Extracting the Surplus in the Common-Value Auction Author(s): R. Preston McAfee, John McMillan, Philip J. Reny Source: Econometrica, Vol. 57, No. 6 (Nov., 1989), pp. 1451-1459 Published by: The Econometric Society Stable URL: http://www.jstor.org/stable/1913717 Accessed: 23/01/2010 14:33 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=econosoc. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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Econometrica,Vol. 57, No. 6 (November,1989),1451-1459

EXTRACTING THE SURPLUS IN TIHECOMMON-VALUE AUCTION ANDPHILIPJ. RENY1 BY R. PRESTON MCAFEE,JOHNMCMILLAN,

1. SELLINGAN ITEM WITH A COMMONVALUE

THESALEof a unique item with a well-defined true value which no one knows CONSIDER (like mineral rights). What selling procedure should the owner choose?2 The mechanism we devise takes advantage of the fact that a common-value sale has an efficient outcome regardless of which of the potential buyers receives the item, since all potential buyers value the item identically ex post. Thus any mechanism that extracts all the rents must be optimal for the seller.3 We consider the following simple mechanism. After each potential buyer has received his private signal about the true value of the item and committed himself to participate in the sale, the seller arbitrarilyselects two of them. The seller then asks one, j, to report his signal, and offers the item to the other, i, for a price z(sj), where sj is j 's report. Since the payoffs to all agents are independent of their own actions, this mechanism is (weakly) incentive compatible. The mechanism satisfies the individual rationality constraints, i.e., the potential buyers are willing to participate, if the price function z is such that they expect nonnegative rents. In addition, if these rents are zero, the mechanism is optimal from the seller's point of view. Does such a price function exist? We shall prove that it is always possible for the seller to extract arbitrarily close to the full expected rents. Moreover, shall we give two sufficient conditions for exactly all of the expected rents to be extractable. We also give an example in which the seller can extract almost all, but not all, of the rents. Simple contracts work: the seller can without loss choose the price function to be a piecewise linear function or a step function. To motivate the results, consider a special case. Suppose the bidders can construct unbiased estimates z(si) of v from their signals si and that si is independent of sj, given v. Then4 (1)

v= fz(sj)f(sjlv)

dsj.

The price function z(sj) extracts all the rents, since

(2)

E[z(sj)IsjI =E[vlsj].

1

We thankMatt Spiegel,two referees,and a coeditorof thisjournalfor comments.McAfeethanks the U.S. Departmentof Justice,AntitrustDivision, for their hospitalitywhile this researchwas completed.McMillanthanksthe NationalScienceFoundationfor researchsupport. 2Most of the literatureon the designof optimalsellingmechanismsassumesindependentprivate values; i.e., valuationsare drawnindependentlyfrom some distributionH(v) (Harrisand Raviv (1981), Maskinand Riley (1984),Matthews(1983),McAfeeand McMillan(1987b),Milgrom(1985), Riley and Samuelson(1981)).The modelof Myerson(1981)is moregeneralthanindependentprivate values in allowing a limited form of correlationamong valuations.Cremerand McLean (1985) showedthatwith correlatedprivatevalues(bidderi's valueis of the formvi(s), wheres is the vector of all bidders'signals) and a finite value space, the seller can extractall of the rents by using a combinationof a lotteryplus a Vickreyauction.For a detailedreviewof this literature,see McAfee and McMillan(1987a). 3It followsthat the mechanismis sensitiveto the common-value assumptionand will not workfor the more generalcase of affiliatedvalues(Milgromand Weber(1982)). 4Equation (1) is a Fredholmequationof the first kind. Generalexistenceof solutionsto such contextby McAfee and Reny (1988); a particular equationsis examinedin the mechanism-design case is examinedby Melamudand Reichelstein(1986)and Caillaud,Guesnerie,and Rey (1988). 1451

1452

R. PRESTON MCAFEE, JOHN MCM1LLAN, AND PHIUP J. RENY

We shall show that in one sense this exampleis typical, in that the common-value environmentfacilitatesrentextraction;but in anothersenseit oversimplifiesthe situation, in that in generalthe sellercan extractalmostall, but not necessarilyall, of the surplus. problem.Whileagent j cannotdo This mechanismsuffersfroma multiple-equilibrium better than reporthis value correctly,he is no worse off if he lies: truth-tellingis only a weakly optimal strategy.This can, however,be correctedat arbitrarilysmall cost to the principal:agent j can be given a strictincentiveto tell the truth.Let 8 > 0 be small and considera revelationgame R in which each bidderhas a strictincentiveto tell the truth (e.g. the revelationequivalentof a second-priceauction).The seller,havingreceivedthe reports, uses them in the revelationgame R with probability8, and in the mechanism describedabove with probability1- 8. This providesa strictincentivefor truth-telling,at a cost to the seller that vanishesas 8 -- 0. 2. RENT EXTRACTION

With the unknowntrue value of the item denotedby v, suppose that each potential buyerhas received(withoutcost) a signalsi independentlygeneratedfromthe distribution F(si Iv). The correspondingdensity,f(si Iv), is assumedto exist and to be strictlypositive and continuouson [0,1] x [0,1].5 Let the uncertaintyabout v be given by a measureG with supportthat is a subsetof [0,1]. The sellerand all potentialbuyersare assumedto be risk neutral. We assumethat, afterhavingobservedhis signal,been told the selihngmechanism,and agreedto participate,the buyer, i, is committedto pay z(sj) whateverthe realizationof the other'sreport s1: thushe cannotback out if the price appearsto him to be atypically high. (This is stricter than take-it-or-leave-itpricing; the buyer must take it.) Since, however,the potentialbuyershave the option of not participating,the mechanismmust offer themrents that arenonnegativeconditionalon theirown signals.The chosenbuyer's expectedrents, givenhis signal si, are )

)

lo[ 010

1?l1f(sI)

f

dG(u)

Remarkably,the problemof extractingthe rents,that is, (4)

(v Si)

'ff(Si)= ?,

can be transformedinto a minimum-norm problem.Considerthe L2([0,1],G) dot product and norm

(5)

(x, y) =

j

x(v)y(v)

dG(v);

lxii= (x,

X)1/2

and the set (6)

X = ( xE

L2([0,1],G)lj3zeL2([0,1],X),

x(v)=

1z(s)f(sjv)ds}

where X is Lebesgue measure,and L2([0,1], X) will be denoted L2(X), and similarly L2(G) for L2([O,1], G). SIf the supportof f( *Iv) is monotonicin v, the rentscan be fully extracted,since the problemof rent extractionreducesto the solutionof a Volterraequation.See Hochstadt(1973).

1453

THE COMMON-VALUE AUCTION

LEMMA:3z E L2(X) satisfying (V si)7r(si) = 0 if and only if

(7)

minlx- vIl

xe

has a solution.6

PRooF: x is the solutionto (7) if and only if (by Theorem1.4.1 of Balakrishnan(1981, p. 9), noting X is a linearsubspace) (v1x G X)

(x -v, x) =O

(V xE X)

j (v-x(v))x(v)

V

z

L2(A)

iff (by (5)) iff (since EX,(6))

dG(v) = 0 z(s)f(slv)

J(v-x(v))

dsdG(v) =O E L1([O,1]2, XX G))

(by Fubini's Theorem, applicable since (v-x(v))z(s)f(slv) v

E L2(A)

iff

dG(v) ds =O

1Z(s)1(v-x(v))f(sIv)

iff

(by the FundamentalLemmaof the Calculusof Variations) a.e. si e [0, 1], a.e. si C=[01],

1[v - x(v)]

dG(v) = 0

f(siv)

iff

(by (6) and x E X)

ds] f(siIv) dG(v)= O

V [v-|z(s)f(sIv)

iff (by continuityof Vs E [0,1], V1Si E=[0, 1],

f z[-jz(s)f(slv)ds

)

iff

f(siIv)dG(v)=0

Q.E.D.

"ff(Si)=0.

The main resultof the paperis that the sellercan alwaysextractalmostall of the rents. V - > 0, 3z E L2(X) such that n(s) THEOREM: buyer's expected rent as defined by (3).

E

[0, E]Vs E [0, 1], where r(s) denotes a

TIhe proof has two steps. In the first, a sequence n,,EeX is constructed, with

4n

convergingto a solution(not necessarilyin X) of (7), with X replacedwith the closureof X. Thereis an associatedsequenceof profitfunctions (8)

a"(u) = |I(v - Ojv))f (ulv) dG(v),

which we prove to be equicontinuousand to convergeuniformlyto zero. In the second step, 4,n and An are used to construct a sequence Zn of price functions, with associated 6v will be used to denoteboth the identityfunction,as in (7), and a dummyof integration,as in

(5).

1454

R. PRESTON MCAFEE, JOHN MCMILLAN, AND PHILIP J. RENY

profit functions inr, which satisfy individualrationalityand also convergeto zero uniformly. PROOF: Let X denotethe closureof X in L2(G). Clearly,X is a closedlinearsubspace (1981,p. 9),(7) has a solutionwith X of L2(G). Hence, by Theorem1.4.1of Balakrishnan replacedby X. Denote this solutionby 4. Hence,for everyx e X, we have (4 - v, x) = 0, i.e.,

(9)

j1x(v)(4o(v)-v)dG=O

VxE X. L2

L2

Now, since 4 EX, 3,,ne X such that on +4 (-* denotesconvergencein L2(G)).By the continuityof the innerproduct,we haveAn(U) -+ Jl(v - 4)(v))f(ulv) dG for all u, where An is given in (8). Let A(u)--JJ(v-40(v))f(uIv)dG so that An(u)-*A(u) for all ue[0,1]. Putting x(v) = Joz(s)f(sIv)ds in (9) above,we have (for any z EL2(X)) (10)

1?

z(s)f(sj v) ds] ( v-S(v))

dG=O,

or, using Fubini'sTheorem(applicablesince z(s)f(slv)(v (11)

v - 4(v))f(sjv)

jz(s4f[|(

-(v))

E L'([O,1]2, Xx

G)),

dGj ds = O,

i.e., (12)

j1z(s)A(s) ds = 0.

V z E L2(X),

Hence, A(s)=O a.e. In fact A(s)=O for all se[0,1] since by definition A(.) is continuous(by the continuityof f( - I - )). Hence, for all u e [0,1], we have (13)

An(u) = 1(v- ),n(V))f(Alv) dG -A(u)

= 0.

Since f is continuous and { v -)n,) is bounded in LN(G), it is easy to show that {n } is uniformlyboundedand equicontinuous(see Hochstadt(1973,p. 51)). In particular,letting ,, 0. Note 8a =n-U [()0,II A(u), and recallingthat A,n(u) -O 0 for all u, we have that 8n that by construction

(14)

aa

-

An (U) + 8An I > 0 for all u E [0,1] and all n. Let

n

min jf(ulv)

dG

uO

(This is well definedsince Jlf( * v) dG is continuouson [0,1] and has a minimum which is bounded away from zero since JJf(uIv) dG>0, Vu 8 [0,1].) Clearly, a,O 0. Define {2,n} c L2(X) as follows: Since (4)n,)5 X, 3{,Zn}C L2(X) such that on(v) = Jz,,(s)f(slv) ds for all n. Put Zn (S)-Zn (S)-aan for all se [0,1], and all n.

1455

THE COMMON-VALUE AUCTION

We now show that Zn is individuallyrationalfor every n. Let rn denote the profits associatedwith Zn,,as givenin (3). (15)

*c >

v,(u)

j

(vjI

Zn(S)A(sIv)

dS)f(UIV)

dG,

where C1a max [ =J

=

(v

j1(Zn(S)

ji(

f-

=1(V

-

-an)f(sIv)

Zn(S)(sIv)

f(ulv)

dG>0,

d) f(ulv) dG

dS)f(UIV) dG+a,n

ff(ulv) dG

j1f(ulv)dG n(v))f(uIv)

dG+ 18n l~~~~~~~min ft(ulv) dG

? >An(U)

+ 16nI >:.

Hence Zn is individuallyrationalfor every n and every u E [0,11. Finally we must show that for largeenough n, 2n extractsan arbitrarilylarge fraction of the surplus.First,observethat fOlzn(s)f(sI ds = I) ds 4 sinceJfOJn(s)f(sIv) , Vn, v. Denoting JJzn(s)f(slv) ds by 4,(v) we then have ,n so that

On(v) - an

-

(16)

nr(u)=J_(v-'n(v))h(u,v)dG, In,,(u)

( v - 0( v)) h( u, v) dG

-

where h(u,v)-

f'f(ulv)dG.

(by continuity of the inner product)

for everyu E [0,1] (since A( u)=O for every u e [0,1]).

-0

Moreover, since {,An} is clearly bounded and h (.,-)

is continuous, we have

{f rn}

is a

uniformlybounded and equicontinuousfamily.Hence rn (u) O-0 uniformly,i.e., (Ve> 0) (3N) such that Vn > N, Vu E [0,1], I?n(u)I < E. Since, from (14), ;7n(u)>O, we have

O< rn (u)

< E, as

desired.

Q.E.D.

By Theorem1.4.1 of Balakrishnan(1981, p. 9), a solutionto (7) exists if X is closed. Note that if the minimizedvalue of the norm in (7) is zero then there is a trivial rent-extractionproblem,as describedin the introduction(equation(1)). The moreremarkable result is that even if the minimumnorm is not zero (i.e. there does not exist an unbiased estimate z(sj) of v) but merely a solution to (7) exists, then there is still a mechanism that extracts exactly all of the rents. In Examples1 and 2, we give two sufficientconditionsfor X to be closed,so that X = X, and hence for the rentsto be fully extractable. EXAMPLE

1: If n

(17)

f(SIv)

a(s)b(v),

= i=l

then 3z E L2(X) satisfying(4).

1456

R. PRESTONMCAFEE,JOHNMCMILLAN,AND PHILIPJ. RENY

PROOF: Viewed as an integraloperatoron L2(G), f(slv) =E)=2jai(s)b,(v) is degener ate, and hence by an obvious modificationof Theorem12 of Hochstadt(1973, p. 60 (modifying from L2(X) to L2(G)),X is finite dimensional.Since X is hence a finite dimensionallinear subspaceof a normedlinearspace (namelyL2(G)), Theorem4.3.2 o, Q.E.D Friedman(1970, p. 132) showsthat X is closed.

2: If G has finitesupport,then 3z E L2(X) satisfying(4). ExAMPLE PROOF: Let

(18)

X={xe

Rilxi= fz(s)f(sIvi)

ds for some z E L2(),

where supp G={v,

(19) (19)

v2

,Vm

and

G(v)=

Eak, k=1 m

and

ak > 0OVk

ak-k1. k=1

In this case, the norm definedin (5) reducesto lxii (ElIa xx2)1/2 Vx e Rm. Since X is normed linearsubspaceof R', (X, 11IDis a finite-dimensional clearlya finite-dimensional Q.E.D. linearsubspaceand X is thereforeclosed. Finally, we give an exampleshowingthat, under the hypothesesof the theorem,no strongerresult exists.Thereare cases in which full rent extractionis impossible;the best the sellercan do, using this type of mechanism,is to extractalmostall the rents. ExAMPLE 3:

Supposethat 1

f(siv)

so that f: [ -r,

=

2

1 +

X

sin ks sin kv + cos ks cos kv

13k=1

k2

k

] X [-7, v]-- RX is continuousand strictlypositiveand

forall vE[-7ri,ri1].

Jf(slv)ds=1

Furthersuppose that (1 g(V)=

/ 2, r'

\0,

v

E[-7T,7T1,

otherwise.

Then full rent extractionis impossibleusing the mechanismof the theorem.

THE COMMON-VALUE AUCTION

1457

Since

PROOF:

77

(a)

1 f(siv) ds=1,

(b)

f__'2k2 sin ksf(s Iv) ds = sin kv,

(c)

f

12k2cos ksf (s Iv) ds = cos kv,

we have: } c X,

{1,sinx,cosx,sin2x,cos2x,...

where L2(X) is now used to denote L2([ -_I, 17],X),and X= xeL 2(X)Ix(v) =f

ds for some zeL2(X).

z(s)f(slv)

Hence, X= L2(X), so that v e X; i.e. argmin,_ kIix- vii= v. On the other hand, v 4 X (shown below) so that min,E xlx - vii has no solution(in X). Since our rent extraction problem,this implies that the sellercannot problemis equivalentto this minimum-norm extractall of the rents. It remains to show that v q X. For this, note that, viewed as an integraloperator, ) is compact and self-adjoint.Hence if {f4}?o' denotes an orthonormalset of f( denotes the set of correspondingeigenvalueswith eigenfunctionsof f( I *) and {u }I'%0 > then Picard's > ..., Theorem(Hochstadt (1973, p. 108)), v e X 1> by [LoI ILI I IA21 implies 2 i=0

Now, correspondingto f( I ) we have 1 sinv cosv sin2v cos2v {fi },o

and A

,-

,r

v/2,r

,g

v,--

1=,

4

(1

1, 3,5,. ...

(i=+1) _rr, . 1 4

.. ), (i = ~~~~~~~~~~2,4,6,.

. _

. so that

2

2 I(v2i)

??Rv)l 2 =O

/i

i odd

-

.1

(

2

k1

2

2V-k2

1458

R. PRESTON MCAFEE, JOHN MCMILLAN, AND PHILIP J. RENY

where the first equality appearssince J'v,v0i(v)dv=0 second equalityis justifiedsince |f voi( v) dv=-

_{

i +l/2

(+)

12A

f

for al i=0,2,4,...,

and the

for all i = 1, 3, 5,...

(the changeof variablek = (i + 1)/2 was also employed).Hence, x,

I( V, 4)

x0

12

2

2

i=0

k=O

yi

which clearlydiverges.We concludethat v i X, as required.

Q.E.D.

3. CONCLUSION AND EXTENSIONS

We have shown that, for the sale of an item with an unknowncommonvalue,the seller can always extract almost all of the rents. In particular,if an optimal auction exists it extractsall of the buyer'ssurplus.The sellerextractsexactlyall of the rentsif eitherthere is a finite number of possible values of the item, or the density function of signals conditionalon true value satisfiesa separabilitycondition.We also illustrated,however, that with the mechanismused here full rent extractionis not alwayspossible. cm) We note that the results continue to hold if a vector of characteristics(cl, determines the common value v (i.e., v(c1,..., Cm)), where G now denotes the joint measureover c = (cl, *-, cm)with support[0,1]m.In addition,each biddermay receivea vectorof signalss = (s1, * , Sm).The mechanismworksas before,exceptnow whencalled upon a potentialbuyerreportshis entirevectorof signals.In this case, a potentialbuyer's rent as a functionof signalsis: (20)

'Z(s)

v(c)

f

[

z(u)f(u1c) du f(slc) dG(c)

f(slu)dG(u).

The functionalform in the statementof Example1 now becomes? If in all the abovewe replaceX by X

{ x e L2(G)Ix(v) = j1z(s)f(slv)

1ai(s)bi(c).

ds forsomezXZ

where Z is eitherthe set of piecewiselinearfunctionson [0,1], or the set of step functions on [0,1], the proofs go throughverbatim.Hence in all cases the price function that the seller announcesin advancecan be a relativelysimplefunction. Three furtherextensionsof the analysisarepossible.First,if the sellerhimselfdrawsa signal from the samedistributionas the potentialbuyers,he need not use the reportof the second potentialbuyer.Instead,he couldmakethe pricea functionof his own signal,and proceedexactly as above,providedhe can crediblycommitnot to misrepresenthis signal. Second,if the sellerhas m units of the good and eachpotentialbuyerwantsone unit, the selleroptimallychoosesm buyersat randomand chargeseachof them z(sj). Third,7the n bidders'conditionaldensitiesneed not be identical.One generalizationrequiresonly that there exist two bidders, say i and j, with conditionaldensities satisfying f,(s Iv) = a(s)fj (b(s) Iv), wherea(s) is boundedawayfromzero and b(s) > 0 on [0,1], and b takes [0,1] onto [0,1]. 7We thank Matt Spiegel for this observation.

1459

THE COMMON-VALUE AUCTION

The mechanismexaminedheresuperficiallyresemblesthe Vickreyauction,as the price paid by the buyer dependsupon anotherpotentialbuyer'sreport.However,the Vickrey auctionis not optimalin the common-valuecase. (Milgromand Weber(1982) show it is dominated by the English auction.) The essential differencebetween the mechanism analyzedhere and the Vickreyauctionis that, in the former,the two biddersare chosen arbitrarily.In contrast,in the latter, the biddersmakingthe highestand second-highest reportsare chosen; and the bidders'knowledgeof this preventsthe full extractionof the surplus.8 Departmentof Economics,Universityof WesternOntario,London,Ontario,NGA SC2, Canada, of California,San Relations& PacificStudies,University GraduateSchoolof International Diego, La Jolla, CA 92093, U.S.A. and Departmentof Economics,Universityof WesternOntario,London,Ontario,NGA SC2, Canada. ManuscriptreceivedNovember,1987; final revisionreceivedApril, 1988. REFERENCES

Analysis,2nd Edition.New York:Springer-Verlag. BALAKRISHAN,A. V. (1981):AppliedFunctional in AdverseSelectionModels," CAILLAUD, B., R. GUESNERIE,AND P. REY (1988):"Noisy Observation Document du Travail No. 8802, Institut National de la Statistique et des Etudes Economiques. CREMER,JACQUES,AND RICHARD P. McLEAN

(1985):"OptimalSellingStrategiesunderUncertainty

for a Discriminating Monopolist When Demands Are Interdependent," Econometrica,53, 345-361. of ModernAnalysis.New York:Dover. FRIEDMAN, AVNER(1970):Foundations HARRIS,M1LTON, AND ARTUR RAvIv (1981): "Allocation Mechanisms and the Design of Auctions," Econometrica, 49, 1477-1499.

New York:Wiley. HOCHSTADT, HARRY (1973):IntegralEquations. KENNEY, Roy W., AND BENJAMINKLEIN (1983):"TheEconomicsof BlockBooking," Journalof Law and Economics, 26, 49-54. MASKIN, ERIC,AND JOHN G. RILEY (1983):"OptimalAuctionswith Risk AverseBuyers,"Economet-

rica, 52, 1473-1519. STEVENA. (1983): "Selling to Risk Averse Buyers with Unobservable Tastes," Journal of MATTHEWS, Economic Theory, 30, 370-400. McAFEE,R. PRESTON,AND JOHN MCMILLAN (1987a):"Auctionsand Bidding,"Journalof Economic Literature, 25, 699-738. -~ (1987b): "Auctions with a Stochastic Number of Bidders," Journal of Economic Theory, 43, 1-19. MCAFEE, R. PRESTON, AND PHILIP J. RENY (1988): "CorrelatedInformationand Mechanism Design," mimeo, University of Western Ontario. MELAMUD, NAHUM, AND STEFAN REICHELSTEIN(1986): "Value of

Communicationsin Agencies,"

Research Paper 895, Graduate School of Business, Stanford University. MILGROM, PAUL R. (1985): "The Economics of Competitive Bidding: A Selective Survey," in Social Goals and Social Organization, ed. by L. Hurwicz, et al. Cambridge: Cambridge University Press. MILGROM, PAUL R., AND ROBERT J. WEBER (1982): "A Theory of Auctions and Competitive Bidding," Econometrica, 50, 1089-1122. MYERSON, ROGER

B. (1981): "OptimalAuction Design," Mathematicsof OperationsResearch,6,

58-73.

Review, RILEY, JOHN G., AND WILLIAMSAMUELSON(1981):"OptimalAuctions,"AmericanEconomic 71, 381-392. 8It is not easy to find actual markets that operate like this full-rent-extraction mechanism, in which buyers sign up in advance, knowing the pricing rule but not the seller's estimate of the item's worth. One market that seems to have this timing and commitment structure is the selling of uncut diamonds by the De Beers monopoly. Since diamonds are purchased for eventual resale, the common-value assumption fits. Each of the invited buyers is offered a single package of diamonds at a nonnegotiable price. The buyer may inspect his package; however, if he rejects it, he is never again invited to participate (Kenney and Klein (1983, pp. 500-502)).