding for men and wives in Babylon

around BC These auctions were unique since bidding sometimes started



at a negative price In ancient Rome auctions were used in commercial trade

to liquidate prop erty and to sell plundered war booty A most notable auction

was held in AD when the Praetorian Guard put the whole empire up for

auction After killing the previous emp eror the guards announced that they

would app oint the highest bidder as the next emp eror Didius Julianus outbid

his comp etitors but after twomonths was b eheaded by Septimius Severus who

seized p ower a terminal case of the winners curse



An interesting Dutch auction in disguise is time discounting In many cities in the

US discounters use this metho d to sell cloth Each item is sold at the price on the tag

minus a discount that dep ends on howmanyweeks the item was on the shelf As time

passes the nal price go es down at the rate of sayperweek until the listed b ottom

price is reached



See Shubik M Auctions bidding and markets an historical sketch in Stark

R ed Auctions Bidding and Contracting Uses and Theory New York University Press

p

Private value auctions

We b egin with the most thoroughly researched auction mo del the symmetric

independent private values SIPV mo del with risk neutral agents In that

mo del

a single indivisible ob ject is put up for sale to one of several bidders unit

auction

b oth the seller and all bidders are risk neutral

each bidder knows his valuation no one else do es private values

the unknown valuations are indep endentrandomvariables drawn from

some continuous probability distribution continuityindependence

all bidders are indistinguishable therefore the probability distribution

of each bidders valuation is the same symmetry

the sellers reservation price is normalized to r

We will mo del bidders b ehavior as a nonco op erative game under incom

plete information One of our goals will b e to rank the four common auctions

To what extent do es institutional detail matter Can the seller get a higher

average price by an oral or a written auction Is comp etition b etween bidders

ercer if the winner pays a price equal to the second highest rather than the

highest bid And which auction if any is strategically easier to handle

Some basic results on Dutch and English auctions

The comparison b etween the four standard auction rules is considerably facil

itated by the fact that the Dutch auction is equivalent to the rstprice and

the English to the secondprice closedseal bid Therefore the comparison of

the four standard auctions can b e reduced to comparing the Dutch and the

English auction Later you will learn the far more surprising result that even

yo equivalent these two auctions are pa

The strategic equivalency b etween the Dutch auction and the rstprice

seal bid is immediatelyobvious In either case a bidder has to decide how

much he should bid or at what price he should shout Mine to claim the

item Therefore the strategy space is the same and so are payo functions

and hence equilibrium outcomes

Similarly the English auction and the secondprice closedseal bid are

equivalent but for dierent reasons and only as long as we stick to the private

values framework Unlike in a seal bid in an English auction bidders can

resp ond to rivals bids Therefore the two auction games are not strategically

equivalent However in b oth cases bidders havetheobvious dominant strategy

to bid an amount or set a limit equal to the own true valuation Therefore

the equilibrium outcomes are the same as long as bidders valuations are not

aected by observing rivals bidding b ehavior whichalways holds true in the



private values framework

The essential prop erty of the English auction and the secondprice closesd

seal bid is that the price paid by the winner is exclusively determined byrivals

bids Bidders are thus put in the p osition of price takers who should accept

any price up to their own valuation The truthrevealing strategy bv v

isadominant strategy The immediate implication is that b oth auctions forms

have the same equilibrium outcome

Revenue equivalence theorem

Using elementary reasoning wehave already established the payo equivalence

of Dutch and rstprice and of English and secondprice auctions Remark

ablypayo equivalence is a much more general feature of auction games

Indeed al l auctions that award the item to the highest bidder are payo

equivalent The four standard auctions are a case in p oint Wenow turn

to pro of of this surprising result and then explore mo dications of the SIPV

framework

We mention that the older literature conjectured that Dutch auctions lead

to a higher average price than English auctions b ecause as Cassady put it in

the Dutch auction each p otential buyer tends to bid his highest demand

price whereas a bidder in the English system need advance a rivals oer by

only one increment Both the reasoning and the conclusion are wrong

Since bidders valuations are indep endent random variables their jointdis

tribution function F is just the pro duct of their separate distribution functions

F v that are in turn identical due to the assumed symmetry

Therefore

F v v v F v F v F v

 n  n

A bidders strategy is a mapping from valuations to bids b v R

Avector of strategies is an equilibrium if it has the mutual b est resp onse

prop erty Since all bidders are alike whatever is an optimal bidding strategy

for one bidder should also b e an optimal strategy for any other bidder This

suggests that the equilibrium should b e symmetric



Obviouslyifrivals bids signal something ab out the underlying common value of the

item each bidders estimated value of the item is up dated in the course of an English auction

No such up dating can o ccur under a seal bid where bidders place their bids b efore observing

rivals b ehavior Therefore if the item has a common value comp onent the English auction

and the secondprice closedseal bid do not have the same equilibrium outcome



Cassady R Auctions and AuctioneeringUniversity of California Press p

The key insight that leads to nice and simple pro ofs is that given rivals

strategies each bidders decision problem can b e viewed as one of cho osing

through his bid bv a probability of winning andanexpectedpayment E

This way the choice of strategies is lo oked at as a selfselection of E Due

to the indep endence assumption b oth and E dep end only on the bid bv

but not directly on the bidders own valuation v

In all results reviewed in this section wecharacterize symmetric equilibria

We assume the SIPV framework and consider the set of auctions that adhere to

the principle of selecting the highest bidder as winner and leave the number

of active bidders the same No other assumptions are made Therefore

our analysis applies to a myriad of auction rules not just the four common

auctions

The rst useful result is

is monotone in Lemma Monotonicity The equilibrium bid strategy b

creasing in v

Pro of Given rivals strategies the exp ected utility of a bidder with valuation v

b is U bv vbv v Ebv Dene the indirect and bidding strategy

utility function U v R

U v b v v Eb v

Since U is a maximum value function the Envelop e Theorem applies and

one has

U v b v

Obviously U is convex in v Therefore

U v b v b v

and hence b v by b v Invoking that the optimal bid is never

the same for dierentvaluations the monotonicityiseven strict

An immediate implication of strict monotonicitycombined with symmetry

is



Notice in this general account it is not assumed that it is only the winner who has to

makea payment

Theauctionrulemay aect the numb er of activebiddersifpayment is not restricted

to the winner For example if the seller requires an entry fee he will lo ose all those whose

valuation is b elow that fee This p ointistaken up on page

The pro of is similar to the pro of of concavity of the exp enditure function in the price

vector that you have learned in your micro course Make sure that you also understand the

intuition of this result Notice that the bidder could always leave the bid unchanged when

v changes as a fall back Therefore U must b e at or ab ove its gradienthyp erplane which

proves convexity

Lemma Pareto optimality The bidder with the highest valuation wins

the auction

This brings us to the amazing revenue equivalence result

Prop osition Revenue equivalence Assume the SIPV framework and

U Then al l auctions that select the highest bidder as winner and that

have the same number of bidders give rise to the same expectedpayos of the

sel ler and bidders

Pro of Integrating one obtains using the fact that U

Z

v

b v dv U v

v must b e equal Since each of the admitted auctions is Pareto ecient b

to the probability that all other bidders valuations are b elow v Therefore by

each bidders payo is the same under all admitted auctions Moreover

the total surplus generated by trade must also b e the same due to the Pareto

optimality of the outcome Therefore the sellers payo must also b e the

same

As you can see from the ingredients to the pro of of this result revenue

equivalence applies not only to the four standard auctions but to all auctions

that select the highest bidder as winner In particular revenue equiv alence

also applies to third and higherprice auctions that are discussed b elow

However the pro of of revenue equivalence assumes the SIPV framework It

remains to b e seen whether all of the SIPV ingredients are crucial

Remark Assuming the SIPV framework revenue equivalence holds for al l

auctions that award the item to the highest bidder This prerequisite seems to

apply to most auction rules However it is already violated when the sel ler

sets a minimum priceabove his reservation value

The case of uniformly distributed valuations

Before wegointo the general account of equilibrium bids and then pro ceed

with various mo dications of the SIPV mo del take a lo ok at the following ex

tensive example in which it is particularly easy to compute equilibrium strate

gies and equilibrium outcomes

The overall surplus or gain from trade is equal to maxfE V r g The sellers

N

payo is the dierence b etween total surplus and bidders payos

Example Suppose al l bidders valuations are uniformly distributedonthe

interval as in example Then the unique equilibrium bidding strategies

are

b v v English auction

i i

v Dutch auction b v

i i

N

In both auctions the expected priceis

N

p

N

Therefore more bidders mean ercer competition between buyers

Wenow explain in detail how these results come ab out using the symmetry

of equilibrium and the monotonicity of equilibrium strategies

Dutch auction If bidder i bids the amount b b v he wins with



probability

b Pr fb V bg

N 

Pr fV bg

N 

N

F b

where bistheinverse of b v which indicates the valuation that leads to

i

bidding the amount b if strategy b is pla yed and V is the i th order statistic

i

of the sample of N random valuations

In equilibrium the bid b must b e a b estresp onse to rivals bids Therefore

b must b e a maximizer of the exp ected gain bv b leading to the rst

i

order condition

bv b b

Using the fact that v b v and inserting one can restate in the

following form

N f b b b b F b

b uniform distribution which simplies to due to F b

N b b b b



To b e careful the condition should read as follows bPrfb V bghowever

N

since the V s are continuous random variables there is no probability mass on single p oints

and therefore you can safely replace the strict bytheweak inequality

N



b Finally This dierential equation has the obvious solution b

N

solve this for b and you have the equilibrium bid strategy

b v v

N



as asserted Notice that the margin of prot that eachplayer allows himself

in his bid decreases as the numb er of bidders increases

Based on this result you can now determine the exp ected pricep N by

the following consideration In a Dutch auction the random price paid is equal

to the highest bid which can b e written as b V where V denotes the

N  N 

highest order statistic of the entire sample of N valuations Therefore bya



known result on order statistics

pN E b V

N 

N

N

E V since b v v

N 

N

N

N

N

as asserted

English auction Recall in an English auction each bidder bids his true



valuation The bidder with the highest valuation wins and pays a price

equal to the second highest order statistic V of the given sample of N

N 

valuations Therefore the exp ected price is

N

pN E V

N 

N



Whyisitobvious Supp ose the solution is linear then you will easily nd the solution

which also proves that there is a linear solution



This b egs the question whether the equilibrium is unique The pro of is straightforward

for N In this case one obtains from d db bd db Therefore



Since bonehas and thus bbv Hence the unique b



equilibrium strategy is b v v In order to generalize the pro of to N apply a



transformation of variables from btoz b where z b Then one can separate

variables and uniquely solve the dierential equation byintegration



Let V b e the r th order statistic of a given sample of N indep endent and identically

r

distributed random variables V with distribution function F v v on the supp ort

i

r N r

r

and Var V See rules Kendall S M and A Stuart Then E V



r r

N N N 

The AdvancedTheory of Statistics Vol I Distribution Theory Charles Grin

Co pp and p



Notice however that the English auction is plagued byamultiplicityproblemFor

example the strategy combination b v b v i N is also an equilibrium

i

even though it prescrib es rather pathological b ehavior However the equilibrium analyzed

in the main text is the only one that is not dominated

Comparison Since the two exp ected prices are the same this example

illustrates the general principle of revenue equivalency Notice however that

actual payos tend to dier even in their risk characteristics Indeed the

Dutch auction leads to a lower price risk Therefore risk averse sellers should

prefer the Dutchover the English auction

To see this clearly compute the variance of the random price P in the

English E and the Dutch D auction

Var P Var V

E

N 

N



N N

Var P Var b V

D

N 



N

Var V

N 



N



N



N N N

N N

you see immediately that the English auction gives rise Since

N N

to a higher variance of price as asserted

Third and higher price auctions Consider a generalization of the

secondprice auction the thirdprice auction where the highest bidder wins

but pays only as much as the third highest bid and more generallythek th

price auction where the winner pays the k th highest bid If you follow the

pro cedure used ab ove to derive the equilibrium bid function under the rst

price auction you will nd the following solution

N

b v v for NkN

N k

Third and higher price auctions have three striking prop erties bids

are higher than the own valuation equilibrium bids diminish as the number

N N

and the variance of the of bidders is increased since

N k N k 

random price P

k N

Var P Var b V

N k 

 

N k N N



Hints Supp ose the equilibrium bid function of the k price auction is linear in the

own valuation b v v Revenue equivalency applies also to the k price auction

k

Therefore the exp ected price must b e the same as under the English auction E V

N

E b V E V Compute the two exp ected values and you havethe

k

N k N k

asserted bid function Once you have a candidate for solution you only need to conrm

that it do es indeed satisfy the mutual b est resp onse requirement

increases in k

Once you have gured out whyitpays to sp eculate and bid higher than

the own valuation it is easy to interpret the second prop erty Just keep in mind

that a rational bidder may get burned and suer a loss b ecause the k th

highest bid is ab ovetheown valuation As the numb er of bidders is increased

it b ecomes more likely that the k th highest bid is in close vicinity to the

own valuation Therefore it makes sense to bid more conservatively when the

numb er of bidders is increased even though this may seem somewhat puzzling

at rst glance

Finally the third prop erty suggests that a risk averse seller should always

prefer lower order k price auctions and therefore should most prefer the Dutch

auction The English auction is always app ealing b ecause of its overwhelming

strategic simplicity Third and higherprice auctions are strategically just

as complicated as the Dutch auction and in addition exp ose the seller to

unnecessary price risk Therefore we conclude that third and higherprice

auctions are strictly dominated byDutch auctions

Nob o dy has ever to our knowledge applied a third or higherprice auc

tion So is this just an intellectual curiosity useful only to challenge your

intuition and technical skills

Exp erimental economists have exp osed inexp erienced sub jects to third

price seal bids and examined their resp onse to a higher numb er of bidders

Amazingly the ma jority of bidders reduced their bid just as the theory

recommends The authors take great pride in this result they takeitas

evidence that inexp erienced sub jects are not as unsophisticated as is often

susp ected by critics of the exp erimental approach The particular ironyisthat

if one asks trained theorists to make a guess they tend to come up with the

wrong hunch concerning the relationship b etween the numb er of bidders and

equilibrium bids until they actually sit down to conrm the computations

Charity or all pay auctions As a last example consider an auc

tion that is frequently used in charities whichiswhywe call it charity or

all pay auction for lack of a b etter name Its p eculiar feature is that every

bidder is required to actually pay his bid Just like in a standard auctions the

item is awarded to the highest bidder But unlike in standard auctions each

bidder must pay his bid even if he is not awarded the item

The charity or all pay auction sticks to the principle of awarding the item

to the highest bidder Therefore revenue equivalence applies But wewantto

conrm it directly and compute equilibrium bids as a further exercise



See Kagel J H and D Levin Indep endent private value auctions bidder

b ehavior in rst second and thirdprice auctions with varying numb ers of bidders

University of Houston Working Pap er

Using a pro cedure similar to the ab ove the equilibrium bid function is

N

N

b v v

N

This is in line with the observation that bidders often bid only nominal amounts

of money for relatively valuable items

In order to determine the exp ected price notice that the seller collects the

following exp ected payment from each bidder

Z

N N

N

v dv E b V

N N N

N

which Take the sum over all bidders and one obtains E Nb V

N

conrms the asserted revenue equivalence

The crucial feature of charity auctions is that they make winners and losers

pay We mention that this is also a feature of the not so charitable seller



optimal auction if bidders are risk averse

Generalization

In many nonco op erative games it is dicult if not imp ossible to nd a closed

form solution of the Nash equilibrium unless one works with particular func

tional sp ecications The SIPV auction game is an exception Indeed the

English and the Dutch auction games have a unique symmetric equilibrium

for all probability distributions of private valuations

Prop osition Equilibrium bidding rules Suppose the distribution func

tion F v is continuous Then the Dutch auction has a unique equilibrium

R

v

N

F v dv

r

b v v

N

F v

where r denotes the sel lers reserve price

Of course the English auction has several equilibria but bv v is the

only one that is not weakly dominated

Pro of The bidding rule for the English auction follows from the fact that

truthtelling is a dominant strategyToprove the bidding rule for the Dutch

The probability of winning is just like under the Dutch auction see eq However

since all bids must actually b e paid the equilibrium bid must b e a maximizer of bv b

hence it must solve the dierential equation v From these two building blo cks it is

easy to compute the equilibrium bid function



See Matthews S Selling to risk averse buyers with unobservable tastes Jour

nal of Economic Theory

auction you havetosolve the dierential equation The details are sp elled



out for example in Milgrom and Web er

Robustness

Assuming the SIPV framework with risk neutral agents wehave derived the

amazing result that al l auctions that award the item to the highest bidder give

rise to the same payos In other words if all participants are risk neutral

many dierent auction forms including the four standard auctions are

nothing but irrelevant institutional detail

This result is in striking contrast to the apparent p opularity of the English

auction Has the SIPV mo del missed something of crucial imp ortance or have

we missed something at work even in this framework

A strong argumentinfavor of the English auction is its strategic simplicity

In this auction bidders need not engage in complicated strategic considerations

Even an unsophisticated bidder should b e able to work out that setting the

limit equal to ones true valuation is the b est one can do Not so under the

Dutch auction This auction game has no dominant strategy equilibriumAsa

result understanding the game is conceptually more demanding not to sp eak

of computational problems

However if the auction cannot b e oral say b ecause it would b e to o costly

to bring bidders together at the same time and place there is a very simple yet

strong reason why one tends to use a rstprice closedseal bid even though

it is strategically much more complicated than the secondprice or Vickrey

auction A secondprice auction can usually b e manipulated by soliciting

phantom bids in close vicinity of the highest submitted bid This suggests

that secondprice closedseal bids should only b e observed if the seller is a

public agency that is not prot oriented

or the English There are several more go o d reasons why the seller should fav

auction But these come up only as we mo dify the SIPV mo del Wenow turn

to this task In addition wewillgive a few examples of auctions that do not

adhere to the principle of awarding the item to the highest bidder and on this

ground fail to b e payo equivalent to the four common auctions

Removing risk neutrality

The simplest variation of the SIPV mo del with risk averse agents is obtained

byintro ducing risk averse bidders Obviously this mo dication do es not aect

the equilibrium strategy under the English auction Therefore the exp ected

price in this auction is unaected as well



Milgrom P and R J Web er A theory of auctions and comp etitive bidding

Econometrica

However strategies and payos change under the Dutch auction In this

auction form bidders always takea chance and shade their bid b elow their



valuation Therefore the more risk averse a bidder the more reluctanthe

will b e to bid far b elow his true valuation and consequently risk averse bidders

tend to bid more conservatively As a result they raise the sellers payo and



alas reduce their own

Prop osition Consider the SIPV model with risk averse bidders The sel lers

payo is higher and the bidders payo lower under the Dutch than under the

English auction

Removing symmetry

The symmetry assumption is central to the revenue equivalence result For if

bidders are characterized by dierent probability distributions of valuations

under the Dutch auction and the equivalent rstprice seal bid it is no longer

assured that the ob ject is awarded to the bidder with the highest valuation

But precisely this prop ertywas needed in the pro of of Prop osition

Example Suppose thereare two bidders A and B characterizedbytheir

a and b b with a b Consider the random valuations with support a

Dutch auction Obviously B could always win and pocket a gain by bidding

the amount a But B can do even better by shadeing the bid further below

a But then the low valuation bidder A wins with positive probability violating

Paretooptimality

An even more imp ortant implication is that Pareto eciency is only assured

by secondprice bids If a public authority uses auctions as an allo cation

device eciency should b e the primary ob jective This suggests that public

authorities should adopt secondprice bids

A rigorous pro of of existence and uniqueness of the equilibrium solution



with heterogenous bidders is in Plum If you are interested in a careful

mathematical study of existence and uniqueness of auction games this contri

bution is a highly recommended source



Recall a bidder can always do b etter than bid his true valuation whichwould give

N

him zero payo In fact in example the equilibrium strategy was b v v which

N

implies shadeing the bid all the waydown to b v v if the numb er of bidders is N





For a formal pro of using sto chastic dominance see Maskin E and J G Riley

Auction theory with private values American Economic Review



Plum M Characterization and computation of Nashequilibria for auctions

with incomplete information International Journal of Game Theory

Intro ducing a minimum price or participation fee

Revenue equivalence is a fairly general prop erty of auction games It applies

to all auctions that adhere to the principle of awarding the item to the highest

bidder This restriction sounds pretty mild Is there any common auction

where it is violated And is ever desirable to deviate from it

Many seemingly inno cuous mo dications of standard auction rules deviate

from the principle of awarding the item to the highest bidder and therefore

violate revenue equivalence For example supp ose the seller sets a minimum

price ab ove his reservation price Then he deviates from this principle b ecause

he only sells to the highest bidder if that bid exceeds the minimum price Of

course as we indicated in our comparison of examples and such deviations

can b e protable for the seller Indeed the theory of optimal auctions tells us

that the seller should always set such a minimum price ab ove his reservation

price and thus leave the domain of revenue equivalence More ab out this in

the section on optimal auctions

Another mo dication of standard auctions that violates revenue equiva

lence o ccurs when a participation fee is added Adding such a fee to the

English auction has also an interesting eect on the numb er of bidders who

actually participate This makes two go o d reasons for briey elab orating on

this mo dication

Example Participation fee Suppose the sel ler requiresaparticipation

fee c from each bidder in an English auction Let therebe two bidders

with uniformly distributed valuations on the support

Then a bidder participates if and only if his valuation is at or above the

p

c The expectedrevenue maximizing entry feeisc and critical level v





the maximum expectedrevenue Therefore for N the participation fee



augmented English auction is moreprotable than the standard English auction

and than the Cournot monopoly approach see examples and

The pro of is sketched as follows In order to computev consider the

marginal bidder with valuationv His cost of participation cmust b e exactly

matched by the exp ected gain from bidding

Pr fwinninggv c

Therefore

p

c v

as asserted

Notice the seller collects the entry fee from b oth bidders if and only

if V v and in that case also collects a price equal to V Whereas if

 

V v V only one bidder participates the entry fee go es all the way

 

down to zero and the sellers only earning is the one entry fee paid the only

one active bidder Therefore the sellers exp ected prot is equal to

PrfV vg E V j V vc PrfV v V gc

    

Of course



Pr fV vg v



and

PrfV v V g v v

 

Hence by all of the ab ove

p p

cc c

solves the rst order Evidently is strictly concaveincand c



p

condition cwhich completes the pro of

Intro ducing numb ers uncertainty

In many auctions bidders are uncertain ab out the numb er of participants

numb ers uncertainty This seems almost comp elling in closedseal bids

where bidders do not convene in one lo cation but make bids each in their own

oce

Notice however that the seller can easily eliminate numb ers uncertainty

if he wishes to do so He only needs to solicit contingent bids where each

bidder makes a whole list of bids each contingent on a dierentnumber of

participating bidders Therefore the seller has a choice The question is is it

in the interest of the seller to leave bidders sub ject to numb ers uncertainty



McAfee Preston and McMillan explored this issue They showed that

if bidders are risk averse and have constant or decreasing absolute risk aver

sion numb ers uncertainty leads to more aggressive bidding in a rstprice

closedseal bid Since numb ers uncertaintyhasobviously no eect on bidding

strategies under the three other auction rules one can conclude from the rev

enue equivalence prop osition that numb ers uncertaintyfavors the rstprice

closedseal bid Incidentally this result is used in exp eriments to test whether



bidders are risk a verse



See McAfee R R Preston and J McMillan Auctions with a sto chastic number

of bidders Journal of Economic Theory



See Dyer D J Kagel and D Levin Resolving uncertaintyaboutthenumber

of bidders in indep endent privatevalue auctions an exp erimental analysis Rand Journal

of Economics

Endogenous quantity

On the impact of endogenous quantity on the ranking of the four common



auctions see Hansen Hansen shows that rstprice auctions lead to a higher

exp ected price therefore revenue equivalence breaks down in this case An

imp ortant application deals with industrial pro curementcontracts in private

industry and government Hansens result explains why pro curement is usually

in the form of a rstprice closedseal bid For a fuller analysis of this matter

see example on p b elow

Multiunit auctions

Supp ose the seller oers more than one unit of a go o d Then everything

generalizes quite easily as long as eachbuyer demands at most one unit But

revenue equivalence fails if the demand is price elastic

Consider the inelastic demand case First of all we need to generalize the

denition of rst and secondprice auctions Supp ose x units are put up for

sale In a secondprice auction a uniform price is set at sucha level that all

but x buyers drop out Each remaining bidder receives one unit and pays that

price In turn in a rst price auction the x highest bidders are awarded the

x items and they pay their own bid

Just like in the single unit case one can show that all auction rules are

revenue equivalent provided they adhere to the principle of awarding each



item to the highest bidder

However this do es not generalize to the case when bidders demand is price

elastic But in view of the previous section this should not come as muchof

a surprise

The theory of multiunit auctions is not very well develop ed Until recently

analysts of applied multiunit auctions av oided to mo del the auction as a non

co op erative game and instead analyzed bidders optimal strategy assuming a



given probability distribution of b eing awarded the demanded items How



ever Maskin and Riley have generalized the theory of optimal auctions to



include the multiunit case And Spindt and Stolz showed that as the num



Hansen R G Auctions with endogenous quantity Rand Journal of Eco

nomics



See Harris M and A Raviv A theory of monop oly pricing schemes with

demand uncertainty American Economic Review



See for example the analysis of treasury bill auctions by Scott J and C Wolf

The ecient diversication of bids in treasury bill auctions Review of Economics and

Statistics Treasury bill auctions are explained on page



Maskin E and J G Riley Optimal multiunit auctions in Hahn F ed

The Economics of Missing Markets Information and Games Clarendon Press



Spindt P A and R W Stolz The exp ected stopout price in a discriminatory

auction Economics Letters

b er of bidders or the quantity put up for auction is increased the exp ected

stopout price the lowest price served go es up which generalizes wellknown

prop erties of singleunit auctions

The most imp ortant applications of multiunit auctions are in nancial

markets For example the US Treasury sells marketable bills notes and

b onds in more than regular auctions p er year using a closedseal bid

multipleprice auction mechanism Some of the institutional details of the

government securities market are sketched in section b elow

Removing indep endence correlated b eliefs

Finally remove indep endence from the SIPV mo del and replace it by the as

sumption of a p ositive correlation b etween bidders valuations Lo osely sp eak

ing p ositive correlation means that a high own valuation makes it more likely

that rival bidders valuations are high as well

The main consequence of correlation is that it makes bidders more con

servativeina Dutch auction Of course correlation do es not aect bidding

in an English auction where truthtelling is always the dominant strategy

Therefore the intro duction of correlation reduces the exp ected price under

the Dutch auction but has no eect on the exp ected price under the English

auction In other words correlation entails that the seller should prefer the

English to the Dutch auction

Prop osition In the symmetric private values model with positively corre

lated valuations the sel lers payo is higher under the English than under the



Dutch auction and that of potential buyers is lower

The intro duction of correlation has a numb er of comparative static impli

cations with interesting public p olicy implications In particular the release

of information ab out the items v alue should raise bids in a Dutch auction In

cidentally this comparative static prop ertywas used by Kagel Harstadt and



Levin to test the predictions of auction theory in exp erimental settings

Auction rings

So far wehave analyzed auctions in a nonco op erative framework But what

if bidders collude and form an auction ring Surely bidders can gain a lot



The formal pro of is in Milgrom P and R J Web er A theory of auctions and

comp etitive bidding Econometrica



Kagel J R M Harstad and D Levin Information impact and allo cation rules

in auctions with aliated private values a lab oratory study Econometrica

by collusive agreements that exclude mutual comp etition But can they b e

exp ected to achieve a stable and reliable agreement And if so which auctions

are more susceptible to collusion than others

To rank dierent auctions in the face of collusion supp ose all p otential

bidders have come to a collusive agreement They have selected their desig

nated winner presumably the one with the highest valuation recommended

him to follow a particular strategy and committed others to abstain from bid

ding However supp ose the ring faces an enforcement problem b ecause ring

memb ers cannot b e prevented from breaching the agreement Therefore the

agreement has to b e selfenforcing Do es it pass this test at least in some

auctions

Consider the Dutch auction or rstprice seal bid Here the designated

winner will b e recommended to place a bid roughly equal to the sellers reserve

price whereas all other ring members are asked to abstain from bidding But

then each of those asked to abstain can gain by placing a slightly higher bid in

violation of the ring agreement Therefore the agreement is not selfenforcing

Not so under the English auction or secondprice closedseal bid Here

the designated bidder is recommended to bid up to his own valuation and

everyone else to abstain from bidding Now no one can gain by breaching the

agreement b ecause no one will ever exceed the designated bidders limit



Therefore

Prop osition Col lusive agreements between potential bidders are selfenfor



cing in an English but not in a Dutch auction

These results give a clear indication that the English auction or second

price closedseal bid is particularly susceptible to auction rings and that the

seller should cho ose a Dutch in lieu of an English auction if he has to deal with

an auction ring that is unable to enforce agreements

Even if auction rings can write enforceable agreements the ring faces the

problem of how to select the designated winner and a void strategic b ehavior

of ring memb ers This is usually done by running a preauction But can

one set it up in suchaway that it always selects the highest valuation ring

memb er

In a preauction every ring memb er is asked to place a bid and the high

est bidder is chosen as the rings sole bidder at the subsequent auction But

if the bid at the preauction only aects the chance of b ecoming designated



This p ointwas made by Robinson M Collusion and the choice of auction

Rand Journal of Economics



To b e more precise under an English auction the bidding game has two equilibria one

where everyb o dy sticks to the cartel agreement and one where it is breached The latter

can however b e ruled out on the ground of payo dominance

winner at no cost each ring member has every reason to exaggerate his val

uation Therefore the preauction problem can only b e solved if one makes

the designated winner share his alleged gain from trade



Graham and Marshall prop osed a simple scheme that resolves the pre

auction problem by an appropriate sidepayment arrangement Essentially

this scheme uses an English auction or secondprice closedseal bid to select

the designated winner If the winner of the preauction also wins the auction

he is required to pay the other ring memb ers the dierence b etween the price

paid and the second highest bid from the preauction bid This wayitis

assured that truthtelling is an optimal strategy at the preauction excluding

strategic b ehavior

Notice however that this solution of the preauction problem works only

if the ring can enforce the agreed up on sidepayments In the absence of

enforcementmechanisms auction rings are plagued by strategic manipulation

so that no stable ring agreementmay get o the ground

Finallywe mention that the seller mayght a susp ected auction ring by

pulling bids o the chandelier that is byintro ducing imaginary bids into

the pro ceedings In the New York City Department of Consumer Aairs

prop osed to outlaw imaginary bids after it had b ecome known that Christies

had rep orted the sale of several paintings that had in fact not b een sold This

prop osed regulation was strongly opp osed bymostNewYork based auction

houses precisely on the ground that it would deprive them of one of their most

p otentweap ons to ght rings of bidders

Optimal auctions

Among all p ossible auctions which one maximizes the sellers exp ected prot

At rst glance this question seems unmanageable There is a myriad of p ossi

ble auction rules limited only by ones imagination Therefore whatever your

orite auction rule howcanyou ever b e sure that someone will not come fav

along and nd a b etter one

On the other hand you may ask what is the issue Didnt weshowthat

virtually all auction rules are payo equivalent

Application of the revelation principle

The breakthrough that made the problem of optimal auction design tractable



is due to Myerson He observed that one may restrict attention without



Graham D A and R C Marshall Collusive bidder b ehavior at singleob ject

secondprice and English auctions Journal of Political Economy



Myerson R B Optimal auction design Mathematics of Operations Re

search and Myerson R B The basic theory of optimal auctions

loss of generality to incentive compatible direct auctions This fundamental



insight applies the famous revelation principlewhichisalsoduetoMyerson

that has b een useful in many areas of mo dern economics from the public go o d

problem to the theory of optimal lab or contracts

Example Auctions that award the item strictly to the highest bidder are

not optimal To show this consider the assumptions of example and let

N Byrevenue equivalence al l auctions that select the highest bidder as

winner arepayo equivalent to the Dutch and English auction Therefore the

expected priceisp Compare this to the Cournotsel ler from example



that sets the takeitorleaveit price p andearns the expected

price p p Evidently the Cournot approach is moreprotable The



immediate implication is that by imposing a minimum bid equal to p the Dutch

and English auction can be improved and be made even moreprotable than

the simple Cournot approach Of course by raising the minimum bid above

the sel lers reservation price the sale fails with positive probability Therefore

the hybrid auction violates Pareto optimality

Direct auctions An auction is called direct if each bidder is only asked to

rep ort his valuation to the seller and the auction rules select the winner and

bidders payments Closed sealbids are direct auctions English and Dutch

auctions are indirect

Incentive compatibility A direct auction is incentive compatible if honest

rep orting of valuations is a nonco op erative Nash equilibrium Many direct

auctions are not incentive compatible For example in a rstprice closedseal

bid every bidder bids less than his valuation Therefore rstprice seal bids

are direct but not incentive compatible In turn in a secondprice closedseal

bid truthtelling is a dominant strategy Therefore secondprice closedseal

e compatible bids are direct as well as incentiv

Since we consider optimality from the p oint of view of the seller incentive

compatibility requires only that buyers reveal their true valuations but we

do not require that the seller rep orts his true reservation price b efore bids

are solicited Again the secondprice closedseal bid is a go o d illustration

Obviouslyitisincentive compatible in the sense that all buyers rep ort their

true valuation But as example indicates it is not truthrevealing with

regard to the sellers reservation price since the seller would always quote a

minimum bid ab ove his true reservation price

in EngelbrechtWiggans R and M Shubik and J Stark Auctions Bidding and

ContractingNewYork University Press



Myerson R B and the bargaining problem Econo

metrica

Revelation principle The revelation principle says that for any equilibrium

of any auction game there exists an equivalent incentive compatible direct

auction Therefore the auction that is optimal among the incentive compatible

direct auctions is also optimal for all typ es of auctions

Toprove the revelation principle consider an equilibrium of some arbitrar

ily chosen auction game We will show that the following pro cedure describ es

an equivalent incentive compatible direct auction

Ask each bidder to rep ort his valuation

Compute each bidders optimal bidding strategy in the given equilibrium

of the assumed auction game

Select the winner and collect payments exactly as in the given equilibrium

of the assumed auction game

Evidently if all bidders are honest this direct auction is equivalent to the

given equilibrium of the assumed auction game Finally this direct auction is

also incentive compatible Because if it were protable for any bidder to lie

in the direct auction it would also b e protable for him to lie to himself in

executing his equilibrium strategy in the original auction game Its as simple

as that

Example Consider the Dutch auction equilibrium from example The

equivalent direct incentive compatible auction is describedbytheN probability

and expectedpayment functions

if v v j i

i j

v v

i N

otherwise

N

v if v v j i

i i j

N

E v v

i N

otherwise

Feasible direct auctions A direct auction is describ ed by a set of N

outcome functions E dened on the supp ort of valuations Thereby

i i

v v denotes the i th bidders probability of winning and E v v

i N i N

his exp ected payment notice the bidder mayhave to makea paymenteven



if he do es not win the auction Tobe feasible these outcome functions have

to satisfy the following three conditions



For example in the so called Al l Pay or each bidder makes an uncon

ditional payment and the item is awarded to the one who makes the highest paymentgo

back to example on page

First b ecause there is only one ob ject to allo cate the probabilities v v

i N

must sum to no more than for all valuations notice the sale may fail with

p ositive probability

N

X

v v budget constraint

i N

i

Second the direct auction has to b e incentive compatible That is if a

bidders true valuation is v rep orting the truth must b e at least as go o d as

i

rep orting any other valuationv v The utilityofanagent whose valuation

i i

is v but who rep orts the valuationv is

i i

U v j v E V v V v

i i i i N i

V v V EE

i N i

The V s are random variables v andv particular realizations Therefore

i i

incentive compatibility requires that for all v v v v

i i i

i

U v j v U v j v incentive compatibility

i i i i

Third participation must b e voluntary Therefore each bidder must b e

oered a nonnegative exp ected utility whatever his valuation

v participation constraint U v j v v v

i i i i

i

The programming problem In view of the revelation principle and the

ab ove feasibility considerations the optimal auction design problem is reduced

to the choice of N outcome functions E i N so that the sellers

i i

exp ected prot is maximized

N N

X X

EE V V E V V r max

i N i N

f E g

i i

i i

sub ject to the budget the incentive compatibility and the partici

pation constraints What lo oked like an unmanageable

problem has b een reduced to a straightforward optimal control problem

In the following we review and interpret the solution If you want to follow



the mathematical pro of you havetolookupMyersons b eautiful contribution



Myerson R B Optimal auction design Mathematics of Operations Research

Solution In order to describ e the solution the so called prioritylevels

asso ciated with a bid play the pivotal role Essentially these prioritylevels

resemble and play the same role as individual customers marginal revenue in

the analysis of monop olistic price discrimination

The prioritylevel asso ciated with the bid v is dened as

i

F v

i i

v v

i i i

f v

i i



We assume that priority levels are monotone increasing in the v s

i

Going back to the Cournot monop oly problem under incomplete informa

tion reviewed in the intro duction see eq v can b e interpreted as

i i

the marginal revenue from oering the item for sale to buyer i at a takeit

orleaveit price equal to p v Therefore the monotonicity assumption

i

means that marginal revenue is diminishing in quantity where quantityis



here the probability of sale Of course the priority level is always less than

the underlying valuation v

i

With these preliminaries the optimal auction is summarized by the follow

ing rules The rst set of rules describ es how one should pick the winner and

the second howmuch the winner shall pay

Selection of winner

Ask each bidder to rep ort his valuation and compute the asso ciated pri

ority levels

Award the item to the bidder with the highest prioritylevel unless it is

b elow the reservation price r

Keep the item if all prioritylevels are b elow the reservation price r even

though the highest valuation may exceed r

Pricing rule To describ e the optimal pricing rule we need to generalize

the secondprice auction rule Supp ose i has the valuation v which leads to

i

the highest prioritylevel v r Now ask byhowmuch could i have

i i

F v

i i



is the inverse of the socalled hazardrate Therefore the assumed Notice

f v

i i

monotonicity of prioritylevels is satised whenever the hazard rate is not decreasing A suf

cientconditionisobviously the logconcavity of the complementary distribution function

F v F v F is called logconcave if ln F is concave This condition holds for

example if the distribution function F is uniform normal logistic chisquared exp onen

tial Laplace See Bagnoli M und T Bergstrom Logconcave probability and its

applications UniversityofMichigan Working Pap er



Myerson also covers the case when prioritylevels are not monotone increasing

rep orted a lower valuation and would still havewon the auction under the

ab ove rules

Let z b e the lowest rep orted valuation that would still lead to winning

i

z minfv j v rv v j ig

i i i j j

Then the optimal auction rule says that the winner shall pay z This com

i

pletes the description of the auction rule

Interpretation Essentially the optimal auction rule combines the idea

of a secondprice or Vickrey auction with that of third degree monop



olistic price discrimination As a rst step customers are ranked by their

F v 

i i

evaluated at the rep orted valuation Since only marginal revenue v

i

f v 

i

one indivisible unit is up for sale only one customer can win The optimal

rule selects the customer who ranks highest in the marginal revenue hierar

chy unless it falls short of the sellers reservation price r In this sense the

optimal selection of winner rule employs the techniques of third degree price

discrimination

But the winner is neither asked to pay his marginal revenue nor his rep orted

valuation Instead the optimal price is equal to the lowest valuation that

would still make the winners marginal revenue equal or higher than that of

all rivals In this particular sense the optimal auction rule employs the idea

of a secondprice auction

Illustrations As an illustration take a lo ok at the following two examples

The rst one shows that the optimal auction coincides with the secondprice

closedseal bid or English auction supplemented by a minimum price ab ove

the sellers reservation price if customers are indistinguishable so that there is

no basis for third degree price discrimination The second example then brings

out the discrimination asp ect assuming that bidders are viewed as dierent

In b oth examples the seller deviates from the principle of selling to the

highest bidder which underlies revenue equivalence and the equilibrium out

come fails to b e Pareto optimal with p ositive probability This indicates that

the four standard auctions are not optimal However the optimal auction it

self p oses a credibility problem that will b e discussed towards the end of this

section



Recall in the theory of Monopoly one distinguishes three kinds of price discrimination

p erfect or rst degree price discrimination imp erfect or seconddegree price discrimination by

means of selfselection devices oering dierent pricequantitypackages and nally third

degree price discrimination based on dierent signals ab out demand for example dierent

national markets

Example Identical bidders SIPV mo del Consider the independent

private values model assume r and let each bidders valuation be uni

formly distributed on the support Then the priority levels are

v

i

v v v

i i i i

By the above optimal auction rules the item is sold if and only if the highest

r r

The winner pays either or the second highest valuation exceeds

   

bid whichever is higher The optimal auction is thus equivalent to a modied

secondprice closedseal bid where the sel ler sets a minimum priceequal to

r r

Because the minimum price is higher than the reservation price

   

r the sel ler must sometimes keep the object even if some buyers valuation

exceeds the reservation price This inecient outcome occurs with probability

r

N

Therefore the optimal auction does not assurePareto optimality



If thereare only two bidders and the reservation priceiszero r as

in many of our examples the optimal minimum price is and the expected





Compare this to the Cournot approach the English revenue is equal to

auction and the participation fee augmented English auction see examples

Example Diering bidders Assume thereare only two bidders A and

B and suppose r Both bidders valuation is a uniform random variable

but the supports dier Let As support be the and B s the interval

Then their priority levels are

v v

A A A

v v

B B B

Obviously if the two happen to have the same valuation bidder A wins In

fact bidder B can only win if his valuation exceeds that of A by The

optimal auction rule discriminates against bidder B it thus handicaps him

as a way to encourage him to bid more aggressively For if bidder B were not

discriminate d against he would never report a valuation greater than and

hence always pay less than in case of winning Whereas under the optimal

auction rule he may have to pay up to to win

Illustration two bidders and twovaluations

This brief intro duction to optimal auctions stressed essential prop erties and

left out pro ofs and computations This mayleaveyou somewhat unsatised



The exp ected revenue is Pr fV gE V j V PrfV V g with



   



and PrfV g PrfV V g E V j V



     

Therefore wenow add a full scale example with two identical bidders and



twovaluations Its main purp ose is to exemplify the computation of optimal

auctions in a particularly simple case where the price discrimination asp ect of

optimal auctions cannot playany role

Supp ose the two bidders are identical in the sense that they have either a

lowv or a high valuation v These valuations o ccur with the probabilities

l h

PrfV v g PrfV v g

l h

A direct auction is completely describ ed by the probabilities of obtaining

the item and bidders exp ected payments E E

h l h l

Optimization problem The optimal auction maximizes the exp ected gain

from each bidder

max G E E

h l

E E

h l h l

y sub ject to the incentive compatibilit

v E v E

h h h l h l

v E v E

l l l h l h

and participation constraints

v E

h h h

v E

l l l

An immediate implication of incentive compatibility is the monotonicity prop

erty

E E

h l h l

Restrictions due to symmetry Having stated the problem wenowmake

it more user friendly The key observation is that symmetry implies three

further restrictions on the probabilities

h l

From the sellers p ersp ectiveeach bidder wins the item with probability

in a symmetri c equilibrium this probability cannot exceed

h l



Therefore

h l

Also in a symmetric equilibrium typ e h must lose with probability or



more whenever he faces a typ e h rival Therefore or

h



equivalently

h



The example is b orrowed from K Binmore Fun and Games A Text on Game

Theory D C Heath and Company pp

or more when he faces a And similarlytyp e l must lose with probability



or equivalently typ e l rival

l



l

Combining these restrictions the set of feasible probabilities is illus

h l

trated in Figure Similarly the set of feasible exp ected payments E E is

h l

illustrated in Figure In b oth diagrams the dashed lines represent the sellers

indierence curves

Why some constraints do not bind As you can see immediately from

Figure only two constraints are binding the truthtelling condition con

cerning typ e h and the participation constraint concerning typ e l Therefore

you can ignore inequalities and replace the inequalities

by equalities

User friendly optimization problem Using these results you can now

eliminate the exp ected paymentvariables and nd the optimal auction as the

solution of the following linear programming problem

max G v v

h h l l l

h l

sub ject to

The only variables are

h l

Solution The solution is easily characterized bythetwo diagrams Takea

tly lo ok at the sellers indierence curves the dashed lines in Figure Eviden

the marginal rate of substitution can b e p ositive or negative dep ending on

the prior probability In particular if typ e l is suciently likely

v v

h l

the marginal rate of substitution is negative the dashed indierence

v

h

v v

h l

curve slop es downwards whereas if it is suciently unlikely the

v

h

marginal rate of substitution is p ositive the dashed indierence curveslopes

upwards Therefore the solution is at one of two corners of the set of feasible

s dep ending up on the size of the prior probability as summarized in Table

Discussion endogenizing bidder participation

We close this intro duction to optimal auctions with a slightly technical note on

an interesting side issue that mayhave already plagued some readers This has

to do with the need to endogenize the numb er of bidders and the robustness

of results with regard to a more meaningful handling of bidder participation

E

l



E E v

l h l h l

E E v

l h h h l

v

l h

v

l l

E E

h l

E

h

v v

h l h h

Figure Optimal auction optimal E s

l



h l



v v G

h h l l l



h



Figure Optimal auction optimal s

E E

l l h h

v v

h l

v

h

v  

h

v v

h l

v v v

l h l

v    

h

Table Optimal auction

Recall in the ab ove optimal auction program it was assumed that all p o

tential bidders do actually participate whenever they can b e sure that they do

not suer a loss from it This seems p erfectly inno cuous But it has the unap

p ealing implication that bidders participate even if they are sure to go empty

handed For example in the symmetric case the optimal auction entailed a

minimum price ab ove the sellers reservation value When that minimum price

is announced all p otential bidders with valuations b elow this minimum price

can b e sure that bidding is absolutely p ointless for them Therefore it would

make all the sense in the world to assume that they withdraw from the auction

and do not participate contrary to what is assumed in the ab ove optimal



auction mo del This waythenumb er of active bidders b ecomes endogenous

And the question comes up do es this endogenizing of the numb er of active

bidders have an impact up on the optimal reserve price that is not accounted

for in the alleged optimal auction

Remarkably the answer is no In the SIPV framework the optimal mini

mum price is independent of the number of bidders

Prop osition Consider a secondprice auction with a minimum priceator

above the sel lers reservation price Then the optimal minimum price p is

independent of the number of bidders it is implicitly dened as the solution of

F p

p

f p

if F is the uniform distribution with support which gives p



Pro of The two order statistics V V have the following joint density

N  N 



This withdrawal would necessarily o ccur if bidding is costlynomatterhow small the

cost Of course in the limiting case of costless bidding it is also part of an equilibrium

conguration to haveeveryb o dy participate even those who are sure to havenochance of

winning



function

N 

N N F x f xf y for y x

g x y

otherwise

Let p denote the minimum price In an English auction the item is sold i

V p and the actual price paid is equal to V i V p and equal

N  N  N 

to p i V p V Therefore the optimal minimum price p solves

N  N 

Z Z Z Z

v y v p

g x y dxdy max xgx y dxdy p

p

p p p

The asso ciated rstorder condition is

Z Z Z

 

p v p

g x p dx p g x y dxdy



p



Its lefthandside LHS can b e rearranged as follows

Z



p

N 

N F x f xdx LHS p Nfp

N

p Nfp F p

Similarly its righthandside RHS is

Z Z



v p

N 

RHS N f y N F x f xdxdy



p

N

NFp F p

Therefore condition simplies to

p f p F p

We conclude that p is indep endentof N for all N as asserted In

particular p for all N ifvaluations are uniformly distributed on

Therefore even if we correct the usual optimal auction program and take

into account that raising the reserve price has an adverse eect on the exp ected

numb er of bidders the optimal reserve price will remain the same



The joint density function of the two order statistics V V rs N is

r s

N

r sr N s

f x y F x f xF y F x f y F y

rs

r s r N s

if x y and equal to zero otherwise See David H A Order Statistics Wileyp

Set r N s N and you have the asserted joint density function

d

 N N 

Notice F x N F x f x

dx

Common value auctions and the winners

curse



In a famous exp eriment Bazerman and Samuelson lled jars with coins and

auctioned them o to MBA students at Boston universityEach jar had a value

of whichwas not made known to bidders The auction was conducted as a

rstprice closedseal bid A large number of identical runs were p erformed

Altogether the average bid was however the average winning bid was

Therefore the average winner suered a loss of alas the winners

were the ones who lost wealth winners curse

What makes this auction dierent is just one crucial feature the ob ject

for sale has an unknown common value rather than known private values The

result is typical for exp eriments So what drives the

winners curse and how can bidders avoid falling prey to it

Most auctions involve some common value element Even if bidders at

an purchase primarily for their own pleasure they are usually also

concerned ab out the eventual resale And even though construction companies

tend to haveprivate values for example due to dierent capacity utilizations

construction costs are also aected by common events such as unpredictable

y of sp ecic tasks and random input weather conditions the unknown dicult

quality or factor prices This suggests that a satisfactory theory of auctions

should cover private as well as unknown common value comp onents

For simplicitywenow fo cus on the other extreme the pure common value

auction In a pure common value auction the item for sale has the same value

for each bidder just like the jar in the ab ove exp eriment contains the same

numb er of coins for each bidder At the time of the bidding this common

value is unknown Bidders mayhave some imp erfect estimate everyone has

his own rule of thumb to guess the numb er of coins in the jar but the items

true value is only observed after the auction has taken place

Tosketch the cause of the winners curse supp ose all bidders obtain an

unbiased estimate of the items value Also assume bids are an increasing

function of this estimate Then the auction will select the one bidder as

winner who received the most optimistic estimate But this entails that the

average winning estimate is higher than the items value

To play the auction right this adverse selection bias must already b e ac

counted for at the bidding stage by shadeing the bid Failure to follow this

advice will result in winning bids that earn less than average prots or even

losses



Bazerman M and W Samuelson I won the auction but dont want the prize

Journal of Conict Resolution



A simple framework Consider the following simplifying assumptions

A The common value V is drawn randomly from a uniform distribution

on the domain v v

A Before bidding each bidder receives a private signal S drawn randomly

i

from a uniform distribution on V V The unknown common

value determines the lo cation of the signals supp ort A lower indicates

greater signal precision



A The auction is rstprice like a Dutch auction or a closedseal bid

As an illustration you may think of oil companies interested in the drilling

rights to a particular site that is worth the same to all bidders Each bidder

obtains an estimate of the sites value from its exp erts and then uses this

information in making a bid

Computing the right exp ected value Just like in the private values

framework each bidder has to determine the items exp ected value and then

strategically shade his bid taking a b et on rival bidders valuations With

out shadeing the bid there is no chance to gain from bidding The dierence

is however that it is a bit more tricky to determine the right exp ected value

in particular since this computation should already account for the builtin

adverse selection bias

For signal values from the interval s v v the expected value of

i



the item conditional on the signal s is

i

s s E V jS

i i i

This estimate is unbiased in the sense that the average estimate is equal to the

sites true value

Nevertheless bids should not b e based on this exp ected value Instead a

bidder should anticipate that he would revise his estimate whenever he actually



Assumptions A and A were widely used by John Kagel and Dan Levin in their

various exp eriments on common value auctions See for example Kagel J H and D Levin

The winners curse and public information in common value auctions American

Economic Review



Secondprice common value auctions are easier to analyze but less instructive A nice

and simple pro of of their equilibrium prop erties based entirely on the principle of iterated

dominance is in Harstad R M and D Levin A class of dominance solvable

commonvalue auctions Review of Economic Studies The pro cedure prop osed

in this pap er requires however that the highest signal is a sucient statistic of the entire

signal vector



Here and elsewhere we ignore signals near corners that is v v v and v

v v

wins the auction As in many other contexts the clue to rational b ehavior is

in thinking one step ahead

In a symmetric equilibrium a bidder wins the auction if he actually received



the highest signal Therefore when a bidder learns that he won the auction

he knows that his signal s was the largest received by all bidders Using this

i

information he should value the item by its exp ected value conditional on hav

ing the highest signal denoted by E V jS s S maxfS S g

max i max N

Evidentlythe updated expectedvalueis lower

N

E V jS s s

max i i

N

E V jS s

i i

Essentially a bidder should realize that if he wins it is likely that the signal

he received was unusually high relative to those received byrival bidders

Notice the adverse selection bias measured by the dierence b etween the

two exp ected values is increasing in N and Therefore raising the number of

bidders or lowering the precision of signals gives rise to a higher winners curse

if bidder are sub ject to judgmental failure and base their bid on E V jS s

i i

rather than on E V jS s

max i

Equilibrium bids The symmetric Nash equilibrium of the common value



auction game was found by Wilson and later generalized by Milgrom and



Web er to cover auctions with a combination of private and common value

elements



In the present framework the symmetric equilibrium bid function is

s bs s

i i i

where

N

 s v

i



s e

i

N

The term s diminishes rapidly as s increases b eyond v Ignoring

i i

it the bid function is approximately equal to bs s and the exp ected

i i

prot of the high bidder is p ositive and equal to N



This standard prop erty follows from the monotonicity of the equilibrium bid function

in the present context it is conrmed by the equilibrium bid function stated b elow



Wilson R B A bidding mo del of p erfect comp etition Review of Economic

Studies



Milgrom P and R J Web er A theory of auctions and comp etitive bidding

Econometrica



Again we restrict the analysis to signal values from the interval s v v

i

Conclusions The main lesson to b e learned from this intro duction to com

mon value auctions is that bidders should shade their bids for twodierent

reasons First b ecause without shadeing bids there can b e no prots in a rst

price auction Second b ecause the auction always selects the one bidder as

winner who received the most optimistic estimate of the items value and thus

induces an adverse selection bias Without shadeing the bid to accountfor

this adverse selection bias the winning bidder regrets his bid and falls prey to

the winners curse

The discount asso ciated with the rst reason for shadeing bids will decrease

with the numb er of bidders b ecause signal values are more congested when

there are more bidders In turn the discount asso ciated with the adverse selec

tion bias increases with the numb er of bidders b ecause the adverse selection

bias b ecomes more severe as the numb er of bidders is increased

Altogether increasing the numberofbiddershastwo conicting eects on

equilibrium bids On the one hand comp etitive considerations require more

aggressive bidding On the other hand accounting for the adverse selection

bias requires greater discounts For lownumb ers of bidders the comp etitive

eect prevails and the bid function is increasing in N but eventually the

adverse selection eect takes over and the equilibrium bid function decreases

in N

Altogether it is clear that common value auctions are more dicult to play

and that unsophisticated bidders may b e susceptible to the winners curse Of

course the winners curse cannot o ccur if bidders are rational and prop erly

account for the adverse selection bias The winners curse is strictly a matter

of judgmental failure rational bidders do not fall prey to it

The exp erimental evidence demonstrates that it is often dicult to avoid

the winners curse Even exp erienced sub jects who are given plenty of time

and opp ortunity to learn often fail to bid b elow the up dated exp ected value

umb er of bidders And most sub jects fail to bid more conservatively when the n

is increased

Similarly many real life decision problems are plagued by p ersistentwin



ners curse eects For example in the eld of b o ok publishing Dessauer

rep orts that most of the auctioned b o oks are not earning their advances



Cap en Clapp and Campb ell claim that the winners curse is resp onsible for

the low prots of oil and gas corp orations on drilling rights in the Gulf of Mex



ico during the s And Bhagat Shleifer and Vishny observe that most of



See Dessauer J P Book PublishingBowker Publ



See Cap en E C and R V Clapp and W M Campb ell Comp etitive bidding

in highrisk situations Journal of Petroleum Technology



See Bhagat S and A Shleifer and R W Vishny Hostile takeovers in the s

the return to corp orate sp ecialization Brookings Papers on Economic Activities Sp ecial

Issue on Micro economics

the ma jor corp orate takeovers that made the nancial news during the s



have tended to actually reduce bidding shareholders wealth

Further applications

We close with a few remarks on the use of the auction selling mechanism

in nancial markets in dealing with the natural monop oly problem and in

oligop oly theoryEach application p oses some unique problem This indicates

that auction theory is still a rich mine of unresolved research problems

Auctions and oligop oly

Recall the analysis of price comp etition and the Bertrand paradoxtwo is

enough for competition in the theory of oligop oly There one usually elab o

rates on various prop osed resolutions of the Bertrand paradoxthattypically

have to do with capacity constrained price comp etition and that culminate

in a defense of the Cournot mo del This resolution is successful though a bit

complicated but unfortunately not quite robust with regard to the assumed



rationing rule

Amuch simpler resolution of the Bertrand paradox can b e found by in

tro ducing incomplete information This explanation requires a marriage of

oligop oly and auction theorywhichiswhywesketch it here in two examples

Example Another resolution of the Bertrand paradox Consider a

simple Bertrand oligopoly game with inelastic demand and N identical

rms Each rm has constant unit costs c and unlimitedcapacity Unlike in

the standard Bertrand model each rm knows only its own cost but not those

of others Rivals unit costs are viewed as identical and independent random

variables described by a uniform distribution with the support

Since the lowest price wins the market the Bertrand game is a Dutch auc

tion with the understanding that we aredealing here with a buyers auction

in lieu of the sel lers auctionconsideredbefore Sincecosts are indepen

dent and identical ly distributed we can employ the apparatus and results of

the SIPV auction model Strategies are price functions that map each rms

unit cost c into a unit price pc



See also Roll R The hubris hyp othesis of corp orate takeovers Journal of

Business



See the ingenious defense of the Cournot mo del by Kreps D and J A Scheinkman

Quantity precommitment and Bertrand comp etition yield Cournot outcomes Bel l

Journal of Economics

The unique equilibrium pricestrategy is characterized by the markup rule

N

p c c

N N



The equilibrium priceisp C the equilibrium expectedprice



pN E p C E C

 

N

and each rms equilibrium expectedprot

N

c

c N

N

Evidently morecompetition leads to more aggressive pricing beginning

with the monopoly price p and approaching the competitive pricing

rule p c as the number of oligopolists becomes very large Similarly the

expected price rule and prot are diminishing in N beginning with p

cwithlim pN and lim N Hence numbers

N N

matter Two is not enough for competition

In order to prove these results all we need to do is to use a transformation

of random variables and then apply previous results on Dutch auctions For

this purp ose dene

b p and v c

this transformation maps the price comp etition game into a Dutch Obviously

auction where the highest bid b the lowest p wins and b v since

N

p c By example weknow that bv v Therefore

N

p b

N

v

N

N

c

N

N

c

N N

as asserted



Notice E p C E C is the familiar revenue equivalence whichholdsforall



distribution functions If the market were second price eachrmwould set p c c

the lowest price would win and trade would o ccur at the second lowest price therefore the

exp ected price at which trade o ccurs would b e equal to E C



Remark Whynumb ers matter By revenue equivalence one has

pN E p C E C

 

Therefore the two is enough for competitionproperty of the Bertrand para

dox survives in terms of expected values Stil l numbers matter essential ly

because E C and E C move closer together as the size of the sample N

 

is increased The gap between these statistics determines the aggressiveness of

pricing because the price function has the form p c cwith

E C

N



N E C



Example Extension to price elastic demand Suppose demand is

adecreasing function of price as is usual ly assumed in oligopoly theory Specif

ical ly assume the simplest possible inverse demand function P X X

Otherwise maintain the assumptions of the previous example Then the equi

librium is characterized by the fol lowing markup rule p

Nc

p c

N

Obviously p and p

N

Compare this to the equilibrium markup rule in the inelastic demand

N c

framework analyzed in the previous example which was p c Ev

N

idently pricing is more aggressive if demand responds to price Of course if

the auction weresecondprice bidding would not beaected since p cc

is a dominant strategy in this case

We conclude if demand is a decreasing function of price a rstprice

auction leads to higher expectedconsumer surplus This may explain why

rstpriceclosedseal bids are the common auction form in industrial and

government procurement

This is just the b eginning of a promising marriage of oligop oly and auction

theory Among the manyinteresting extensions an exciting issue concerns

the analysis of the rep eated play Bertrand game The latter is particularly

interesting b ecause it leads to a dynamic price theory where agents succes

sively learn ab out rivals costs whichmakes the information structure itself

endogenous

Natural gas and electric p ower auctions

As a next example recall the natural monop oly problem In many practical

applications it has often turned out that the natural monop oly characteristic

applies only to a small part of an industrys activity In these cases vertical

disintegration may help to reduce the natural monop oly problem to a mini

mum

The public utility industry gas and electric p ower has traditionally b een

viewed as a prime example of natural monop oly However the presence of

economies of scale in the pro duction of energy has always b een contested An

unambiguous natural monop oly exists only in the transp ortation and lo cal

distribution of energy This suggests that the natural monop oly problem in

public utilities can b est b e handled when one disintegrates pro duction trans

p ortation and lo cal distribution of energy

In the UK the electric p ower industry has b een reshap ed in recentyears

by the privatization of pro duction combined with the intro duction of electric

power auctions These auctions are run on a daily basis by the National Grid

CompanyAt am every day the suppliers of electric p ower makea bidfor

each of their generators to b e op erated on the following day By pm the

National Grid Company has nalized a plan of action for the following dayin

the form of merit order ranked by bids from low to high Dep ending up on

the random demand on the following day the sp ot price of electric p ower is

then determined in suchawaythatdemandismatched by supply according

to the merit order determined on the previous day

Similar institutional innovations have b een explored dealing with the trans

p ortation of natural gas in longdistance pip elines and in the allo cation of

airp ort landing rights in the US Interestingly these innovations are often



aluated in lab oratory exp eriments b efore they are put to a real life test ev

Treasury bill auctions

Eachweek the US Treasury uses a discriminatory auction to sell Treasury



bills Tbills On TuesdaytheTreasury announces the amountofday

and day bills it wishes to sell on the following Monday and invites bids

for sp ecied quantities On Thursday the bills are issued to the successful

bidders Altogether in scal year the Treasury sold over trillion of

marketable Treasury securities bills notes and b onds

Prior to the early s the most common metho d of selling Tbills was

that of a subscription oering The Treasury xed an interest rate on the

securities to b e sold and then sold them at a xed price A ma jor deciency



See for example Rassenti S J and V L Smith and R L Buln A combinatorial

auction mechanism for airp ort time slot allo cation Bel l Journal of Economics

and McCab e K and S Rassenti and V Smith Designing smart computerassisted

markets an exp erimental auction for gas networks Journal of Political Economy



For more details see Tucker J F Buying Treasury securities at Federal Reserve

banks Federal Reserve Bank of Richmond Feb

of this metho d of sale was that market yields could change b etween the an

nouncement and the deadline for subscriptions

The increased market volatility in the s made xedprice oerings

to o risky for the Treasury Subsequently the Treasury switched to an auction

technique in which the coup on rate was still preset bytheTreasury and bids

were made on the basis of price The remaining problem with this metho d was

that presetting the coup on rate still required forecasting interest rates with

the risk that the auction price could deviate substantially from the par value

of the securities

In the Treasury switched to auction coup on issues on a yield basis

Thereby bids were accepted on the basis of an annual p ercentage yield with

the coup on rate based the weighted average yield of accepted comp etitive

tenders received in the auction This freed the Treasury from having to preset

the coup on rate

Another sale metho d was used in several auction of longterm b onds in



early s This was the closedseal bid uniformprice auction metho d

Here the coup on rate was preset bytheTreasury and bids were accepted in

terms of price All successful bidders were awarded securities at the lowest

price of accepted bids

In the currently used auction technique two kinds of bids can b e submitted

competitive and noncompetitive Comp etitive bidders are typically nancial

intermediaries who buy large quantities A comp etitive bidder indicates the

numb er of bills he wishes to buy and the price he is willing to pay Multiple

bids are p ermitted Noncomp etitive bidders are typically small or inexp eri

enced bidders Their bids indicate the numb er of bills they wish to purchase

up to with the understanding that the price will b e equal to the

quantityweighted average of all accepted comp etitivebids

When all bids have b een made the Treasury sets aside the bills requested

by noncomp etitive bidders The remainder is allo cated among the comp etitive

bidders b eginning with the highest bidder until the total quantityisplaced

The price to b e paid by noncomp etitive bidders can then b e calculated

Tbill auctions are unique discriminatory auctions b ecause of the distinct

treatment of comp etitive and noncomp etitive bids Compared to the standard

discriminatory auction there is an additional element of uncertainty b ecause

comp etitive bidders do not know the exact amount of bills auctioned to them

Another distinct feature is the presence of a secondary afterauction mar

ket and of preauction trading on a whenissued basis which serves a

pricediscovering purp ose and where dealers typically engage in shortsales



In the nancial community a singleprice multipleunit auction is often called Dutch

except by the English who call it American Be aware of the fact that in the academic

literature one would never call it Dutch Because of its secondprice qualityonewould

p erhaps call it English

In manycountries central banks use similar discriminatory auctions to sell



short term repurchase agreements to provide banks with short term liquidity

Several years ago the German Bundesbank switched from a uniformprice

auction to a multipleprice discriminatory auction The uniformprice auction

had induced small banks to place very high bids b ecause this gave them sure

access to liquidity without the risk of having a signicant impact on the price to

b e paid Subsequently large banks complained that this pro cedure put them

at a comp etitive disadvantage and the Bundesbank resp onded by switching to

a discriminatory auction However this problem could also have b een solved

without changing auction pro cedures simply byintro ducing a ner grid of

feasible bids

In recentyears the currently used discriminatory auction pro cedures b e

came the sub ject of considerable public debate Many observers claimed that

discriminatory auctions invite strategic manipulations and p erhaps paradoxi

cally lead to unnecessarily low revenues Based on these observations several

prominent economists prop osed to replace the discriminatory by uniformprice

auction pro cedures

The recent p olicy debate was triggered by an inquiry into illegal bidding

practices by one of the ma jor security dealers Salomon Brothers Apparently

during the early s Salomon Brothers rep eatedly succeeded to corner

the market by buying up to of a security issue This was in violation

of US regulations that do not allow a bidder to acquire more than



of an issue Such cornering of the market tends to b e protable b ecause

many security dealers engage in shortsales during the time when an issue is

es them vulnerable to announced and the time it is actually issued This mak

a short squeeze

Typically a short squeeze develops during the whenissued p erio d b efore

a security is auctioned and settled During this time dealers already sell the

so ontob eavailable securities and thus incur an obligation to deliver at the

issue date Of course dealers must later cover this p osition either by buying

back the security at some p oint in the whenissued market or in the auction

or in the p ostauction secondary market or any combination of these If those

dealers who are short do not bid aggressively enough in the auction they may

have diculties to cover their p ositions in the secondary market



The pro cedures used bytheFederal Reserve are describ ed in The Federal Reserve Sys

tem Purposes and Functions Board of Governors of the Federal Reserve System Wash

ington DC



Salomon Brothers circumvented the lawby placing unauthorized bids in the names of

customers and employees at several Treasury auctions When these practices leaked the

Treasury security market suered a substantial loss of condence For a detailed assess

ment of this crisis and some recommended p olicy changes consult the Joint Report on the

Government Securities Market U S GovernmentPrinting Oce Washington DC

Are discriminatory auctions the b est choice Or should central banks and

the Treasury go back to singleprice auction pro cedures This is still an ex

citing and imp ortant research issue Already during the early s Milton



Friedman made a strong case in favor of a uniformprice closedseal bid

He asserted that this would end cornering attempts by eliminating gains from

market manipulation And p erhaps paradoxically he also claimed that total

revenue would go up by surrendering the p ossibility to pricediscriminate Es

sentially b oth claims are based on the exp ectation that the switch in auction

rules would completely unify the primary and secondary markets and induce



bidders to reveal their true willingness to pay

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O Ashenfelter How auctions work for wine and art Journal of Economic

Perspectives

M Bagnoli and T Bergstrom Logconcave probability and its applica

tions Working pap er UniversityofMichigan

M H Bazerman and W F Samuelson I won the auction but dont want

the prize Journal of Conict Resolution

S Bhagat A Shleifer and R W Vishny Hostile takeovers in the s

the return to corp orate sp ecialization Brookings Papers on Economic

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K Binmore Fun and Games A Text on Game Theory D C Heath and

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W J Boyes and S K Happ el Auctions as an allo cation mechanism in

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See Friedman M Comment on collusion in the auction market for Treasury

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How to sell government securities Wal l Street Journal August



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N F Brady R C Breeden and A Greenspan Joint Report on the

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