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AUCTIONS

An Introduction�

Elmar Wolfstettery

April ��Humboldt�Universit�at zu BerlinInstitut f� Wirtschaftstheorie I

Wirtschaftswissenschaftliche Fakult�atSpandauerstr� �� BerlinGermany

e�mail� wolf�wiwi�hu�berlin�de

Abstract

This is a fairly detailed review of auction theory� It begins withbasic results on private value auctions with particular emphasis on thegenerality and limitations of the revenue equivalence of a large class ofdistinct auction rules� The basic framework is then gradually modi�edto admit for example risk aversion a minimum price entry fees andother �xed costs of bidding multi�unit auctions and bidder collusion�There follows an introduction to the theory of optimal auctions andto common value auctions and the associated winner�s curse problem�It closes with a sample of applications of auction theory in economics such as the regulation of natural monopolies the theory of oligopoly and the government securities market�

�Diese Arbeit ist im Sonderforschungsbereich �� Quanti�kation und Simulation�Okonomischer Prozesse�� HumboldtUniversit�at zu Berlin� entstanden und wurde auf seineVeranlassung unter Verwendung der ihm von der Deutschen Forschungsgemeinschaft zurVerf�ugung gestellten Mittel gedruckt

yComments by Friedel Bolle� Uwe Dulleck� Peter Kuhbier� Michael Landsberger�Wolfgang Leininger� Georg Merdian� and in particular by Dieter Nautz are gratefullyacknowledged

�

Contents

� Introduction �

� 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 � � � � � � � � � � � � � � � � � � � � �� Introducing a minimum price or participation fee � � � � �� Introducing �numbers uncertainty � � � � � � � � � � � � �� Endogenous quantity � � � � � � � � � � � � � � � � � � � � �� Multi�unit auctions � � � � � � � � � � � � � � � � � � � � � �� Removing independence� correlated beliefs � � � � � � � � ��

� Auction rings ��

� Optimal auctions �� Application of the �revelation principle � � � � � � � � � � � � � �� Illustration� two bidders and two valuations� � � � � � � � � � � � �� Discussion� endogenizing bidder participation� � � � � � � � � � � ��

� Common value auctions and the winner�s curse ��

� Further applications �� Auctions and oligopoly � � � � � � � � � � � � � � � � � � � � � � � �� Natural gas and electric power auctions � � � � � � � � � � � � � � �� Treasury bill auctions � � � � � � � � � � � � � � � � � � � � � � � � ��

�

�Men prize the thing ungained more than it is��Shakespeare �Troilus and Cressida�

� Introduction

Suppose you inherit a unique and valuable good � say a painting by the latevan Gogh or a copy of the �� edition of Adam Smith�s Wealth of Nations�For some reason you are in desperate need for money� and decide to sell� Youhave a clear idea of your reservation price� Of course� you hope to earn morethan that� You wonder� isn�t there some way to reach the highest possibleprice� How should you go about it�

If you knew the potential buyers and their valuations of the object for sale�your pricing problem would have a simple solution� You would only need tocall a nonnegotiable price equal to the highest valuation� and wait for the rightcustomer to come and claim the item� It�s as simple as that�

Information problem The trouble is that you� the seller� usually have onlyincomplete information about buyers� valuations� Therefore� you have to �gureout a pricing scheme that performs well even under incomplete information�Your problem begins to be interesting�

Basic assumptions Suppose there are N � � potential buyers� each ofwhom knows the object for sale well enough to decide how much he is willingto pay� Denote buyers� valuations by V �� V�� VN �� Since you� the seller�do not know V � you must think of it as a random variable� distributed by somejoint probability distribution function �CDF� F � �v�� v�� vN � �vN � �� F�v�� vN� �� Pr�V� � v�� VN � vN�� Similarly� potential buyersknow their own valuation but not that of others� Therefore� buyers also viewvaluations as a random variable� except their own�

Cournot monopoly approach Of course� you could stick to the �take�it�or�leave�it pricing rule� even as you are subject to incomplete informationabout buyers� valuations� You would then call a nonnegotiable price p at orabove your reservation price r� and hope that some customer has the rightvaluation and is willing to buy� Your pricing problem would then be reducedto a variation of the standard Cournot monopoly problem� The only unusualpart would be that the trade�o� between price and quantity is replaced by onebetween price and the probability of sale�

Alluding to the usual characterization of Cournot monopoly� you can evendraw a �demand curve � with the nonnegotiable price p on the �price axis

�

and the probability that at least one bidder�s valuation exceeds the listed pricep

��p� � ��G�p�� G�p� �� F�p� � � � � p��

on the �quantity axis �as in Figure � below�� And you would then pick thatprice p that maximizes the expected value of your gain from trade ��p��p� r��

Equivalently� you can state the decision problem in quantity coordinates�as is customary in the analysis of Cournot monopoly� where �quantity is hererepresented by the probability of sale q �� G�p�

maxq

�P �q�� r�q� P �q� �� G�� q��

Computing the �rst�order condition� the optimal price p� can then be char�acterized in the familiar form as a relationship between �marginal revenue �MR� and �marginal cost �MC��

MR �� p � � �G�p�

G��p�� r �� MC� ��

q

p

v

v

p�

r

�

��

��

��

q � ��G�p� �demand�

�p � ��G�p�

G��p� �marginal revenue�

Figure �� Optimal take�it�or�leave�it price

�As always� this condition applies only if the maximization problem is wellbehavedWellbehavedness requires that revenue is continuous and marginal revenue strict monotoneincreasing in p

�

Example � Suppose valuations Vi are independent and uniformly distributedover the �� interval� and the reservation price is r � �� Then� for allp �� G�p� � F �p�N � pN � Hence� the optimal take�it�or�leave�it price is

p��N� � N

s�

N � ��

The resulting expected price paid �price asked times probability of success� is

�p�N� �� p��N�� G�p��N�� p��N�N

N � ��

Both p��N� and �p�N� are strict monotone increasing in N � starting fromp�� p�� and approaching � as N ��

Take�it�or�leave�it pricing is one way to go� But generally you can dobetter by setting up an auction� This brings us to the analysis of auctions�and the design of optimal auction rules� Interestingly� the Cournot monopolyprice p� will continue to play a role as optimal minimum price in the optimalauction setting� as you will learn in the section on optimal auctions�

Auctions what� where� and why An auction is a bidding mechanism�described by a set of auction rules that specify how the winner is determinedand how much he has to pay� In addition� auction rules may restrict partici�pation and feasible bids and impose certain rules of behavior�

Auctions are widely used in many transactions � not just in the sale ofart and wine� Every week� the Treasury auctions�o� billions of dollars ofbills� notes and Treasury bonds� Governments and private corporations solicitdelivery�price o�ers of products ranging from o�ce supplies to tires or con�struction jobs� The Department of the Interior sells the rights to drill oil andother natural resources on federally owned properties� And private �rms sellproducts ranging from fresh owers� �sh and tobacco to diamonds and realestate� Altogether� auctions account for an enormous volume of transactions�

Essentially� auctions are used for three reasons�

� speed of sale�

� to reveal information about buyers� valuations�

� to prevent dishonest dealing between the seller�s agent and the buyer�

�For a simple proof of monotonicity� work with ln �p� which is obviously a monotonetransformation of �p

The latter agency role of auctions is particularly important if government agen�cies are involved in buying or selling� As a particularly extreme example justthink of the current privatization programs in Eastern Europe� Clearly� if theagencies involved� such as the Treuhandanstalt that is in charge of privatizingthe state run corporations of former communist East Germany� were free tonegotiate the terms of sale� the lucky winner probably would be the one whomade the largest bribe or political contribution� However� if assets are put upfor auction� cheating the taxpayer is much more di�cult and costly and henceless likely to succeed�

Popular auctions There are many di�erent auction rules� Actually� theword itself is something of a misnomer� Auctio means increase� but not allauctions involve calling out higher and higher bids�

One distinguishes between oral and written auctions� In oral auctions�bidders hear each other�s bids� and can make countero�ers! each bidder knowshis rivals� In a written or closed�seal bid� bidders submit their bids simultane�ously without revealing them to others! often bidders do not even know howmany rival bidders participate�

The best known and most frequently used auction is the ascending price orEnglish auction� followed by the �rst�price closed�seal bid or Dutch auction�and the second�price closed�seal bid �also known as Vickrey auction��

Oral seal�Bid

ascending price second�price�English�

descending price �rst�price�Dutch�

Table �� The most popular auctions

In the ascending price or English auction� the auctioneer seeks increasingbids until all but the highest bidder�s� are eliminated� If the last bid is at orabove the reserve price� the item is awarded or knocked down to the remainingbidder�s�� If a tie bid occurs� the item is awarded for example by a chancerule� In one variation� used in Japan� the price is posted on a screen and raisedcontinuously� Any bidder who wants to be active at the current price pushesa button� Once the button is released� the bidder has withdrawn altogether�

�Many variations of these basic auctions are reviewed in Cassady� R � �� Auctions

and Auctioneering� University of California Press

Instead of letting the price rise� the descending price or Dutch auctionfollows a descending pattern� The auctioneer begins by asking a certain price�and gradually lowers it until some bidder�s� shout �Mine to claim the item�Frequently� the auctioneer uses a mechanical device called the �Dutch clock �This clock is started� and the asking price ticks down until someone calls out�The clock then stops and the buyer pays the indicated price� Again� provisionsare made to deal with tie bids��

Finally� in a written auction bidders are invited to submit a closed�sealbid� with the understanding that the item is awarded to the highest bidder�Under the �rst�price rule the winner actually pays as much as his own bid�whereas under the second�price rule the price is equal to the second highestbid�

Of course� this terminology is not always used consistently in the aca�demic� and in the �nancial literature� For example� in the �nancial commu�nity a multiple�unit� single�price auction is termed a Dutch auction� and amultiple�unit closed�seal bid auction is termed an English auction �except bythe English� who call it an American auction�� Whereas in the academic litera�ture� the labels English and Dutch would be exactly reversed� In order to avoidconfusion� keep in mind that here we adhere to the academic usage of theseterms� and use English as equivalent to ascending and Dutch as equivalent todescending price�

Early history of auctions Probably the earliest report on auctions is foundin Herodotus in his account of the bidding for men and wives in Babylonaround �� B�C� These auctions were unique� since bidding sometimes startedat a negative price�� In ancient Rome� auctions were used in commercial trade�to liquidate property� and to sell plundered war booty� A most notable auctionwas held in �� A�D� when the Praetorian Guard put the whole empire up forauction� After killing the previous emperor� the guards announced that theywould appoint the highest bidder as the next emperor� Didius Julianus outbidhis competitors� but after two months was beheaded by Septimius Severus whoseized power � a terminal case of the winner�s curse�

�An interesting Dutch auction in disguise is �time discounting� In many cities in theUS discounters use this method to sell cloth Each item is sold at the price on the tagminus a discount that depends on how many weeks the item was on the shelf As timepasses� the �nal price goes down at the rate of say � � per week� until the listed bottomprice 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 begin with the most thoroughly researched auction model� the symmetricindependent private values �SIPV� model with risk neutral agents� In thatmodel�

� a single indivisible object is put up for sale to one of several bidders �unitauction�!

� both the seller and all bidders are risk neutral!

� each bidder knows his valuation � no one else does �private values�!

� the unknown valuations are independent random variables� drawn fromsome continuous probability distribution �continuity�independence�!

� all bidders are indistinguishable! therefore� the probability distributionof each bidder�s valuation is the same �symmetry�!

� the seller�s reservation price is normalized to r � ��

We will model bidders� behavior as a non�cooperative game under incom�plete information� One of our goals will be to rank the four common auctions�To what extent does institutional detail matter� Can the seller get a higheraverage price by an oral or a written auction� Is competition between bidders�ercer if the winner pays a price equal to the second highest rather than thehighest bid� And which auction� if any� is strategically easier to handle�

�� Some basic results on Dutch and English auctions

The comparison between the four standard auction rules is considerably facil�itated by the fact that the Dutch auction is equivalent to the �rst�price andthe English to the second�price closed�seal bid� Therefore� the comparison ofthe four standard auctions can be reduced to comparing the Dutch and theEnglish auction� Later you will learn the far more surprising result that eventhese two auctions are payo� equivalent�

The strategic equivalency between the Dutch auction and the �rst�priceseal bid is immediately obvious� In either case� a bidder has to decide howmuch he should bid or at what price he should shout �Mine to claim theitem� Therefore� the strategy space is the same� and so are payo� functions�and hence equilibrium outcomes�

Similarly� the English auction and the second�price closed�seal bid areequivalent� but for di�erent reasons� and only as long as we stick to the privatevalues framework� Unlike in a seal bid� in an English auction bidders canrespond to rivals� bids� Therefore� the two auction games are not strategically

�

equivalent� However� in both cases bidders have the obvious dominant strategyto 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 nota�ected by observing rivals� bidding behavior� which always holds true in theprivate values framework��

The essential property of the English auction and the second�price closesd�seal bid is that the price paid by the winner is exclusively determined by rivals�bids� Bidders are thus put in the position of price takers who should acceptany price up to their own valuation� The �truth�revealing strategy b�v� � vis a dominant strategy� The immediate implication is that both auctions formshave the same equilibrium outcome�

�� Revenue equivalence theorem

Using elementary reasoning we have already established the payo� equivalenceof Dutch and �rst�price and of English and second�price auctions� Remark�ably� payo� equivalence is a much more general feature of auction games�Indeed� all auctions that award the item to the highest bidder are payo�equivalent� The four standard auctions are a case in point� We now turnto proof of this surprising result� and then explore modi�cations of the SIPVframework�

We mention that the older literature conjectured that Dutch auctions leadto a higher average price than English auctions� because� as Cassady put it� inthe Dutch auction �� each potential buyer tends to bid his highest demandprice� whereas a bidder in the English system need advance a rival�s o�er byonly one increment� Both the reasoning and the conclusion are wrong�

Since bidders� valuations are independent random variables� their joint dis�tribution function F is just the product of their separate distribution functionsF � �� v� � �� that are in turn identical� due to the assumed symmetry�Therefore�

F�v�� v�� vn� � F �v��F �v�� F �vn��

A bidder�s strategy is a mapping from valuations to bids� b � �� v� � R�A vector of strategies is an equilibrium� if it has the mutual best responseproperty� Since all bidders are alike� whatever is an optimal bidding strategyfor one bidder should also be an optimal strategy for any other bidder� Thissuggests that the equilibrium should be symmetric�

�Obviously� if rivals� bids signal something about the underlying common value of theitem� each bidders� estimated value of the item is updated in the course of an English auctionNo such updating can occur under a seal bid� where bidders place their bids before observingrivals� behavior Therefore� if the item has a common value component� the English auctionand the secondprice closedseal bid do not have the same equilibrium outcome

�Cassady� R � �� Auctions and Auctioneering� University of California Press� p ��

�

The key insight that leads to nice and simple proofs is that� given rivals�strategies� each bidder�s decision problem can be viewed as one of choosingthrough his bid b�v� a probability of winning� �� and an expected payment E��This way� the choice of strategies is looked at as a self�selection of �� E�� Dueto the independence assumption� both � and E depend only on the bid b�v�but not directly on the bidder�s own valuation v�

In all results reviewed in this section we characterize symmetric equilibria�We assume the SIPV framework� and consider the set of auctions that adhere tothe principle of selecting the highest bidder as winner� and leave the numberof active bidders the same�� No other assumptions are made� Therefore�our analysis applies to a myriad of auction rules� not just the four commonauctions�

The �rst useful result is�

Lemma � �Monotonicity The equilibrium bid strategy b� is monotone increasing in v�

Proof Given rivals� strategies� the expected utility of a bidder with valuation vand bidding strategy b is� U�b�v�� v� �� b�v��v�E�b�v�� De�ne the indirectutility function U� � �� v�� R

U��v� �� b��v��v � E�b��v��

Since U� is a maximum value function� the Envelope Theorem applies� andone 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 neverthe same for di�erent valuations� the monotonicity is even strict� �

An immediate implication of strict monotonicity� combined with symmetry�is�

�Notice� in this general account it is not assumed that it is only the winner who has tomake a payment

The auction rule may a�ect the number of active bidders if payment is not restrictedto the winner For example� if the seller requires an entry fee� he will loose all those whosevaluation is below that fee This point is taken up on page �

�The proof is similar to the proof of concavity of the expenditure function in the pricevector� that you have learned in your micro course Make sure that you also understand theintuition of this result Notice� that the bidder could always leave the bid unchanged whenv changes� as a fall back Therefore� U� must be at or above its gradient hyperplane� whichproves convexity

��

Lemma � �Pareto optimality The bidder with the highest valuation winsthe auction�

This brings us to the amazing revenue equivalence result�

Proposition � �Revenue equivalence Assume the SIPV framework� andU�� Then� all auctions that select the highest bidder as winner and thathave the same number of bidders give rise to the same expected payos of theseller and bidders�

Proof Integrating �� one obtains� using the fact that U��

U��v� �Z v

��b��"v��d"v� ��

Since each of the admitted auctions is Pareto e�cient� ��b��v�� must be equalto the probability that all other bidders� valuations are below v� Therefore� by�� each bidder�s payo� is the same under all admitted auctions� Moreover�the total surplus generated by trade must also be the same� due to the Paretooptimality of the outcome�� Therefore� the seller�s payo� must also be thesame� �

As you can see from the ingredients to the proof of this result� revenueequivalence applies not only to the four standard auctions� but to all auctionsthat select the highest bidder as winner� In particular� revenue equivalencealso applies to third� and higher�price auctions� that are discussed below�However� the proof of revenue equivalence assumes the SIPV framework� Itremains to be seen whether all of the SIPV ingredients are crucial�

Remark � Assuming the SIPV framework� revenue equivalence holds for allauctions that award the item to the highest bidder� This prerequisite seems toapply to most auction rules� However� it is already violated when the sellersets a minimum price above his reservation value�

�� The case of uniformly distributed valuations

Before we go into the general account of equilibrium bids� and then proceedwith various modi�cations of the SIPV model� take a look 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�N�� r� �g The seller�spayo� is the di�erence between total surplus and bidders� payo�s

��

Example � Suppose all bidders� valuations are uniformly distributed on the�� interval� as in example �� Then� the unique equilibrium bidding strategiesare

b��vi� � vi �English auction� ��

b��vi� � ��

N�vi �Dutch auction��

In both auctions� the expected price is

�p �N � �

N � ��

Therefore� more bidders mean �ercer competition between buyers�

We now explain in detail how these results come about� using the symmetryof equilibrium and the monotonicity of equilibrium strategies�

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

��b� � Prfb��V�N�� bg� PrfV�N�� b�g� F ��b��N��

where ��b� is the inverse of b��vi�� which indicates the valuation that leads tobidding the amount b if strategy b� is played� and V�i� is the i �th order statisticof the sample of N random valuations�

In equilibrium the bid b must be a best�response to rivals� bids� Therefore�b must be a maximizer of the expected gain ��b��vi � b�� leading to the �rst�order condition

��b��v � b�� b� � ��

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

�N � ��f��b�� b�� b��b�� F ��b��

which simpli�es to� due to F ��b�� b� �uniform distribution�

�N � ��b�� b��b�� b� � ��

��To be careful� the condition should read as follows� ��b� � Prfb��V�N�� bg� however�since the V �s are continuous random variables� there is no probability mass on single points�and therefore you can safely replace the strict by the weak inequality

��

This di�erential equation has the obvious solution ��b� � N

N��b�� Finally�

solve this for b� and you have the equilibrium bid strategy

b��v� � ��

N�v� ��

as asserted�� Notice that the margin of pro�t that each player allows himselfin his bid decreases as the number of bidders increases�

Based on this result you can now determine the expected price �p�N�� bythe following consideration� In a Dutch auction� the random price paid is equalto the highest bid� which can be written as b��V�N�� where V�N� denotes thehighest order statistic of the entire sample of N valuations� Therefore� by aknown result on order statistics��

�p�N� � E�b��V�N��

�N � �

NE�V�N�� since b��v� � N��

Nv�

�N � �

N � ��

as asserted�

English auction Recall� in an English auction each bidder bids his truevaluation�� The bidder with the highest valuation wins� and pays a priceequal to the second highest order statistic V�N�� of the given sample of Nvaluations� Therefore� the expected price is

�p�N� � E�V�N�� N � �

N � ��

��Why is it obvious� Suppose the solution is linear� then you will easily �nd the solution�which also proves that there is a linear solution

��This begs the question whether the equilibrium is unique The proof is straightforwardfor N � � In this case� one obtains from � �� d� � �db � bd� � d��b� Therefore��b � �

�� Since b�� one has �� and thus ��b� � �bv Hence� the unique

equilibrium strategy is b��v� � ��v In order to generalize the proof to N � �� apply a

transformation of variables� from �� b� to �z� b�� where z �� b Then one can separatevariables and uniquely solve the di�erential equation by integration

��Let V�r� be the rth order statistic of a given sample of N independent and identicallydistributed random variables Vi with distribution function F �v� � v on the support ��

Then� E�V�r�� r

N � and Var �V�r�� r�N�r ��

�N ��N �� See rules Kendall� S M and A Stuart

� �� The Advanced Theory of Statistics � Vol I �Distribution Theory�� Charles Gri�n �Co� pp �� and p � �

��Notice� however� that the English auction is plagued by a multiplicity problem Forexample� the strategy combination b��v� � � bi�v� � �� i � �� N � is also an equilibrium�even though it prescribes rather pathological behavior However� the equilibrium analyzedin the main text is the only one that is not �dominated�

��

Comparison Since the two expected prices are the same� this exampleillustrates the general principle of revenue equivalency� Notice� however� thatactual payo�s tend to di�er� even in their risk characteristics� Indeed� theDutch auction leads to a lower price risk� Therefore� risk averse sellers shouldprefer the Dutch over the English auction�

To see this clearly� compute the variance of the random price P � in theEnglish �E� and the Dutch �D� auction

Var �P �E � Var �V�N��

��N � ��

�N � ��N � ��

Var �P �D � Var�b��V�N��

�

��N � ��

N�Var �V�N��

��N � ��

N�N � ��N � ��

Since N��N

� N

N� �� you see immediately that the English auction gives rise

to a higher variance of price� as asserted�

Third and higher price auctions� Consider a generalization of thesecond�price auction� the �third�price auction� where the highest bidder winsbut pays only as much as the third highest bid� and more generally� the �k�thprice auction� where the winner pays the k�th highest bid� If you follow theprocedure used above to derive the equilibrium bid function under the �rstprice auction� you will �nd the following solution�

b��v� �N � �

N � k � �v� for N � �� k � �� N� ��

Third� and higher price auctions have three striking properties� �� bidsare higher than the own valuation! �� equilibrium bids diminish as the numberof bidders is increased �since N��

N�k� � N

N�k��! and �� the variance of therandom price P

Var �P � � Var�b��V�N�k��

��

k�N � ��

�N � k � ��N � ��N � ��

��Hints� Suppose the equilibrium bid function of the kprice auction is linear in theown valuation� b��v� � kv Revenue equivalency applies also to the kprice auctionTherefore� the expected price must be the same as under the English auction� E�V�N�� E�b��V�N�k �� kE�V�N�k �� Compute the two expected values� and you have theasserted bid function �� Once you have a candidate for solution� you only need to con�rmthat it does indeed satisfy the mutual best response requirement

��

increases in k�Once you have �gured out why it pays to �speculate � and bid higher than

the own valuation� it is easy to interpret the second property� Just keep in mindthat a rational bidder may get �burned � and su�er a loss� because the k�thhighest bid is above the own valuation� As the number of bidders is increased�it becomes more likely that the k�th highest bid is in close vicinity to theown valuation� Therefore� it makes sense to bid more conservatively when thenumber of bidders is increased� even though this may seem somewhat puzzling�at �rst glance�

Finally� the third property suggests that a risk averse seller should alwaysprefer lower order k�price auctions� and therefore should most prefer the Dutchauction� The English auction is always appealing because of its overwhelmingstrategic simplicity� Third� and higher�price auctions are strategically justas complicated as the Dutch auction� and in addition expose the seller tounnecessary price risk� Therefore� we conclude that third� and higher�priceauctions are strictly dominated by Dutch auctions�

Nobody has ever� to our knowledge� applied a third� or higher�price auc�tion� So is this just an intellectual curiosity� useful only to challenge yourintuition and technical skills�

Experimental economists have exposed inexperienced subjects to third�price seal bids� and examined their response to a higher number of bidders�Amazingly� the majority of bidders reduced their bid� just as the theoryrecommends�� The authors take great pride in this result! they take it asevidence that inexperienced subjects are not as unsophisticated as is oftensuspected by critics of the experimental approach� The particular irony is thatif one asks trained theorists to make a guess� they tend to come up with thewrong hunch concerning the relationship between the number of bidders andequilibrium bids� until they actually sit down to con�rm the computations�

�Charity� or �all pay� auctions� As a last example consider an auc�tion that is frequently used in charities� which is why we call it �charity or�all pay auction� for lack of a better name� Its peculiar feature is that everybidder is required to actually pay his bid� Just like in a standard auctions� theitem is awarded to the highest bidder� But unlike in standard auctions� eachbidder 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 itemto the highest bidder� Therefore� revenue equivalence applies� But we want tocon�rm it directly� and compute equilibrium bids� as a further exercise�

��See Kagel� J H and D Levin� � �� Independent private value auctions� bidderbehavior in �rst� second� and thirdprice auctions with varying numbers of bidders��University of Houston� Working Paper

�

Using a procedure similar to the above� the equilibrium bid function is��

b��v� �N � �

NvN � ��

This is in line with the observation that bidders often bid only nominal amountsof money for relatively valuable items�

In order to determine the expected price� notice that the seller collects thefollowing expected payment from each bidder�

E �b��V �� N � �

N

Z �

vNdv �

N � �

N�N � ��

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

� whichcon�rms the asserted revenue equivalence�

The crucial feature of charity auctions is that they make winners and loserspay� We mention that this is also a feature of the not so charitable selleroptimal auction� if bidders are risk averse��

�� Generalization�

In many non�cooperative games it is di�cult if not impossible to �nd a closed�form solution of the Nash equilibrium� unless one works with particular func�tional speci�cations� The SIPV auction game is an exception� Indeed� theEnglish and the Dutch auction games have a unique symmetric equilibrium�for all probability distributions of private valuations�

Proposition � �Equilibrium bidding rules Suppose the distribution function F �v� is continuous� Then� the Dutch auction has a unique equilibrium

b��v� � v �R vr F �"v�

N��d"v

F �v�N��

where r denotes the seller�s reserve price�Of course� the English auction has several equilibria but� b�v� v is the

only one that is not �weakly� dominated�

Proof The bidding rule for the English auction follows from the fact thattruth�telling is a dominant strategy� To prove 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 be paid� the equilibrium bid must be a maximizer of ��b�v � b�hence it must solve the di�erential equation ��v � From these two building blocks it iseasy to compute the equilibrium bid function

�See Matthews� S � �� Selling to risk averse buyers with unobservable tastes�� Jour�nal of Economic Theory� �� ! ��

�

auction� you have to solve the di�erential equation �� The details are spelledout for example in Milgrom and Weber��

�� Robustness

Assuming the SIPV framework with risk neutral agents we have derived theamazing result that all auctions that award the item to the highest bidder giverise to the same payo�s� In other words� if all participants are risk neutral�many di�erent auction forms � including the four standard auctions � arenothing but irrelevant institutional detail�

This result is in striking contrast to the apparent popularity of the Englishauction� Has the SIPV model missed something of crucial importance� or havewe missed something at work even in this framework�

A strong argument in favor of the English auction is its strategic simplicity�In this auction bidders need not engage in complicated strategic considerations�Even an unsophisticated bidder should be able to work out that setting thelimit equal to one�s true valuation is the best one can do� Not so under theDutch auction� This auction game has no dominant strategy equilibrium� As aresult� understanding the game is conceptually more demanding� not to speakof computational problems�

However� if the auction cannot be oral� say because it would be too costlyto bring bidders together at the same time and place� there is a very simple yetstrong reason why one tends to use a �rst�price closed�seal bid� even thoughit is strategically much more complicated than the second�price or Vickreyauction� A second�price auction can usually be manipulated by solicitingphantom bids in close vicinity of the highest submitted bid� This suggeststhat second�price closed�seal bids should only be observed if the seller is apublic agency� that is not pro�t oriented�

There are several more good reasons why the seller should favor the Englishauction� But these come up only as we modify the SIPV model� We now turnto this task� In addition we will give a few examples of auctions that do notadhere to the principle of awarding the item to the highest bidder� and on thisground fail to be payo� equivalent to the four common auctions�

�� Removing risk neutrality

The simplest variation of the SIPV model with risk averse agents is obtainedby introducing risk averse bidders� Obviously� this modi�cation does not a�ectthe equilibrium strategy under the English auction� Therefore� the expectedprice in this auction is una�ected as well�

��Milgrom� P and R J Weber � �� A theory of auctions and competitive bidding��Econometrica� "�� ! ��

��

However� strategies and payo�s change under the Dutch auction� In thisauction form bidders always take a chance and �shade their bid below theirvaluation�� Therefore� the more risk averse a bidder� the more reluctant hewill be to bid far below his true valuation� and consequently risk averse bidderstend to bid more conservatively� As a result� they raise the seller�s payo� and�alas� reduce their own��

Proposition � Consider the SIPV model with risk averse bidders� The seller�spayo is higher and the bidders� payo lower under the Dutch than under theEnglish auction�

�� Removing symmetry

The symmetry assumption is central to the revenue equivalence result� For ifbidders are characterized by di�erent probability distributions of valuations�under the Dutch auction �and the equivalent �rst�price seal bid� it is no longerassured that the object is awarded to the bidder with the highest valuation�But precisely this property was needed in the proof of Proposition ��

Example � Suppose there are two bidders� A and B� characterized by theirrandom valuations� with support �a� �a� and �b��b� with �a � b� Consider theDutch auction� Obviously� B could always win and pocket a gain by biddingthe 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� violatingPareto�optimality�

An even more important implication is that Pareto e�ciency is only assuredby second�price bids� If a public authority uses auctions as an allocationdevice� e�ciency should be the primary objective� This suggests that publicauthorities should adopt second�price bids�

A rigorous proof of existence and uniqueness of the equilibrium solutionwith heterogenous bidders is in Plum�� If you are interested in a carefulmathematical study of existence and uniqueness of auction games� this contri�bution is a highly recommended source�

��Recall� a bidder can always do better than bid his true valuation �which would givehim zero payo�� In fact� in example � the equilibrium strategy was b��v� � N��

Nv� which

implies shadeing the bid all the way down to b��v� � ��v if the number of bidders is N � �

��For a formal proof using stochastic dominance see Maskin� E and J G Riley � ��"��Auction theory with private values�� American Economic Review� �"� "�! ""

��Plum� M � �� Characterization and computation of Nashequilibria for auctionswith incomplete information�� International Journal of Game Theory � �� ! �

��

�� Introducing a minimum price or participation fee

Revenue equivalence is a fairly general property of auction games� It appliesto all auctions that adhere to the principle of awarding the item to the highestbidder� This restriction sounds pretty mild� Is there any common auctionwhere it is violated� And is ever desirable to deviate from it�

Many seemingly innocuous modi�cations of standard auction rules deviatefrom the principle of awarding the item to the highest bidder� and thereforeviolate revenue equivalence� For example� suppose the seller sets a minimumprice above his reservation price� Then he deviates from this principle� becausehe only sells to the highest bidder if that bid exceeds the minimum price� Ofcourse� as we indicated in our comparison of examples � and �� such deviationscan be pro�table for the seller� Indeed� the theory of optimal auctions tells usthat the seller should always set such a minimum price above his reservationprice� and thus leave the domain of revenue equivalence� More about this inthe section on optimal auctions�

Another modi�cation of standard auctions that violates revenue equiva�lence occurs when a �participation fee is added� Adding such a fee to theEnglish auction has also an interesting e�ect on the number of bidders whoactually participate� This makes two good reasons for brie y elaborating onthis modi�cation�

Example � �Participation fee Suppose the seller requires a participationfee c �� from each bidder� in an English auction� Let there be two bidders�with uniformly distributed valuations� on the support ��

Then� a bidder participates if and only if his valuation is at or above thecritical level #v ��

pc� The expected revenue maximizing entry fee is c� � �

� � andthe maximum expected revenue �

�� Therefore� for N � �� the participation fee

augmented English auction is more pro�table than the standard English auctionand than the Cournot monopoly approach �see examples � and ��

The proof is sketched as follows� �� In order to compute #v� consider themarginal bidder� with valuation #v� His cost of participation� c� must be exactlymatched by the expected gain from bidding

Prfwinningg#v � c � ��

Therefore�#v �

pc� ��

as asserted�� Notice� the seller collects the entry fee from both bidders if and only

if V�� #v� and in that case also collects a price equal to V�� Whereas ifV�� #v � V�� only one bidder participates� the entry fee goes all the way

��

down to zero� and the seller�s only earning is the one entry fee paid the onlyone active bidder� Therefore� the seller�s expected pro�t is equal to

$ �� PrfV�� #vg�E�V�� j V�� #v� � �c

�� PrfV�� #v � V��gc� ��

Of course�PrfV�� #vg � �� #v��

andPrfV�� #v � V��g � �#v�� #v��

Hence� by all of the above�

$ ��pc��c�

pc� ��

��

�� Evidently� $ is strictly concave in c� and c� � �� solves the �rst order

condition �� pc� which completes the proof�

�� Introducing �numbers uncertainty�

In many auctions bidders are uncertain about the number of participants��numbers uncertainty �� This seems almost compelling in closed�seal bids�where bidders do not convene in one location� but make bids each in their owno�ce�

Notice� however� that the seller can easily eliminate numbers uncertainty�if he wishes to do so� He only needs to solicit contingent bids� where eachbidder makes a whole list of bids� each contingent on a di�erent number ofparticipating bidders� Therefore� the seller has a choice� The question is� is itin the interest of the seller to leave bidders subject to numbers uncertainty�

McAfee� Preston and McMillan�� explored this issue� They showed thatif bidders are risk averse and have constant or decreasing absolute risk aver�sion� numbers uncertainty leads to more aggressive bidding in a �rst�priceclosed�seal bid� Since numbers uncertainty has obviously no e�ect on biddingstrategies under the three other auction rules� one can conclude from the rev�enue equivalence proposition � that numbers uncertainty favors the �rst�priceclosed�seal bid� Incidentally� this result is used in experiments to test whetherbidders are risk averse��

��See McAfee� R� R Preston and J McMillan � �� Auctions with a stochastic numberof bidders�� Journal of Economic Theory� �� ! �

��See Dyer� D� J Kagel and D Levin � �� Resolving uncertainty about the numberof bidders in independent privatevalue auctions� an experimental analysis�� Rand Journal

of Economics� ��

��

�� Endogenous quantity

On the impact of endogenous quantity on the ranking of the four commonauctions see Hansen�� Hansen shows that �rst�price auctions lead to a higherexpected price! therefore� revenue equivalence breaks down in this case� Animportant application deals with industrial procurement contracts in privateindustry and government� Hansen�s result explains why procurement is usuallyin the form of a �rst�price closed�seal bid� For a fuller analysis of this matter�see example �� on p� �� below�

�� Multiunit auctions

Suppose the seller o�ers more than one unit of a good� Then everythinggeneralizes quite easily� as long as each buyer demands at most one unit� Butrevenue equivalence fails if the demand is price elastic�

Consider the inelastic demand case� First of all we need to generalize thede�nition of �rst and second�price auctions� Suppose x units are put up forsale� In a second�price auction� a uniform price is set at such a level that allbut x buyers drop out� Each remaining bidder receives one unit and pays thatprice� In turn� in a �rst price auction� the x highest bidders are awarded thex items� and they pay their own bid�

Just like in the single unit case� one can show that all auction rules arerevenue equivalent� provided they adhere to the principle of awarding eachitem to the highest bidder��

However� this does not generalize to the case when bidders� demand is priceelastic� But in view of the previous section� this should not come as much ofa surprise�

The theory of multi�unit auctions is not very well developed� Until recently�analysts of applied multi�unit auctions avoided to model the auction as a noncooperative game� and instead analyzed bidders� optimal strategy� assuming agiven probability distribution of being awarded the demanded items�� How�ever� Maskin and Riley� have generalized the theory of optimal auctions toinclude the multi�unit 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 monopoly pricing schemes with

demand uncertainty�� American Economic Review� � � � �!��"�See for example the analysis of treasury bill auctions by Scott J and C Wolf � ��

�The e�cient diversi�cation 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 expected stopout price in a discriminatory

auction�� Economics Letters� � � ��

��

ber of bidders or the quantity put up for auction is increased� the expectedstop�out price �the lowest price served� goes up� which generalizes well�knownproperties of single�unit auctions�

The most important applications of multi�unit auctions are in �nancialmarkets� For example� the U�S� Treasury sells marketable bills� notes� andbonds in more than �� regular auctions per year� using a closed�seal bid�multiple�price auction mechanism� Some of the institutional details of thegovernment securities market are sketched in section �� below�

�� Removing independence� correlated beliefs

Finally� remove independence from the SIPV model� and replace it by the as�sumption of a positive correlation between bidders� valuations� Loosely speak�ing� positive correlation means that a high own valuation makes it more likelythat rival bidders� valuations are high as well�

The main consequence of correlation is that it makes bidders more con�servative in a Dutch auction� Of course� correlation does not a�ect biddingin an English auction� where truth�telling is always the dominant strategy�Therefore� the introduction of correlation reduces the expected price underthe Dutch auction� but has no e�ect on the expected price under the Englishauction� In other words� correlation entails that the seller should prefer theEnglish to the Dutch auction�

Proposition � In the symmetric private values model with positively correlated valuations� the seller�s payo is higher under the English than under theDutch auction� and that of potential buyers is lower��

The introduction of correlation has a number of comparative static impli�cations with interesting public policy implications� In particular� the releaseof information about the item�s value should raise bids in a Dutch auction� In�cidentally� this comparative static property was used by Kagel� Harstadt andLevin�� to test the predictions of auction theory in experimental settings�

� Auction rings

So far we have analyzed auctions in a non�cooperative framework� But whatif bidders collude and form an auction ring� Surely� bidders can gain a lot

��The formal proof is in Milgrom� P and R J Weber � �� A theory of auctions andcompetitive bidding�� Econometrica� "�� ! ��

��Kagel� J� R M Harstad and D Levin � �� Information impact and allocation rulesin auctions with a�liated private values� a laboratory study�� Econometrica� ""� ��"! ��

��

by collusive agreements that exclude mutual competition� But can they beexpected to achieve a stable and reliable agreement� And if so� which auctionsare more susceptible to collusion than others�

To rank di�erent auctions in the face of collusion� suppose all potentialbidders have come to a collusive agreement� They have selected their designated winner� presumably the one with the highest valuation� recommendedhim to follow a particular strategy� and committed others to abstain from bid�ding� However� suppose the ring faces an enforcement problem because ringmembers cannot be prevented from breaching the agreement� Therefore� theagreement has to be self�enforcing� Does it pass this test at least in someauctions�

Consider the Dutch auction �or �rst�price seal bid�� Here� the designatedwinner will be recommended to place a bid roughly equal to the seller�s reserveprice� whereas all other ring members are asked to abstain from bidding� Butthen each of those asked to abstain can gain by placing a slightly higher bid� inviolation of the ring agreement� Therefore� the agreement is not self�enforcing�

Not so under the English auction �or second�price closed�seal bid�� Herethe designated bidder is recommended to bid up to his own valuation� andeveryone else to abstain from bidding� Now� no one can gain by breaching theagreement� because no one will ever exceed the designated bidder�s limit�

Therefore��

Proposition � Collusive agreements between potential bidders are self�enforcing in an English� but not in a Dutch auction��

These results give a clear indication that the English auction �or second�price closed�seal bid� is particularly susceptible to auction rings� and that theseller should choose a Dutch in lieu of an English auction if he has to deal withan auction ring that is unable to enforce agreements�

Even if auction rings can write enforceable agreements� the ring faces theproblem of how to select the designated winner� and avoid strategic behaviorof ring members� This is usually done by running a pre�auction� But canone set it up in such a way that it always selects the highest valuation ringmember�

In a pre�auction� every ring member is asked to place a bid� and the high�est bidder is chosen as the ring�s sole bidder at the subsequent auction� Butif the bid at the pre�auction only a�ects the chance of becoming designated

��This point was made by Robinson� M � ��"� �Collusion and the choice of auction��Rand Journal of Economics� �� ! "

��To be more precise� under an English auction the bidding game has two equilibria� onewhere everybody sticks to the cartel agreement and one where it is breached The lattercan� however� be 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 pre�auction problem can only be solved if one makesthe designated winner share his alleged gain from trade�

Graham and Marshall�� proposed a simple scheme that resolves the pre�auction problem by an appropriate side�payment arrangement� Essentially�this scheme uses an English auction �or second�price closed�seal bid� to selectthe designated winner� If the winner of the pre�auction also wins the auction�he is required to pay the other ring members the di�erence between the pricepaid and the second highest bid from the pre�auction bid� This way it isassured that truth�telling is an optimal strategy at the pre�auction� excludingstrategic behavior�

Notice� however� that this solution of the pre�auction problem works onlyif the ring can enforce the agreed upon side�payments� In the absence ofenforcement mechanisms� auction rings are plagued by strategic manipulation�so that no stable ring agreement may get o� the ground�

Finally� we mention that the seller may �ght a suspected auction ring by�pulling bids o� the chandelier � that is by introducing imaginary bids intothe proceedings� In �� the New York City Department of Consumer A�airsproposed to outlaw imaginary bids� after it had become known that Christie�shad reported the sale of several paintings that had in fact not been sold� Thisproposed regulation was strongly opposed by most New York based auctionhouses� precisely on the ground that it would deprive them of one of their mostpotent weapons to �ght rings of bidders�

� Optimal auctions

Among all possible auctions� which one maximizes the seller�s expected pro�t�At �rst glance� this question seems unmanageable� There is a myriad of possi�ble auction rules� limited only by one�s imagination� Therefore� whatever yourfavorite auction rule� how can you ever be sure that someone will not comealong and �nd a better one�

On the other hand� you may ask� what is the issue� Didn�t we show thatvirtually all auction rules are payo� equivalent�

�� Application of the �revelation principle

The breakthrough that made the problem of optimal auction design tractableis due to Myerson�� He observed that one may restrict attention� without

��Graham� D A and R C Marshall � �� Collusive bidder behavior at singleobjectsecondprice 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 fundamentalinsight applies the famous revelation principle� which is also due to Myerson��

that has been useful in many areas of modern economics� from the public goodproblem to the theory of optimal labor contracts�

Example � Auctions that award the item strictly to the highest bidder arenot optimal� To show this� consider the assumptions of example �� and letN � �� By revenue equivalence� all auctions that select the highest bidder aswinner are payo equivalent to the Dutch and English auction� Therefore� theexpected price is �p� � �� Compare this to the Cournot�seller from example

� that sets the take�it�or�leave�it price p� � ��

� � and earns the expectedprice �p� � p�� Evidently� the Cournot approach is more pro�table� Theimmediate implication is that by imposing a minimum bid equal to p�� the Dutchand English auction can be improved and be made even more pro�table thanthe simple Cournot approach� Of course� by raising the minimum bid abovethe seller�s 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 toreport his valuation to the seller� and the auction rules select the winner andbidders� payments� Closed seal�bids are direct auctions � English and Dutchauctions are indirect�

Incentive compatibility A direct auction is incentive compatible if honestreporting of valuations is a non�cooperative Nash equilibrium� Many directauctions are not incentive compatible� For example� in a �rst�price closed�sealbid every bidder bids less than his valuation� Therefore� �rst�price seal bidsare direct but not incentive compatible� In turn� in a second�price closed�sealbid truth�telling is a dominant strategy� Therefore� second�price closed�sealbids are direct as well as incentive compatible�

Since we consider optimality from the point of view of the seller� incentivecompatibility requires only that buyers reveal their true valuations� but wedo not require that the seller reports his true reservation price� before bidsare solicited� Again� the second�price closed�seal bid is a good illustration�Obviously� it is incentive compatible in the sense that all buyers report theirtrue valuation� But� as example indicates� it is not truth�revealing withregard to the sellers reservation price� since the seller would always quote aminimum bid above his true reservation price�

in� Engelbrecht!Wiggans� R and M Shubik and J Stark � �� Auctions� Bidding� andContracting� New York University Press� �! ��

��Myerson� R B � �� Incentive compatibility and the bargaining problem�� Econo�metrica� �� !��

�

Revelation principle The revelation principle says that for any equilibriumof any auction game� there exists an equivalent incentive compatible directauction� Therefore� the auction that is optimal among the incentive compatibledirect auctions is also optimal for all types of auctions�

To prove the revelation principle� consider an equilibrium of some arbitrar�ily chosen auction game� We will show that the following procedure describesan equivalent incentive compatible direct auction�

�� Ask each bidder to report his valuation�

�� Compute each bidder�s optimal bidding strategy in the given equilibriumof the assumed auction game�

�� Select the winner and collect payments exactly as in the given equilibriumof the assumed auction game�

Evidently� if all bidders are honest this direct auction is equivalent to thegiven equilibrium of the assumed auction game� Finally� this direct auction isalso incentive compatible� Because if it were pro�table for any bidder to liein the direct auction� it would also be pro�table for him to lie to himself inexecuting his equilibrium strategy in the original auction game� It�s as simpleas that�

Example � Consider the Dutch auction equilibrium from example �� Theequivalent direct incentive compatible auction is described by the �N probabilityand expected payment functions

�i�v�� vN � �

�� if vi � vj��j �� i� otherwise

Ei�v�� vN� ��

N��N

vi if vi � vj��j �� i� otherwise�

Feasible direct auctions A direct auction is described by a set of �Noutcome functions� �i� Ei� de�ned on the support of valuations� Thereby��i�v�� vN� denotes the i�th bidder�s probability of winning� and Ei�v�� vN�his expected payment �notice� the bidder may have to make a payment evenif he does not win the auction�� To be feasible� these outcome functions haveto satisfy the following three conditions�

�For example� in the so called All Pay or Charity Auction each bidder makes an uncon!ditional payment� and the item is awarded to the one who makes the highest payment �goback to example on page "�

�

First� because there is only one object to allocate� the probabilities �i�v�� vN�must sum to no more than �� for all valuations �notice� the sale may fail� withpositive probability�

NXi��

�i�v�� vN� � � �budget constraint��

Second� the direct auction has to be incentive compatible� That is� if abidder�s true valuation is vi� reporting the truth must be at least as good asreporting any other valuation #vi �� vi� The utility of an agent whose valuationis vi but who reports the valuation #vi is

U�#vi j vi� � � E ��i�V�� #vi� � � � � VN�vi� ��

�E �Ei�V�� #vi� � � � � VN ��

�The V �s are random variables� vi and #vi particular realizations�� Thereforeincentive compatibility requires that� for all vi� #vi �vi� �vi�

U�vi j vi� � U�#vi j vi� �incentive compatibility��

Third� participation must be voluntary� Therefore� each bidder must beo�ered a nonnegative expected utility� whatever his valuation

U�vi j vi� � �� vi �vi� �vi� �participation constraint��

The programming problem In view of the revelation principle� and theabove feasibility considerations� the optimal auction design problem is reducedto the choice of �N outcome functions ��i� Ei�� i � �� N � so that the seller�sexpected pro�t is maximized

maxf�i�Eig

NXi��

E �Ei�V�� VN ��NXi��

E ��i�V�� VN ��r� ��

subject to the budget �� the incentive compatibility �� and the partici�pation constraints �� What looked like an unmanageable mechanism designproblem has been reduced to a straightforward optimal control problem�

In the following we review and interpret the solution� If you want to followthe mathematical proof you have to look up Myerson�s� beautiful contribution�

�Myerson� R B � �� Optimal auction design�� Mathematics of Operations Research�� "�!��

��

Solution In order to describe the solution� the so called �priority levels associated with a bid play the pivotal role� Essentially� these priority levelsresemble and play the same role as individual customers� marginal revenue inthe analysis of monopolistic price discrimination�

The �priority level associated with the bid vi is de�ned as

i�vi� �� vi � �� Fi�vi�

fi�vi��

We assume that priority levels are monotone increasing in the v�is��

Going back to the Cournot monopoly problem under incomplete informa�tion� reviewed in the introduction �see eq� �� i�vi� can be interpreted asthe marginal revenue from o�ering the item for sale to buyer i at a take�it�or�leave�it price equal to p � vi� Therefore� the monotonicity assumptionmeans that marginal revenue is diminishing in �quantity � where quantity ishere the probability of sale�� Of course� the priority level is always less thanthe underlying valuation vi�

With these preliminaries� the optimal auction is summarized by the follow�ing rules� The �rst set of rules describes how one should pick the winner� andthe second how much the winner shall pay�

Selection of winner

�� Ask each bidder to report his valuation and compute the associated pri�ority levels�

�� Award the item to the bidder with the highest priority level� unless it isbelow the reservation price r�

�� Keep the item if all priority levels are below the reservation price r �eventhough the highest valuation may exceed r��

Pricing rule To describe the optimal pricing rule we need to generalizethe second�price auction rule� Suppose i has the valuation vi which leads tothe highest priority level i�vi� � r� Now ask� by how much could i have

��Notice ��Fi�vi�fi�vi�

is the inverse of the socalled hazard rate Therefore� the assumed

monotonicity of priority levels is satis�ed whenever the hazard rate is not decreasing A suf!�cient condition is obviously the logconcavity of the complementary distribution function�F �v� �� F �v� � �F is called logconcave� if ln �F is concave� This condition holds� forexample� if the distribution function F is uniform� normal� logistic� chi!squared� exponen!tial� Laplace See Bagnoli M und T Bergstrom � �� Logconcave probability and itsapplications�� University of Michigan� Working Paper

��Myerson � �� also covers the case when priority levels are not monotone increasing

��

reported a lower valuation� and would still have won the auction under theabove rules�

Let zi be the lowest reported valuation that would still lead to winning

zi �� minf"v j i�"v� � r� i�"v� � j�vj�� j �� ig� ��

Then� the optimal auction rule says that the winner shall pay zi� This com�pletes the description of the auction rule�

Interpretation Essentially� the optimal auction rule combines the ideaof a �second�price �or Vickrey� auction with that of �third degree monop�olistic price discrimination�� As a �rst step� customers are ranked by theirmarginal revenue vi� ��Fi�vi�

f�vi�� evaluated at the reported valuation� Since only

one indivisible unit is up for sale� only one customer can win� The optimalrule selects the customer who ranks highest in the marginal revenue hierar�chy� unless it falls short of the seller�s reservation price r� In this sense� theoptimal selection of winner rule employs the techniques of third degree pricediscrimination�

But the winner is neither asked to pay his marginal revenue nor his reportedvaluation� Instead� the optimal price is equal to the lowest valuation thatwould still make the winner�s marginal revenue equal or higher than that ofall rivals� In this particular sense� the optimal auction rule employs the ideaof a �second�price auction�

Illustrations As an illustration� take a look at the following two examples�The �rst one shows that the optimal auction coincides with the second�priceclosed�seal bid �or English auction�� supplemented by a minimum price abovethe seller�s reservation price� if customers are indistinguishable� so that there isno basis for third degree price discrimination� The second example then bringsout the discrimination aspect� assuming that bidders are viewed as di�erent�

In both examples� the seller deviates from the principle of selling to thehighest bidder �which underlies revenue equivalence�� and the equilibrium out�come fails to be Pareto optimal� with positive probability� This indicates thatthe four standard auctions are not optimal� However� the optimal auction it�self poses a credibility problem that will be discussed towards the end of thissection�

��Recall� in the theory of Monopoly one distinguishes three kinds of price discrimination�perfect or �rst degree price discrimination� imperfect or second�degree price discrimination bymeans of selfselection devices �o�ering di�erent pricequantity packages�� and �nally third

degree price discrimination based on di�erent signals about demand �for example� di�erentnational markets�

��

Example � �Identical bidders � SIPV model Consider the independentprivate values model� assume r �� and let each bidders� valuation be uniformly distributed on the support �� Then� the priority levels are

i�vi� � vi � �� vi�

� �vi � ��

By the above optimal auction rules� the item is sold if and only if the highestvaluation exceeds �

�� r

�� The winner pays either �

�� r

�or the second highest

bid� whichever is higher� The optimal auction is thus equivalent to a modi�ed�second�price� closed�seal bid� where the seller sets a minimum price equal to�� r

�� Because the minimum price �

�� r

�is higher than the reservation price

r� the seller must sometimes keep the object even if some buyer�s valuationexceeds the reservation price� This ine�cient outcome occurs with probability��r

��N � �� Therefore� the optimal auction does not assure Pareto optimality�If there are only two bidders� and the reservation price is zero� r � �� as

in many of our examples� the optimal minimum price is �� and the expectedrevenue is equal to �

�� Compare this to the Cournot approach� the English

auction� and the participation fee augmented English auction �see examples ��

Example � �Di�ering bidders Assume there are only two bidders� A andB� and suppose r � �� Both bidders� valuation is a uniform random variable�but the supports dier� Let A�s support be the �� and B�s the �� interval�Then� their priority levels are

A�vA� � �vA � �

B�vB� � �vB � ��

Obviously� if the two happen to have the same valuation� bidder A wins� Infact� bidder B can only win if his valuation exceeds that of A by �� Theoptimal 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 notdiscriminated against� he would never report a valuation greater than �� andhence always pay less than � in case of winning� Whereas under the optimalauction rule he may have to pay up to �� to win�

�� Illustration two bidders and two valuations�

This brief introduction to optimal auctions stressed essential properties� andleft out proofs and computations� This may leave you somewhat unsatis�ed�

��The expected revenue is� PrfV�� gE�V�� j V��

�� PrfV��

�� V��g

�� with

E�V�� j V��

�� and PrfV��

��g �

�� PrfV��

�� V��g �

��

��

Therefore� we now add a full scale example with two identical bidders andtwo valuations�� Its main purpose is to exemplify the computation of optimalauctions in a particularly simple case where the price discrimination aspect ofoptimal auctions cannot play any role�

Suppose the two bidders are identical� in the sense that they have either alow �vl� or a high valuation �vh�� These valuations occur with the probabilitiesPrfV � vlg � �� PrfV � vhg � ��

A direct auction is completely described by the probabilities of obtainingthe item �h� �l� and bidders� expected payments Eh� El�

Optimization problem The optimal auction maximizes the expected gainfrom each bidder

max�h��l�Eh�El

G �� Eh � �El� ��

subject to the �incentive compatibility

�hvh � Eh � �lvh � El ��

�lvl � El � �hvl � Eh� ��

and �participation constraints

�hvh � Eh � � ��

�lvl � El � ��

An immediate implication of incentive compatibility is the monotonicity prop�erty

�h � �l� Eh � El� ��

Restrictions due to symmetry Having stated the problem� we now makeit more �user friendly � The key observation is that symmetry implies threefurther restrictions on the probabilities �h� �l�

From the seller�s perspective each bidder wins the item with probability�� h � ��l! in a symmetric equilibrium this probability cannot exceed �

� �Therefore�

�� h � ��l � �

��

Also� in a symmetric equilibrium type h must lose with probability �� or

more whenever he faces a type h rival� Therefore� � � �h � �� or

equivalently

�h � � ��

��

�

��

��The example is borrowed from K Binmore � �� Fun and Games� A Text on Game

Theory D C Heath and Company� pp "��"��

��

And similarly� type l must lose with probability �� or more when he faces a

type l rival� �� l � �� or equivalently

�l � ��

��

Combining these restrictions� the set of feasible probabilities �h� �l is illus�trated in Figure �� Similarly� the set of feasible expected payments Eh� El isillustrated in Figure �� In both diagrams� the dashed lines represent the seller�sindi�erence curves�

Why some constraints do not bind As you can see immediately fromFigure �� only two constraints are binding� the truth�telling condition con�cerning type h� and the participation constraint concerning type l� Therefore�you can ignore inequalities �� and replace the inequalities �� by equalities�

User friendly optimization problem Using these results� you can noweliminate the expected payment variables� and �nd the optimal auction as thesolution of the following linear programming problem

max�h��l

G �� vh��h � �l� � �lvl� ��

subject to ��The only variables are �h� �l�

Solution The solution is easily characterized by the two diagrams� Take alook at the seller�s indi�erence curves �the dashed lines� in Figure �� Evidently�the marginal rate of substitution can be positive or negative� depending onthe prior probability �� In particular� if type l is su�ciently likely �� vh�vlvh

�� the marginal rate of substitution is negative �the dashed indi�erence

curve slopes downwards�� whereas if it is su�ciently unlikely �� vh�vlvh

�� the

marginal rate of substitution is positive �the dashed indi�erence curve slopesupwards�� Therefore� the solution is at one of two corners of the set of feasible��s� depending upon the size of the prior probability �� as summarized in Table��

�� Discussion endogenizing bidder participation�

We close this introduction to optimal auctions with a slightly technical note onan interesting side issue that may have already plagued some readers� This hasto do with the need to endogenize the number of bidders� and the robustnessof results with regard to a more meaningful handling of bidder participation�

��

�hvl �hvh

�lvl

�lvh

�

�

El

Eh��

��

��

��

��

��

��

��

��

��

��

��

��

��

�

El � Eh � vl��h � �l�

El � Eh � vh��h � �l�

�� Eh � �El

� � � � � � � � � � � � � � � � � � � � � � � ��

Figure �� Optimal auction� optimal E�s

��

� � ��

�

�

�l

�h

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

�

�

�

�

�� h � ��l ��

�� vh��h � �l� � �lvl � G

� � � � � � � � � � � � � � � � � � � � � � � ��

Figure �� Optimal auction� optimal ��s

��

� �l El �h Eh� � vh�vl

vh� � �

��

�� vh

� � vh�vlvh

��

��vl

��

��vh � �vl�

Table �� Optimal auction

Recall� in the above optimal auction program it was assumed that all po�tential bidders do actually participate whenever they can be sure that they donot su�er a loss from it� This seems perfectly innocuous� But it has the unap�pealing implication that bidders participate even if they are sure to go emptyhanded� For example� in the symmetric case� the optimal auction entailed aminimumprice above the seller�s reservation value� When that minimumpriceis announced� all potential bidders with valuations below this minimum pricecan be sure that bidding is absolutely pointless for them� Therefore� it wouldmake all the sense in the world to assume that they withdraw from the auctionand do not participate � contrary to what is assumed in the above optimalauction model�� This way� the number of active bidders becomes endogenous�And the question comes up� does this endogenizing of the number of activebidders have an impact upon the optimal reserve price� that is not accountedfor in the alleged optimal auction�

Remarkably� the answer is no� In the SIPV framework the optimal minimum price is independent of the number of bidders�

Proposition � Consider a second�price auction� with a minimum price at orabove the seller�s reservation price� Then� the optimal minimum price p� isindependent of the number of bidders it is implicitly de�ned as the solution of

p� �� F �p��

f�p��

which gives p� � �� if F is the uniform distribution with support ��

Proof The two order statistics V�N�� V�N�� have the following joint density

��This withdrawal would necessarily occur if bidding is costly� no matter how small thecost Of course� in the limiting case of costless bidding� it is also part of an equilibriumcon�guration to have everybody participate� even those who are sure to have no chance ofwinning

��

function�

g�x� y� �

�N�N � ��F �x�N��f�x�f�y� for y � x� otherwise�

��

Let p denote the minimum price� In an English auction the item is sold i�V�N� � p� and the actual price paid is equal to V�N�� i� V�N�� p� and equalto p i� V�N�� p � V�N�� Therefore� the optimal minimum price p� solves

maxp

Z �v

p

Z y

px g�x� y�dxdy � p

Z �v

p

Z p

g�x� y�dxdy� ��

The associated �rst�order condition is

p�Z p�

g�x� p��dx �

Z �v

p�

Z p�

g�x� y�dxdy� ��

Its left�hand�side �LHS� can be rearranged as follows��

LHS �� p�N f�p��Z p�

�N � ��F �x�N�� f�x�dx

� p�Nf�p��F �p��N��

Similarly� its right�hand�side �RHS� is

RHS �� NZ �v

p�f�y�

Z p�

�N � ��F �x�N�� f�x�dxdy

� N F �p��N�� F �p��

Therefore� condition �� simpli�es to

p� f�p�� F �p��

We conclude that p� is independent of N for all N � �� as asserted� Inparticular� p� � �� for all N � if valuations are uniformly distributed on ��

Therefore� even if we correct the usual optimal auction program� and takeinto account that raising the reserve price has an adverse e�ect on the expectednumber of bidders� the optimal reserve price will remain the same�

��The joint density function of the two order statistics V�r�� V�s�� r � s � N � is

frs�x� y� �N #

�r � �#��s� r � �#�N � s�#F �x�r��f�x��F �y� � F �x��s�r��f�y�� F �y��N�s�

if x � y� and equal to zero otherwise See David� H A � �� Order Statistics � Wiley� p� Set r � N � � s � N � and you have the asserted joint density function

��Notice� ddxF �x�N�� N � �F �x�N��f�x�

�

� Common value auctions and the winner�s

curse

In a famous experiment� Bazerman and Samuelson�� lled jars with coins� andauctioned them o� to MBA students at Boston university� Each jar had a valueof %�� which was not made known to bidders� The auction was conducted as a�rst�price closed�seal bid� A large number of identical runs were performed�Altogether� the average bid was %��! however� the average winning bid was%�� Therefore� the average winner su�ered a loss of %��! alas� the winnerswere the ones who lost wealth ��winner�s curse ��

What makes this auction di�erent is just one crucial feature� the objectfor sale has an unknown common value rather than known private values� Theresult is typical for common value auction experiments� So what drives thewinner�s curse! and how can bidders avoid falling prey to it�

Most auctions involve some common value element� Even if bidders atan art auction purchase primarily for their own pleasure� they are usually alsoconcerned about the eventual resale� And even though construction companiestend to have private values� for example due to di�erent capacity utilizations�construction costs are also a�ected by common events� such as unpredictableweather conditions� the unknown di�culty of speci�c tasks� and random inputquality or factor prices� This suggests that a satisfactory theory of auctionsshould cover private as well as unknown common value components�

For simplicity� we now focus on the other extreme� the pure common valueauction� In a pure common value auction� the item for sale has the same valuefor each bidder �just like the jar in the above experiment contains the samenumber of coins for each bidder�� At the time of the bidding� this commonvalue is unknown� Bidders may have some imperfect estimate �everyone hashis own rule of thumb to guess the number of coins in the jar�! but the item�strue value is only observed after the auction has taken place�

To sketch the cause of the winner�s curse� suppose all bidders obtain anunbiased estimate of the item�s value� Also assume bids are an increasingfunction of this estimate� Then� the auction will select the one bidder aswinner who received the most optimistic estimate� But this entails that theaverage winning estimate is higher than the item�s value�

To play the auction right� this adverse selection bias must already be ac�counted for at the bidding stage� by shadeing the bid� Failure to follow thisadvice will result in winning bids that earn less than average pro�ts or evenlosses�

�Bazerman� M and W Samuelson � �� I won the auction but don�t want the prize��Journal of Con�ict Resolution � �� !��

�

A simple framework Consider the following simplifying assumptions��

�A� The common value V is drawn randomly from a uniform distributionon the domain �v� �v��

�A� Before bidding� each bidder receives a private signal Si� drawn randomlyfrom a uniform distribution on �V � � V � �� The unknown commonvalue determines the location of the signals� support� A lower indicatesgreater signal precision�

�A� The auction is �rst�price� like a Dutch auction or a closed�seal bid��

As an illustration you may think of oil companies interested in the drillingrights to a particular site that is worth the same to all bidders� Each bidderobtains an estimate of the site�s value from its experts� and then uses thisinformation in making a bid�

Computing the right expected value Just like in the private valuesframework� each bidder has to determine the item�s expected value� and thenstrategically �shade his bid� taking a bet on rival bidders� valuations� With�out shadeing the bid� there is no chance to gain from bidding� The di�erenceis however that it is a bit more tricky to determine the right expected value�in particular since this computation should already account for the built�inadverse selection bias�

For signal values from the interval si �v � � v � �� the expected value ofthe item conditional on the signal si is��

E�V jSi � si� � si� ��

This estimate is unbiased in the sense that the average estimate is equal to thesite�s true value�

Nevertheless� bids should not be based on this expected value� Instead� abidder 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 theirvarious experiments on common value auctions See for example Kagel� J H and D Levin� �� The winner�s 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 proof of their equilibrium properties� based entirely on the principle of �iterateddominance�� is in Harstad� R M and D Levin � ��"� �A class of dominance solvablecommonvalue auctions�� Review of Economic Studies� "��"�"!"�� The procedure proposedin this paper requires� however� that the highest signal is a sucient statistic of the entiresignal 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 behavior isin thinking one step ahead�

In a symmetric equilibrium� a bidder wins the auction if he actually receivedthe highest signal�� Therefore� when a bidder learns that he won the auction�he knows that his signal si was the largest received by all bidders� Using thisinformation� he should value the item by its expected value conditional on hav�ing the highest signal� denoted by E�V jSmax � si�� Smax �� maxfS�� SNg�

Evidently� the updated expected value is lower

E�V jSmax � si� � si � N � �

N � �� E�V jSi � si��

Essentially� a bidder should realize that if he wins� it is likely that the signalhe received was unusually high� relative to those received by rival bidders�

Notice� the adverse selection bias� measured by the di�erence between thetwo expected values� is increasing in N and � Therefore� raising the number ofbidders or lowering the precision of signals gives rise to a higher winner�s curse�if bidder are subject to judgmental failure� and base their bid on E�V jSi � si�rather than on E�V jSmax � si��

Equilibrium bids The symmetric Nash equilibrium of the common valueauction game was found by Wilson�� and later generalized by Milgrom andWeber�� to cover auctions with a combination of private and common valueelements�

In the present framework� the symmetric equilibrium bid function is��

b�si� � si � � ��si��

where

��si� ��

N � ��e�

N

��si��v��

The term ��si� diminishes rapidly as si increases beyond v � � Ignoringit� the bid function is approximately equal to b�si� � si � � and the expectedpro�t of the high bidder is positive and equal to ��N � ��

��This standard property follows from the monotonicity of the equilibrium bid function�in the present context� it is con�rmed by the equilibrium bid function stated below

�� Wilson� R B � �� A bidding model of perfect competition�� Review of Economic

Studies� �" !" �� Milgrom� P and R J Weber � �� A theory of auctions and competitive bidding��

Econometrica� "�� ! ��Again� we restrict the analysis to signal values from the interval si � �v � � �v � �

��

Conclusions The main lesson to be learned from this introduction to com�mon value auctions is that bidders should shade their bids� for two di�erentreasons� First� because without shadeing bids there can be no pro�ts in a �rstprice auction� Second� because the auction always selects the one bidder aswinner who received the most optimistic estimate of the item�s value� and thusinduces an adverse selection bias� Without shadeing the bid to account forthis adverse selection bias� the winning bidder regrets his bid and falls prey tothe winner�s curse�

The discount associated with the �rst reason for shadeing bids will decreasewith the number of bidders� because signal values are more congested whenthere are more bidders� In turn� the discount associated with the adverse selec�tion bias increases with the number of bidders� because the adverse selectionbias becomes more severe as the number of bidders is increased�

Altogether� increasing the number of bidders has two con icting e�ects onequilibrium bids� On the one hand� competitive considerations require moreaggressive bidding� On the other hand� accounting for the adverse selectionbias requires greater discounts� For low numbers of bidders the competitivee�ect prevails� and the bid function is increasing in N ! but� eventually theadverse selection e�ect takes over� and the equilibrium bid function decreasesin N �

Altogether it is clear that common value auctions are more di�cult to play�and that unsophisticated bidders may be susceptible to the winner�s curse� Ofcourse� the winner�s curse cannot occur if bidders are rational� and properlyaccount for the adverse selection bias� The winner�s curse is strictly a matterof judgmental failure! rational bidders do not fall prey to it�

The experimental evidence demonstrates that it is often di�cult to avoidthe winner�s curse� Even experienced subjects who are given plenty of timeand opportunity to learn often fail to bid below the updated expected value�And most subjects fail to bid more conservatively when the number of biddersis increased�

Similarly� many real life decision problems are plagued by persistent win�ner�s curse e�ects� For example� in the �eld of book publishing� Dessauer�

reports that �� most of the auctioned books are not earning their advances� Capen� Clapp and Campbell�� claim that the winner�s curse is responsible forthe low pro�ts of oil and gas corporations 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 Publishing� Bowker Publ��See Capen� E C and R V Clapp and W M Campbell � �� Competitive 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 corporate specialization�� Brookings Papers on Economic Activities� SpecialIssue on Microeconomics� !�

��

the major corporate takeovers that made the �nancial news during the ��shave tended to actually reduce bidding shareholder�s wealth��

� Further applications

We close with a few remarks on the use of the auction selling mechanismin �nancial markets� in dealing with the natural monopoly problem� and inoligopoly theory� Each application poses some unique problem� This indicatesthat auction theory is still a rich mine of unresolved research problems�

�� Auctions and oligopoly

Recall the analysis of price competition� and the Bertrand paradox ��two isenough for competition�� in the theory of oligopoly� There one usually elabo�rates on various proposed resolutions of the Bertrand paradox that typicallyhave to do with capacity constrained price competition� and that culminatein a defense of the Cournot model� This resolution is successful� though a bitcomplicated � but unfortunately not quite robust with regard to the assumedrationing rule��

A much simpler resolution of the Bertrand paradox can be found by in�troducing incomplete information� This explanation requires a marriage ofoligopoly and auction theory� which is why we sketch it here� in two examples�

Example � �Another resolution of the Bertrand paradox Consider asimple Bertrand oligopoly game� with inelastic demand� and N � � �identical��rms� Each �rm has constant unit costs c� and unlimited capacity� Unlike inthe standard Bertrand model� each �rm knows only its own cost� but not thoseof others� Rivals� unit costs are viewed as identical� and independent randomvariables� described by a uniform distribution with the support ��

Since the lowest price wins the market� the Bertrand game is a Dutch auction� with the understanding that we are dealing here with a �buyers� auction�in lieu of the �seller�s auction� considered before� Since costs are independent and identically distributed� we can employ the apparatus and results ofthe SIPV auction model� Strategies are price functions that map each �rm�sunit cost c into a unit price p�c��

�See also Roll� R � �� The hubris hypothesis of corporate takeovers�� Journal ofBusiness� "�� !� �

��See the ingenious defense of the Cournot model by Kreps� D and J A Scheinkman� �� Quantity precommitment and Bertrand competition yield Cournot outcomes�� BellJournal of Economics� � ��!��

��

The unique equilibrium price strategy is characterized by the mark�up rule

p��c� ��

N�

�N � ��

Nc� ��

The equilibrium price is p��C�� the equilibrium expected price��

�p�N� �� E�p��C�� E�C��

N � ��

and each �rm�s equilibrium expected pro�t

��c�N� �� c�N

N� ��

Evidently� more competition leads to more aggressive pricing� beginningwith the monopoly price p�� and approaching the competitive pricingrule p � c as the number of oligopolists becomes very large� Similarly� theexpected price rule and pro�t are diminishing in N � beginning with �p�� c� with limN�� p�N� � � and limN�� N� � �� Hence� numbersmatter� �Two is not enough for competition��

In order to prove these results all we need to do is to use a transformationof random variables� and then apply previous results on Dutch auctions� Forthis purpose� de�ne

b �� p and v �� c� ��

Obviously� this transformation maps the price competition game into a Dutchauction� where the highest bid b �the lowest p� wins� and b� v �� sincep� c �� By example �� we know that b�v� �� N��

Nv� Therefore�

p� � �� b

� �� N � �

Nv

� �� N � �

N�� c�

��

N�

N � �

Nc� ��

as asserted�

��Notice� E�p��C�� E�C�� is the familiar �revenue equivalence�� which holds for alldistribution functions �If the market were �second price�� each �rm would set p��c� � c�the lowest price would win� and trade would occur at the second lowest price� therefore� theexpected price at which trade occurs would be equal to E�C��

��

Remark � �Why numbers matter By revenue equivalence one has

�p�N� �� E�p��C�� E�C��

Therefore� the �two is enough for competition� property of the Bertrand paradox survives in terms of expected values� Still� numbers matter� essentiallybecause E�C�� and E�C�� move closer together� as the size of the sample Nis increased� The gap between these statistics determines the aggressiveness ofpricing� because the price function has the form p��c� � �� c� with

� ��N � �

N�

� � E�C��

� � E�C��

Example � �Extension to price elastic demand� Suppose demand isa decreasing function of price� as is usually assumed in oligopoly theory� Specifically� assume the simplest possible inverse demand function P �X� �� X�Otherwise� maintain the assumptions of the previous example� Then� the equilibrium is characterized by the following mark�up rule p� � ��

p��c� �� Nc

N � ��

Obviously� p�� and p�� N� �

Compare this to the equilibrium mark�up rule in the inelastic demandframework analyzed in the previous example� which was p��c� � ��N��c

N� Ev

idently� pricing is more aggressive if demand responds to price� Of course� ifthe auction were second�price� bidding would not be aected� since p��c� � cis a dominant strategy in this case�

We conclude� if demand is a decreasing function of price� a �rst�priceauction leads to higher expected consumer surplus� This may explain why�rst�price closed�seal bids are the common auction form in industrial andgovernment procurement�

This is just the beginning of a promising marriage of oligopoly and auctiontheory� Among the many interesting extensions� an exciting issue concernsthe analysis of the repeated play Bertrand game� The latter is particularlyinteresting� because it leads to a dynamic price theory� where agents succes�sively learn about rivals� costs� which makes the information structure itselfendogenous�

�� Natural gas and electric power auctions

As a next example� recall the natural monopoly problem� In many practicalapplications� it has often turned out that the natural monopoly characteristic

��

applies only to a small part of an industry�s activity� In these cases� verticaldisintegration may help to reduce the natural monopoly problem to a mini�mum�

The public utility industry �gas and electric power� has traditionally beenviewed as a prime example of natural monopoly� However� the presence ofeconomies of scale in the production of energy has always been contested� Anunambiguous natural monopoly exists only in the transportation and localdistribution of energy� This suggests that the natural monopoly problem inpublic utilities can best be handled when one disintegrates production� trans�portation� and local distribution of energy�

In the U�K�� the electric power industry has been reshaped in recent yearsby the privatization of production� combined with the introduction of electricpower auctions� These auctions are run on a daily basis by the National GridCompany� At �� a�m� every day� the suppliers of electric power make a bid foreach of their generators to be operated on the following day� By � p�m�� theNational Grid Company has �nalized a plan of action for the following day� inthe form of merit order� ranked by bids� from low to high� Depending uponthe random demand on the following day� the spot price of electric power isthen determined in such a way that demand is matched by supply� accordingto the merit order determined on the previous day�

Similar institutional innovations have been explored� dealing with the trans�portation of natural gas in long�distance pipelines� and in the allocation ofairport landing rights in the U�S�� Interestingly� these innovations are oftenevaluated in laboratory experiments� before they are put to a real life test��

�� Treasury bill auctions

Each week the U�S� Treasury uses a discriminatory auction to sell Treasurybills �T�bills�� On Tuesday the Treasury announces the amount of ��dayand ��day bills it wishes to sell on the following Monday and invites bidsfor speci�ed quantities� On Thursday� the bills are issued to the successfulbidders� Altogether� in �scal year �� the Treasury sold over % �� trillion ofmarketable Treasury securities �bills� notes� and bonds��

Prior to the early ��s� the most common method of selling T�bills wasthat of a subscription oering � The Treasury �xed an interest rate on thesecurities to be sold� and then sold them at a �xed price� A major de�ciency

��See for example Rassenti� S J and V L Smith and R L Bul�n � �� A combinatorialauction mechanism for airport time slot allocation�� Bell Journal of Economics� �� ! ��and McCabe� K and S Rassenti and V Smith � �� Designing �smart� computerassistedmarkets� an experimental auction for gas networks�� Journal of Political Economy� �� "�!��

��For more details see Tucker� J F � ��"� �Buying Treasury securities at Federal Reservebanks�� Federal Reserve Bank of Richmond� Feb ��"

��

of this method of sale was that market yields could change between the an�nouncement and the deadline for subscriptions�

The increased market volatility in the ��s made �xed�price o�eringstoo risky for the Treasury� Subsequently� the Treasury switched to an auctiontechnique in which the coupon rate was still preset by the Treasury� and bidswere made on the basis of price� The remaining problem with this method wasthat presetting the coupon rate still required forecasting interest rates� withthe risk that the auction price could deviate substantially from the par valueof the securities�

In �� the Treasury switched to auction coupon issues on a yield basis�Thereby� bids were accepted on the basis of an annual percentage yield� withthe coupon rate based the weighted average yield of accepted competitivetenders received in the auction� This freed the Treasury from having to presetthe coupon rate�

Another sale method was used in several auction of long�term bonds inearly ��s� This was the closed�seal bid� uniform�price auction method��

Here� the coupon rate was preset by the Treasury� and bids were accepted interms of price� All successful bidders were awarded securities at the lowestprice of accepted bids�

In the currently used auction technique� two kinds of bids can be submitted�competitive and noncompetitive� Competitive bidders are typically �nancialintermediaries who buy large quantities� A competitive bidder indicates thenumber of bills he wishes to buy� and the price he is willing to pay� Multiplebids are permitted� Noncompetitive bidders are typically small or inexperi�enced bidders� Their bids indicate the number of bills they wish to purchase�up to % �� with the understanding that the price will be equal to thequantity weighted average of all accepted competitive bids�

When all bids have been made� the Treasury sets aside the bills requestedby noncompetitive bidders� The remainder is allocated among the competitivebidders� beginning with the highest bidder� until the total quantity is placed�The price to be paid by noncompetitive bidders can then be calculated�

T�bill auctions are unique discriminatory auctions because of the distincttreatment of competitive and noncompetitive bids� Compared to the standarddiscriminatory auction� there is an additional element of uncertainty� becausecompetitive bidders do not know the exact amount of bills auctioned to them�

Another distinct feature is the presence of a secondary after�auction mar�ket� and of pre�auction trading on a �when�issued basis� which serves a�price�discovering purpose� and where dealers typically engage in short�sales�

��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 academicliterature one would never call it Dutch Because of its secondprice quality one wouldperhaps call it �English�

��

In many countries� central banks use similar discriminatory auctions to sellshort term repurchase agreements to provide banks with short term liquidity��

Several years ago� the German Bundesbank switched from a uniform�priceauction to a multiple�price� discriminatory auction� The uniform�price auctionhad induced small banks to place very high bids� because this gave them sureaccess to liquidity without the risk of having a signi�cant impact on the price tobe paid� Subsequently� large banks complained that this procedure put themat a competitive disadvantage� and the Bundesbank responded by switching toa discriminatory auction� However� this problem could also have been solvedwithout changing auction procedures simply by introducing a �ner grid offeasible bids�

In recent years� the currently used discriminatory auction procedures be�came the subject of considerable public debate� Many observers claimed thatdiscriminatory auctions invite strategic manipulations� and perhaps paradoxi�cally� lead to unnecessarily low revenues� Based on these observations� severalprominent economists proposed to replace the discriminatory by uniform�priceauction procedures�

The recent policy debate was triggered by an inquiry into illegal biddingpractices by one of the major security dealers� Salomon Brothers � Apparently�during the early ��s� Salomon Brothers repeatedly succeeded to �corner the market� by buying up to � & of a security issue� This was in violationof U�S� regulations that do not allow a bidder to acquire more than � &of an issue�� Such �cornering of the market tends to be pro�table becausemany security dealers engage in short�sales during the time when an issue isannounced and the time it is actually issued� This makes them vulnerable toa short squeeze�

Typically� a short squeeze develops during the �when�issued period beforea security is auctioned and settled� During this time� dealers already sell thesoon�to�be�available securities and thus incur an obligation to deliver at theissue date� Of course� dealers must later cover this position either by buyingback the security at some point in the �when�issued market� or in the auction�or in the post�auction secondary market or any combination of these� If thosedealers who are short do not bid aggressively enough in the auction� they mayhave di�culties to cover their positions in the secondary market�

��The procedures used by the Federal Reserve are described in The Federal Reserve Sys�

tem� Purposes and Functions Board of Governors of the Federal Reserve System� Wash!ington DC

��Salomon Brothers circumvented the law by placing unauthorized bids in the names ofcustomers and employees� at several Treasury auctions When these practices leaked� theTreasury security market su�ered a substantial loss of con�dence For a detailed assess!ment of this crisis and some recommended policy changes consult the Joint Report on the

Government Securities Market � �� U S Government Printing O�ce� Washington� DC

�

Are discriminatory auctions the best choice� Or should central banks andthe Treasury go back to single�price auction procedures� This is still an ex�citing and important research issue� Already during the early ��s MiltonFriedman�� made a strong case in favor of a uniform�price� closed�seal bid�He asserted that this would end cornering attempts by eliminating gains frommarket manipulation� And� perhaps paradoxically� he also claimed that totalrevenue would go up by surrendering the possibility to price�discriminate� Es�sentially� both claims are based on the expectation that the switch in auctionrules would completely unify the primary and secondary markets� and inducebidders to reveal their true willingness to pay��

References

�� O� Ashenfelter� How auctions work for wine and art� Journal of EconomicPerspectives� ��

�� M� Bagnoli and T� Bergstrom� Log�concave probability and its applica�tions� Working paper� University of Michigan� ��

�� M� H� Bazerman and W� F� Samuelson� I won the auction but don�t wantthe prize� Journal of Con�ict Resolution� ��

�� S� Bhagat� A� Shleifer� and R� W� Vishny� Hostile takeovers in the ��s�the return to corporate specialization� Brookings Papers on EconomicActivity� Special Issue on Microeconomics��

�� S� Bikchandari and C��F� Huang� Auctions with resale markets� An ex�ploratory model of Treasury bill markets� Journal of Financial Studies��

�� K� Binmore� Fun and Games� A Text on Game Theory� D� C� Heath andCompany� ��

�� W� J� Boyes and S� K� Happel� Auctions as an allocation mechanism inacademia� the case of faculty o�ces� Journal of Economic Perspectives��

��See Friedman� M � �� Comment on collusion in the auction market for Treasurybills�� Journal of Political Economy� �� and more recently Friedman� M � �� How to sell government securities�� Wall Street Journal � August ��

�For a discussion of Friedman�s ideas� see Goldstein� H � �� The Friedman proposalfor auctioning Treasury bills�� Journal of Political Economy � �� !�� and Bikchandari�S and C!F Huang � �� Auctions with resale markets� an exploratory model of Treasurybill markets�� The Review of Financial Studies � �� !��

�

�� N� F� Brady� R� C� Breeden� and A� Greenspan� Joint Report on theGovernment Securities Market� U� S� Government Printing O�ce� ��

�� J� Bulow and J� Roberts� The simple economics of optimal auctions�Journal of Political Economy� ��

�� E� C� Capen� R� V� Clapp� and W� M� Campbell� Competitive bidding inhigh�risk situations� Journal of Petroleum Technology� ��

�� R� Cassady� Auctions and Auctioneering� University of California Press��

�� J� Cox� V� Smith� and J� Walker� Theory and behavior of multiple unitdiscriminative auctions� Journal of Finance� ��

�� H� A� David� Order Statistics� Wiley� ��

�� J� P� Dessauer� Book Publishing� Bowker� N�Y��

�� D� Dyer� J� Kagel� and D� Levin� Resolving uncertainty about the num�ber of bidders in independent private�value auctions� an experimentalanalysis� Rand Journal of Economics� ��

�� M� Friedman� Comment on 'Collusion in the auction market for treasurybills�� Journal of Political Economy� ��

�� H� Goldstein� The Friedman proposal for auctioning Treasury bills� Journal of Political Economy� ��

�� D� A� Graham and R� C� Marshall� Collusive bidder behavior at single�object second�price and English auctions� Journal of Political Economy��

�� R� G� Hansen� Auctions with endogenous quantity� Rand Journal ofEconomics� ��

�� M� Harris and A� Raviv� A theory of monopoly pricing schemes withdemand uncertainty� American Economic Review� ��

�� R� M� Harstad and D� Levin� A class of dominance solvable common�valueauctions� Review of Economic Studies� ��

�� J� Kagel� R� M� Harstad� and D� Levin� Information impact and alloca�tion rules in auctions with a�liated private values� a laboratory study�Econometrica� ��

��

�� J� H� Kagel and D� Levin� The winner�s curse and public information incommon value auctions� American Economic Review� ��

�� J� H� Kagel and D� Levin� Independent private value auctions� bidderbehavior in �rst�� second�� and third�price auctions with varying numbersof bidders� Economic Journal� ��

�� S� M� Kendall and A� Stuart� The Advanced Theory of Statistics� volume I�Charles Gri�n ( Co��

�� D� M� Kreps and J� A� Scheinkman� Quantity precommitment andBertrand competition yield Cournot outcomes� Bell Journal of Economics� ��

�� E� Maskin and J� G� Riley� Auction theory with private values� AmericanEconomic Review� ��

�� E� Maskin and J� G� Riley� Optimal multi�unit auctions� In F� Hahn� ed�itor� The Economics of Missing Markets� Information� and Games� pages�� Clarendon Press� ��

�� S� Matthews� Selling to risk averse buyers with unobservable tastes� Journal of Economic Theory� ��

�� R� McAfee� R� Preston� and J� McMillan� Auctions with a stochasticnumber of bidders� Journal of Economic Theory� ��

�� K� McCabe� S� Rassenti� and V� Smith� Designing �smart computer�assisted markets� an experimental auction for gas networks� Journal ofPolitical Economy� ��

�� P� Milgrom� Auctions and bidding� a primer� Journal of Economic Perspectives� ��

�� P� Milgrom and R� J� Weber� A theory of auctions and competitive bid�ding� Econometrica� ��

�� P� R� Milgrom� Auction theory� In T� F� Bewley� editor� Advances inEconomic Theory� pages �� Econometric Society Monographs No� ��Cambridge University Press� Cambridge� ��

�� R� B� Myerson� Incentive compatibility and the bargaining problem�Econometrica� ��

�� R� B� Myerson� Optimal auction design� Mathematics of Operations Research� ��

��

�� R� B� Myerson� The basic theory of optimal auctions� In R� Engelbrecht�Wiggans� M� Shubik� and J� Stark� editors� Auctions� Bidding� and Contracting� pages �� New York University Press� New York� ��

�� M� Plum� Characterization and computation of Nash�equilibria for auc�tions with incomplete information� International Journal of Game Theory��

�� S� J� Rassenti� V� L� Smith� and R� L� Bul�n� A combinatorial auctionmechanism for airport time slot allocation� Bell Journal of Economics��

�� J� G� Riley� Expected revenue from open and sealed bid auctions� Journalof Economic Perspectives� ��

�� M� Robinson� Collusion and the choice of auction� Rand Journal ofEconomics� ��

�� R� Roll� The hubris hypothesis of corporate takeovers� Journal of Business� ��

�� M� H� Rothkopf� T� J� Teisberg� and E� P� Kahn� Why are Vickrey auctionsrare� Journal of Political Economy� ��

�� J� Scott and C� Wolf� The e�cient diversi�cation of bids in Treasury billauctions� Review of Economics and Statistics� ��

�� M� Shubik� Auctions� bidding� and markets� an historical sketch� InR� Engelbrecht�Wiggans� M� Shubik� and J� Stark� editors� Auctions� Bidding� and Contracting� pages �� New York University Press� NewYork� ��

�� P� A� Spindt and R� W� Stolz� The expected stop�out price in a discrim�inatory auction� Economics Letters� ��

�� J� F� Tucker� Buying treasury securities at federal reserve banks� FederalReserve Bank of Richmond� ��

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�� R� B� Wilson� A bidding model of perfect competition� Review of Economic Studies� ��

��

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