Correlated Information and Mecanism Design Author(s): R. Preston McAfee and Philip J. Reny Source: Econometrica, Vol. 60, No. 2 (Mar., 1992), pp. 395-421 Published by: The Econometric Society Stable URL: http://www.jstor.org/stable/2951601 Accessed: 23/01/2010 14:06 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. 60, No. 2 (March,1992),395-421

CORRELATED INFORMATION AND MECHANISM DESIGN BY R. PRESTON MCAFEE AND PHILIP J. RENY1 In most modelsof asymmetricinformation,possessionof privateinformationleads to rents for the possessors.This tends to inducemechanismdesignersto distortawayfrom efficiency.We show that this is an artifact of the presumptionthat informationis independentlydistributed.Rent extractionin a largeclass of mechanismdesigngamesis analyzed,and a necessaryand sufficientconditionfor arbitrarilysmall rents to private informationis provided.In addition,the two personbargaininggame is shownto have an efficient solution under first order stochasticdominanceand a hazard rate condition. Similarconditionslead to full rent extractionin Milgrom-Weberauctions. KEYwORDS: Surplus extraction,efficiency,mechanism,correlatedinformation,auctions, privateinformation.

1. INTRODUCTION IN MOST MODELS OF PRIVATE OR ASYMMETRIC INFORMATION,

possessors of

private informationreceive rents or profits. For example, in the independent privatevalues auction,the winningbuyerpaysless for the item for sale than it is worth to him, even when the auction is designed to maximizethe price paid to the seller.2 Milgromand Weber (1982) show by examplethat these rents result from the privacyof the informationratherthan its accuracy.Basically,if the information is held by two players, it has no value to either player. One can think of a Bertrand competition set up by a third player (the mechanism designer) to extract the information.More generally,when players' private informationis jointly distributedin a perfectlycorrelatedmanner,it is easily renderedpublic and hence providesits possessorsno rents. Manyapplicationsof the mechanismdesign paradigminclude the assumption that the informationheld by the players is jointly independentlydistributed. This has the implicationthat the informationis purelyprivate,in the sense that one learns nothing about one player'sinformationfrom anotherplayer'sinformation. Hence the kind of Bertrand competition which reveals the private informationwhen it is perfectlycorrelatedfails to do so here. As a result, the independenceassumptionoften leads to positive rents accruingto the possessors of privateinformation.Under the assumptionthat agents are risk neutral, we find that introducingarbitrarilysmall amountsof correlationinto the joint distributionof private information among the players is enough to render privateinformationvalueless,in the sense that its possessorsearn no rents. It is worth noting that while this result does depend upon the agents' attitudes 1We thank Andreu Mas-Colell for helpful discussions, and an anonymous referee who both suggested that we pursue the characterization result (Theorem 2), and dramatically improved the proof of Corollary 3. -2 Myerson (1981), Riley and Samuelson (1981). See McAfee and McMillan (1987) for a survey of the voluminous auction and mechanism design literature.

395

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R. PRESTON MCAFEE AND PHILIP J. RENY

towardrisk (riskneutralityis heavilyused), it applies to virtuallyall mechanism design environmentsof interest. Furthermore,we provide a condition on the joint distributionof agents' private informationwhich is both necessary and sufficientfor reducing the value of this informationto zero.3 This condition essentiallyasks that each agent'sprivateinformationnot be entirelyuninformative about other agents' private information,and thus rules out the case of independence. Cremer and McLean (CM) (1985) and (1988) motivated the present study. The (1985) paper provides a condition on the joint (conditional)distribution over consumers'(uncertain)characteristicssufficientfor a price-discriminating monopolistto extractfrom them the full surplus.Their (1988) paper focuses on the privatevalues auctionenvironment(each agent'svalue of the item for sale is known to, and only to, that agent), where the agents'values are correlated.A conditionis providedon the joint (conditional)distributionover agents'values that is both necessaryand sufficientfor allowingthe auctioneerto extractall of the surplusfrom the bidders. One of our main objectives is to demonstrate that these results go well beyond the auctionenvironment.The resultswe obtain applyto all mechanism design environmentswith risk neutral bidders in which an agent's type affects his payoff in a continuousmanner.In particular,one can apply our results to problemsinvolvingthe allocationof publicgoods, optimaltaxationschemes,and a varietyof agencyand regulatoryenvironments. Both of CM's papers make heavy use of not only agents' risk neutrality,but also the assumed finite state space. In CM (1985), each consumer's utility functionis characterizedby a parameterthat can take on at most finitelymany distinctvalues. In CM (1988), each bidder'svalue of the item is one of finitely manyfixed possiblevalues. Another of our objectivesis to extend CM's results to the case where the agents (bidders,consumers,bargainers,etc.) may have infinitelymany possible types. This is not merely an exercise in mathematicalcompleteness.Note that CM'sresult impliesthat we cannot explainthe predominantuse of the standard auctionforms based on a revenue maximizingseller, since these auctionsleave bidders with positive rents. However, the finite values model they employ to obtain their result is only appropriateif the biddersand auctioneerswe wish to model, explicitlytake into accountthe fixedand finite numberof possiblevalues when makingdecisions. If, on the other hand, the bidders and auctioneerswe wish to model are alwayswilling to admit that "one more"value distinctfrom the (finitelymany) others currentlydeemed possible, is also a possibility,then the appropriatemodel is not one with finitely many values, but one with infinitelymany.Consequently,if CM'sresult does not hold in the infinitevalues model and the standardauctionsare revenuemaximizingthere for "many"joint distributions(allowing correlation)over bidders'values, then not only do we regain a revenue maximizationbased explanation of the emergence of the The conditionwe presentis the continuumanalogof that givenin Cremerand McLean(1988).

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standardauction forms, we also (because of its inabilityto explain the facts) have some basis for rejecting the finite values model outright. Thus, it is a matterof some importanceto pursuethis line of researchthroughto the infinite types case. To understandCM's (1988) result, it suffices to consider the case of two bidders. The bidders (knowing only their own value) must simultaneously choose whetheror not to participatein a Vickreyauction(a sealed bid auction in which the item goes to the highest bidder at a price equal to the second highest bid). If they choose not to participate,they get zero. If they choose to participate,then they must agree to pay a participationfee. The participation fee is allowed to be random. In particular,bidder l's participationfee may depend upon bidder 2's bid in the Vickrey auction to follow and vice versa. Since honesty is a dominant strategy in a Vickrey auction, and since each bidder'sfee is independentof his own reportedvalue in the Vickreyauction, reportinghonestlyremainsa dominantstrategy.Hence, in equilibrium(if both are willingto participate),each bidder'sultimateparticipationfee is a random variable,the outcome of which depends upon the other bidder'svalue. From now on, we shall refer to this randomvariableas a participationfee schedule.In fact, the mechanism is just slightly more complicated than this. Instead of presentingeach bidderwith a single participationfee schedule, the auctioneer presents each with a finite set (a different set for each bidder perhaps) of participationfee schedules.After learningtheir own value, the bidders decide whetheror not they wish to participatein the upcomingVickreyauction.If so, they mustchoose one of the availableparticipationfee schedules.They are then committedto payingthe fee associatedwith the outcome of that schedule. If both biddersare willingto participate,then as before honestyis a dominant strategyin the Vickrey auction to follow. Since a Vickrey auction is ex post efficient (the item goes to the bidder with the highest value), the mechanism (auction plus participationfees) will be optimal from a revenue maximizing point of view if the participationfee schedules can be constructedto recover any bidder'sexpectedprofitsfrom the Vickreyauction(leavinga bidderwith no surplus,regardlessof his value;biddersare thereforewillingto participate).We now show how CM (1988),with an appropriaterestrictionon the joint distribution of bidders'values,were able to constructthe requiredsets of participation fee schedules. Let v1, .. ., v,, be each bidder's set of possible values, and let P be the matrix

of bidderl's conditionalprobabilities.Thus, the ijth entry,pij, of P denotes the probabilitythat bidder2 has value vj giventhat bidder1 has value vi. Denote by Pi, the ith row of P. Finally, let wi be bidder l's expected profit from the Vickreyauction(excludingany participationfees) when his value is vi. Consider now CM's restrictionon P: for all i = 1,..., n, pi. co{pk}k 0. That is, the vector of conditionalprobabilitiescorrespondingto any possible value of bidder 1 is not in the convex hull of the vectors of conditional probabilitiescorrespondingto his other possible types. With the conditional distributionsatisfyingthis condition,the auctioneercan extractall of bidder l's

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surplusas follows:4For each i = 1,..., n, there is a hyperplanexi E Rn separating pi and Co{PkJ so that xi -pi = 0 and xi *Pk> Ofor all k 0 i. Now, for each m = 1, .. ., n constructthe participationfee schedule (for bidder 1) Zm(I) = Tm +a xmj, where a > 0 will be specified below. Thus, if bidder 1 wishes to participatein the Vickreyauction,he must first(knowinghis own value) choose a participationfee schedule Zm(*) say, therebyagreeingto pay Zm(i) if player2 announcesa value of vj. Since l's payoffin the auctionitself is independentof the participationfee schedule he chooses, he will choose that scheduleyielding the lowest expected fee. That is, bidder-1, given that his value is i, will choose m = 1, . . , n to minimizepi* Zm = Wm + api.* Xm. Now, since pi* Xm> 0 whenever m # i and pi xi = 0, we may choose a > 0 so that for every i, pi Zm is minimized when m = i. Hence for every i = 1,..., n, if bidder 1 has value vi he will optimallychoose fee schedule zi(*) and earn an expectedsurplusof zero. Using a similarlyconstructedset of fee schedules for bidder 2, the auctioneercan in this way extractthe full surplus. Since P satisfiesthe conditiondescribedabove for almost every distribution of values (in Lebesguemeasure),full rent extractionis "usually"possible. Note that for such distributions,the precise manner in which P determines the bidder's rents in the original Vickery auction need not be considered to conclude that an optimal auction in this environmentmust extractall of these rents. It is this observationof Cremerand McLean'sthat we wish to exploit in Section 2. Now consider the continuum analogue to CM's result. Let f(sit) be the density of s conditionalon an agent's type t E [0, 1], and suppose this agent anticipatesprofits wr(t)on average from participationin the Vickery auction. The analogousfull rent extractionproblem for the seller is: Constructfinitely manyparticipationfee schedules z1(k),... ZN(') So that for all t E [0,1] 11 (1.1)

w(t) =

min

1
0

zj(s)f(sIt) ds.

If such schedules exist, and the agent is risk neutral,then the agent'srents can be extracted. There are severalsimple observationsto be made. First,(1.1) is not generally solvable.If f does not depend on t (i.e. s and t are independent),and v is not a constantfunction,then (1.1) has no solution. Second, if the supportof f(- It) is monotonic in t, then (1.1) reduces to a Volterra equation and is always solvable.5We shall assume only that the support of f is contained in [0, 1]2. Third,solutionsto (1.1) never exist for all continuousw, if f is continuous.That is, when fe C[0, 1]2 one can alwaysfind a rweC[0, 1] such that (1.1) has no solution. 4 We are grateful to an anonymous referee for providing this argument. See Hochstadt (1973). The support of f is monotonic if it is of the form [a(t), b(t)] and a' and b' > 0, f(b(t)) > 0. See Demougin (1987) for an economic application.

0

CORRELATED

INFORMATION

399

Thus, unlike the finite dimensionalcase it appears as though we can never conclude that an optimal auction in an environmentyielding a continuous conditionaldensity f(slt) must extract all of bidder l's (receiver of signal t) rents, without investigatingthe precise mannerin which f determinesw, since the v we are faced with may be one of those for which (1.1) is not solvable. Suppose howeverthat given f, the followingwere true: (1.2)

VE > O,

O< 7T(t) -

E C[O, 1],

3zl,...,ZN

E

C[O,"] such that Vt E [O,1]

min f1Zj(s)f(sIt) ds
1 n
Then, regardlessof the v determinedby f as a result of the Vickeryauction there is a participationcharge which does not induce bidder 1 to refuse to participategiven his type (the first inequalityin (1.2)) and extractsall but E of his rentswhere E is arbitrarilysmall.Hence, if an optimalauctionexists,it must extractall of bidder l's rents regardlessof his type. Thus, it is enough for f to satisfy(1.2) in order that we may extend CM's result to the continuumcase. In additionto showingthat the continuumanalogueof the conditionon the conditionaldensityprovidedby CM is both necessaryand sufficientfor full rent extraction,we shall provideremarkablysimple sufficientconditionson f under which the mechanismdesigner can extract almost all of the rents (up to an arbitraryE > 0) for all type realizationst. The techniques developed in Section 2 apply not only to auction environments but to a large class of mechanismdesign problems,and do not impose much structure on the environment(only properties of the density f). As previouslymentioned, these techniques apply to Groves mechanismsfor the allocation of a public good, taxationschemes, agency and regulatoryenvironments, and generally to environmentswhere the presence of information correlated to private information is reasonable. However, when a specific environmentis given, somewhat sharperresults obtain, because propertiesof the relationshipbetween the rent function v and f can be utilized. This is illustratedin Sections 3 and 4, where specific environmentsare described.In particular,conditions leading to complete rent extraction,rather than almost complete rent extraction,are given. Myersonand Satterthwaite(1983) considerthe followingbargainingproblem with two sided asymmetricinformation:Suppose that a seller's cost s and a buyer's value t are known only to the respective agents, are independently distributed,and that the supportsof the densities overlap,so that the decision of whether to trade is nontrivial.It then turns out that any ex-post efficient tradingmechanismrequiressubsidiesfrom outside, that is, there is no efficient tradingmechanismthat "breakseven." We shall show in Section 3 (Theorem 3) that the combinationof first order stochasticdominance(increasesin the buyer'svalue tend to increasethe seller's opportunitycost of sale) and a "hazard rate" assumptionon the cumulative

400

R. PRESTON MCAFEE AND PHILIP J. RENY

distributionfunctionof the seller's reportedvalue given the buyer'stype imply the existence of efficient solutions to the bargainingproblem. Moreover,one such solution has the property that the buyer pays a positive participation charge and in additionpays a price less than his value for the good, provided that trade is efficient.The mechanismis constructedto be incentivecompatible and for the buyer to exactlybreak even. Moreover,the seller obtains all of the rents (he gets to sell the good at a price equal to the buyer'svalue, making honest reporting a dominant strategy), and only honest reporting survives iterativeeliminationof dominatedstrategies. Finally, we consider rent extractionin the Milgrom-Weber(1982) auction environment,and providean alternativeto the Cremer-McLeanresults,outside environmentswith finitely many privatevalues. A condition analogousto that used in the bargainingenvironmentleads to full rent extractionin the auction environment. The conclusionexploresthe implicationswe wish to drawfrom this analysis. Althoughthe paper developstools for solvingmechanismdesign problemswith correlatedinformation,the results(full rent extraction)cast doubt on the value of the currentmechanismdesign paradigmas a model of institutionaldesign.

2. SURPLUS EXTRACTFION

We shall focus on extractingrents from a particularagent, who has type t known only to himself. We take the view that the agent is participatingin a game which leaves the agent with rents equal to IT(t) on average.This game might be an auction, a bargaininggame, or any other game involvingprivate information.We assume that the agent's type falls in [0, 1] and that r is continuous,the latter being a feature of any mechanismdesign game in which type enters the payofffunctionscontinuously.6Finally,we assumethat the agent can achieve a payoff of zero by not participatingin the game in question by defining w(t) as the surplusin excess of the value of nonparticipation.By the revelationprinciple,we restrictattention(withoutloss of generality)to incentive compatible mpchanisms.Thus, in the game under consideration,we focus attentionon equilibriain which all participantsreporttheir types truthfully. We also assumethat there is a mechanismdesignerwho maychargethe agent a participationfee for the right to play the game. Thus, for example, an auctioneer may charge for the right to bid, or an arbitratorto a bargaining problemmightchargeboth agents some amount,etc. In addition,the participation fee maybe a function z of some randomvariables, the realizationof which the agent does not know at the time he makes the decision to participate,

6 For most of the resultsof this

section, s and t couldbe membersof convex,compactsubsetsof Euclideanspace. In particular,this holds for Theorems1 and 2, Corollaries2 and 3, and Lemma1, usingminorvariationsof the proofs.

CORRELATED

401

INFORMATION

althoughthe agent does know t, his own type. Call such a z(Q) a participation fee schedule.The randomvariables, which determineshis participationfee (i.e. a particularvalue z(s) of z( )), mightbe anotherbidder'sreportedvalue in an auction,the other party'sreportedvalue in a bargainingenvironment,etc. What is importantis that the realizationof s is not influencedby the agent'sreported value of his type. The agent is assumedto be risk neutral. Let f: [0, 1]2 -- R be the continuousconditionaldensity of s, given t. Given the participationfee schedule z(Q), y(t) = fJz(s)f(s t) ds is the agent's (expected) participationchargegiven that his type is t. Supplementingthe original mechanismby adding this kind of participationcharge, and assuming(for the moment)that the agent chooses to participatefor every realizationof his type, again renders truth-tellingas an equilibriumsince the agent's participation charge is independent of his report. Let R(f) denote the set of all such participationcharges.Hence, R(f) = (y: (3z e C[0, 1])(Vt ( [0, 1])y(t) = f1z(s)f(slt)

ds}

C C[O, 1].

Note that R(f ) is a linear subspaceof C[0,1]. Althoughwe restrictattentionto continuousparticipationfee schedules z, none of the resultschange if we allow for,instance z E L1[0,1]. As suggested by (1.2), we employ the supnorm IIy =max 0 11y(t)1. For any A c C[0,1], we shall denote the closureof A under this normby A. Hence, y e A if there exist arbitrarilygood uniformapproximationsx e A: (Vt (VE > 0)(3x c=-A)

E-[0,

1])

Iy(t)

-x(t)l

<

E

As noted in the introduction (and by Cremer and McLean (1988)), the mechanismdesigner also has availableparticipationchargesthat are independent of the agent's report and are not contained in R(f ). These charges are constructedas follows:Let N be a finite set of indices, and let zn be a member of C[0,1] for every n e N. Present the agent with a choice of participation chargesfrom R(f ). That is, the agent selects n E N, and is then charged zn(s) when s is realized.The agent of type t will select n minimizingthe participation charge: f1Zn(s)f(sIt)

ds.

If the agent's choice of n is not used in the game to follow, the participation chargegiven by f z(s)f(sIt) y(t) = min nO

ds

402

R. PRESTON MCAFEE AND PHILIP J. RENY

is independentof his reportedvalue in the game to follow.We denote the set of such participationchargesby r(f ) DR(f ). Thus, (2.1)

r(f)

= {y: (3N)(Vt E [O,1])y(t)

= 1
Zn(S

stds

cC[0,1]. The followingfacts are easily established: (2.2)

Y1,Y2Er(f) =y1

(2.3)

yE r(f),

(2.4)

ylE..yker(f)

(2.5)

1,-l

a >O

+Y2 Er(f),

ayE r(f)I

min Eynr(f)

E r(f ),

Y1Er(f), Y2 ER(f) =y1-Y2 (2.6) Er(f) Now, as outlined in the introductionfor the special case of an auction environment,our goal is to establish conditionsunder which (1.2) is satisfied. Even in our more generalenvironment,if this is the case, then regardlessof the r determinedby f and the equilibriumof the given mechanismbeing played, for any E> 0 there is a participationchargein r(f ) which inducesthe agent to participatein the original game (playingthe original equilibriumthere), and which extracts all but E of the agent's rents. Since (1.2) is equivalentto the conditionthat r(f) is dense in C[0,1] (with respect to 11i), the problemof full rent extractionis equivalentto findingconditionsupon the conditionaldensity f( I *) so that Kf) = C[O,1]. The rest of this section is devoted to preciselythis issue.

Our firstresultprovidesconditionssufficientfor subsetsof C[0,1] to be dense in C[0,1]. It is thereforeanalogousto the classicalStone-Weierstraussapproximation theorem (see Friedman(1970, p. 116). Our proof follows similarlines, even though they assume that the closure of their class of functions is closed under the taking of both minima and maximawhereas we assume that it is closed only under minima.This accountsfor our additionalhypothesis(2.11). Before statingthe theoremwe define for any E > 0, 8 > 0, and toE [0,1],the set U(E, 8, to) of (8, 8) u-shaped functions at to as follows: u E C[0,1] is in U(, , to) if and only if

(i)

u(t) > O

for all

(ii)

U(to)?<,

and

t Ec [0, 1],

u(t) > 1 whenever It- toI > s. Note that if E < Eo and 8 < 80, then U(8, 8, to) c U(Eo, 80, U(E, 8, to) is convexwith a nonemptyinterior.7 (iii)

to).

Also note that

7To see that U(E, 6, to) has nonempty interior, note that it contains the 8/2 ball centered at 8/2+

It-to/j6.

THEOREM

1:

403

INFORMATION

CORRELATED

SupposeA c C[O,1] satisfies

x+yE A,

(2.7)

x,Y eA

(2.8)

xeA, a >0=axcA,

(2.9)

X1*E,x

(2.10)

1,-1 eA,

(2.11)

for all ?,

yeA,

A, y(t) =min {x1(t),..*,xn(t)}

$> 0 and every t E [0, 1], U(E, $,t) nTAz 0.

Then A = C[O,1]. (e C[O,1], and E > 0. Let a be a positive constant satisfying PROOF: Fix a > 2111T11. Choose now any to E [0,1], and correspondingto it a 5,0> 0 such that

17(t) - 7(t')I < ?/2 for all t, t' E

[to

-St

to +

8to]. Next, choose utoE

U(E/(2a), Sto to) nA, and let xt (t) = a(ut (t) - u, (to)) + nr(to).Finally, choose

This is -qto>0 such that Ixto(t)- 7n(t)I 0, xtj EA, and -it,> 0 with the propertiesendowedthem by the construction. Now, xt(t) < r(t) + E/2 for all t E (ti - -qt' ti + -1t.). Hence, letting y(t) = minl< i < k{xt(t)) for every t E [0, 1], we have that y(t) < d(t) + 8/2 for every t E [0, 1]. Also, for each i, and every t E [0,1], ,m(t) -xt,(t)

= (v7(t) - 7(ti)) < 7r(t)

since

ut e U(8/(2),

- uti(t))

+ ar(utj( + e/2,

-m(tj)

if t

St,, ti). Hence,

e [ti -

t.,t,

ti+ 8t] we have

n(t)

-

xt(t) < E/2 + c/2 = E. On the other hand, if t 0 (ti - Si, ti + 8i], xt(t) > a E/2 + w(ti) (since ut E U(E/2a, 3t,j tj)).

Puttingthese togetheryields: xt(t) > 7(t)

-8,

if

t E [ti

-

8,, ti + 8,]

and xj(t) >a-s8/2

+ 7(tj),

if

t E [0,1]\[t

-St.,

ti +

StiJ.

But since a was chosen so that a > 211r11,we have a - 8/2 + 7r(ti) > n(t) - E for all t E [0, 1]. Hence, for every i = 1,..., k, xt(t) > n(t) - 8 for all t E [0, 11. This implies then that y(t) > w(t) - E for all t e [0, 1], so that IIY- 711I < E. Q.E.D. Since E > 0 was arbitraryand y EA, we concludethat T c=A. REMARK 1: For the set A = r(f), (2.7)-(2.10) are satisfied(see (2.2)-(2.5)). Thus, we need only ensure that (2.11) holds in order to produce r(f ) = C[O, 1].

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R. PRESTON MCAFEE AND PHILIP J. RENY

Also, Theorem 1 continuesto hold if s and t varyover a compactmetricspace. In particularit covers the finite types case as well. We now present our main result which provides a necessary and sufficient condition (the continuumanalogue of that in Cremerand McLean(1988)) for (almost)full rent extraction. Let [0, 1] denote the set of probabilitymeasures on the Borel subsets of [0, 1]. We have the followingtheorem. 2: (f) = C[O,1] if and only if thefollowingconditionholds: For everyto E [O,1] and every,E [O, 1],

THEOREM

(*)

Au({to})s1

impliesthat f(
PROOF: We firstprove the necessityof (*) for K(f) = C[O,1]. So, supposethat r(f ) = C[O,1], and that to E [0, 1] and ,u E [0, 1] satisfyff Ito)= Jfof( It),u(dt).

We will show that (A{to}) = 1.

Let y(t) = (t - t0)2 for every t E [0, 1]. Hence, y E C[O,1] = Kf ). There must therefore be a sequence {y,}7=1 of functions in r(f) so that yn '->y. Since each YnE

r(f) we have Yn(t) M~t)

= 1mimm n lw 1n( t), {w1t,

for every n, and every t

E

i in.t. Wmn(

[0, 1], where forsome z E C[O,1].

W/1(t)=f|zi(s)f(s1t)ds 0

Thus, for each n and every t E [0, 1], Yn(t) = w(n t)(t) for some i(n, t) < mn. In particular, Yn(t0)

= Wj(n to)(tO)

=

f

ds

t ZI(f,to)(s)f(sIto)

= ffZI(fltO)(5)f(51t)l/L(dt) in(n,

to)(st),u (sd

= Wn(l,

(t)lJi(dt).

=

ds

Since yn(tO) y(tO) = 0, this implies that the last integral convergesto zero. Now, by definition,yn(t) < wi(n,to)(t) so that (since ,u E [0, 1]) eYn(nt)efy(dt)

<

l|

Hence, O> foly(t),u(dt) = fol(t

n _

to)(dt) so t) to)2A

O

(dt), so that li(fto)) = 1.

405

INFORMATION

CORRELATED

We turn now to sufficiency,and proceed by way of contradiction.Suppose that r(f) = C[0,1] and that (*) holds. Since hypotheses(2.7)-(2.10) of Theorem 1 are satisfiedwhen A = r(f), it must be the case (by Theorem 1) that (2.11) fails when A is replaced by r(f). Thus, there exist eo, 80 > 0, and to E [0,1] such that U(e0, 80, to) n (f) = 0. Since R(f) c r(f) we have a fortiori that U(eo, 8o, t) n R(f) = 0.

Now, R(f) is convex (being a linear subspace) and as previouslynoted, U(E0,80, to) is convex and has a nonempty interior. So, by the separating hyperplanetheorem (Dunford and Schwartz(1958, 1988; Theorem 8, p. 417), there is a continuous linear functional on C[0,1] separating R(f) and U(e0, 80, to). Equivalently,by the Riesz representationtheorem (Dunford and Schwartz(1958, 1988;Theorem3, p. 265)),there is a regular,countablyadditive, signed measure ,u = 0 on the Borel subsets of [0,1] and a constant c E DRsuch that (2.12)

f1x(t),u(dt)

< c

for all

xE R(f),

(2.13)

f1x(t),u(dt)

>c

for all

xE U(c0,80, to).

and

Since R(f) is a linear subspace, we must therefore have fJolx(t),u(dt)= 0 for every x E R(f). (Otherwisethere is an x0 E R(f) with fJ0xo(t)A(dt)0 0, and a > c, violating(2.12) since ax0 E suitable choice of a E DRyields fJolaxo(t),u(dt) R(f).) Hence, c can be taken to be zero withoutloss of generality. Combining(2.12) and the definitionof R(f) we then have f (f1z(s)f(sIt)

ds}.(dt)

=0

for every z E C[0,1].

By Fubini'stheorem,this is equivalentto f0z(s)

[f f(slt),.L(dt)

ds=O

forevery

zEcC[0,1].

Hence, the continuous function of s in square brackets is identically zero. That is (2.14)

f1f(

1t)t(dt)

= 0.

By the Jordandecompositiontheorem (Cohn (1980, Corollary4.1.5, p. 125)), we may write ,u as the difference between two positive measures ,u+ and ,u- at

least one of which is finite. Furthermore,there are disjoint Borel subsets of [0,1], A+ and A-, such that ,u+ (A-)=,

-(A+)=0,

and A+uA-=[0,1].

Thus (2.14) becomes (2.15)

+f( It)pA (dt)

=

ff

It)A(dt).

Regardingboth sides of (2.15) as functionsof s E [0, 1], integratingover s (with

406

R. PRESTON MCAFEE AND PHILIP J. RENY

respect to Lebesgue measure) and using Fubini's theorem yields JA+ di+= d- = 1, where the second equality is without loss of generality.Hence, JAboth ,A+ and ,u- are in A[0, 1]. Combining (*), (2.15), and the fact that ,u = 0, yields that neither ,A'+ nor ,uis a point mass on to. In particular, since ,u-Ee A[0, 1] is regular (see Billingsley

(1968, Theorem 1.1)), there is a closed subset B of A-, and a 8 E (0, 8o] such that B n (to - 8, to + 8) = 0 and Au-(B) > 0. Choose K > 1/,A -(B) > 1, and define the step function x on [0, 1] as follows: (0, K,

x(t)=

1.1,

if te(to-8,to+8) if teB,

otherwise.

Hence, fJ0x(t)1i(dt) < 1 - Kui -(B) < 0.

Now, using Theorem 1.2 of Billingsley,it is straightforwardto construct a sequence of continuousfunctions{x(}x=1on [0,1] such that for every n, (i) X,,(t) > 1, for every t O-(to - 5, to + 5), (ii)

X,,(t) > 0, forevery t E- [0, 1],

(iii)

Xn(tO)

(iv)

for every te [0,1],

=0,

xn(t) -*x(t),

for every t E=[0,1], (v) Xn(t) fJox(t),u(dt)< 0. Thus for n largeenough, fJ0xn(t)1,(dt)< 0, contradicting(2.13). Q.E.D.

To better understandcondition (*), consider,u to be a prior on the agent's type t. The induced distributionon the signal s is then given by JolJfIt)O(dt). Hence, if f Ito)= Jolf( It)(dt), then learning that the agent's type is to providesno new informationabout the signal s. Condition(*) asks that unless one's prior is already concentrated on an agent's type to say, learning the agent'stype isgalwaysinformativeaboutthe signal s. In particular,when s is the reportedvalue of another agent (which in equilibriumis a truthfulreport),(*) asks that each agent's privateinformationnot be entirelyuninformativeabout other agents'privateinformation. REMARK 2: Note that if for every to E [0, 1], there is an xto E r(f) taking a minimum uniquely at to, then setting y(t) = xt(t) - xto(tO)in the proof of necessityabove is enough to show that (*) holds and hence (by sufficiency)that r(f) = C[0, 1]. This observationis at the heart of the three corollarieswhich folloW.8Like Theorem 1, Theorem 2 also holds if [0, 1] is replaced by any 8Equivalently, if for every to E [0, 1], such an x, e (f) exists, then r(f) satisfies (2.11) and hence all the hypotheses of Theorem 1, by Remark 1. Again this yields r(f) = C[O,1].

CORRELATED

407

INFORMATION

compactmetricspace. In particular,the theoremand proof givenhere cover the finite types case. COROLLARY

1: Suppose x E R(f), y E r(f ) satisfy

(Vt)x'(t) > 0, (Vt ) y' ( t ) /x'( t ) is strictly increasing in t. Then (f)

=

C[0, 1].

PROOF: As noted in Remark 2, we need only find for each to0E[0,1] a functionin r(f) takinga minimumuniquelyat to. Let

- y(t)

q(t) =y(t)

By (2.6), q E r(f ). Moreover q'(t) =y'(t)

-

y'(to)

x'(to)

x'(t) > < O as

y'(t)

x(t) 't<

> y'(to)>

t

( ?t ,t)o

Thus q(t) achieves a minimum uniquely at t = to.

Q.E.D.

of F, the conditionalc.d.f. REMARK 3: Considera monotonictransformation of s given t: G(slt) =F(p(s)Ii/(t))

= /(O) = O, p(l) = q/(l) = 1. Then R(g) = {x: x(t)= A(O)

where Sp', '> 0, y (o-'(t)),

y eR(f)}.

To see this, note

y(t)f()= | 1z(s)g(slt) z(s)g(slt)

ds

=

du

| z(u)f((p(u)If(t))
z f1z(6'(s))f(sIl(t))

Thus, if y E R(f), y(-r1(t)) E R(g) and vice versa. Now suppose, x, y E R(f) satisfy the hypotheses of Corollary 1. Then x(+- 1(t)), y(-/

dt

(dt t)) -x(tf'(t))=

d y'((f1(t)) x(v()-

1(t)) E R(g), and

[du x'(u) ] ux()q()>0

Thus, if Corollary1 applies to f, it applies to a rescalingof f. By Corollary1, it is straightforwardto show that combinedwith first order stochastic dominance, a sufficient condition for r(f) = C[O,1] is that E[s2IE(s It) = A] be a strictly convex function of ,u. The following example illustratesthis.

408

R. PRESTON MCAFEE ANDPHILIP J. RENY

EXAMPLE1: f(s It) = tst'. ,U= E(s It) = t/t + 1 implies that t = u/(1 - ,). Also, E(s2It) = t/t + 2, so that E{s2IEs = ,u} = ,A/(2 - A), a convexfunctionof /i E [0, 1/2]. Thus, letting x(t) = E(s It) = t/t + 1 and y(t) = E(s21t) = t/t + 2,

we have that y'(t)/x'(t) = 2(t + 1/t + 2)2 is increasingin t. Since by firstorder stochastic dominance x'(t) > 0, Corollary 1 can be directly applied to conclude

that r(f) = C[O,1]. In general, with first order stochastic dominance and E[s2IE(sIt)= A] a convex function of ,u, x(t)=E(sjt) and y(t)=E(s21t) will satisfy the hypotheses of Corollary 1. Furthermore, in this case the participation

fee schedules zn(s), can be chosen to be quadraticin s. These conditions are satisfied for many common distributions, in particular those with mean and

varianceincreasingin t. The lemma to follow establishesa useful equivalencefor placingfunctionsin R(f). [A] denotes the linear span of A. The proof of Lemma 1 is in the Appendix. LEMMA1: R(f)=

[{f(sI .):0
2: Supposethat (2.16) (Vt)(3s)(Vt' + t) f(s It) >f(s It'). Thenr(f ) = C[O,1]. COROLLARY

PROOF: By Lemma 1, -f(sI ) c r(f). By (2.16), for each t, there exists an s with -f(s *) taking a minimumuniquely at t. In light of Remark 2, r(f) = C[O,1]. Q.E.D. Figure 1 presents a density satisfyingthe hypotheses of Corollary2. The conditionon f expressedin Corollary2 has a numberof interpretations.The first is the direct one, namelythat for each of a player'stypes t, there exists a value of the signal s, so that t maximizesthe likelihoodof s. Alternatively,one can relate the hypotheses of Corollary 2 to a strengtheningof first order stochasticdominance.For instance,supposethat in additionto f satisfyingfirst orderstochasticdominance,for every t0ot1 E [0,1] there exists a uniques E [0,1] such that f(sIto) =f(sIt1) and that fixing to, this s is strictlymonotonicin t1. (Figure 2 below illustratesthis.) It is not hard to show that in such a case the hypothesesof Corollary2 must be satisfied. The final result of this section providesfurtherconditionsfor rent extraction which, in some instances,are simple to verify.

(2.17)

3: Suppose there exists a set S c [0,1] such that (Vs E S) f(slt) is strictlyconcavein t, and

(2.18)

(Vt0, t1 E [0, 1])(3s E S)

COROLLARY

Then r(f ) = C[O,1].

f(sIt0) >f(sIt1).

409

CORRELATED INFORMATION 1 0.8 0.6 0.

0 2 1.5 1

0.5

'A

0.8

0 FIGURE

f(.tJ

1.-f(s

It) = 2 min {s/t,

(1 -s)/(l

-t).

fEIt)

f( it)

0

0

s(t0St1)s(t0t) FIGURE

2.

1

2

410

R. PRESTON MCAFEE AND PHILIP J. RENY

PROOF: Suppose (2.17) and (2.18) hold, but (*) fails. Then there exists a to E [0,1] and A E A[0,1] not a point mass on to, with ff Ito)= ff(. It),u(dt). Define tI = ft,u(dt). Since f(s It) is strictlyconcave in t, for all s E S, we have, by Jensen's inequality, (Vs E S)

f(sIt0)

=

ff(slt)p,(dt)


Q.E.D.

whichcontradicts(2.18).

We gratefullyacknowledgethat a referee providedus with this dramatically improvedproof. 4: If the set S in the hypotheses of Corollary3 is compact and convex, it follows that for all to E [0, 1], there exists an s E S so that f(s It) is maximizedat t = to) in which case the hypothesis of Corollary2 is satisfied. However, there are examples where S is not convex and the hypothesis of Corollary2 fails to hold, even though the hypothesesof Corollary3 hold. REMARK

REMARK 5: The conclusion of Corollary3 continues to obtain if, in the hypotheses,"concave"is replacedwith "convex." REMARK 6: One case of interest occurs when f(s0 It) is strictlyconcave and decreasingin t, while f(s1 It) is strictlyconcaveand increasingin t. In this case, S = {so,s1l suffices, as (2.18) follows immediately.Thus, we can establish the desired propertywithout any informationabout "most"of f. REMARK 7: Corollaries2 and 3 continueto obtainwhen s and t are members of convex,compactsubsetsof Euclideanspace, which includesthe case of many players.Thus, there is nothing special about the one dimensionalcase, at least for these results.

We now show by example that the combination of first order stochastic dominance and affiliationis not sufficientto guarantee (f) = C[0,1]. As the exampleillustrates,the combinationof these propertiesadmitsan f with R(f) comprisedof only linear functionsand no u-shapedfunctions. 2: f(slt) = 1 + (2s- 1)t. Note Jff(sIt)ds = 1 + (S2 - s)tjO EXAMPLE f(s It) > 1 t > 0, so f is an admissible conditional density. du =S + t(S2 -s),

F(slt) =ff(ult) F,(slt)

=S2-S

<

0

for s E (0,1),

=

1, and

CORRELATED

411

INFORMATION

so F satisfiesstrictfirst order stochasticdominance.Also, a2 dsdt

logf(slt)

=-

2s-1

a

as 1 +(2s-1)t

> 0, ?

so f is affiliated (see Milgrom and Weber (1982)). Equivalently,f has the monotone likelihoodratio property. Finally, R(f) = [(1,t}], the set of linear functions. (1 indicates the constant function, t the identity.)It is easily seen that r(f) is then the set of concave functions,and is thus a strict subset of C[O,1]. We mention briefly that the results of this section can be extended in a straightforwardmanner to the unbounded support case. This may require allowingplayers to choose from among countably(rather than finitely) many participationcharges,so in what immediatelyfollows r(f) is: r(f)

= (y(t)

EC(1R)Iy(t)

=

minfzJ(s)f(sIt)ds

for some countable subset

{ Zn}n=

of C(R)

Corollary1 where C(lM)denotes the set of bounded continuousfunctionson WR. above goes throughverbatim. COROLLARY

then (f) =C

1': If xe R(f), ye r(f), x'> 0, and y'/x' is strictly increasing,

.

Corollary2 above also admitsa naturalcounterpartnamely,Corollary2': COROLLARY

(i)

2': Suppose

VtEI , 3sEI R such that:

(a) f(sIt) >f(slt') (b) f(slt)

Vt' # t,

> limsupf(sIt'), It'I -?00

and

(ii)

fQ* *) is uniformlycontinuouson WR2

Then (f) =C(R). Both Corollaries1' and 2' are of particularinterestsince their hypothesesare satisfiedwhen s and t are jointlynormallydistributed,with nonzerocovariance. The proofs of Corollaries 1' and 2' follow from suitable modificationsof Theorem1 and Lemma1. The hypothesesof Corollary1' are also satisfiedwhen the agent's type and the signal are additively related (i.e. f(s It) = h(s - t)), and

to cases in which s=x0+x1

and t=xO+x2

where the xi's are independent

412

R. PRESTON MCAFEE AND PHILIP J. RENY

drawsfrom gammadistributionswith parameters(ai, ,i), i = 0, 1, 2. In each of these cases, participationfee schedules zn(s) that are quadraticin s suffice. We now return to the case of bounded support and end this section with a brief application.(A more detailed applicationis providedin the next section.) Consider a principaldesigning a contract for a risk neutral agent possessing private information t E [0, 1]. The principal knows he can receive signal s correlatedto t, sometime in the future. What is the value of s? Considerthe full informationgains from trade G, and the solution to the informationally constrainedcontractdesignproblem,which gives the principalprofitsof G'. We have shown that if an efficientmechanismexists, then, for many densities, the value of the correlatedinformationis G - G'. This follows since the principal can set up a mechanismwhich is full-informationefficient,producingrents G, and then extract those rents via a participation charge zn(s). That is, the

principal"sells the agency"to the agent for zn(s). We believe that, in many economic problems, the presence of correlated informationis natural, and destroysthe "inefficienciesresultingfrom privateinformation"so often cited in the literature(see McAfee and McMillan(1987a)for references).The thirdand fourth sections providetwo such examples. 3. BARGAINING

MECHANISMS

Consider a buyer with value t, known only to himself, of an item and a potential seller, who privatelyobserveshis own opportunitycost of sale, s. It is commonknowledgethat s and t were drawnfrom a joint density g(s, t) with support[0, 1]2. Both buyer and seller are risk neutral. As Myerson and Satterthwaite(1983) showed, if s and t are independent, then there is no efficient mechanismfor arrangingtrades in this environment that does not lose money on average. As we shall show, however, under alternativeconditions,there is an efficientmechanism.We think it is plausible that the determinantsof the buyer's value may also influence the seller's opportunitycost. Thus when the buyer'svalue is high (e.g. due to an increased estimate of resale value) the seller's opportunitycost will typicallybe higher than usual. Thus, independenceof values is likelythe exceptionratherthan the rule. We now show how the resultsobtainedin the last section (Corollaries1, 2, and 3) can be used to demonstratethe existenceof an efficientmechanismwhen the values are correlated. Considerfirst the following "pre"-mechanismwhich includes a risk neutral third party who acts solely as a budget balancer when necessary. Let ra, rp denote the seller's, buyer's reported signal respectively.If r., < rp, then the seller receives rp for the good and the buyerpays r,. The differencerp - r, is made up by the budget balancer. If r. > rp, then the good is unsold and no paymentsare made. Honesty is a dominantstrategyfor both buyer and seller here and in equilibriumthe budget balanceris expectedto lose G=

f'ft(t 00o

- s)g(s,

t) dsdt,

CORRELATED

413

INFORMATION

the expected gain from trade,whichwe assumeis positive.This pre-mechanism is clearly ex-post efficient althoughit requires "outside"money. Myersonand Satterthwaiteshowed that any ex-post efficient mechanism requires outside money (in the sense that a budget balancermust be includedand would expect to lose money) when the valuationsof the buyer and seller are independently distributedand'theirsupportsintersectin an interval. However,let h(s It)be the conditionaldensityof the seller'svalue given that the buyer'svalue is t and let k(tis) be the conditionaldensity of the buyer's value given the seller's value is s, and suppose that both h and k satisfy (*) (ruling out independence, in particular). Then, letting rr, 70 denote the seller's, buyer'srent function(a functionof their respectivevalue of the good) respectivelyobtainedfrom participationin the game definedby the pre-mechanism, we have, for (s, t) E [0, 112,

f'(t - s)k(tls) dt,

(3.1)

(s)'

(3.2)

-|( 7rO3(t)=

ds.

t -s)h(sIt)

Now, by assumption r(k) = r(h) = C[A,1]. Hence, given any ? > 0 there exist one for the buyer finite sets of participationfee schedules {zf }neN {zi E -andone for the seller, such that for all (s, t) [O,1]2, ,

(3.3)

0 < w'v(s)

min f1z'(t)k(tis) 0

-

dt
neNN,

and (3.4)

0O< r(t) - min fZ-'(s)h(sIt) n

ds
n E Nps

Hence, if we supplement the pre-mechanismabove by offering the seller a choice among participationfee schedules from {zn},n,eN and his value is s, he will expect to be charged c(s) "()n

=

min f1z(t)k(tIs) n

dt,

E-N,,

since by choosing n E N, he will be charged z"(t) if the buyerreportsvalue t. His rents therefore become wa(s)

-

c'f(s) and lie between 0 and E by (3.3). The

buyer must similarlychoose among {Z!)}neNp and his rents become wr(t) C(t) E [0, E] when his value is t and where c(t) is defined analogouslyto c'T(s).All revenue generatedby these participationfees is given to the budget balancer.

414

R. PRESTON MCAFEE AND PHILIP J. RENY

Hence, the budget balancer'snet expected revenuebecomes: R

f1f'co(s)g(s, 00

t) dsdt + f1'1cP(t)g(s, ~~~~~~~00

t) dtds-G

>(G-e) + (G-e) -G -G-22 where the inequality arises from the definitions of G, c'() and c-(), and (3.1)-(3.4). So, for e small enough, R > 0. Finally,choose aEE [0,11and require the budget balancer to give the seller aR and the buyer (1 - a)R (independentlyof reports),so that the budgetbalancernow expectsto breakeven. Since none of the chargeswe have introducedaffectthe buyer'sor seller'sincentiveto reveal his true value, this mechanism(i.e. pre-mechanismplus net participation charge) is ex-post efficient and allows the budget balancer to break even on average.Note that since R -- G as E -* 0, by choosing a E [0,1] appropriately we may give the seller SG of the gains from trade for any 8 E (0,1). On the other hand, givingall of the gains from trade to either the seller or the buyer may not be possiblewith participationchargesof this sort. As mentionedin the introduction,the analysisof the rent extractionproblem is greatlysimplifiedwhen one need not take into account the precise relationship between the conditionaldensity f and a player'srent function 7r.On the other hand in some environmentsthis more detailed analysisis tractableand providesstrongerresults;namely, all rents ratherthan almost all rents can be extracted and more importantlythe mechanismwhich extracts the rents can actuallybe constructedratherthan simplyshownto exist. As we now show, the Myerson and Satterthwaitebargainingenvironmentis amenable to this more detailed approach. As before, g(s, t) is the joint density between the buyer's and seller's valuationof the good. Let f(sit)

=g(s,t)/

g(u, t)du

so that f is the conditionaldensity,and let s

F(slt)

=

ult) du

be the distributionfunctionof s, conditionalon t. Let F2(sIt) = d/t F(s It). By taking advantageof our explicit descriptionof this mechanismdesign environment, we get the followingresult: THEOREM

3: Suppose V(s, t)

(3.5)

F2(Slt) < 0,

(3.6)

d

E (0,1)2,

1>0.

Then there exists an efficient trading mechanism giving all of the rents to the seller.

INFORMATION

CORRELATED

415

PROOF: We displaythe mechanism.The buyerand seller make reports(r and respectively)of theirvalues t and s, respectively.The seller receives r for the item if r > s, and otherwiseno trade occursand the seller gets nothing.Honesty is a dominantstrategyfor the seller, and should the buyerchoose to be honest (as he will in equilibrium),all the rents go to the seller. The buyer is awarded the good if r > s, and is requiredto pay

s,

F(rir) r+

F(_r_r)

F2(rlr)

- F(rlr)2 F2(rlr)

if

r
f

r s.

ifr

.

Since the seller is honest, this providesa buyerwith value t who reports r rents equal to )2r

u(r, t)

-

F2(rlr) +

t-r-

F(rlr)

jF(rlt)

as s < r with probabilityF(rIt). Since u(t, t) = 0, the buyeris willingto participate,as he can obtain at least 0. We need only verifythat he can do no better than 0 to complete the proof: r >
as

F(rlr) F(rlt) < r+ F2(rlr) F2(rIt)

t+ F r

(y36) (by (3.6))

F(rIt) ? 0 FF(rlr) t1 r-+ F2(rlr) JF2(rlt)>

as [t -r-r -

F'j

]F2(rlt) +F(rlt) < O (by (3.5)),

as u2(r, t) > O. Thus u2(r, t) > 0 as r > t. We concludethat u(r, t) > 0, since u(r, t) increasesin t for t < r up to u(r, r)

0, and then decreases in t > r. Consequently,

incentivecompatibilityis satisfied.

Q.E.D.

REMARK 8: The mechanismrequires a budget balancer, since paymentsby the buyer equal paymentsto the seller on average,but not for all realizations.

We wish to argue that the hypothesesof Theorem 3 are plausible.The first hypothesis,firstorder stochasticdominance,merelyrequiresthat the aspectsof the environmentthat increasethe buyer'svalue also tend to increasethe seller's opportunitycost. Thus there is a "commonvalue" aspect to sale: increases in the buyer'svalue also increase the seller's use value of the item. The second condition(3.2), looks like a hazardrate condition (see McAfee and McMillan

416

R. PRESTON MCAFEE AND PHILIP J. RENY

(1987a)for an intuitiveexplanation),and is equivalentto a2

1

(at)2 F(sIt) This conditionmay be related to the "usual"hazardrate condition, d/ds(s + F(s)/f(s)) > 0, by observingthat the latter is equivalentto a2

1

(aS)2 F(s)

> 0.

Thus, (3.6) requires the standardhazard propertyto hold for t instead of s. This condition is satisfied for many examples. Example 1 in the preceding section satisfiesit, so that, although r(f) contains only concave functions,and the profitsnet of participationchargesto the buyer in efficientmechanismsare strictly convex, the rents may still be extracted. This paradox is resolved by noting that the mechanismdescribedhere is not of the Section 2 form (which consist of participationchargesalone) but links the participationcharge to the sale of the item. Therefore, by exploiting aspects of the game, a mechanism designer can attain allocations which cannot be attained using unconnected participationchargesand gambles. To summarize,under reasonable assumptionson the distributionof values, which implya certainamountof correlation,there exists an efficientsolutionto the bilateralbargainingproblemwith asymmetricinformation. Next we show that Milgromand Weber (1982) auctionenvironmentsare also amenableto the more detailed analysisjust applied to the bargainingproblem. This supplementsthe implicationsof our analysisin Section 2 that under the hypothesesof either Theorems2, 4, or 5 an optimalauctionextractsall bidders' rents by providing conditions under which an optimal auction exists and a constructionof such an auction. 4. OPTIMAL AUCTIONS

Milgromand Weber (1982) present a general model of the auction environment, allowing for correlation among valuations, and valuations which are viewed by the bidders as random.For simplicity,we shall consider the case of two bidders. It is assumedthat the seller values the item at zero. The bidders receivesignals s and t respectively,privatelyobserved,which are generatedby a density g(s, t) with support [0, 1]2. We assume g is symmetricand focus on bidder "1"who has signal t. Let f(st )

and

-a)

f0g(u, t) du s

F(slt) = |f(ult)

du.

CORRELATED

417

INFORMATION

Even if bidder 1 knows s and t, his valuationmaybe random,so we let u(t, s) be the expectedvalue of the objectto bidder 1 given s and t. Symmetrically, we assumethat the other biddervalues the item at u(s, t). Finally,we assumethat u is strictlyincreasingin its first argument,and that t > s -* u(t, s) > u(s, t) so that the agent with the highest signal is the efficientconsumerof the item. Define v(r, t)

s)f(slt) ds E{u(t, s)Is < r}F(rlt),

= fu(t,

which is the expectedvaluationto bidder 1 if his signal is t and he obtainsthe item whenevers < r. As before, subscriptsdenote partialderivatives. THEOREM

4: Suppose V(s, t)

and

F2(sIt)
a

E (0, 1)2

V2(s, t)

dt F2(slt) Then there exists an efficient mechanism which extracts exactly all of the bidders' surplus, and hence is optimal from the seller's point of view. PROOF: Supposebidder 2 reportshis signal honestly,and bidder 1 has signal t and reports r. The mechanismawardsbidder 1 the item if r > s, with a charge of v2(r,

v(r, r) -

r)F(rlr)

Fs(rlr) v2(r,

ur, r)-

5>

r)F(rlr)

F2(rlr)

+

r,

V2(r, r)

F2(rlr)

s

r.

Bidder l's profitis -Tr(r,t) =v(r,

v

v2(r, r)(F(rlr) -F(rlt)) t) - v(r, r) +F2(rlr

Clearly ir(t, t) = 0, so individual rationality is satisfied. 'r2(r,t) =v2(r,t) v2(r,t) F2(rlt)

<

v2(r,r) - F2(rlr)

-

>2(rr)F2(r)

F2(rlr)

0

<0

as

a

418

R. PRESTON MCAFEE AND PHILIP J. RENY

Thus, Tr(r,t) is maximized,as a functionof t, at t = r, that is, ir(r,t)


=0.

Consequently,incentive compatibilityis satisfied.The mechanismis then efficient, incentivecompatible,and extractsall the rents. Q.E.D. 9: If the auction is a private values one, u(s, t) = t and v(r, t)

REMARK

=

tF(rl t). Thus

a

v2(r,

t)

dt F2(rlt)

a

t + F(rIt)

dt

F2(rlt)

J

which is the hazard condition (31) from the previous section. Generally,this conditionsays that the expectedvalue of receivingthe item whenevers < r is a convexfunctionof F(r It), which is not (at least to us) an intuitiverequirement. For the commonvalue environment,where u(t, s) = u(s, t), and where there is no issue of efficiency(awardingthe item to an agent chosen at random is efficient), a stronger result holds. As long as s and t are independently distributedconditional on the true valuation (which is unknown),all but an arbitrarilysmallfractionof the rents maybe extracted.This resultmaybe found in McAfee, McMillan,and Reny (1989). 5. CONCLUSION

We have examined the robustness of mechanism design solutions when independenceof informationdoes not hold. We found that privateinformation is often worthless;it does not lead to rents for its possessors in a variety of contexts. A commonreactionto this paper focuses attentionon the heavyuse made of the agents'risk neutrality,and arguesthat only together do the assumptionsof risk neutral agents (RNA) and independentlyand identicallydistributedinformation(iid) combineto make a good proxyfor the more difficultreal worldcase of risk averse agents and correlatedinformation.Considerthe auctionenvironment. With RNA and iid, any of the usual auctionforms maximizethe seller's revenue.This is taken as corroborationof the mechanismdesign paradigm.On the other hand, we know that when either RNA (Maskin and Riley (1984), Matthews(1983)) or identicallydistributedinformation(Myerson(1981)) fail, the usual auctionforms do not maximizethe seller's revenue.From this paper, if independence fails, the usual auction forms do not maximize the seller's revenue. Thus, if either RNA or iid fail to hold, we do not have a mechanism design explanationof the usual auctions.In light of this, it is difficultto believe that if both RNA and iid fail, the mechanismdesign solutionwill be similarto the case in which they both hold. We consider this a strong argumentthat mechanismdesign has no reasonable explanationof the usual auction forms, and that some other criteriamust be invoked.

CORRELATED

419

INFORMATION

This begs a substantiallymore difficultquestionwhichwe have not addressed. It is a remarkablefact that the English auction (with reserve price) is the solution to a mechanism design problem, maximizingthe seller's expected profits,in the independentprivatevalues framework.However,minorperturbations of the environment destroy this result. To see this, note that if f(s It) is a

conditionaldensity of s given t satisfying(*) or the hypothesesof any one of e) + ef(s It) is also a Corollaries 1, 2, or 3, then for any e E (0,1) g(s It) conditional density of s given t and satisfies the hypotheses of one of our theorems.Furthermore,g8 convergesin sup norm to the independencecase as E goes to zero. This indicates(at least to us) that the prevalenceof the English auction in selling items whose value is uncertainis almost certainlynot due to the fact that sellers are maximizingexpected revenue. The English auction does possess some important features. Milgrom and Weber (1982) showed that because the Englishauctionrevealsa lot of information as bidders drop out of the bidding, prices are pushed higher on average. Moreover, we suspect that the English auction does well in a variety of circumstancespreciselybecause it does not depend, as a selling mechanism,on informationabout the specificenvironment,such as densities of valuations,etc. We are not surprisedthat the mechanismsdescribedin this paper are not in common use, because these mechanisms(the z functions) will generally be -(

-

sensitive to the environment's description (e.g. f).9 Thus, this paper is really

more about economists'models of asymmetricinformationthan about asymmetric informationitself, since generallythe descriptionof the environment,at the level of detail requiredby mechanismdesigners,is absurd. Therefore, a reasonablequestion for the mechanismdesign literatureis how to capture the importanceof robustness.Specifically,we think the answer to questionslike "underwhat circumstancesare Englishauctionsused?"has much to do with the need for an institution to perform "well" in a variety of circumstances.Indeed, one might well imagine that the circumstancesare at least partially determined by the institution. That is, English auctions will attractbuyers who prefer the English auction over another selling institution, and thus the choice of mechanism-affectsthe distributions.One cannot hold the distributionsfixed in the experimentof choosingthe mechanism.'0 These concerns have led several authors (Holmstromand Milgrom(1987), McAfee and McMillan(1987b),Laffontand Tirole (1985)) to look for environments in which "simple"contractsor mechanismsare optimalin a wide variety of circumstances.These papers share a theme that a mechanism designer's desire to use complicatedmechanismswhich exploit aspectsof the environment (utilityfunctions,distributions)is reduced by enlargingan agent's action space. Indeed, it can be easily shown that the "optimal"mechanism,where optimalmaximizesone agent'srents, is not continuousin the densityof values. l?A glimmerof this idea may be found in McAfee and McMillan(1987c), which examines optimal auctions when participationis costly, and the participationdecision is made after the mechanismis chosen. This destroysthe seller's incentiveto post a reserveprice, commonto the literature,and exchangeis efficient.

420

R. PRESTON MCAFEE AND PHILIP J. RENY

Effectively,an agent can thwartthe mechanismby exertingeffortat a smallcost. The lesson of this paper is that asymmetricinformation,when combinedwith risk neutrality,plays a small or nonexistent role in such a research program. Generally, in environmentswith correlated information, the importance of private information is near zero. Department of Economics, Universityof Texas, Austin, TX 78712, U.S.A. and Department of Economics, University of Western Ontario, Social Science Center, London N6A5C2, Canada ManuscriptreceivedAugust, 1988; final revision receivedJuly, 1991.

APPENDIX PROOF OF LEMMA 1:

(s)

(D): Fix so E [0, 1] and E> 0. For s e [0,1], let

1/2aY

if

0O

otherwise,

s0- a

sso0+

a,

and choose a so that Is- sol

(this is feasiblesince f is continuouson a compactset, and hence uniformlycontinuous).Then

fI1z(s)f(sit) 0 _

=|1

ds -f(so It) 2af(SIt)

iso-a 2a "'' < 2

1so+a

2a

sl t)-f(solt))

dsff(SoSt)

(sI)

lfif()d

If(s Sit )f

t ( s 1|tds

so-a

ds

so-a

so+a e ds =E < ese f~~~~so-a

.

Thus, for so E [0,1], f(so I )e R(f). Since R(f) is closed under linear combinations,we have establishedone inclusion. Sk such that for all t e [0, 1], (c5): Since z, f are continuous,Ve > 03s,, k

fJz(s)f(sIt)ds-1/kEz(sj)f(sjIt) 0 .

Thus flz(s)f(s,


E1

)dsE-[{f(sl

)IsE-[0,1])]

as desired.

Q.E.D.

REFERENCES

Measures.New York:John Wiley and Sons. of Probability P. (1968):Convergence D. (1980):MeasureTheory.Boston:,Birkhaiuser.

BILLINGSLEY, COHN, CREMER,

J., AND R. McLEAN (1985): "Optimal Selling Strategies Under Uncertainty for a Discriminating Monopolist When Demands Are Interdependent," Econometrica, 53, 345-361. (1988): "Full Extraction of the Surplus in Bayesian and Dominant Strategy Auctions," Econometrica, 56, 1247-1257. D. (1987): "A Principal-Agent Model with Ex Post Pareto Optimality," mimeo, DEMOUGIN, University of Toronto.

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N., AND J. T. SCHWARTZ (1958; 1988): Linear Operators, Part I: General Theory. New York: John Wiley and Sons; Interscience. FRIEDMAN, A. (1970): Foundations of Modern Analysis. New York: Dover. HOLMSTROM, B., AND P. MILGROM (1987): "Aggregation and Linearity in the Provision of Intertemporal Incentives," Econometrica, 55, 303-328. HOCHSTADT, H. (1973): Integral Equations. New York: John Wiley and Sons. J. J., AND J. TIROLE (1985): "Auctioning Incentive Contracts," mimeo, MIT. LAFFONT, MASKIN, E., AND J. RILEY (1984): "Optimal Auction with Risk Averse Buyers," Econometrica, 52, 1473-1518. MATTHEWS, S. (1983): "Selling to Risk Averse Buyers with Unobservable Tastes," Journal of Economic Theory, 30, 370-400. MCAFEE, R. P., AND J. MCMILLAN (1987a): "Auctions and Bidding," Journal of Economic Literature, 25, 699-738. (1987b): "Competition for Agency Contracts," Rand Journal of Economics, 18, 296-307. (1987c): "Auctions with Entry," Economics Letters, 23, 343-347. AND P. RENY MCAFEE, R. P., J. MCMILLAN, (1989): "Extracting the Surplus in Common Value Auctions," Econometrica, 57, 1451-1459. P., AND R. WEBER (1982): "A Theory of Auctions and Competitive Bidding," EconometMILGROM, rica, 50, 1089-1122. R. (1981): "Optimal Auction Design," Mathematics of Operations Research, 6, 58-73. MYERSON, MYERSON, R., AND M. SATTERTHWAITE (1983): "Efficient Mechanisms for Bilateral Trading," Journal of Economic Theory, 29, 265-282. AND W. RILEY, J., SAMUELSON (1981): "Optimal Auctions," American Economic Review, 71, 381-392. DUNFORD,