the simple economics of optimal auctions jeremy bulow and ...dm121/phd auction... · the simple...

The Simple Economics of Optimal Auctions

Jeremy Bulow and John RobertsStanfordUniversity

We show that the seller’s problemin devisingan optimal auction isvirtually identical to the monopolist’sproblemin third-degreepricediscrimination. More generally,manyof the important resultsandeleganttechniquesdevelopedin the field of mechanismdesigncanbereinterpretedin the languageof standardmicro theory.We illus-tratethis by consideringtheproblemof bilateral exchangewith pri-vately known values.

I. Introduction

An “optimal auction” is abidding mechanismdesignedto maximizeaseller’sexpectedprofit. The literature on optimal auctionshasmush-roomed in recent years (see, e.g., Harris and Raviv 1981a, 1981b;Myerson1981; Riley andSamuelson1981; Milgrom andWeber 1982;Matthews 1983; Maskin and Riley 1984a,1984b)andhasprovidedthebasis for the more generalstudy of efficient mechanismdesign.’

Unfortunately, this field hasbeena difficult one for most econo-mists,seeminglybearinglittle relation to traditional pricetheory.Thispapergreatly simplifies the analysisof optimal auctionsby showing

Much of the work reportedhere was donewhile Bulow was visiting the GraduateSchoolof Businessat the University of Chicago.We wish to thankMilton Harris, PaulMilgrom, Kevin M. Murphy, Roger Myerson, Hugo Sonnenschein,Sherwin Rosen,Bob Wilson, andan anonymousrefereefor valuablediscussionsandcomments.Thefinancial support of the Sloan Foundationand the National ScienceFoundation isgratefully acknowledged.

Examplesof researchemployingmechanismdesign techniquesincludestudiesofbilateralmonopolyandbargaining(MyersonandSatterthwaite1983),multilateralpar-tial equilibrium exchange(Wilson 1985), regulationof a monopolistwithout knowingits costs(Baron and Myerson 1982), and dissolutionof a partnership(Cramton,Gib-bons,and Klemperer 1987).

io6o

OPTIMAL AUCTIONS 1 o6 1

that it is essentiallyequivalentto the analysisof standardmonopolythird-degreepricediscrimination. The auctionsproblem can there-fore besolvedby applyingtheusuallogic of marginalrevenueversusmarginal cost. The samelogic also clarifies thecelebratedresultsonthe revenueequivalenceof variousauctionsand manyof theresultsconcerningbilateral monopoly.

Our primary purposeis not to obtain new results,though we dosomewhatextendearlier work, but ratherto connectexisting resultsto familiar ideas.We hope that this reinterpretationof optimal auc-tion theorywill give readersabetterunderstandingof awide rangeofprivateinformation problemsand,thus,perhapsleadto newinsights.

II. The Optimal Auction Problem

The literatureon optimal auctionsbeginswith the work of Vickrey(1961).He consideredthe following situation.A sellervaluesanitemat zero. Shehas theoption to sell it to oneof n risk-neutralbidders.Each bidder alone knows the value, v, that he placeson the good.Becausethesellerandtheotherbuyersareuncertainaboutthisvalue,it appearsto them to be a randomvariable. It is assumedto be com-mon knowledgeamong all the buyersand theseller that everyoneviews thebuyers’valuesasindependentdraws from acommondistri-bution F(v), with F(v) = 0 andF(iJ) = 1. That is, everyone(includingi) agreesthat theprior probability that v~ is lessthanv is givenby F(v),everyoneknowsthateveryoneknowsthis, everyoneknowsthatevery-one knows that everyoneknows this, ad infinitum. The seller is con-sideringtwo alternativeways to sell the item: a sealed-bidfirst-priceauction,in which the high bidder wins and paysthe amounthe bid,and asealed-bidsecond-priceauction,in which thehigh bidderwinsbutpaysonly theamountbid by thesecond-highestbidder. The issueis, Which of thesemethodsshould the seller use to maximize herexpectedprofit?

Eitherchoiceinducesagameamongthepotentialbuyers,with eachhaving to decidehow much to bid as a function of his valuationforthegood. Vickrey notedthat whenthesecond-priceauctionis used,eachbidder has a dominant strategyof bidding his true valuation.This is becausealteringhis bid from his valuechangestheoutcomeoftheauction in only two cases:whenunderbiddingcauseshim not tobe the high bidder although his value is highest (so he gets a zerosurplusinsteadof a positive one), and when bidding more thanhisvalue causeshim to get thegood butpaymorethan it is worth to him(becausethenext-highestbid exceedshis actual valuation).

Given that eachbidder will announcehis true value in thesecond-priceauction, it is clear that an individual with valuev will win with

1062 JOURNAL OF POLITICAL ECONOMY

probability ~ 1~(v) [F(v)]~~~ 1), which isjust theprobability thatallothers will have values (and bids) below v. It is also clear that hisexpectedpaymentwill be

P(v) wdF~~ 1~(w) — vF(r~ ‘~(v) — j F~’~ 1~(w)dw

and that his expectedpayment,contingenton winning, will be

B(v) • P(v) •f F~ - 1~(w)dw— 1 ~(v) — 1 ~(v)

which is the expectedvalue of the second-highestvaluation, giventhat v is the highest.

With a first-price auction,thestrategicinteractionamongthebid-dersis not trivial; thereis anincentivefor eachbidderto shadehis bidbelowhis value,tradingoff a reducedprobabilityof winning againstalower paymentif he wins. Vickrey showedthat, in fact, it is equilib-rium behavior for a bidder with value v to bid the amount B(v)definedabove.(A largely graphical treatmentof theseresultsis pro-vided in the Appendix.) Vickrey’s celebratedrevenueequivalenceresult follows immediately from B(v) being the optimal bid. Givensymmetric,risk-neutralbidderswith independentvaluations,theex-pectedrevenuefrom the first- andsecond-priceauctionsis the samebecausethe bidder with the highestvalue always wins under eitherauction(note that B(v) is increasing)and thewinner’s expectedpay-ment conditional on winning is thesameamount,B(v), undereitherauction.

Although therewas a steadystreamof work on auctionsover thetwo decadesfollowing Vickrey’s paper,the literatureexpandeddra-matically during the 1980s. A particularly seminal piece—bothfortheresultsobtainedandfor themethodsintroduced—isthat of My-erson(1981).Myersonextendedearlier work in two important ways.The first was to consider the caseof asymmetricbidders, that is,bidderswhose(privately known) valuationsof the objectaredrawnfrom independent,but not necessarilyidentical,probability distribu-tions. Thesedistributionsareassumedto be commonknowledge,sothat all biddersand thesellerknow thedistribution from which eachbidder’svalue is drawn. With resaleexcluded,thisallows theseller todiscriminateamongbidders.Thesecondextensionwasto considerallpossiblewaysof selling thegood ratherthanjust a prespecifiedsetofalternativeauction forms as Vickrey had done.In thiscontext,Myer-son formulated and solved the “optimal auction designproblem”:amongall possibleways of selling the good, which one should theseller use if shewants to maximizeher expectednet revenues?

OPTIMAL AUCTIONS 1063

Myerson’ssolutiongivesan explicit formula for theoptimalauctionwith asymmetricbidders.He alsoextendedVickrey’s revenueequiva-lenceresult to showthat any two mechanismsthat alwaysleadto thesameallocation of thegood (and meet one further trivial condition)would yield thesameexpectedrevenue.Finally, the methodshe in-troducedhavesincebeenwidely appliedto otherproblemsinvolvingprivate information.

Note that the optimal auction designproblem is extremelycom-plex. There area myriad of possiblewaysof allocatingandchargingfor the good: a simple postedprice; the variouscommon forms ofauctions, including first- and second-pricesealed-bidauctions,andascendingand descendingoral auctions (each of which could bemodified by using reservation prices and fees charged for theprivilegeof bidding); lesscommonauctionformssuchasthe“all-pay”auction, in which the highest bidder wins but everyonepays theamounthe bid2—the list seemslimited only by the imagination.

The crucial breakthroughin this researcheffort was to identify amethodof formulating theoptimal auctionproblemthat immenselysimplified its solution.The key insight is to usethe“revelationprinci-ple” (seeMyerson[1981] for an exposition).This statesthatin search-ing for an optimal methodof selling thegood it is sufficient to con-sideronly “direct revelationmechanisms.”In these,biddersareaskedto announcetheir valuationsdirectly, and theseller commitsherselfto usingrules for allocatingthegood andfor chargingthebuyersthatensureboth thatthebuyerswill bewilling to participateandthateachwill find it in his interestto announcehis true valuation. Given thisresult, the problem of selectingan optimal way to sell the good re-ducesto a relatively simpleconstrainedmaximizationproblem:max-imize theseller’sexpectedrevenuesby thechoiceof functionsgivingthe probability of allocatingthegood to eachbuyerandthe paymentto be made by each (both as functions of the announcedvalues),subjectto the“participationconstraints”thateachbidderreceivenon-negativeexpectedsurplus and the “incentive constraints”that it beequilibrium behaviorfor biddersto revealtheir true valuations.

While this is an immensesimplification of theproblem,its solutionis still complex,and the literatureon optimal auctionshasremaineddifficult despitesuch excellentsurveys as thoseof Milgrom (1985,1987, 1989) and McAfee andMcMillan (19b7).

In thenext sectionwe presenta simpleand easily interpretedrec-ipe for constructingtheoptimal auction.This approachconnectsdi-rectly to the standardmonopolyproblemof third-degreepricedis-

2 Lobbying and other rent-seekingactivities and warsof attrition areexamplesof

economicsituationsrelevantly modeledas all-pay auctions.


crimination. In SectionIV, we showthat this recipeactuallyworks byarguingthat Myerson’soptimal auctionproblem is essentiallymathe-matically identical to the price discriminationproblem,and so theirsolutionsarealsoin closecorrespondence.This connectionbetweenthe two problemscastslight on both. In SectionV we then show howformulaederivedin SectionIV for the expectedrevenuesof a mo-nopolistor a seller in an auctionyield a strongrevenueequivalenceresult. Section VI is concernedwith a technical problem corre-spondingto the marginalrevenuecurve’s not beingeverywherede-creasing.SectionVII containsadiscussionof bilateralexchangewhenneitherthe buyernor thesellerknowstheother’svalue.This section

builds on the methodsdevelopedearlier and illustratestheir wider Iapplicability. SectionVIII is a brief conclusion.

III. A Recipefor Solving the Optimal AuctionProblem in the “Regular Case”

Assumethat therearen risk-neutralpotentialbuyersfor agood.Eachbuyeri valuesthegood at an amountv~, knownonly to him, which isconsideredby everyone(including the buyer) to have beendrawnfrom- a distribution with cumulativedensityF~, F~(v~) 0, F~(~~) 1.The draws for thedifferent individuals areindependent.The differ-ing distributions from which buyer values are drawn are commonknowledgeamongall buyersand the seller. The seller has a valueknown to be zero. (Generalizationto the casein which the seller’svalueis a known,positiveamount,vo, is trivial.) Whatsalesmechanismwill maximizethe seller’sexpectedprofits?

As a specificexample,assumethatA’s value is drawnfrom a distri-bution with uniform densitybetweenzeroand 10; B’s value is drawnfrom a uniform distributionbetween10 and30. If thesearetheonlytwo bidders, a traditional English (open, ascending-bid)auctionwould lead to B’s alwayswinning andpayinganexpectedpriceof five(whereA will dropout on average).A minimumbid of 10 will causeBalways to bid 10, raisingexpectedrevenueto 10. A minimum bid of15 will leadto no bidsone-quarterof the time (when B’s value is lessthan 15) and to a bid of 15 three-quartersof the time. Expectedrevenuebecomes¾x 15 — 11.25. Our questionis, How doesoneconstructa mechanismthat extractsthe maximumpossibleexpectedrevenue?The answeris givenbelow.

1. For eachbidder, graphtheinverseof his cumulativedistributionfunction,with valuev on the Y or “price axis” andtheprobability thatthebuyer’svalueexceedsa certainvalue, 1 — F~(v) q, on theX or“quantity axis.” For eachbidder,we thenhavesomethingthat lookslike a demandcurve, with thebidder’s maximumpossiblevalue being

OPTIMAL AUCTIONS io6~

the price at a quantity of zero and the bidder’s minimum possiblevaluebeingthepriceat a quantityof one.

2. From thedemandcurvefor eachbidder drawa marginalrevenuecurve,calculatedthewaywe alwayscalculatemarginalrevenuecurvesfrom demandcurves: Multiply “quantity,” q 1 — F~(v), times“price,” v = FJ 1(1 — q), and take the derivative with respectto“quantity”:

dq , (l—q)+q dq

= FJ’(1 — q) — q— q))

Now expressthis asa function of V:3

MR~(v) = 1 — F2(v) (1)

f~(v)

We define MR~(v) = — ~ for all v < v~.3. Conduct the following “secondmarginal revenueauction”: (a)

Have each bidder announcehis value. (We shall show below thatbidderswill not havean incentiveto lie.) (b) Translateeachbidder’svalueto amarginalrevenue,which we shallassumefor themomentismonotonically increasingin thebuyer’s value. Considertheseller tobea bidderwith a valueandmarginalrevenueof zero.(c) Award thegood to thebidderwith thehighestmarginalrevenue.The pricepaid isthe lowestvaluethebiddercould haveannouncedwithout losing theauction.If no bidder hasa positivemarginalrevenue,thenthesellerwins the auction and there is no sale. If only one bidder’s value

translatesto apositivemarginalrevenue(saythatof bidder 1), then 1getstheitem at apriceequalto his valuev~ at which MRi(v~) = 0, thatis, at the lowest value he could have had and still had a positivemarginal revenue.If more than onebidder announcesa value thattranslatesto a positive marginal revenue,the one with the highestmarginalrevenue(saybidder 1) getstheitem ata pricedeterminedbytakingthesecond-highestmarginalrevenue,sayM2, andchargingthewinner thevalue,MRr ‘(M2), thathe would havehadif his marginalrevenuehad beenM2.

Note that thebidder with thehighestvalue will not always win theauctionbecausehe will not alwayshavethe highestmarginalrevenue.Note too that honestreportingof valuesis a dominantstrategyforbidders. If a bidderwins theauction,theamounthepaysis indepen-

~Thosefamiliar with the optimal auctionsliterature will recognizethis expressionasMyerson’s “virtual utility.” Similar formulae often appearin incompleteinformationanalysesand bearthe interpretationgiven here.

1 o66 JOURNAL OF POLITICAL ECONOMY

V

4) P4)L.

C

L.(U

4V

00

Fic. 1.—Constructionof an optimal auction

dent of his report sinceit is alwaysthe lowestamounthe could havebid and still havewon. Any lie that changestheoutcomeof theauc-tion will reducethe liar’s utility, while any lie that doesnot affect theoutcomealsodoesnot affect theamountpaid by theliar and so doesnot changethe liar’s utility. Finally, note that the value at which abidder’smarginalrevenueequalszerocanbeinterpretedasareserva-tion price for this biddersinceit is the lowestprice he would everbechargedfor thegood,andthereis no point to his biddingif his actualvalue is lessthan this level.

In figure 1 a buyer hasa value distributedbetweenv and V. Thereservationpricefor him is setat r, whichcorrespondsto his marginalrevenueequalingzero.(In thecasein which theseller’svalueis vo> 0,the reservationprice would be set at the point at which the buyer’smarginalrevenueequalsvo.) If this bidder has thehighestmarginalrevenueandthebidderwith thenext-highestmarginalrevenuehasamarginalrevenueof M2, thenthis bidderwill win theauctionandpaya priceofP, thelowestvaluehecould havehadandstill havehadthemaximalmarginalrevenue.The otherbuyerspaynothing. For sym-metric biddersthis is clearly identical to a second-priceauction; forasymmetricbidders,it is possiblefor one bidder to havethehighestvalue and anotherthe highestmarginal revenue,just as in conven-tional price discriminationacrossmarkets.

In thenumericalexampleconsideredabove,buyerA with a valuedistributeduniformly betweenzeroand 10 can be thoughtof as hay-

q=1-F(v)

1(1 - F(v)

)

q=v- f(v)


ing theinversedemandcurvePA 10 — lOqA. His marginalrevenuecurve(in termsof quantity)is PA = 10 —

2OqA. Similarly, B’s demandcurveisPB 30 — 2OqB,andhis marginalrevenuecurveisPB = 30 —

4OqB. Substitutingfrom thedemandfunctions,we seethatthecorre-spondingMR, functions are

MRA(vA) 10~20(l0 vA = 2vA —10,

10 /

= 30 — 40(30 — vB) = 2vB — 30.

ThusMRA =O~vA =5,MRB =O0vB = 15, andMRA = MRB~VA

> vB — 10. Consequently,if vA = 5 andvA > vB — 10, A wins andpaysthe largerof his reservationprice, five, andvB — 10. If vB = 15andvB > vA ± 10, B is thewinnerat thepricemax(15,vA + 10). If vA

K 5 and vB K 15, no saleis made.For example,if A announcedhisvalue asvA = 8, then MRA = 6. If B announcedhis valueasvB = 17,thenMRB = 4. So A would havethehighermarginal revenue,eventhoughvA K vB, andhe would win theauction.He would paysevensinceif vA were seven,then MRA would be four, the actual valueofMRB(vB).

The major remainingdetail is to discusswhat happenswhen themarginalrevenuecurveof anindividual is not monotonicallydecreas-ing in quantity or, equivalently,increasingin his value. This casewillbe discussedin SectionVI.

Finally, it is easy to generalizeto the optimal mechanismfor anauctionin which thesellerhask identicalgoodsto sell, but eachbuyerstill wantsonly one unit. Simply run a “(k + 1)st marginalrevenue~’auction,with eachof the winners (the k individualswith thehighestmarginal revenues)paying the price on his demandcurve corre-spondingto thegreaterof the (k + I)st marginalrevenueand zeromarginalrevenue.Reservationprices remainas before if theseller’svalue of eachitem is zero. The generalizationto a seller who canproduceextra itemsat increasingmarginalcostshouldalsobe clear.Sell k units to thebuyerswith thek highestmarginalrevenues,wherethekth-highestmarginalrevenueis greaterthan themarginalcostofthekth unit and the (k + l)st marginalrevenueis lessthanthecorre-spondingmarginal cost. Eachof thek buyerspays thevalue on hisdemandcurveassociatedwith thegreaterof thekth marginalcostandthe (k + 1)st marginal revenue,in other words, theminimum value

thebuyerwould havehad to haveto qualify to receivea unit.This rule for theoptimalauctionrevealssomeinterestingfeatures.

First, theallocationof the good may notbe optimal ex post: thegoodmay stay with the seller who valuesit at zero when a buyer places

io68 JOURNAL OF POLITICAL ECONOMY

positive valueon it, and, as noted, it may be allocatedto onebuyerwhenanothervaluesit more highly. Second,somesimple compara-tive statics are possible.For example,raising the seller’s valuationfrom zero to v0 > 0 raiseseachindividual’s reservationprice, fromMR ‘(0) to MR7 ‘(vo). Therefore,theprobability of a salefalls. Sup-pose, however, that for a given set of buyers’ values, a sale is stillmade.Then it will be madeto thesamebidderasbefore.Moreover,the price he payswill be affectedonly if the next highestmarginalrevenueis lessthanv0, in whichcasethepricerisesto thebuyer’snewreservationprice. As anotherexample,considerany changein thedistributionF~ of buyer i’s valuesthat decreasesMR~ asa function ofv~. This shift will raisebidder i’s reservationprice. Further, for anyvalueof v~, the shift will decreasei’s probability of winning thegoodand increasethe price he payswhen he wins. In fact, it is easytoconstructexamplesin which a buyer’s demandcurveshifts upwardand yet thebuyer’s probability of winning theauctionandexpectedsurplusareboth decreased.(Analogously,amonopolistmay respondto an increasein demandby reducingquantity sold and consumersurplus.)

This recipe in terms of demandand marginal revenuecurvesisobviously reminiscentof the familiar solution of the third-degreepricediscriminationproblem.Someof thecommonelementsdeservespecialcomment.

1. BecauseMR = — cc for all values below the lower end of thesupport of a buyer’s valuation, buyerscan neverpay less than thelowestvaluesin their distributions.Justasin thestandardmonopolyproblem,if the monopolistknows that no customerhasa value lessthanv, thenevenif sheshoulddecideto sell to every customerin amarket, shestill will not chargea pricebelow v.

2.Justasin themonopolyproblem,goodsareallocatedto buyersinthe priority establishedby their marginal revenues.Just as in themonopolyproblem,thepriceabuyerpaysis thelowestvaluehecouldhavehadandstill endup buying thegood. Finally, no buyerfrom thepartof thedemandcurveor distributionwith negativemarginalreve-nueis evergivena unit, evenif thecapacityconstraintis notbinding.

3. Justasin themonopolypricediscrimination problem,we haveimplicitly ruled out resaleif someoneother than the highest-valuebuyer wins the auction,

4and to avoid the durablegoodsmonopoly

In the monopoly context,we usuallyassumethat if resaleis possible,the bestthesellercando is setthe samepricein eachmarket—equivalentto runningasecond-priceauctionin the auctionscase.However, whenresaleis allowed, the monopoly/auctionsanalogy fails. The reason is that, with monopoly, we assumethat if prices are setdifferently in different markets so that resale will occur, the resalemarket will becompetitive. In the auctionsproblem,resaleinvolvesa problemof bilateralmonopoly.


problem,we haveassumedthat thesellercanmakea promisenot tohold anotherauctionif thereis no salethefirst time.Thepoint hereisnotjust to note thesemajor assumptions,but to emphasizethat foreveryassumptionwe makein oneproblem,we makeacorrespondingassumptionin the other. For example,our analysisof auctionsas-sumesrisk neutrality throughout.The correspondingassumptioninthe theory of monopoly is that either a monopolist’scustomersarerisk neutralor themonopolistis constrainedto simply settinga pricein eachmarket(for, if not, standardmonopolypricing doesnotmax-imize the profits of a monopolist).This linking of assumptionsis aconsequenceof themathematicallinkagebetweenthemonopolyandoptimal auctionsproblems,which we now demonstrate.

IV. Price Discriminating Monopoly andOptimal Auctions5

In this sectionwe first developan unconventionalrepresentationofthethird-degreemonopolypricediscriminationproblem.Despitetheunfamiliarity of the resultingmathematics,thesolutionis, of course,just theusualone.We thenshowthattheoptimalauctionsproblembyMyersoncanbe describedby thesameobjectivefunction andan onlyslightly different constraint,so thecorrespondencebetweenthe twobecomesclear.

Consideramonopolistwho sellsin n different markets.Eachprice-taking consumerin eachmarket is interestedin buying at most oneunit of thegood.Customersin marketi havevaluesbetweenv, andV~,

with F~(v) customershavinga value lessthanor equalto v. TakeF~(V~)— 1, SO that the massof potentialconsumersin eachmarket is one.Marginal revenueis assumedto be downwardsloping in all markets,and themonopolisthasconstantmarginalcostsof zeroup to a capac-ity of Q units, whereQ is a randomvariablebetweenQ and Q thattakeson the value Q with probability h(Q). We assumethat the mo-nopolist cansetprices(and, implicitly, marketquantities)contingenton the realized valueof Q and that sheseeksto maximizeexpectedprofits by her choiceof price in eachmarket.

Given that pricescandependon Q, we definep~(v, 93 as theproba-

Milgrom (1987)discussesauctionswith resaleandwithout reservationpricesin thecasein which all buyers’valuesare commonknowledge.In fact, theno-resaleassumptionisnot necessaryin mostmodelswith symmetricbuyers,such as thoseof Vickrey (1961)andRiley andSamuelson(1981).However,an implicit no-resaleassumptionis madeinMyerson(1981)andin Milgrom andWeber’s(1982)analysisof the “generalsymmetriccase,”which treatsnonindependentvalues.

~This section is relatively technical and might be skimmed or skipped on firstreading.


bility that a buyer in market i with valuev will acquirea unit if themonopolist’s capacity is Q. We shall need this general formulation(with p dependingon Q) in SectionV, wherewe considermarginalrevenuecurvesthatarenoteverywheredownwardsloping. However,for nowp~(v, 93 is oneif themonopolistwith capacityQ will setapricelessthanor equalto v in marketi andp2(v, 93 is zero if themonopolistwill chooseapriceabovev (aslong asbuyersactasprice takers,whichis rational whenthereis an atomlessmassof them, andthemonopo-list setsprices so that total demanddoesnot exceedQ). Further,wedefine

p~(v) Jpi(v, Q)h(Q)dQ

as theunconditionalprobability thata buyer in marketi with value vwill getaunit (or equivalentlythatpricewill belessthanor equalto v).

Sincethe expectedsocial valueof the units sold is the sumof thebuyers’ expectedconsumersurplusand the monopolist’sexpectedrevenues,the monopolistcan be thoughtof as maximizing the ex-pectedsocial valueof theunits sold minus the expectedconsumerssurplus.

The expectedsocial valueof units sold can be written as

J h(93 >j J vf(v)p~(v,

- z=I ~

or, equivalently,

I i vf1(v)p~(v)dv, (2)i~J

wheref~(v) dF~(v)/dv> 0 is the number(density) of customersinmarket i with valuev. Note thatexpectedunit salesin market i equal

j f~(v)p~(v)dv.

By notingthatabuyerin marketi with avalueof v, will receivezeroexpectedconsumer’ssurplus,sincetherecan neverbe any incentiveto set the price in market i below yi, we are able to calculate theexpectedconsumer’ssurplusfrom agraphsuchasthatin figure 2. Inthe figure, we considera simpleexampleof a marketin whichbuyervaluesaredistributedbetweenzeroand 100andin which, with proba-bility .5, themonopolistchargesa priceof 50 and,with probabilityx,shechargesapriceof lOOx or less,.5 <x ~ 1. (This might happen,for


(100) v~

(80) V

(50) y~

0p~ Lv1)

(.50)

FIG. 2

p1 (vi)

1

example,if an optimizing monopolistsold in onemarket with a de-mand curve p~ = 100 — qi and the monopolist’scapacity was uni-formly distributedbetweenzeroand 100.) Thus a buyerwith avalueof 80 can expect,with probability .5, to buy a unit for 50 and, withprobability .3, can expectto buy a unit for a price uniformly distrib-utedbetween50 and80. Soj3~(v) is zero for v < 50 andis .Olv for 100_ = 50. Integratingalong thevalueaxisandnoting that with onemarket p2(v) = p~(v), we see that a buyer with a value of v has anexpectedconsumer’ssurplusof f~ fi~(x)dx.

Using this formula for theconsumer’ssurplusof abuyerwith valuev, we find that total expectedconsumer’ssurplusis

( LYJ h(Q) ~ Jfz(v) p~(x, Q)dxdvdQ- Z] ~

or, equivalently,

> Jfi(v) JPz(x)dxdv.

The monopolist’srevenuescan now be written as (2) minus(3):

1 1 vf~(v)p~(v)dv — I f~(v) J

(3)

(4)


andher problem becomesoneof maximizing (4) subjectto the feasi-bility constrainton salesrelative to capacity,

Z J fi(v)p~(v, Q)dv .~ Q V Q, (5),= I Yi

by choosingpricesin eachmarketor, equivalently,thevalueat whichp~(v, Q) switchesfrom zeroto one.This maximizationis alsosubjecttotheadditional conditionsthat 0 ~ pXv, 93 ~ 1 andthatp, be nonde-creasing in v. This latter conditionreflectstheimpossibilityof findingaprice thatwill makeahigher-valuedbuyertakeaunit lessoftenthana lower-valuedbuyer in thesamemarket.

We now notetwo featuresof (4) and (5) thatwill give this problemsomeof thesimplicity andfamiliarity it deserves.First, sincethemo-nopolist is allowed to make her allocationrule contingenton Q, wecan solvetheoptimizationproblemseparatelyfor eachQ. Therefore,we can simply maximize

ir(QJ = > J vf(v)p~(v,QJdv — > Jf(v) f p~(x, Qjdxdv (4’)

subjectto

I J f(v)pt(v, QJdv ~ Q. (5’)i~-1 !A

Second,defining z(v, x) to bezero for x> v andonefor x =v, we canrewrite the last term in (4’) by noting that

Vj t~

jf~(v) Lvpz(x, L L.Q)dxdv = f~(v)p,(x, Q)z(v,x)dxdv

— J p~(v, Q) jf,(x)dxdv

— [F~(V)— F~(v)Ijp~(v, Q)dv

— j [1 — F1(v)]p~(v, Q)dv.

With this substitution,expectedrevenues(4’) can be expressedas

n 1 —F(v~ 1

ER=Z J [v — ‘ c”’ f~(v)p~(v,Q)dv.

,= I ~‘U L ia(v) J(6)


However, the expressionin bracketsis just MR~(v). Therefore, ex-pectedrevenuesare

I J V~ MR~(v)f~(v)p~(v, Q)dv, (7)i~1 YU

and, subjectto the constraints,thesellerwants to selectthep, func-tions to maximize theexpectedmarginal revenuesof thosereceivingthegood.

Sincef~(v) > 0 enters in both the objective and the constraintlinearly and both are linear in p~(v, Q), thesolution is clear: put asmuchweight as possible(p~(v, Q) = 1) on high valuesof MR~(v) andnoneon low values.SinceMR, is assumedincreasing,the constraintdp~(v, Q)/r3v = 0 is not binding. So for eachQ, themonopolistshould(1) sell to all buyers in eachmarket for whom MR~(v) = 0 if thequantityconstraint(5’) is notbindingand(2) if theconstraintis bind-ing, allocateunits to thebuyerswith thehighestmarginal revenues,by choosingprices to equatethe marginal revenuesof the lowest-valuedbuyersactuallysupplied in eachmarket.

We shall now show that the optimal auctionsproblem solved byMyerson(1981)is virtually identicalto this formulationof themonop-oly problem.In theauctioncontext,n is thenumberof biddersratherthanthenumberof markets,andF~(v) is theprobability that buyeri’svalueis lessthanv ratherthan thenumberof buyersin marketi withvaluesbelow v. Furthermore,in theauctionscasetheprobability thatabuyerwith valuev in marketi will win theauctionmustbewritten asafunctionp~(v1, . , v,~) of thevaluesofall thebiddersratherthanasafunction of his value and thequantity sold. However,defining P~(v)for theauctionsproblemasE(p~(v1, . . . , v~ ~, v, ~ 1,... , v,~)), we stillhave~~(v) asthe unconditionalprobability that the buyerwith distri-butionF, (from marketi) will receivethegood,justasin themonopolyproblem.

Defining S~(v)astheexpectedsurplusof the ith bidderif his valueisv, Myersonnotesthat equilibrium in thegameinducedby anysetofaction rulesrequiresthat

OS~(v) = —

av p~(v) ~ 0. (8)

The reasonis thata buyerwith a valueof v + dv canachieveat leastthesurplusof a buyerwith avalueof v, plus~(v)dv, simply by imitat-ing thebiddingstrategyof abuyerwith avalueof v. Similarly, abuyerwith a value of v can achieveat least S~(v + dv) — ~~(v + dv)dv.Therefore,surplusmustbeincreasingin v at a rateof~~(v) in orderto


induce correct revelationof values, while the revelationprinciplemeansthat correctrevelationcanbe assumedto be induced.

Equation(8) implies that buyerswith lower valueswill havelowersurpluses, so the lowest surplus in any market i will be achievedby abuyerwith value ~,. Since the seller’s expected revenueis theexpectedvalueof thegood to theauctionwinner lessexpectedsurplus,underthe optimal auctionthesellerwill obviouslysetS~(y~) = 0, the lowestvaluepossible.Therefore,thesurplusto a buyeri with valuev, S,(v),equals

0 ± fv[as()] = f p2(x)dx,

just as in themonopolyproblem.Total expectedconsumer’ssurpluscan be written then as (3), if we bearin mind thatp~(v1) is now theexpectedvalueoverv~ = (v1,. . . , v~ ~, v~+ ~,- . . , v,) ofp~(v1,. . .

Myerson’sobjective function in theoptimal auctionsproblemis thenjust (4) or, equivalently,(7), preciselyas in themonopolyproblem.

Theseller in an auctionmaximizes(4) or (7) subjectto thecapacityconstraint

n

i~= I

which saysthat the seller cannotpromiseto deliver more than oneunit. Again, therearealsoconstraintsthat 0 ~ p~(v1, v2, . . . , vi,) = 1andap1(v1, ... , v~,,)/Ov~= 0. Thesecondarisesbecauseit is impossibletodesignamechanism(pickp, functions)thatwill makeabuyer follow astrategythatleadsto a lower probabilityof winning theauctionwhenhe has a higher value. In any given mechanism,buyers trade offraising their probability of receivinga good againstraisingexpectedpayments. If a low-value buyer finds it worth paying some extraamount to raise the probability of receiving the good, then a high-valuebuyer,whosehigher valuationmakesthe increasedprobabilityeven more valuable,would certainly do so as well. Therefore,if alltypes of buyers maximize utility, a higher-value type will alwayschooseastrategythatleadsto at leastashigh aprobability of receivingthegood as a lower-valuetype.

Thus the mathematicalproblemfacing theauctiondesigneris es-sentially equivalentto themonopolist’sproblem.Thereare two dif-ferences:in themonopolyproblemthechoicevariablesaretheproba-bilities p~(v~, Q), while thosein themonopoly problemarep~(v1, . . .

vs); thecapacityconstraintin the monopolyproblem is


while the corresponding constraint in the auctions problem is~ p~(v1, . . . , v~) ~ 1. Note, however,that eventhesedifferencesarelessthan what might appear.First, theargumentsof thep, functionsotherthanv2—namely,Q andv— i—areboth randomvariablesdistrib-utedindependentlyof v~. Second,whenwe considerp~, thesevariablescan be taken as given and known: Q by assumptionandv~ becausethe other agentswill be inducedto revealtheir valuationscorrectly.Third, as noted, both constraintsbeara capacityinterpretation. Infact, the only important differenceis the absenceof thef~(v) in theauction constraint.But eventhis doesnot changethe natureof thesolution.

To solve theauctionproblem,substitutein (4) asaboveto obtain

Iandnow write this as

n -

~ f. J MR~(v~)fi(v1). . .f~(v,~)p~(vi, . . . , v~)dv1 . . . dv,~. (10)

This formulation makesclear the linearity of the objective functionin p~.

Thustheexpectedcontribution to profit from increasingtheprob-ability of allocatingthegood to buyer i wheneverthevectorof buyervalues is (v1, . . . , v,~) is the probability that buyerswill haveexactlythosevalues,H~1 f(v2), times MRS. Combinedwith (9), this obviouslymeans that, for each vector (v1, . . , v,~), profits aremaximizedbyallocating thegood to thehighestmarginal revenuebidderwith prob-

ability one in all cases,as long as the highestmarginal revenueisgreaterthan zero. As in the monopolyproblem,MR~(v~) is indepen-dentof thep functions and of thevaluesof anybidder other than i.

Finally, we note that the derivationleading to (10) is valid for allmechanismsthat meetincentivecompatibility and participationcon-straints, regardlessof whethersuch a mechanismis optimal. Theparticipation constraints’ being binding was usedonly in showingS~(v~) = 0. The implication is that the expectedrevenuefrom anymechanismcanbe written as

ER= >3 f_ MR,(v)f~(v)~~(v)dv— Ka ___

(11)= expectedMR of winning bidder — K,

whereK is thesumof theexpectedsurplusesof then biddersif eachhad the lowest value in the supportof his distribution. Obviously a


monopolistor auctionsellerwill neverchooseto give buyerswith theminimum possiblevalueanyconsumersurplus,andsohewill setK =

0. If we restrict ourselvesto mechanismsfor whichK = 0, aswe shalldo for the remainderof the paper, it is clear from (11) that anymechanismis an optimal auction mechanismif it alwaysallocatesthegood to thebidder with thehighestpositivemarginal revenue,withno sale occurringwhenall biddershavenegativemarginalrevenues.

Note that this solutiondoesnot specifyhow to chargefor thegoodin theauction,just how to assignit. Of course,whatevermethod is

usedto determinepaymentsmust meet the incentiveand participa-tion constraints,so thateachbidderwill reporthis valuehonestlyandwill participate no matter what his value. One such is the systemdiscussedbefore of charging the lowest value that thewinner couldhaveannouncedandstill havehadthehighestpositivemarginalreve-nue: it givesexpectedsurplusof zero to a bidder with valuev, =

eachbidder’s expectedsurplusis anincreasingfunctionof his valua-tion, and,asarguedabove,it hasthepropertythat no onecangainbymisrepresentation.However, we shall see in the next section thattheremay be otheroptimal paymentschemes.

To summarize,just asin themonopoly problem,theseller in theauctionhasan objectivefunction that can be interpretedasmaximiz-ing the expectedmarginal revenueof the consumerswho receiveunits subjectto a capacityconstraint.Just as in monopoly,the prob-lem is solved by allocating the unit(s) to the buyer(s)with the highestmarginal revenue(s).Finally, the solutionsto the two optimizationproblemsaresimilar: in both, unitsareallocatedto buyerswith prior-ity basedon marginal revenueuntil either quantity is exhaustedorthereareno buyersleft with positive marginalrevenue.

Oneother essentiallytechnical point remains:Becauseweassumedthat marginal revenuewasdownwardsloping in quantity within eachmarket, we could be sure that if the monopolist wants to sell to aparticularbuyer in any givenmarket,sheshouldalsowant to sell to allhigher-valuebuyers.What if marginalrevenueis notdownwardslop-ing? According to thecurrentsetupof theproblem,onewould like tosell to somebuyers with a lower value but higher marginal revenuebefore otherbuyerswith higher valuesbut lower marginal revenues.But higher-valuebuyerswill alwaysbewilling to buy wheneverlower-valuebuyerswant to buy, so it is impossibleto make the probabilitythat a high-value buyer will acquirea good less than the probabilitythat a low-valuebuyer will acquire the good. Mathematically,if mar-ginal revenue is not always downward sloping, the constraint that

= 0 becomesbinding. This constraintis the concernof Sec-tion VI, where we explain what Myersoncalls the “general case.


V. RevenueEquivalence

Weare now ready to explore further the single most important resultin Myerson’s paper, the calculation of the expected revenue from theoptimal—or any other—auction. This result in turn yields a strongrevenue equivalence result.

Consider a monopolist with a random capacity constraint who sellsin several markets. Then, as shown in the preceding section, regard-less of what algorithm the monopolist usesfor determining prices6(and thereforequantities)in eachmarketasa function of total avail-ablequantity, expectedrevenuecan be written as

i~I -~

wherep~(v) is the probability that a buyer in market i with valuev willbe assigneda unit of the good and f~(v) is the density of buyers inmarket i with value v. That is, expectedrevenuecan be found bysumming,overall buyersin all markets,themarginalrevenueofeachbuyer times the probability that thebuyerwill receivea unit. (Whenprobabilitiesareall oneandzero, this isjust sayingthat total revenueis the sum of the marginal revenues of all units sold.) This formula,which is just (7) averagedover all possiblequantityconstraints,holdsregardlessof whetherthemonopolistoptimizesor not.

Another way of restating(12) is that expectedrevenueequalstheexpectationof the product of the numberof salesand the averagemarginal revenueof those who buy. For auctions we think of thesellerasabuyerwith amarginalrevenueof zero.This implies that thenumberof salesis alwaysone andthe averagemarginal revenueofthosewhobuy isjust themarginalrevenueof theonewinning bidder.Therefore,we caninterpret (12)assayingthat theexpectedrevenuefromanyfeasibleauction is the expectedmarginal revenueof the winningbidder.

This formula is central. Most obviously, it makesclear why theoptimal auction gives the good to the bidder with the highestmar-ginal revenue.Moreover, it carriesimmediateimplications for com-paring expectedrevenuesacrossauctions.

If thewinner underonesuchauctionrule is alsothewinner undera secondauction rule, then the two auctionswill yield the sameex-pectedrevenue.Therefore,with symmetricbuyers,first- andsecond-

6 Provided sheneverchargesany buyer lessthan the lowest value he could possibly

placeon the good.The correspondingqualificationin interpretingthis equationin theauction context,which we shall do below, is that eachbidder i receivezero expectedsurpluswhen v, = v

2.


priceauctionsalwaysyield thesameexpectedrevenue:sincethebid-der with the highest value will always win in both auctions, theexpectedmarginalrevenueof thewinning bidderis thesamein both.It is alsoclearthat theoptimalreservationpriceoccurswhenMR = 0for eachbuyerand that this reservationprice is independentof thenumberof bidders.Furthermore,anymechanismthatalwaysawardsthe good to thehighestmarginalrevenuebuyer(including theselleras a buyer with MR = 0) yields thesameexpectedrevenue.There-fore, suchamechanismis optimal. For symmetricbidders,both first-and second-priceauctions with appropriately chosen reservationpricesareoptimal, providedmarginalrevenueis downwardsloping.

If buyersarenot symmetric, thenfirst- andsecond-priceauctionswill not always have the samewinner becausesometimesa lower-valued buyer can win a first-price auction,while the highest-valuedbidder always wins under the second-priceauction. Therefore,ex-pectedrevenuesarealsogenerallydifferent. While neitherauctionisoptimal,whetherafirst-priceor second-priceauctionwill havehigherexpectedrevenuefor asymmetricbuyersdependson whether,whena buyerwins the first-priceauctionwithout having thehighestvalue,that buyeron averagehasa higher or lower marginal revenuethanthebuyerwith the highestvalue.

VI. Optimal Auctions in the General Case:“Ironing Out” the Marginal RevenueCurve

In thissectionwe dealwith thetechnicalproblemof how to designanoptimal auctionwhenv — {[ 1 — F~(v)]/f(v)} is not necessarilymono-tonic. Myerson’ssolution of this generalcaseagain turns out to beequivalentto the solution to the monopoly problem when marginalrevenueis not downwardsloping. We begin by discussingthis latterproblem.

Consideraconventionalmonopolistwhosemarginalrevenuecurveis not always downward sloping. For example, figure 3 graphsthemarginalrevenuecurvefor a monopolistfacingademandcurvep =

lOO—2qforO=q=2Oandp=70—.5qfor2o=q=loo.ThenMR= 100 — 4q for 0 = q ~ 20 andjumpsdiscontinuouslyto MR = 70 —

q for q> 20.Supposethatthefirm hasconstantmarginalcosts.In our numerical

example,if costsexceed50 per unit or are lessthan 20 per unit, themonopolist’sproblemis simple:Thereis only onepoint at which MR— MC. However,if costsarein themidrange,theproblemis harder.In figure 3, wherewe assumeconstantmarginalcostsof 40, the mo-nopolist would like to collect the marginal revenueof the first 15buyers, who have values greater than 70 and marginal revenues


6

50

40

MR,MC 30

20

10

0

0 10 15 40FIG. S.—”Ironing” a marginalrevenuecurve

greaterthan40, skip thenext five buyerswith valuesbetween70 and60 and marginal revenuesbetween40 and 20, and then sell to thebuyers with values between 60 and 55 and marginal revenues between50 and40. Unfortunately, it is impossibleto devisea mechanisminwhich a buyer with a value between55 and 60 will receive a goodmoreoften thanabuyerwith a valuebetween60 and 70. In termsofSectionIV, we facea constraintthat ap1/av = 0)’

In the caseof figure 3, the losseson units 15—20 (areaA) exactlyequal the profits on units 20—30 (areaB). So the monopolistwithconstantmarginal costs of 40 is indifferent to selling 15 units at apriceof 70 or 30 unitsat a priceof 55. If costswere greaterthan40,areaA would be greaterthanB andthefirm would producelessthan15; with costs less than 40, the firm would producemore than 30units. So the firm operatesas though it is facing an “ironed-out”marginal revenuecurvethat hasMR = 40 for 15 = q ~ 30.8

What if thefirm haszeromarginalcostsup to 20 units andthenhitsa capacityconstraint?Ideally, the monopolistwould like to sell to the14 buyerswith the highest reservationprices, skip six buyers,andthen sell to the next six. However,this is impossiblebecauseof theap~/av= 0 constraint.Thebestthemonopolistcan do is sell to the top

‘ Myerson establishesthis condition as a necessarycondition for obtainingcorrectrevelation.

8 If marginal costs equaled40, a monopolist facing the ironed marginal revenue

curvewould be indifferent to selling anyquantitybetween15 and30. A firm facingthemarginalrevenuecurve in fig. 3 will also be indifferent to anyquantity from 15 to 30,but only if it usesthe lottery proceduredescribedin the next two paragraphs.

MR = 100-4q~6

MR70-q

20 25 30 35quantity

io8o JOURNAL OF POLITICAL ECONOMY

15 buyers and give all the buyers on the ironed part of the marginalrevenuecurvea V3 chanceof receivinga unit, throughalottery,whichwill make the shadowvalue to the monopolist of the last five unitsequal to the ironed marginal revenueof 40, times five.9

The sellerwishesto sell to the top 15 buyersand give eachof thenext 15 a 1/3 chanceof buying. Shethereforegives buyersa choice:eitherparticipatein a lottery or payahigher priceandacquirea unitfor sure.In orderto attractthethirtieth-highest-valuedbuyerinto thelottery, a winnerof thelottery mustbe givenaunit for his valueof v= 70 — .5(30) = 55. The priceof gettinga unit for suremustbejustlow enoughso that thefifteenth buyer,with avalueof 70, will decideto buy. That buyercan havean expectedsurplusof Y3(70 — 55) = 5from enteringthe lottery since5A5 of all lottery participantswill win.Therefore, he must be offered a certain unit at a price of 65 toguaranteethesamesurplus.

By runningthis lottery, themonopolistearns65 x 15 + 55 x 5 =

1,250versus60 x 20 = 1,200 if shejust chargesa market-clearingprice of 60. The differenceof 50 (areaA in fig. 3) is just thediffer-ence, for q = 20, of the areaunder the ironed marginal revenuecurve and abovethe original marginal revenuecurve.

Myerson’sanalysisof optimal auctionsin thegeneralcaseallows forthe possibility that marginal revenueis not downwardsloping. Hissolution irons the marginal revenuecurve for each buyer just asabove,althoughthe exposition is in terms of the total revenueas afunction of v rather than marginalrevenueas a function of q.

With ironedmarginalrevenuecurves,thereis someprobability thatone or moreof the top two buyerswill be on the ironed partof thecurve. For example,consideran auction with symmetricbuyers,foreachofwhomF(v) = .02(v — 20),20 = v =60, andF(v) = .5 + .005vfor 60 ~ v ~ 100. In termsof our analogy,eachof thesebuyershasademandcurve identical to the onein figure 3, with quantitiesmulti-plied by .01. Buyerswith valuesbetween55 and70 would thenbeonthe ironed segment.If both buyers’ valuesput them on the ironedpart, Myerson specifies that a lottery occur, with the winner of thelottery paying theequivalentof 55 in theexampleabove.

If only thesecond-highestmarginal revenuebidder is on the flatpartof thecurve,thebidderwith thehighestmarginalrevenuepaysaprice thatwould havemadethe lowest-valuebuyeron thepartof themarginalrevenuecurvebeforetheflat (thebuyerwith thevalueof 70in our example)indifferent to buying and participatingin a lottery.

~The readershould be able to confirm that changingthe lottery to averageovermoreor lessthan the 15 buyerson the ironedpart of the marginalrevenuecurve willreducethe averagemarginal revenueof the lottery participants.

OPTIMAL AUCTIONS io8i

Note that while Myerson specifiesthat the lotteriesbe “fair,” thereisno needfor this: Theauctionslotterycouldalwaysawardthegood tothebuyerwhosenamecamefirst in thealphabet,asystemBulow likesbetter than Roberts. Becausethe ironed marginal revenueof allbuyerswho entera lottery is the same,discriminationamongthesebuyersdoesnot alter either theexpected(ironed)marginal revenueof thewinning bidder or theexpectedrevenuefrom the auction.’0

VII. Bilateral Exchangewith Private Values

The solution methodsdevelopedin the optimal auctions literatureare widely applicableto problemsinvolving private information. Inthissectionweillustratehow themarginalrevenue/marginalcostlogicdevelopedabovecan makethesolutionto suchproblemsmoreintui-tive. We focuson theproblemof socialefficiency in bilateralmonop-oly whenboththebuyer’sandtheseller’s valuationsfor theobjectareprivate information.

Note that the “optimality” of optimal auctionsrelatesto the max-imization of seller surplus rather than social efficiency.” However,designinga socially efficient auction mechanismwhen the seller’svalueis known is easy:asecond-priceauctionwith areservationpriceequalto theseller’s valuedoesthe trick, andif both sellerandbuyervaluesareknown, theneventhemechanismthat is privatelyoptimalfor the seller (essentiallychargingthe highest-valuedbuyer his fullvalue,assumingthat it exceedstheseller’svalue) will be socially effi-cient.However,whenboth buyerandsellervaluesareprivate knowl-edgeandit is unclearwhethergainsfrom tradeexist,it is not possibleto designa mechanismthat results in transfer of the good exactlywhenthegains from tradearepositive. This resultwas first shownbyMyerson and Satterthwaite(1983). In this sectionwe use the logicdevelopedearlier to derive both their impossibility result and theirmechanismfor maximizing expectedgains from tradesubjectto theconstraintsarising from theprivate informationaboutvaluesandtheconcomitantopportunitiesto misrepresentthem strategically.

Assumethata singlebuyerhasavalue,v, drawnfrom adistributionF(v), F(v) = 0, F(V) = 1, dF/dv = f(v) > 0, for v = v =V. The potentialseller has a value, s, drawn independentlyfrom a distribution G(s),

10 It shouldbe notedthat PrattandZeckhauser(1986)treatpreciselythis problemin

the monopoly contextand presentthe ironing procedure.Earlier, Mussaand Rosen(1978) werethe first to fully developthe ironing procedurefor marginal revenueintheir analysis of product quality. And Hotelling (1931), in discussingmonopoly inexhaustibleresources,also did a numerical exampleof ironing a marginal revenuecurve, although he did not usethe then-newterminology of “marginal revenue.”

~ We should note, however,that in Vickrey’s original work he was primarily con-cernedwith issuesof social efficiency.


also on [v, V], with dG/dS= g(s) > 0 for v ~ s =V. Thatis, therangeofpossible values is the same for the buyer and seller. Gains from trademay or maynot be possible, depending on the actual values of s andv.Is there any mechanism that can guarantee ex post efficient trade,and, if not, how can expectedgainsfrom tradebe maximized?

The problem of whetheran efficient mechanismexistsis mosteas-ily solved by creating a fictitious, risk-neutral broker whose job it is tomaximize expected profits subject to the constraint of having tradetake place if and only if v = s. If the expected profit-maximizingmechanism, which maximizes expected buyer payments less expectedseller receipts, yields negative expected profits, then clearly the buyerand seller cannot achieve efficiency on their own (e.g., without a sub-sidy from some third party).

The broker’s job is considerably simplified by her ability to choosemechanisms that make the buyer’s payment contingent on the seller’svalue and similarly make the seller’s revenue contingent on thebuyer’s value, while ensuringthatbotharereportedcorrectly.There-fore, the broker can solve separatelythe problems of maximizingexpected revenue subject to efficiency constraints contingent onknowing the seller’svalue,andminimizingexpectedpaymentsto theseller contingenton knowing the buyer’s value, with the broker’sprofit being the difference betweenbuyer paymentsand seller re-ceipts.

From (4’) we find that thebroker’s objective function when facedwith onebuyercan be written as

max J vf(v)p(v, s)dv — j f(v) J p(x, s)dxdv

or, equivalentlyfrom (7),

max f MR(v)f(v)p(v,s)dv, (13)

where

MR(v) = — 1 — F(v

)

f(v)

subject to the efficiency constraints

p(v,s) = 1 ifv=s, (14)

p(v,s) = 0 if v<s. (15)

The constraint that trade must always and only occur when thebuyer’s value equals or exceeds the seller’s renders the maximization


trivial. Simply put, if the seller’s value is s, thebrokercando no betterthan offer the good to the buyer at a take-it-or-leave-itprice of s,therebygiving the buyerall thegainsfrom trade.

But thebrokermustalsoinducethesellerto sell whenevers= v. Ifwe regard the broker as a monopsonistrelative to the seller in thesameway sheis a monopolistrelativeto thebuyer,thebroker’sprob-lem with theseller is to minimize

J MC(s)g(s)p(v,s)ds, (16)

where

MC(s) = s + G(s

)

g(s)

subject to

p(v,s) = 1 ifs~v, (17)

p(v,s)=0 ifs>v. (18)

The simplest—andessentiallythe only—way of meeting the effi-ciency constraint of having the seller agree to sell whenever s ~ v is tooffer theselleratake-it-or-leave-itpriceequalto v, giving all thegainsfrom tradeto theseller.

Combining the buyer and seller results, we find that the brokermust give each party all thegains from trade.To do this, thebrokermustrun alossequalto theexpectedgainsfrom trade,implying thata subsidyof like amount is requiredif efficiency is to be achieved.

Thus ex postefficiency cannotbe achievedin bilateral tradewithprivate information on both sidesif thebuyer’sandseller’srangesofpossiblevaluescoincide. Further,generalizationto the caseof over-lapping butnot identical rangesfor buyerand sellervaluesis simple.If buyerscan havevaluesas high asb> V, the highestpossiblevaluefor s, we know that all suchbuyerswill haveto be treatedlike buyersof value V: they can alwaysimitate thosebuyers,who will receivethegood with probability one, and so there is no way to preventsuchmimicry. Similarly, if sellerscan havevaluesaslow ass < v, all sellerswith valuess < y areindistinguishable,for thepurposeof thisanaly-sis, from sellerswith valuesv.

If full ex post efficiency is unattainable,what then is the bestthatcan be done?

The mechanismthat maximizesefficiency subject to a feasibilityconstraint is derived in two parts. First, we find the most efficientmechanismthat givesthe brokerzeroexpectedprofits. Second,stan-


dard mechanism design tricks are used to create an equivalent mecha-nism that yields the broker zero actual (rather than expected) profits,allowing the buyer and seller to dispense with the broker altogether.

Expectedgains from tradecanbe written as

jj. (v — s)f(v)g(s)p(v,s)dvds. (19)

Combining(13), which gives thebuyer’sexpectedpayments,and(16),which gives the seller’s expected receipts, yields an expected profitconstraint of the form

f: £ [MR(v) — MC(s)]f(v)g(s)p(v,s)dvds= 0. (20)

This inequality was obtained by Myerson and Satterthwaite as neces-sary and sufficient for optimality of a mechanism that calls for tradewith probability p(v, s).

The problem of solving (19) subject to (20) is just a simple Ramseypricing problem. The solution is familiar: specify first that trade willoccur whenever MR> MC. For cases in which MR< MC, assignpriority based on the ratio of (v — s)/(MC — MR) (“efficiency” gainedto “profits” lost). Allow trade in as many cases as possible using thispriority scheme up to the point at which (20) equals zero.’2 A domi-nant strategies mechanism that achieves the result begins by specify-ing that trade will occur when and only when the ratio of (v — s)/(MC— MR) exceeds the critical level corresponding to (20) equal to zero.Each participant is asked to reveal his or her true value. If trade doesnot occur, no money changes hands. If trade occurs, the buyer paysthe minimum value he could have announced, given the seller’s value,for trade to have occurred. Similarly, if trade occurs, the seller re-ceives the maximum value she could have announced, given thebuyer’s value, for trade to have occurred. Note that, no matter whatthe values of v and s, both the buyer and the seller receive nonnega-tive surplus, so the participation constraint is met. Further, now-familiar arguments show that correct revelation is a dominant strat-egy for each party, so the incentive constraints are also met.

Note that if we have chosen p(v, s) in such a way that (20) equalszero, the expected receipts of the seller exactly equal the expectedpayments of the buyer. However, there is no guarantee that actual

12 If the distributions of buyer and sellervaluationsare not ~welI behaved”sothat

marginal revenueand marginal cost curves are not downward and upward sloping,respectively,we must usean ironing techniquesimilar to that employedin Sec.V.


receiptsand paymentsareequalfor all v and s. Thus we still must gofrom this mechanismto onein which actualseller receiptsequalac-tual buyerpayments.

To do this, we createanothergamewith a BayesianNashequilib-rium outcomethat is the sameas the equilibrium in the dominantstrategies game. Wehave the buyer write down his expected paymentin the dominant strategy game, inferring his value from what hewrites. Similarly, the seller writes down her expected receipts, and weinfer her value from what she writes. Finally, we calculate the ex-pected receipts of the seller unconditional on her value, which equalsthe expected payments of the buyer unconditional on his value.

Our new mechanism specifies payments by the buyer equal to whatthe buyer has written plus what the seller has written, less the un-conditional estimate of payments/receipts. The good is transferredwhenever the inferred buyer and seller values would imply trade inthe earlier game. Note that the expected payments of any buyer areexactly those in the game with dominant strategies, since from hispoint of view the expectation of the seller’s expected receipts condi-tional on the seller’s value equals the unconditional estimate of theseller’s receipts. The same reasoning for the seller implies that bothparties will have the same expected payoffs from any announcementof their value that they have in the game with dominant strategies,provided that the other party always truthfully reveals his or hervalue. Therefore, buyers and sellers announce the same values as inthe dominant strategies game, and we have created a mechanism thatmaximizes efficiency subject to the constraint that buyer paymentsequal seller receipts.

An example of a mechanism that meets the requirements of thissection was given by Chatterjee and Samuelson (1983) for the casein which both buyer and seller have values drawn from the sameuniform distribution. Have each write down a price. If the seller asksfor more than the buyer bids, no trade occurs. Otherwise, trade oc-curs at the average of the bid and ask prices. In this uniform andidentical distribution case, the Chatterjee-Samuelson mechanismleads to the identical payoffs and outcomes as the revelation mecha-nism described in the last few paragraphs.

VIII. Conclusion

Wehave shown how some of the central results of the theory of mar-kets under incomplete information can be interpreted and under-stood in terms of familiar microeconomic theory. Wehope and expectthat this will both broaden the impact of these results and lead to abetter understanding of other problems of this nature.


Appendix

RevenueEquivalencein Auctions with Symmetric Bidders

This Appendix focuseson a graphical treatmentof the resultsof Vickrey(1961) discussedin SectionI. We begin by analyzinga second-priceauction.The dominantstrategyfor eachbidder is to bid his true value.This is becausea buyer’saltering his bid from its true value, v, hasno effect on whetherhewins and what hepays,exceptin two cases,and in theseit is harmful. In thefirst case,thehighestotherbid, b, exceeds v and the buyer bids morethan b.Thenhe getsa surplusof v — b < 0 insteadof thezerohewould havegottenfrom bidding v. In the second,b is lessthan v and the buyer underbidsbyenoughthat hisbid is lessthanb. Thenhegetszerosurplusinsteadof v —

0. Therefore,over- or underbiddingcanreducebut cannotincreaseabuyer’ssurplus.

Givencorrectrevelation,the expectedpaymentsandprofits of abidderin asecond-priceauctioncan be seenin figure Al. This figure graphsa bidder’spossiblevaluationof theauctionedgoodagainstthelikelihood of winning theauction. In the specialcasein which there are only two bidders, eachwithvalues independentlyand uniformly distributed betweenzero and 100, thecurvebecomesa straight line from (0, 0) to (1, 100). With symmetricbidders,abidder with value v will win the auctionwheneverhis value is highest,andthat will occurwith probability [F(v)]~ F’~ ‘(v); thatis, the probability thatthatbidder will win is theprobability that theothern — 1 bidderswill all haveavalueof v or less. In thenumericalexample,abidder with avalueof v willhavea probability of winning of .Olv.

V

V

C)

L..C)>1

V

0

Probability of winning theauction0 Fn-l(V) 1

FIG. Al


With eachbidderbiddinghis true value,theexpectedpaymentofa bidderwith valuev is

— vF~1(v) — fFfl’(x)dx,

thatis, theareaP(v) in figure Al. Theexpectedpaymentcontingenton winningis the expectedsecond-highestvaluecontingenton v beinghighest,B(v)P(v)/F~ ‘(v). In the numericalexample,if v = 80, the bidder will win theauctionwith a probability of .8 (F~ 1(v) = .8), payan averageof 40 (B(v) =

40) when he wins, andmakeexpectedpaymentsof 32.Note that themarginal cost to the bidderof winning the auctionwhen the

highestvalueof any otherbidderis v canbe calculatedasthe derivativeofexpectedpaymentswith respectto the probabilityof winning whenthehigh-est-valuetypeofbidderyou aredefeatinghasa valueofv, dP(v)/dF~ ‘(v) = v.This is a necessarycondition for a symmetricequilibrium for any type ofauctionthat alwaysawardstheobjectto thehighestbiddersincea bidderwitha valueof v must bejust indifferent to biddingenoughto beata competitorwith thesamevalue.The expectedsurplusof a bidderis theareaS(v),whichequals the bidder’s value times his probability of winning the auction lessexpectedpayments.Inspectionof figure Al indicatesthefollowing formulae:

S(v) = f:Ffl~l(X)dX, (Al)

P(v) vF~’(v) — S(v) = {xdF~’(x), (A2)

P(v) f~F~1(x)dxB(v) = _______ = v — F~1 (A3)

F~ ‘(v)

We now considerfirst-price auctions.Assumethat eachbidder with valuev bids an amountB(v), which, in termsof figure A2, equatesareaC and areaD. In the numericalexample,the bidderwith a valueof 80 bids 40, paying

morein thefirst-priceauctionthanin thesecond-priceauctionwheneverthesecond-highestbidder’s valueis below 40 (thedifferencerepresentedby areaC) and less whenthe second-highestbidder’s valueis above40 (representedby D).

If everyone’sbidding B(v) is an equilibrium, “revenue equivalence”willhold: thebidder with the highestvaluewill win both auctionsand makethesameexpectedpaymentsin either becauseB(v) is the expectedbid in thesecond-priceauction.However,to prove that this is anequilibrium, we mustshowthat if all (n — 1) otherbiddersfollow this strategy,then it will paythenth bidderto follow the samestrategy.

Note first that under the specifiedstrategies,thebidderwith valuev winswith probability F~ ‘(v) and gets expectedsurplus [v — B(v)]F~ 1(v) =

vF~ (v) — P(v). Now considerwhat happensif he decidesto deviatebybiddingless thanthespecifiedamount.Let thedeviationbe to bid only B(v’),V <v, as in figure A3.

By makingthelower bid, which will succeedonly with probabilityEn— ‘(v’),thebidderreduceshis expectedpaymentto B(v’)F~ ‘(v’) = P(v’), a reductionofH + I + J or, equivalently,G + H. However,his expectedgrossbenefitsfall by G + H + F, so thebuyer’snetsurplusis now only vF~ ‘(v’) — P(v’), anetreductionof areaF. Sothedeviationis notworthwhile. In theexample,if

V

V

B(v)

V

0

Probability of winning the auctionFIG. A2

V

V

0 F ~‘ (v’) F”1 (v) 1

Probabiiity of winning the auction

a)

a-4)

0 Fn-l (v) 1

V

B(v)

B(v’)

a)

a-a)

V

0

FIG. A3


a bidder with a value of 80 actedlike a bidderwith value70 andsobid 35, hisexpected payments would drop from .8 x 40 = 32 to .7 >( 35 = 24.5.However, net surplus would fall from .8(80) — 32 = 32 to .7(30) — 24.531.5. Analogously, expectedsurplus is reduced by bidding more than B(v).Thusthespecifiedstrategiesconstitutea (Bayesian)Nashequilibrium and sorevenueequivalenceobtains.

Theanalysisabovecanbe easilymodifiedto allow for theseller’s imposing a

common reservation price or minimum bid. Insteadof graphingv againstF~ 1(v), graph M(v, r) = max(v, r) againstF~ ‘(v). The termM(v, r) can bethoughtof as the marginal cost of winning the auction againsta bidderwithvalue v, or M(v, r) = dP(v, r)/dF~ ‘(v). In a second-priceauction, a bidderwith a valueof v =r will bid v andmakeexpectedpaymentsof P(v, r). In afirst-price auction, sucha bidder will offer B(v, r) = P(v, r)/F~ ‘(v). In theexample,a bidderwith a valueof 80 facinga reservationprice of 50 will bid80 in a second-priceauctionandmakeexpectedpaymentsof 50 )< .5 + 65 X.3 + 0 x .2 = 44~5,i3or 44.5/.8= 55.625 = B(80, 50)contingenton winningthe auction. In the first-price auction theequilibrium bid is 55.625.

It is alsoeasyto derivea revenueequivalenceresultgraphicallyfor multi-pIe-goodsauctions,wherethe seller is offering k identical goodsand eachbuyerwantsonly oneunit. Simply graphv (or M(v, r) to allow for reservationprices) against F% ‘(v), where ~ ‘(v) is the probability that the kth-highestvalue among the n — 1 other bidders is less than or equal to v (and sorepresentsthe probability that a buyerwith a valueofv will make a successfulbid in a symmetricequilibrium). If we applythe sameanalysisasbefore,it iseasyto show revenueequivalencebetween an auction that allocatesitems tothe top k bidders who all pay the (k + 1)st-highestbid and an auctionthatallocatesitems to the top k bidderswho eachpaytheir own bid.

Finally, the samegraphical techniquecan be employed to show revenueequivalencefor other forms of auctions,such as the all-pay auction in whichall bidderspaytheir bid regardlessofwhethertheywin. The equilibrium all-paybid for a bidderwith a valuegreaterthanr is just P(v, r)—the expectedpaymentthe bidderwould makein a first- or second-priceauction.

References

Baron,David P., and Myerson,RogerB. “Regulatinga Monopolistwith Un-known Costs.”Econometrica50 (July 1982): 911—30.

Chatterjee,Kalyan, andSamuelson,William. “BargainingunderIncompleteInformation.” OperationsRes. 31 (September—October1983): 835—61.

Cramton,Peter;Gibbons,Robert; and Klemperer,Paul. “Dissolving a Part-nershipEfficiently.” Econometrica55 (May 1987): 615—32.

Harris,Milton, and Raviv,Artur. “Allocation Mechanismsand theDesignofAuctions.”Econometrica49 (November 1981): 1477—99. (a)

“A Theoryof MonopolyPricingSchemeswith DemandUncertainty.”A.E.R. 71 (June 1981): 347—65. (b)

Hotelling, Harold. “The Economics of ExhaustibleResources.”J.P.E. 39(April 1931): 137—75.

13 Half thetime theotherbidderwill haveavalueof less than 50, so thebidder willpay50; 30 percentof the time theother bidder’svaluewill be between50 and80, andthebidderwill payan averageof65; and20 percentof thetime thebidderwill lose theauction and pay nothing.


McAfee, R. Preston,and McMillan, John. “Auctions and Bidding.” J. Econ.Literature 25 (June 1987): 699—738.

Maskin, Eric S., and Riley, John G. “Monopoly with IncompleteInforma-tion.” RandJ.Econ. 15 (Summer1984): 171—96. (a)

“Optimal Auctions with Risk Averse Buyers.” Econometrica 52(November 1984): 1473—1518.(6)

Matthews, Steven A. “Selling to Risk Averse Buyers with UnobservableTastes.”J.Econ. Theory30 (August 1983): 370—400.

Milgrom, Paul R. “The Economicsof Competitive Bidding: A SelectiveSur-vey.” In Social GoalsandSocialOrganization:Essaysin MemoryofElishaPazner,editedby Leonid Hurwicz, David Schmeidler,and Hugo Sonnenschein.Cambridge:CambridgeUniv. Press, 1985.

“Auction Theory.” In Advancesin EconomicTheory:F~fth World Con-gress, edited by Truman F. Bewley. Cambridge:CambridgeUniv. Press,1987.

“Auctions and Bidding: A Primer.”J. Econ. Perspectives3 (Summer1989).

Milgrom, PaulR., andWeber,RobertJ.“A Theoryof AuctionsandCompeti-tive Bidding.” Econometrica50 (September1982): 1089—1122.

Mussa, Michael, and Rosen,Sherwin. “Monopoly and ProductQuality.” J.Econ. Theory 18 (August 1978): 301—17.

Myerson,RogerB. “Optimal Auction Design.”Math. OperationsRes.6 (Febru-ary 1981): 58—73.

Myerson, RogerB., and Satterthwaite,Mark A. “Efficient MechanismsforBilateral Trading.”]. Econ. Theory29 (April 1983): 265—81.

Pratt,John,andZeckhauser,Richard.“Smoothingthe Bumpsin Profit Max-imization: Pro RataTenders,Monopolist’s Roulette,and Other RationedTransactions.”Manuscript. Cambridge,Mass.: Harvard Univ., KennedySchoolGovernment,1986.

Riley, John G., and Samuelson,William F. “Optimal Auctions.” A.E.R.71(June 1981): 381—92.

Vickrey, William. “Counterspeculation,Auctions, and Competitive SealedTenders.”]. Finance 16 (March 1961): 8—37.

Wilson, RobertB. “Incentive Efficiency of Double Auctions.” Econometrica53(September1985): 1101—15.

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