Optimal Pricing Mechanisms with Unknown Demandweb.stanford.edu/~isegal/pricing.pdfOptimal Pricing Mechanisms with Unknown Demand By ILYA SEGAL* The standard proﬁt-maximizing multiunit

Optimal Pricing Mechanisms with Unknown Demand

By ILYA SEGAL*

The standard profit-maximizing multiunit auction intersects the submitted demandcurve with a preset reservation supply curve, which is determined using thedistribution from which the buyers’ valuations are drawn. However, when thisdistribution is unknown, a preset supply curve cannot maximize monopoly profits.The optimal pricing mechanism in this situation sets a price for each buyer on thebasis of the demand distribution inferred statistically from other buyers’ bids. Theresulting profit converges to the optimal monopoly profit with known demand as thenumber of buyers goes to infinity, and convergence can be substantially faster thanwith sequential price experimentation. (JEL D42, D44, D82, D83)

Recent advances in information technology—most notably the Internet—have enabled the useof economic allocation mechanisms that had beenimpractical before. Many goods traditionally soldat posted prices are now sold using auction-likemechanisms, in which buyers express their pref-erences by making bids. Some Internet web sites,such as eBay.com, use traditional auction mecha-nisms, such as the English auction. Other websites have developed new mechanisms. For exam-ple, so-called “demand aggregation” sites, such asMercata.com, LetsBuyIt.com, and eWinWin.com,obtain the price by intersecting the demand curveformed by the buyers’ bids with a downward-sloping “price curve.”

What is the profit-maximizing pricing mech-anism, and does it improve upon posted pricing?1

The present paper examines this question in thecontext of selling multiple homogeneous unitsto buyers with unit demands. First the paperadopts the standard assumption of auction the-ory that the seller knows the distribution fromwhich the buyers’ valuations are drawn. Underthis and other standard assumptions, the optimalauction can be represented by intersecting asupply curve submitted by the seller with thedemand curve revealed by the buyers’ bids, andselling to those buyers whose bids are above theintersection. The seller’s profit-maximizingsupply curve depends on her cost function aswell as on the distribution of buyers’ valuations.Furthermore, in two important special cases theseller cannot improve upon a posted price. Onesuch case is when the seller’s marginal cost isconstant, and so her optimal supply curve isperfectly elastic. The other case is when thenumber of buyers is large, and by the Law ofLarge Numbers the seller can predict the aggre-gate demand curve and the price at which itintersects the optimal supply curve.

The problem ignored by this standard analy-sis is that in reality, the seller may not know thedistribution from which buyers’ valuations aredrawn, and thus may be unable to calculate theoptimal reservation supply curve. A typical ex-ample is the sale of tickets or subscriptions to aone-of-a-kind concert or sporting event. Eventhough there are many identical units for sale,such units have not been sold before and so theseller does not know the potential demand. Asemphasized in microeconomic textbooks, in this

* Department of Economics, Stanford University, LandauEconomics Building, 579 Serra Mall, Stanford, CA 94305(e-mail: [email protected]). I am especially indebtedto Paul Milgrom, who raised the question motivating thisresearch and offered several invaluable suggestions. I amalso grateful to Susan Athey, Guido Imbens, JonathanLevin, Eric Maskin, Steven Tadelis, Hal Varian, EdwardVytlacil, Michael Whinston, and the participants of semi-nars at Duke University, New York University, the Univer-sity of Maryland, the University of Pennsylvania, PrincetonUniversity, Stanford University, Yale University, and the2002 Decentralization Conference and 2002 Stanford Insti-tute for Theoretical Economics Summer Workshop for theirinvaluable comments. Vinicius Carrasco and David Millerprovided excellent research assistance.

1 While this paper focuses on the profit objective, theanalysis is also applicable to a socially minded seller whoneeds to cover a fixed cost of production.

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situation “a monopolistic market does not havea supply curve” (Robert S. Pindyck and Daniel L.Rubinfeld, 1995), because the profit-maximizingprice depends on the overall shape, and in partic-ular the elasticity, of the demand curve.2

This paper proposes a new pricing mechanismthat maximizes the seller’s profit without requir-ing prior knowledge of demand. The mechanismis based on the idea that buyers’ bids reveal infor-mation about the distribution of their valuations.While standard auctions ignore this information,the optimal mechanism uses it for pricing.When the number of buyers is large, the sellerlearns the distribution precisely, and can priceoptimally given the revealed distribution.

To ensure that a buyer cannot obtain a betterprice by misreporting his valuation, he shouldface a price that depends only on other buyers’bids, and not on his own. Formally, such mech-anisms are the only ones satisfying dominant-strategy incentive compatibility and ex postindividual rationality. This paper characterizesthe expected profit-maximizing mechanism sat-isfying these requirements. In the simple case inwhich the seller’s marginal cost is constant, theoptimal mechanism offers each buyer the opti-mal monopoly price against the demand curveinferred from other buyers’ bids.

The proposed mechanism improves substan-tially upon posted pricing, but is qualitativelydifferent from standard auctions. The key dif-ference is that each buyer’s bid has an informa-tional effect: it affects other buyers’ allocationsdirectly, rather than through his own allocation.In particular, such a mechanism cannot be rep-resented with a supply curve.3

With a small number of buyers, the seller’sBayesian prior affects her posterior beliefsabout the distribution of valuations, and thereby

optimal pricing. The optimal mechanism is thusstill not completely “detail-free” in the sense ofRobert B. Wilson (1987)—the dependence onthe seller’s prior is simply pushed to a higherlevel. However, as the number n of buyersgrows, the information revealed by buyers’ bidsoverwhelms the seller’s prior. The paper showsthat for any consistent estimation of demandand its elasticity, as n 3 �, the seller’s ex-pected profit converges to the maximum profitachievable with the knowledge of the true de-mand distribution. In particular, this holds forBayesian estimation provided that the prior’ssupport includes the true distribution. This alsoholds for classical statistical estimation, bothparametric and nonparametric. For example, theseller can use the reported empirical distributionof the valuations of all buyers other than i as anestimate of the distribution of buyer i’s valua-tion, and offer buyer i the optimal monopolyprice against this distribution.4

With a large number of buyers, there aremany alternative ways to learn demand andattain the optimal monopoly profit asymptoti-cally. For example, the seller can survey a smallproportion of buyers and use their reported val-uations to set the optimal price to the remainingbuyers. Alternatively, the seller can experimentby pricing to different buyers sequentially andupdating the price using purchase history (see,e.g., Philippe Aghion et al., 1991; Leonard J.Mirman et al., 1993; Yongmin Chen and RuquWang, 1999; and Godfrey Keller and SvenRady, 1999). However, both these strategies seta price to each buyer utilizing less informationthan the optimal mechanism derived in this pa-per. In particular, the price offered to a buyerdepends only on the information received fromthe preceding buyers, but not from the subse-quent buyers. This “ informational inefficiency”may slow down convergence to the optimalmonopoly profit, sometimes quite dramatically.

It should be noted that relaxing the “ex post”constraints of dominant-strategy incentive com-patibility and ex post individual rationality tothe corresponding “ interim” constraints of

2 A similar motivation underlies the analysis of AndrewV. Goldberg et al. (2001). A key difference is that theyassume complete ignorance of the buyers’ valuations, whilein this paper these valuations are drawn from the same,although unknown, distribution. Also, they maximize theworst-case revenue (relative to that from the optimal postedprice), for which purpose randomized mechanisms strictlydominate deterministic ones.

3 Multiunit auctions that cannot be represented with asupply curve have also been considered by Yvan Lengwiler(1999) and David McAdams (2002), though with a differentmotivation.

4 This mechanism is also suggested by Sandeep Baligaand Rakesh Vohra (2002), in independent and contempora-neous work.

510 THE AMERICAN ECONOMIC REVIEW JUNE 2003

Bayesian incentive compatibility and interimindividual rationality would allow the seller toextract buyer surplus using mechanisms sug-gested by Jacques Cremer and Richard P.McLean (1985, 1988). However, such mecha-nisms are not “detail-free”— they are sensitiveto the buyers’ knowledge about the distributionand each other’s valuations, and a seller who isignorant of the extent of such knowledge maynot want to use them.

The paper is organized as follows: Section Idescribes the model and the class of mecha-nisms being considered. Section II characterizesthe optimal auction with a known demand dis-tribution, and shows that it can normally berepresented as the Vickrey-Groves-Clarkemechanism in which the seller manipulates hersupply curve in a way that depends on thedemand distribution. The section also examinescircumstances under which the seller can do justas well with a posted price. Section III derivesthe optimal pricing mechanism when the sellerdoes not know the distribution of demand buthas a Bayesian prior over it, and so the buyers’valuations are correlated from her viewpoint.Section IV shows that the seller’s expectedprofit converges to the maximum monopolyprofit achievable with known demand as thenumber of buyers goes to infinity. Section Vexamines the rate of convergence and comparesit to that achieved by sequential experimenta-tion mechanisms. Section VI motivates therestriction to ex post mechanisms. SectionVII concludes and discusses several potentialextensions.

I. Setup

A monopolistic seller faces n buyers, each ofwhom has unit demand.5 Each buyer i � 1, ... ,n privately observes his own valuation vi; thisvaluation is not observed by the seller or by theother buyers. Buyers’ valuations are indepen-dently drawn from a distribution F on [0, v�)(where v� � � is allowed), with a positive con-tinuous density function f(v) � F�(v) and a

finite expectation E[v]. Section II will considerthe standard case in which the distribution F iscommon knowledge, while subsequent sectionswill suppose that F is not known. Section III, inparticular, will suppose that the seller has aBayesian prior over possible distributions,which makes the buyers’ valuations correlatedfrom her viewpoint.

An outcome is described by the allocation ofthe good and the buyers’ payments to the seller.An allocation of the good is a vector x �( x1, ... , xn) � �n , where � � {0, 1} is the setof a buyer’s possible purchases from the seller.The buyers’ payments to the seller constitute avector t � (t1, ... , tn) � �n. All buyers’utilities as well as the seller’s profit are quasi-linear in the payments. The seller’s cost ofproducing X units is C(X).

By the Revelation Principle, the seller canrestrict attention to direct revelation mecha-nisms, which ask each buyer to bid his valua-tion, and which ensure that all buyersparticipate and bid truthfully in equilibrium. Tosimplify exposition, we focus on deterministicmechanisms, which specify an allocation rule x:[0, v� )n 3 �n and a payment rule t: [0, v� )n 3�n. While the seller may sometimes gain byconditioning the outcome on a public random-ization, this will prove not to be the case in thesettings considered in this paper.

It is customary to impose the Bayesian Incen-tive Compatibility (BIC) and Interim IndividualRationality (IIR) constraints on the mechanism.Formally, the constraints say that for any buyeri and any vi , vi � [0, v� ),

�BIC� Ev�i�vi�vi xi �v� � ti�v��

� Ev�i�vi�vi xi �vi , v�i � � ti �vi , v�i ��,

�IIR� Ev�i�vi�vi xi �v� � ti �v�� 0.

Mechanisms satisfying these constraints will becalled interim mechanisms.

This paper, however, will focus on mecha-nisms satisfying the stronger requirements ofDominant-strategy Incentive Compatibility(DIC) and Ex post Individual Rationality (EIR).Formally, for any buyer i, any valuation profilev � [0, v� )n, and any vi � [0, v� ),

5 With obvious alterations the analysis could be appliedto the problem of procuring from n sellers, each of whomhas unit supply.

511VOL. 93 NO. 3 SEGAL: OPTIMAL PRICING MECHANISMS WITH UNKNOWN DEMAND

�DIC� vi xi �v� � ti �v�

� vi xi �vi , v�i � � ti �vi , v�i �,

�EIR� vi xi �v� � ti �v� � 0.

These constraints require that each buyer’s in-centives to participate and bid truthfully aresatisfied ex post (for any possible realization ofother buyers’ valuations), rather than just in-terim (on expectation over these valuations).Mechanisms satisfying (DIC) and (EIR) will becalled ex post mechanisms. Ex post mechanismsare also studied by Kim-Sau Chung and JeffreyC. Ely (2001), in the more general case ofinterdependent valuations.

In the standard auction theory setup in whichbuyers’ valuations are independently drawnfrom a known distribution F, the restriction toex post mechanisms typically does not reducethe seller’s expected profit, as explained in Sec-tion II below. However, when the distribution Fis unknown, and so the valuations are correlatedfrom the seller’s viewpoint, the restriction doesreduce the expected profit. Yet, ex post mecha-nisms have the important advantage of beingrobust to the buyers’ beliefs about each other’svaluations. A motivation for ex post mecha-nisms along these lines is offered in Section VI.

Ex post mechanisms have a very simplecharacterization:

LEMMA 1: A deterministic mechanism x(�),t(�) is an ex post mechanism if and only if foreach buyer i there exist functions pi , si : [0,v� )n � 1 3 �� such that for every valuationprofile6 v � [0, v� )n ,

xi �v� � �1 if vi � pi �v�i �0 if vi � pi �v�i �, and

ti �v� � pi �v�i � xi �v� � si �v�i �.

PROOF:The “ if ” part is easy to verify. The “only if”

part follows from the Taxation Principle (see,

e.g., Bernard Salanie, 1997), which under (DIC)allows us to represent the mechanism faced bybuyer i for any given profile v�i of other buy-ers’ bids as a nondecreasing tariff Ti(�, v�i) :� 3 �. Let si(v�i) � �Ti(0, v�i) andpi(v�i) � Ti(1, v�i) � Ti(0, v�i). (EIR) forvi � 0 requires that si(v�i) � 0.

The mechanism described in Lemma 1 offerseach buyer i a lump-sum subsidy si(v�i) � 0and a price pi(v�i) � 0 that depend on otherbuyers’ reports. Buyer i receives a unit at thisprice if and only if the price is below his re-ported valuation. Such mechanisms will becalled pricing mechanisms, and the functionspi(�) and si(�) will be called the pricing andsubsidy rule, respectively.

Lemma 1 implies that any allocation rule thatis implemented in a deterministic ex post mech-anism satisfies the following monotonicity con-dition:

�M� xi �vi , v�i � is nondecreasing in vi

for all i, all v�i � �0, v� �n � 1.

In the unique pricing rule implementing such anallocation rule, the price to each buyer equalsthe minimum bid procuring him a unit:

(1) pi �v�i �

� inf �vi � �0, v� � : xi �vi , v�i � � 1 .

As for the subsidies, a profit-maximizing sellerwill set them identically to zero.

One example of a pricing mechanism is theVickrey-Groves-Clarke mechanism, in whichpi(v�i) equals the externality that giving buyeri a unit imposes on the others. Another exampleis a posted-price mechanism, in whichpi(v�i) � p* for all i, i.e., buyers face a singleprice that does not depend on any bids.

Observe that any pricing mechanism could inprinciple be implemented with a two-stage pro-cedure, in which (1) buyers report their valua-tions (v1, ... , vn), and (2) each buyer i decideswhether to purchase at price pi(v�i). Since abuyer’s stage 1 report has no effect on the pricehe faces in stage 2, truthtelling is a weak equi-librium at stage 1. However, there are several

6 The consumption xi(v) for vi � pi(v�i) is left inde-terminate, which is not important because the probability ofthis occurrence is zero.


concerns with such implementation. One con-cern is that a buyer who has an (arbitrarilysmall) cost of learning his valuation would notexpend the cost at stage 1, expecting to avoid itwhen offered a very high or very low price atstage 2. Another concern is that arbitrarily smallbribes could induce collusion at stage 1. Forthese (unmodeled) reasons, it is preferable toeliminate the buyers’ discretion at stage 2, in-stead determining their purchases on the basisof their reported valuations. This makes truth-telling uniquely optimal for a buyer who facessufficient uncertainty about other buyers’ reports.

II. The Optimal Mechanism with a KnownDistribution

This section describes the optimal mecha-nism when the distribution F is known by theseller. This problem was first analyzed by RogerB. Myerson (1981) for a single-unit setting,and the analysis was extended to the multiunitcase by Jeremy I. Bulow and John Roberts(1989). Here we offer new characterizations ofthe optimal mechanism for important specialcases, and provide a useful benchmark for thesubsequent analysis of the case in which F isunknown.

A. The Virtual Surplus Representation

By the Revenue Equivalence Theorem, theallocation rule x(�) fully determines the infor-mation rents of buyers in any Bayesian incen-tive-compatible mechanism in which theparticipation constraints of zero-valuation buy-ers bind. The seller’s expected profit can beexpressed as the difference between the ex-pected social surplus and the sum of buyers’expected information rents. Upon integration byparts, this difference can be written as the ex-pectation of the virtual surplus:

J�x, v� � �i

m�vi �xi � C� �i

xi� ,

where m�vi � � vi �1 � F�vi �

f�vi �.

The difference between m(vi), called buyer i’s

virtual valuation, and his true valuation vi ac-counts for the buyer’s information rent that isnot captured by the seller. The function m(�) iscalled the marginal revenue function.7

If the seller can use an interim mechanism,she chooses an allocation rule to maximize theexpected virtual surplus EvJ(x(v), v) subject toEv�i

xi(v) being nondecreasing in vi for all i,which is the monotonicity constraint stemmingfrom Bayesian incentive compatibility. If thesolution turns out to satisfy the stronger mono-tonicity condition (M), then, according toLemma 1, it can be implemented with a pricingmechanism, which yields the same expectedrevenue by the Revenue Equivalence Theorem.Therefore, the restriction to ex post mechanismscan hurt the seller only by strengthening themonotonicity constraint to (M).8 In the casesconsidered below, this restriction does not re-duce the seller’s expected profit.

B. When a Reservation Supply Curve IsOptimal

We begin by considering the case in whichthe marginal revenue function m(�) is increas-ing9 and the cost function C(�) is convex. Forthis case, we describe the allocation rule thatmaximizes the virtual surplus in each state v.Since this allocation rule turns out to satisfy(M), it solves the seller’s problem.10 Since thecost function C(�) is convex, the virtual surplusis maximized by allocating units to buyers indescending order of their virtual valuations

7 This name, suggested by Bulow and Roberts (1989),comes from the following parallel to the monopoly pricingproblem. Thinking of D( p) � 1 � F( p) as the expecteddemand curve for a given buyer, R(X) � D�1(X) � X is therevenue function, and m( p) � R�(D( p))—i.e., the mar-ginal revenue expressed as a function of price.

8 A more general version of this argument is given byDilip Mookherjee and Stefan Reichelstein (1992).

9 A sufficient condition for this is the monotonicity of thehazard rate f(v)/[1 � F(v)], which holds for many impor-tant distributions (see Mark Bagnoli and Ted Bergstrom,1989).

10 This also implies that a randomized mechanism wouldnot provide an improvement, since the seller’s expectedprofit in such a mechanism can still be written as theexpected virtual surplus, while the proposed deterministicmechanism maximizes the virtual surplus for any givenvaluation profile v.


m(vi), proceeding while these virtual valuationsexceed the incremental cost C(X) � C(X � 1).Since the marginal revenue function m(�) isincreasing, the buyers receive units in descend-ing order of their true valuations while the val-uations exceed m�1(C(X) � C(X � 1)) �S(X). The function S(X), obtained by trans-forming the seller’s incremental cost curveC(X) � C(X � 1) upward with the inversemarginal revenue function m�1(�), can be in-terpreted as the seller’s inverse reservation sup-ply curve. Intersecting this curve with theinverse demand curve reported by the buyersyields the optimal quantity X*. Formally, X* isdescribed by

(2) v �X*� � S�X*� if X* � 0, and

v �X* � 1� � S�X* � 1� if X* � n,

where v(X) denotes the Xth-highest order statis-tic of vector v. Note that a buyer is more likelyto receive a unit when he has a higher valuation,therefore the described allocation rule indeedsatisfies (M).

By Lemma 1, the ex post mechanism imple-menting the described allocation rule is a pric-ing mechanism, whose pricing rule is uniquelydetermined by formula (1).11 This implies thateach buyer receiving a unit pays the price equalto his highest bid that would entail either notproducing his unit or giving it to the first runner-up, buyer X* � 1:

(3) p � max�S�X*�, v�X* � 1� .

These conclusions are summarized as follows:

PROPOSITION 1: Suppose that the buyers’valuations are independently drawn from aknown distribution F, the marginal revenuefunction m(�) is increasing, and the cost func-tion C(�) is convex. Then any optimal mechanism(under either ex post or interim constraints)allocates units to buyers in descending order oftheir valuations while the valuations exceedS(X) � m�1(C(X) � C(X � 1)). The optimal

quantity X* is thus described by (2). In theoptimal ex post mechanism, losers do not pay,and all winners pay price (3).

The optimal mechanism is thus equivalent tothe Vickrey-Groves-Clarke mechanism inwhich each buyer pays the externality he im-poses on others, except that the seller misrepre-sents his incremental cost to be S(X) � C(X) �C(X � 1). The mechanism is depicted in Fig-ure 1.

Many features of this characterization extendto the case in which the cost function C(�) is notconvex. Namely, the virtual surplus-maximizingallocation rule still satisfies (M),12 and it stillallocates units to buyers in descending order oftheir valuations. The optimal quantity X* muststill satisfy the seller’s “discrete first-order con-ditions” (2), for otherwise she would prefer tosell either one more or one less unit. Thus, theoptimal quantity still lies at an intersection ofthe reservation supply curve S(�) and the de-mand curve revealed by the buyers. Observethat when the seller’s cost function is not con-vex, her optimal supply curve is not upwardsloping. Auctions with downward-sloping sup-ply curves have been implemented by “demandaggregation” web sites such as Mercata.com,LetsBuyIt.com, and eWinWin.com, presumablyreflecting sellers’ economies of scale.13

11 The same allocation rule can be implemented by manyinterim mechanisms.

12 This can be seen using the Monotone Selection The-orem of Paul R. Milgrom and Chris Shannon (1994).

13 Complications arise when the revealed demand curveintersects the reservation supply curve more than once, inwhich case the reservation supply curve alone cannot de-

FIGURE 1. STANDARD OPTIMAL AUCTION


C. When a Posted Price Is Optimal

When the seller’s marginal cost is a constantc, her inverse reservation supply curve S(X) �m�1(c) is horizontal. Then the optimal mecha-nism derived in Proposition 1 reduces to aposted price p* � m�1(c), which maximizesthe expected per capita profit:

(4) p* � arg maxp � �0,v� �

�� p�, �* � �� p*�,

where �� p� � � p � c��1 � F� p��

� �p

v�

�m�v� � c�f�v� dv.

While the above argument relies on the as-sumption of increasing marginal revenue, theoptimality of posted pricing is more general:

PROPOSITION 2: If the buyers’ valuations areindependently drawn from a known distribution F,and C(X) � cX, the posted price p* is an optimalmechanism among all interim mechanisms.

PROOF:Since the seller’s program is additively sep-

arable across buyers, she can restrict attention tomechanisms in which each buyer i’s allocationxi(v) depends only on his own valuation vi. Inparticular, in this case the interim constraintsare equivalent to the ex post constraints. Ran-domized mechanisms are not useful because theseller’s problem of choosing a nondecreasingrandomized allocation rule xi : [0, v� ) 3 [0, 1]to maximize the expected virtual surplus onbuyer i is a linear program, whose solution is

attained at an extreme point, all of which aredeterministic allocation rules. By Lemma 1, theseller can then use a pricing mechanism, with aprice pi to each buyer independent of others’announcements. Finally, by (4), p* is an opti-mal price offer to each buyer.

Even if the marginal cost is not constant, aposted price becomes optimal asymptotically asthe number n of buyers goes to infinity. For nor-malization across n, consider the asymptotic set-ting in which the set of each buyer’s possiblepurchases is � � {0, 1/n}. This ensures that theexpected demand at any posted price p is 1 � F(p)for any n. As n3 �, by the Strong Law of LargeNumbers the empirical demand at price p con-verges to 1 � F(p) almost surely, and the resultingprofit converges almost surely to p(1 � F(p)) �C(1 � F(p)) [provided that the cost function C(�)is continuous]. The asymptotically optimalposted price can then be defined as

(5) p* � arg maxp � �0,v� �

�� p�, �* � �� p*�,

where �� p� � p�1 � F� p�� C�1 � F� p��.

The asymptotic optimality of posted pricingis again the easiest to see in the case of increas-ing marginal revenue and nondecreasing mar-ginal cost, the optimal mechanism for which isdescribed in Proposition 1. Intuitively, as n 3�, the reported demand curve converges to 1 �F( p) and the reservation supply curve con-verges to m�1(C�(X)); therefore the price atwhich they intersect converges to p* (see Fig-ure 2). Thus, the optimal mechanism asymptot-ically reduces to posting price p*. Thisconclusion carries over to a more general setting:

PROPOSITION 3: Suppose that the buyers’valuations are independently drawn from aknown distribution F, and let D( p) � 1 �F( p) and R(X) � D�1(X) � X (the revenuefunction). Suppose that the cost function C : [0,1] 3 � is continuous, and that for X* �D( p*), there exists � �� such that14

termine the optimal quantity. (This does not happen atdemand aggregation web sites, because their dynamic bid-ding procedures stop once the first intersection is achieved.)Furthermore, in this case some buyers may be “pivotal,”meaning that without them it would be optimal to drop someother buyers so as to switch to a lower intersection point(perhaps even to shut down to save a fixed cost). In thepricing mechanism implementing the optimal allocationrule, pivotal buyers face prices that are different from (3).See Francesca Cornelli (1996) for a characterization of theoptimal mechanism in the setting with a fixed cost and aconstant marginal cost.

14 Note that when X* � (0, 1), condition (6) implies that �R�(X*) � C�(X*) (provided that the latter derivative exists).


(6) X* � arg maxX � �0,1�

�R�X� � X�,

X* � arg maxX � �0,1�

�X � C�X��.

Then the seller’s expected profit in any interimmechanism with n buyers and � � {0, 1/n}cannot exceed �*, while her profit from post-ing price p* converges to �* almost surely asn 3 �.

PROOF:Divide the seller’ s expected profit into two

terms, one being as though her marginal costwere constant and equal to , and the otherbeing E[X � C(X)] (where X is the quantitysold by the mechanism). By Proposition 2, thefirst term is maximized by a posted-pricemechanism, and the first line in (6) impliesthat p* is an optimal posted price, yieldingthe maximum value ( p* � ) X*. As for thesecond term, by the second line in (6) itcannot exceed X* � C(X*). Adding up, wesee that the seller’ s expected profits cannotexceed p*X* � C(X*) � �*. On the otherhand, as noted above, by the Strong Law ofLarge Numbers the profit from posting pricep* converges to �( p*) � �* almost surelyas n 3 �.

Condition (6) says that the graphs of R(X) �R(X*) and C(X) � C(X*) are separated with astraight line passing through the point (X*, 0)

(see Figure 3).15 By the Separating HyperplaneTheorem, this condition is weaker than the con-cavity of the revenue function R(�) and theconvexity of the cost function C(�), which areassumed in Proposition 1 [as noted in footnote7, R�(X) � m(D�1(X))].

The asymptotic setting considered in Propo-sition 3, in which the aggregate expected de-mand is held fixed, should be distinguishedfrom the setting in Dov Monderer and MosheTennenholtz (2001) and Zvika Neeman (2001),in which demand grows proportionally to n(e.g., � � {0, 1} for any n). In the latter setting,the expected demand curve in the limit becomesperfectly elastic at price v�. By posting a pricejust slightly below v� and optimally rationingdemand at the price, the seller can extract nearlyall buyer surplus, while realizing almost allavailable total surplus as the number of buyersgoes to infinity.16 Monderer and Tennenholtz(2001) and Neeman (2001) instead propose theVickrey-Groves-Clarke mechanism, which theseller can use to achieve the same profit asymp-totically even without knowing v�. The present

15 The first line in (6) can also be interpreted as sayingthat the “ ironed-out” marginal revenue curve coincides withR�(X) at X* [since ironing corresponds to the convexifica-tion of R(�)]. If this does not hold, then profit maximizationrequires convexification, as discussed in Bulow and Roberts(1989). With a large n, this convexification can be achievedby posting two different prices to different groups of buyers, andso the seller again need not resort to bidding mechanisms.

16 If v� � �, the seller’s profits would be unbounded.

FIGURE 2. ASYMPTOTICS WITH KNOWN DISTRIBUTION

FIGURE 3. THE SEPARATION CONDITION


model offers a better approximation of real-lifesituations with many buyers in which the ag-gregate demand is well known and is downwardsloping, while the individual buyers’ valuationsare not observed by the seller. In such situa-tions, the Vickrey-Groves-Clarke mechanismasymptotically reduces to posting the competi-tive equilibrium price, which is clearly subop-timal when demand is not perfectly elastic.

D. Extension to Asymmetric Buyers

While we have assumed that all buyers are apriori identical, much of the analysis extends tothe case where the valuations of different buyersare independently drawn from different (andknown) distributions. In this case, the optimalmechanism may no longer be representable with asupply curve, because a low-valuation buyer canhave a higher virtual valuation than another buyerwith a higher valuation, and so the units may notbe allocated in the order of the buyers’ true valu-ations. Thus, Proposition 1 no longer holds. Weabstain from describing the optimal mechanismfor the asymmetric case, because an even moregeneral setting with correlated valuations is exam-ined in Section III, subsection A, below.

On the other hand, Propositions 2 and 3 canbe generalized to the asymmetric case. With aconstant marginal cost, the optimal mechanismmay post different prices to different buyers—i.e., engage in third-degree price discrimina-tion—but still does not use buyers’ bids. Thesame is true with a general cost function, pro-vided that there is only a finite number of ob-servable buyer types, and there are sufficientlymany buyers of each type with independentlyand identically distributed (i.i.d.) valuations sothat their total demand is predictable by the Lawof Large Numbers. If the demand of each typesatisfies the first separability condition in (6)and the cost function satisfies the second con-dition in (6), then asymptotically it is againoptimal to use (discriminatory) posted pricingrather than a bidding mechanism.

III. The Bayes Optimal Mechanism withUnknown Distribution

Now we turn to the mechanism design prob-lem in which the distribution F from which the

buyers’ valuations are drawn is unknown. Thissection considers the case in which the seller isendowed with a Bayesian prior over possibledistributions. For example, the seller may knowthat the distribution belongs to a parametricfamily {F(�� )} � �, and have a prior over theparameter . Note that the buyers’ valuations(v1, ... , vn), while independent conditional onF, are correlated from the seller’s viewpoint.

A. The Case of a General Joint Distributionof Valuations

We begin by deriving the optimal mechanismfor the case where the buyers’ valuations havean arbitrary joint distribution, without regard tothe source of their correlation. We again restrictattention to ex post mechanisms, though this isno longer without loss (the restriction is moti-vated in Section VI below). Just as in the inde-pendent value case, dominant-strategy incentivecompatibility and the binding ex post participa-tion constraints of zero-valuation buyers pindown the information rents of each buyer i.Upon integration by parts, the seller’s expectedprofit can be expressed as the expectation of thevirtual surplus

(7) J�x, v� � �i

mi �v�xi � C� �i

xi� ,

where mi �v� � vi �1 � Fi �vi�v�i �

f i �vi�v�i �.

The only difference from the independent valuecase is that buyer i’s virtual valuation mi(vi ,v�i) is calculated using the conditional distri-bution and density functions Fi(��v�i) andfi(��v�i) respectively, and so it depends on otherbuyers’ valuations as well as his own. Theseller’s problem again takes the form of maxi-mizing the expected virtual surplus subject tothe monotonicity constraint (M).

When the marginal cost is constant, the sell-er’s program is additively separable across buy-ers. With an appeal to Proposition 2, the optimalmechanism reduces to setting the optimal priceto each buyer using the information gleanedfrom other buyers’ bids:


PROPOSITION 4: If C(X) � cX, then theBayes optimal ex post mechanism is a pricingmechanism with the pricing rule

(8)

pi �v�i � � arg maxp � �0,v� �

� p � c� � �1 � Fi � p�v�i ��.

For more general cost functions, we identifyconditions under which the monotonicity con-straint (M) does not bind and so the optimalallocation rule is obtained by maximizing thevirtual surplus in each state:17

PROPOSITION 5: Suppose that (i) mi(v) isincreasing in vi , (ii) mi(v) � mj(v) implies thatmi(v�i , v�i) � mj(v�i , v�i) for all v�i � vi , and(iii) C(�) is convex. Then any Bayes optimal expost mechanism allocates units to buyers indescending order of their virtual valuationswhile they exceed the incremental cost. Thus,the optimal quantity X* is described by

m�X*��v� � C�X*� � C�X* � 1� if X* � 0,

m�X*�1��v� � C�X* � 1� � C�X*� if X* � n.

The losers in the mechanism do not pay, and theprice pi paid by a winner i satisfies

(9) mi�pi , v�i �

� max�C�Xi � � C�Xi � 1�, m�Xi�1��pi , v�i � ,

where Xi is the largest optimal quantity for thevaluation profile ( pi , v�i).

PROOF:Due to (iii), the proposed allocation rule max-

imizes the virtual surplus in every state. Theallocation rule satisfies (M), because raising abuyer’s valuation increases both his virtual val-uation by (i) and its rank among all virtual

valuations by (ii), thus making him more likelyto receive a unit. (This implies that a random-ized mechanism would not be useful, by thesame argument as in footnote 10.) According to(1), each winner i in the mechanism pays theprice pi equal to his lowest bid that wouldprocure him a unit. When buyer i bids exactlypi , the seller is indifferent between serving himand either giving his unit to the first runner-upor not producing it at all. This is described in(9), with Xi representing the optimal quantitywhen buyer i is still served in this situation.

Proposition 5 generalizes Proposition 1 to thecase of a general joint distribution of valuations.The new condition (ii) ensures that buyer i’smarginal revenue function crosses buyer j’s atmost once, and from below, as buyer i’s valu-ation increases. With independent valuations, abuyer’s virtual valuation depends only on hisown valuation, and so condition (ii) is impliedby condition (i).18

The calculation of the optimal quantity isstraightforward, and in the special case of sym-metrically and independently distributed valua-tions studied in Proposition 1 it agrees with (2).However, the calculation of prices is more com-plicated than in Proposition 1. In the case ofindependent valuations, a reduction in buyer i’sbid from vi to pi does not affect the allocationsof other buyers as long as buyer i is still served.Thus we could take Xi � X* in the pricingformula (9), which yields formula (3) for thesymmetric case. In the general correlated case,however, a reduction in buyer i’s bid affectsother buyers’ virtual valuations and thus thequantity sold, even if buyer i still receives aunit. For this reason, identifying the price tobuyer i now requires solving a system of twoequations with two unknowns, Xi and pi.

Note that in the optimal mechanism, a buy-er’s bid vi has an informational effect: it affectsother buyers’ allocations x�i(v) even when itdoes not affect his own allocation xi(v). This

17 Giuseppe Lopomo (2001) offers a related character-ization of the profit-maximizing ex post mechanism forselling a single object, the buyers’ valuations for which maybe interdependent.

18 Observe also that when the joint distribution of valu-ations is symmetric with respect to the buyers (while notnecessarily a product distribution), mi(v) � mj(v) when-ever vi � vj , and therefore condition (ii) simply says thatthe buyers’ virtual valuations are ordered in the same way astheir true valuations.


informational effect cannot arise in auctions thatare optimal for the standard case of independentvaluations. In such auctions, as well as in mostmechanisms observed in real life (such as thoserepresented with a supply curve), a buyer’s bidcan affect other buyers’ allocations onlythrough his own allocation.

The informational effect identified here alsoarises in efficient mechanisms in the case wherethe buyers’ valuations are interdependent, stud-ied by Lawrence M. Ausubel (1999), ParthaDasgupta and Eric Maskin (2000), and MottyPerry and Philip J. Reny (2002). For example,with interdependent valuations, the efficient al-location of a unit between buyers i and j maydepend on the information of a losing buyer k,and this dependence cannot be implementedwith a standard auction. Though the present modelhas purely private values, a buyer’s valuation doesconvey information about other buyers’ valuationsto the seller, which gives rise to the interdepen-dence of the buyers’ virtual valuations, thus cre-ating an informational effect of messages in theprofit-maximizing mechanism.19

B. The Case of Affiliated Valuations

The analysis is simplified when the buyers’valuations are affiliated, as defined by Milgromand Robert J. Weber (1982). In this case, con-dition (ii) of Proposition 5 can be dispensedwith. Indeed, with affiliated valuations, an in-crease in v�i (weakly) increases the conditionaldistribution Fi(vi�v�i) in the monotone likeli-hood ratio order, and therefore (weakly) reducesthe distribution’s hazard rate fi(vi�v�i)/(1 �Fi(vi�v�i)) (see Louis Eeckhoudt and ChristianGollier, 1995, Lemmas 1, 2). This implies that

buyer i’s virtual valuation mi(v) is nonincreas-ing in v�i , which, together with the proposi-tion’s condition (i), implies its condition (ii).

With affiliated valuations, we can also saymore about the pricing rule in the optimal mech-anism described in Proposition 5. Since an in-crease in v�i reduces buyer i’s virtual valuation,while raising the other buyers’ virtual valua-tions by the proposition’s condition (i), buyer ibecomes less likely to receive a unit for anygiven vi. By (1), this implies that the pricingrule pi(v�i) is nonincreasing in v�i. Intuitively,an increase in v�i raises in a stochastic sensethe posterior distribution of buyer i’s valuation,which, coupled with the fact that the other buy-ers are now more deserving of the good, makesit optimal to raise the price to buyer i.

When the buyers’ valuations are drawn inde-pendently from an unknown distribution, theyare affiliated if the family {F(�� )}�� of pos-sible distributions is ordered in the monotonelikelihood ratio order. By symmetry, in this casethe pricing rule pi(�) is now the same for allbuyers. This implies that for any two buyerswith valuations vi � vj in a given valuationprofile v, buyer i faces a lower price than buyerj because v�i � (vj , v�i � j) is lower thanv�j � (vi , v�i � j). When the valuations arestrictly affiliated, a buyer’s virtual valuation isstrictly decreasing in the others’ valuations,and the same chain of arguments implies thathigher-valuation buyers pay strictly lower prices.Contrast this with the case where valuations aredrawn independently from a known distributionF, in which, by Proposition 1, all winners paythe same price.

C. A Parametric Example

Let the buyers’ valuations be drawn indepen-dently from an exponential distribution:20

F(v� ) � 1 � e�v, with the hazard parameter � 0 not known by the seller. To simplifyanalysis, suppose that the seller’s prior over lies in the conjugate family to exponential dis-tributions, which, according to Morris H. De-Groot (1970, p. 166), consists of gamma

19 To make the analogy precise, imagine that, with vi

representing buyer i’s private signal, his valuation is givenby mi(v). Then the function J(x, v) in (7) describes thesocial surplus, and so Proposition 5 gives the surplus-maximizing allocation rule. Under the proposition’s as-sumptions, this allocation rule satisfies (M), and it isimplementable in an ex post mechanism (as defined byChung and Ely, 2001, for the interdependent value case).However, this mechanism differs from that described inProposition 5: since buyer i’s valuation is now mi(v), histruthful reporting of vi is induced by charging him pricemi( pi , v�i), where pi is the minimum report procuring hima unit (characterized in Proposition 5).

20 This demand model has been considered by Jeffrey M.Perloff and Steven C. Salop (1985).


distributions. A gamma distribution of is de-fined by a density function of the form21

��, � � �

�� 1e� ,

with parameters �, � 0.

More precisely, if the prior distribution of is agamma distribution with parameters (�0 , 0),then its posterior conditional on a vector v�i ofn � 1 independent draws from F(�� ) is also agamma distribution, with parameters (�, ) �(�0 � n � 1, 0 � ¥j � i vj). The posteriordistribution of vi�v�i can then be calculated as

Fi �v i�v�i � � �0

�

F�vi��v�i � d

� 1 � �

� � vi �� .

It is easy to verify that the marginal revenueof this distribution is increasing in vi , thuscondition (i) of Proposition 5 is satisfied. Itscondition (ii) is also satisfied by the argument insubsection B, because the family of exponentialdistributions is ordered in the monotone likeli-hood ratio order. Thus, the optimal ex postmechanism for this setting with a convex costfunction is described in Proposition 5.

When the marginal cost is a constant c, theoptimal price to each buyer i solves (8), whichyields

pi �v�i � ��c �

� � 1

� ��0 � n � 1

�0 � n � 2�c � 0 � ¥j � i vj

�0 � n � 2.

In particular, if the seller lacks any prior infor-mation about the demand parameter , shecould use the improper uniform prior on ��

given by parameters (�0 , 0) � (1, 0), whichyields the pricing rule

pi(v�i) �n

n � 1c �

1

n � 1¥j � i vj .

Observe that a similar pricing rule obtains if,instead of updating a Bayes prior, the seller usesmaximum likelihood estimation of parameter .The log-likelihood of a vector v�i is

�n � 1�log � ¥j � ivj ,

which is maximized by

�v�i � � � 1

n � 1¥j � i vj��1

.

If the seller assumes that buyer i’s valuationis distributed according to the estimated pa-rameter, i.e., takes Fi(��v�i) � F(��(v�i)),then program (8) yields the pricing rule

pi�v�i� � c � 1/ � c �1

n � 1¥j � i vj .

That is, each buyer is offered a price equal to themarginal cost plus the average of other buyers’ bids.

Note that as n3 �, under both Bayesian andmaximum likelihood estimation the prices con-ditional on a “ true” parameter value 0 convergeto the optimal monopoly price for this parame-ter value. Indeed, by the Strong Law of Large

Numbers,1

n � 1¥j� i vj converges almost surely

to E[v�0] � 1/0, and therefore pi(v�i) con-verges almost surely to c � 1/0, which is theprice solving the profit-maximization program(4) for the true distribution F(��0). This im-plies that, as n 3 �, the seller’s expected profitconverges to the maximum profit from monop-oly pricing with known demand.

IV. Convergence

The optimal mechanism derived in SectionIII depends on the seller’s prior. However, asthe example in subsection C illustrates, for alarge n the prior is overwhelmed by the infor-mation obtained from the buyers’ bids. As n 3�, the seller learns the distribution F fromwhich the buyer’s valuations are drawn, theprices converge to the optimal posted price forF, and the resulting profit converges to theoptimal monopoly profit given F. General for-21 Where �(�) � �0

� z� � 1e�z dz.


mulations of the convergence result are given inthis section.

We adopt the “ frequentist” approach of clas-sical statistics, assuming the existence of a“ true” distribution to be estimated, and examin-ing convergence conditional on this distribu-tion.22 This approach allows us to dispense withthe prior altogether, letting Fi(��v�i) stand forany consistent estimator of the true distribu-tion F, and not necessarily the Bayes poster-ior distribution. For simplicity, from now onwe restrict attention to the symmetric case inwhich the same estimating function F(��) isused for all buyers. The simplest convergenceresult obtains for the case of constant marginalcost:23

PROPOSITION 6: Suppose that C(X) � cX;that for each vi � [0, v� ), F�vi�v�i � ¡

pF�vi � as

n3 �; and that vi �1 � F�vi�v�i �� ¡p

0 as vi ,n 3 �.24 Then as n 3 �, the expected percapita profit in the pricing mechanism solvingprogram (8) converges to the maximum ex-pected per capita profit �* achievable with Fknown [given by (4)].

For more general cost functions, a similarconvergence result can be established for theasymptotic setting in which each unit containsquantity 1/n, under the assumptions of Propo-sition 5 ensuring that the optimal mechanismmaximizes the virtual surplus state-by-state:

PROPOSITION 7: Suppose that as n 3 �,for each v i � [0, v� ), F�v i�v�i � ¡

pF�v i �,

f�vi�v�i � ¡p

f�vi �, and mi(vi , v�i) � vi �(1 � F(v i�v�i))/ f(v i�v�i) is asymptoticallyuniformly integrable.25 Suppose also that condi-tions (i)–(iii) of Proposition 5 hold and C: [0,

1] 3 � is continuous. Then as n 3 �, theexpected profit in the mechanism described inProposition 5 with n buyers and � � {0, 1/n}converges to the maximum expected profit �*achievable asymptotically with F known [givenby (5)].

When the distribution estimator F(��v�i) isa posterior distribution obtained by Bayes up-dating of a parameter whose prior distributionis �, the consistency assumptions of Proposi-tions 6 and 7 are verified for �-almost all pa-rameter values using Doob’s ConsistencyTheorem. The Theorem states that the Bayesposterior distribution �v�i converges to thetrue parameter value 0 weakly, in probability,as n3 �. This in turn implies that the posteriordistribution and density functions, F(vi�v�i) �E�v�i

F(vi� ) and f(vi�v�i) � E�v�if(vi� ), are

consistent estimators of the true distribution anddensity functions, respectively.

Propositions 6 and 7 are also applicable tonon-Bayesian estimation. For example, themaximum likelihood estimator of the parame-ter,

(10) �v�i � � arg max � �

�j � i f�vj��,

is consistent under standard assumptions, lead-ing to the consistent distribution and densityestimators F(� �v�i) � F(� �(v�i)) and f(� �v�i) �f(� �(v�i)), respectively. Alternatively, the sellercan use nonparametric estimation, the simplestexample of which is given by the empiricaldistribution of v�i:

(11) F�v�v�i � �1

n � 1��j � i : vj � v �.

Consistency of this estimator is established bythe Glivenko-Cantelli Theorem.

Application of Proposition 7 to nonparamet-ric estimation may be problematic because theestimation would typically yield virtual valua-tion estimates mi(�) that fail assumptions (i) and(ii) of Proposition 5, in which case the proposedallocation rule may fail (M). For example, anincrease in vj can raise the hazard rate of thedistribution estimate F(v�v�i), thus raisingbuyer i’s virtual valuation mi(v) to such an

22 If convergence is uniform across possible distribu-tions, then it also implies the convergence of the uncondi-tional expectation of profit given any Bayesian prior overpossible distributions.

23 The statistical concepts and results used below can befound in A. W. van der Vaart (1998). The proofs of thissection’s results are given in the Appendix.

24 The last assumption is vacuous when v� � �.25 The last assumption holds, in particular, when for each

vi , Ev�i�F(mi(v))2 is uniformly bounded across n. This canbe easily verified in the parametric example in Section III,subsection C.


extent that it becomes optimal to reallocatebuyer j’s unit to buyer i, violating (M) for buyerj. The problem can be avoided using instead thefollowing mechanism, inspired by Goldberg etal. (2001) and Baliga and Vohra (2002): parti-tion buyers into two equal-sized subsets S1, S2,and offer each subset S � S1, S2 an optimalprice against the distribution estimate using thebids from the other subset:

pS �vN � S � � arg maxp � �0,v� �

�p�1 � F� p�vN � S ��

� C�1 � F� p�vN � S ��.

Provided that F(��vN � S) is a consistent estima-tor of the true distribution F, and the profit-maximizing price p* defined in (5) is unique, bythe Theorem of the Maximum the pricespS(vN � S) to both groups S � S1, S2 converge top* in probability, and therefore the expectedprofit converges to �*. However, this pricingmechanism is not informationally efficient, forin setting each price it ignores the informationreceived from half of the buyers.

V. Rates of Convergence

Convergence to the optimal per capitaprofit �* is not the only useful asymptoticcriterion. In fact, approximating �* with alarge number n of buyers is not at all hard.For example, the seller could experiment onsome buyers by offering them differentprices, as in Aghion et al. (1991) and Kellerand Rady (1999). Alternatively, she could asksome buyers to report their valuations, re-fraining from selling to them to ensure truth-ful reporting. Either experimentation on orsurveying of a sufficiently large “ test group”of buyers would reveal the demand curve andenable the seller to set an approximately op-timal price to the remaining buyers. At thesame time, when n is large, the size of the“ test group” can be small relative to n, ensur-ing that the per capita profit approaches �*.This section compares the asymptotic perfor-mance of mechanisms such as surveying andexperimentation to that of the optimal mech-anism, using as the criterion the rate of con-

vergence to the optimal monopoly profit �*as n 3 �.26

It is clear that no mechanism can achieveuniformly faster profit convergence on a set ofdistributions F than the Bayes optimal mecha-nism for a prior concentrated on this set (forotherwise it would achieve a higher expectedprofit than the supposedly optimal mechanismfor n large enough). In fact, surveying and ex-perimentation mechanisms are likely to have aslower convergence rate than the optimal mech-anism because they both ignore useful infor-mation in setting prices. For example, bothmechanisms are sequential pricing mechanisms,which set the price to a given buyer usingonly information obtained from the preced-ing buyers, rather than from all the otherbuyers. In addition, while the optimal sequentialpricing mechanism would offer each buyer ithe optimal price pi (v1, ... , vi � 1) given thepreceding buyers’ reported valuations,27 bothexperimentation and surveying sacrifice profitson the first buyers (by setting a suboptimalprice to them or not selling to them at all) inorder to acquire information about theirvaluations.

We examine this intuition in the simple casein which the marginal cost is a constant c, andso the seller’s maximum expected profit �* isgiven by (4). The expected loss on a givenbuyer i when his price p(v�i) is determinedfrom n � 1 other buyers’ bids is

Ln � �* � Ev�i�F �� p�v�i ��.

By Proposition 6, Ln 3 0 as n 3 �. Weexamine the rate of this convergence using the

26 The same asymptotic criterion for mechanism designwith many agents is adopted by Thomas A. Gresik andMark A. Satterthwaite (1989) and Tymon Tatur (2001), buttheir objective is efficiency rather than the designer’sprofits.

27 For example, each buyer i could be asked to report hisvaluation after deciding whether to buy at the quoted pricepi(v1, ... , vi � 1). However, recall from the discussion at theend of Section I that for buyer i to have a strict incentive toreport truthfully, the price pi(v1, ... , vi � 1) should not berevealed to him until after his report, and he should receivethe good at the revealed price if and only if his reportedvaluation exceeds the price.


following terminology: Two sequences {�n}n�1�

and { n}n�1� have the same convergence rate if

there exist two positive numbers a� , a� such that�n/ n � [a� , a� ] � � for n large enough. Thetwo sequences satisfy the stronger property ofbeing asymptotically proportional, written as�n � n , if �n/ n 3 a � (0, ��) as n 3 �.

The convergence rate of Ln will depend onhow large the family {F(�� )} � � of possibledemand distributions is. We consider threecases in turn: (1) hypothesis testing, in which �is a finite set of parameters (“simple hypothe-ses” ), (2) parametric estimation, in which � isa Euclidean (finite-dimensional) parameterspace, and (3) nonparametric estimation, inwhich � is an infinite-dimensional space (forexample, including all distribution functions ofsufficient smoothness). Suppose without loss ofgenerality that all distributions in the family aredistinct, and let 0 denote the true parametervalue, so that the true distribution is F(��0) �F(�).

A. Hypothesis Testing

In this case, the optimal mechanismachieves exponential convergence to the op-timal monopoly profit as n 3 �. For exam-ple, the maximum likelihood estimator givenby (10) selects a false hypothesis (v�i) � 0with an exponentially small probability (thisfollows from Chernoff ’ s Theorem—see, e.g.,Robert J. Serfling, 1980, Sec. 10.3). There-fore, offering buyer i the optimal pricep(v�i) � p*((v�i)) against this estimator,where

(12) p*�� arg maxp � �0,v� �

� p � c��1 � F� p��,

yields exponentially small expected loss. Sincethis pricing mechanism is also available to aBayesian decision maker, the expected loss inthe Bayes optimal mechanism must converge tozero at least as fast conditional on each positive-probability parameter value . In fact, the ex-pected loss in the Bayes optimal mechanismis exponentially small because for any full-support prior, the expected posterior probabili-ties of false hypotheses shrink exponentially(see, e.g., Erik N. Torgersen, 1991, Sec.

1.4).28 Thus, under both Bayesian and maxi-mum likelihood estimation, the expected percapita loss Ln satisfies

(13) log Ln � �n.

On the other hand, the expected per capita lossin any sequential pricing mechanism is at least ofthe order 1/n, because the mechanism sets a priceto buyer 1 without the benefit of any information.Thus, sequential pricing mechanisms convergeexponentially slower than the optimal mechanism.The optimal sequential pricing mechanism in factachieves convergence rate n�1, because settingprice p(v1, ... , vi�1) � p*((v1, ... , vi�1)) to eachbuyer i yields expected loss Li on this buyer, hencethe total expected loss is bounded above by¥i � 1

� Li � �, due to (13).Experimentation can only perform worse

than the optimal sequential pricing mechanismbecause it uses only past buyers’ purchasesrather than their reported valuations. Here, how-ever, optimal experimentation achieves thesame convergence rate as the optimal sequentialpricing mechanism, under the generic conditionF( p*(�)�0) � F( p*(�)� ) for all , � � �.To see this, note that the seller can offer eachbuyer the optimal price p*( ) given the maxi-mum likelihood estimate of that uses only thepast purchase observations at the most fre-quently set price. Since there are only �� pos-sible prices, buyer i’s price will be based on atleast (i � 1)/�� i.i.d. purchase observations.A Chernoff ’s Theorem argument then againimplies that the expected loss Li on buyer i isexponentially small in i, and therefore ¥i�1

� Li� �, hence the per capita expected loss1

n¥i � 1

n Li converges to zero at rate n�1.

B. Parametric Estimation

Assume that p* � p*(0) � 0 is a uniquesolution to the expected profit-maximization

28 Under the second-order Taylor expansion (14) below,the minimization of Bayesian expected loss yields a priceerror p(v�i) � p*(0) that is asymptotically proportional tothe seller’s posteriors on false hypotheses, hence the ex-pected loss is proportional to the square of these posteriors.


program (4). The first-order condition for the pro-gram can be written as m(p*) � c, and the second-order condition as m�(p*) � 0. Assume that in factm�( p*) � 0, in which case the second-orderTaylor expansion of � around p* yields

(14)

�* � �� p� � A� p � p*�2 � o�� p � p*�2�,

where A � �12

�� p*� � m�� p*�f� p*� � 0.

This implies that the expected loss on buyer i isasymptotically proportional to the squared priceerror, E( p(v�i) � p*)2.

Suppose that the seller offers buyer i theoptimal price (12) against the maximum likeli-hood estimator (10) of the parameter: p(v�i) �p*((v�i)). Suppose also that the functionp*( ) is uniquely defined in a neighborhood of � 0 , with the gradient p*(0) � 0.29 It iswell known that under standard regularity con-ditions, �n((v�i) � 0) is asymptoticallynormal with zero mean and a nondegeneratecovariance matrix (see van der Vaart, 1998,Sec. 5.5). By the “delta method” based on thefirst-order Taylor expansion of p*( ) around 0(see van der Vaart, 1998, Theorem 3.1),�n( p(v�i) � p*) is also asymptotically nor-mal with zero mean and a positive variance. Bythe Bernstein-von Mises Theorem, the sameasymptotic normality holds also for the Bayesoptimal price p(v�i), which can be viewed asthe Bayes point estimate of the optimal price p*with the loss function (14). Therefore, in bothcases, E( p(v�i) � p*)2 is asymptotically pro-portional to 1/n, hence by (14), the per capitaexpected loss is

Ln � n�1.

The optimal sequential pricing mechanismhas a slower convergence rate. Indeed, since theexpected loss on buyer i in this mechanism isLi , the per capita expected loss is

1

n �i � 1

n

Li �1

n �i � 1

n 1

i�

1

n �1

n di

i�

log n

n.

(The first proportionality is by Cesaro’s Theo-rem and the second by the Integral Test forseries—see Thomas John I’Anson Bromwich,1931.) Thus, here sequentiality slows downconvergence by the factor log n. Optimal ex-perimentation may in fact achieve this conver-gence rate: Intuitively, even if the seller sets themyopically optimal price to each buyer on thebasis of past purchase observations, the pricewill eventually arrive in a neighborhood of theoptimal price p* in which the partial derivativeF( p� ) is bounded away from zero, and so theamount of information about received from apurchase observation is bounded below. Thenthe expected loss on buyer i is asymptoticallyproportional to 1/i, yielding again the expectedper capita loss of the order of n�1log n.

C. Nonparametric Estimation

The simplest nonparametric distribution esti-mator F(v�v�i) is the empirical distribution ofthe other buyers’ valuations, given by (11). Theprice p(v�i) solving program (8) against thisdistribution is an “M-estimator” of the correctprice p* (see van der Vaart, 1998). KislayaPrasad (2001) shows that the distribution ofn1/3( p(v�i) � p*) converges to a distributionwith a finite positive variance. Under the as-sumptions of the previous subsection, (14) impliesthat the expected per capita loss Ln � n�2/3.

Faster convergence rates can be achieved us-ing kernel estimation of the density function f,provided that f is smooth. For example, CharlesJ. Stone (1983) shows that if f is known to be rtimes continuously differentiable, then the op-timal uniform probabilistic convergence rate ofthe kernel density estimator f(��v�i) to the truedensity f is (n�1log n)r/(2r � 1). This impliesthat the optimal price against the estimated dis-tribution converges in probability to p* at leastas fast, and therefore by (14) the expected percapita loss satisfies

Ln � O�� n

log n�� , where � �

2r

2r � 1� 1.

29 By the Implicit Function Theorem, both assumptionshold when m( p*�0) � 0, where m(v� ) � v � (1 �F(v� ))/f(v� ).


Optimal sequential pricing mechanisms may infact achieve the same convergence rate. For ex-ample, suppose that Ln � (n/log n)�� or Ln � n��,with � � (0, 1) (recall that empirical distributionestimation yields the latter with � � 2⁄3 ). In bothcases, Cesaro’s Theorem implies that

limn3�

nLn

¥i � 1n Li

� limn3�

�n � 1�Ln � 1 � nLn

Ln � 1

� 1 � � � 0.

Thus,1

n¥i � 1

n Li � Ln ; i.e., the expected per

capita loss in the optimal sequential mechanismconverges at the same rate in as in the fullyoptimal mechanism.

The optimal experimentation mechanismwould be very difficult to characterize in thissetting. Intuitively, it appears that its conver-gence rate may be slower, because the earlypurchases at prices that are far from p* willprove useless for fine-tuning the price around p*.

VI. Justifying Ex Post Mechanisms

If the ex post constraints (DIC) and (EIR) arerelaxed to the corresponding interim constraints(BIC) and (IIR), the seller is typically able toextract all buyer surplus, while implementingthe surplus-maximizing allocation. Cremer andMcLean (1988) show how this can be done,even while satisfying (DIC) [but not (EIR)].Namely, the seller can employ the Vickrey-Groves-Clarke mechanism, but in additioncharge each buyer i a participation fee �i(v�i)that depends on other buyers’ reports. For ageneric joint distribution of valuations, the feefunction �i(�) can be chosen so that the ex-pected interim payoff of buyer i is zero nomatter what valuation vi he has.30

Neeman (2002) notes that the surplus extrac-

tion mechanisms of Cremer and McLean (1988)exploit a one-to-one correspondence between abuyer’s own valuation and his belief about theothers’ types. In a more general information struc-ture, two different types of buyer i with differentvaluations may share the same beliefs about theothers’ types, in which case it is impossible tofully extract the information rents of both typesof buyer i. In the extreme case in which a buyer’svaluation does not constrain his beliefs aboutothers, any mechanism that is robust to thebuyers’ beliefs (as Wilson, 1987, calls it, detail-free) must be an ex post mechanism, which isformally shown by John O. Ledyard (1978) andDirk Bergemann and Stephen Morris (2001).

To be sure, if buyers’ beliefs stem from theirinformation about each other’s valuations, the“second-best” optimal mechanism, rather thanbeing detail-free, will elicit these beliefs. Forexample, if buyer i knows the distribution Ffrom which other buyers’ valuations are drawn,the mechanism can ask this buyer to set theoptimal monopoly price to the other buyers.However, the seller may be wary of using thismechanism if she is not sure how well-informedbuyer i is about F. For the same reason, shemight also be wary of using the Cremer-McLean mechanism described above. Moregenerally, a seller who is “ ignorant” about thebuyers’ knowledge of each other’s valuations(while being confident that they are drawn in-dependently from an unknown distribution)may be concerned with the mechanism’s worst-case performance over all information structures. Iconjecture that such worst-case performance ismaximized by an ex post mechanism that elicitsonly the buyers’ valuations and not their beliefs.

VII. Conclusion

This paper has examined the profitability ofbidding mechanisms relative to posted pricing.The advantage of bidding mechanisms is thatthey create interdependence among buyers,whereby one buyer’s bid vi affects other buyers’allocations x�i. In the standard auction theorysetting in which the buyers’ valuations are in-dependently drawn from known distributions,interdependence is desirable to the extent thatthe seller’s cost is nonseparable across buyers(in the extreme case, the seller has a capacity

30 For example, consider the parametric setting of Sec-tion III, subsection C, with C(X) � 0 (so that the Vickrey-Groves-Clarke mechanism gives the good for free to allbuyers). It can be verified that in this setting, charging buyeri the fee �i(vj) � �0vj � 0 (which depends on the reportof another buyer j � i) ensures that his expected surplus inthe mechanism is vi � E[�i(vj)�vi] � 0 for all vi.


constraint). Indeed, a buyer’s bid vi affects hisallocation xi , which due to interactions in theseller’s cost function affects the other buyers’optimal allocations x�i. However, this reason-ing does not apply when the seller’s marginalcost is either constant or little affected by asingle buyer (e.g., when there are many smallbuyers). In these practically important cases,interdependence is not useful, hence optimalauctions do not improve upon posted pricing inthe standard setting.

Interdependence becomes useful, however,when the buyers’ valuations are correlated fromthe seller’s viewpoint, and in particular whenthey are drawn independently from an unknowndistribution. In this case, one buyer’s bid viconveys information to the seller about otherbuyers’ valuations v�i , and therefore affectstheir optimal allocations x�i even when it doesnot affect the buyer’s own allocation xi. Theoptimal mechanism is thus qualitatively differ-ent from standard auctions (in particular, it can-not be represented with a supply curve). Rather,it resembles (but differs from) the efficientmechanisms suggested by Ausubel (1999), Das-gupta and Maskin (2000), and Perry and Reny(2002) for the case of interdependent values.

The mechanisms suggested in the present papersatisfy Wilson’s (1987) desideratum of being“detail-free,” i.e., robust to buyers’ beliefs abouteach other’s valuations. This is ensured by impos-ing the “ex post” constraints of dominant-strategyincentive compatibility and ex post individual ra-tionality, which rule out the surplus extractionschemes proposed by Cremer and McLean (1988).

Another dimension of detail-freeness is ro-bustness to the seller’s beliefs about buyers’valuations. The rationale for this kind of robust-ness is not as strong: if the seller has some priorinformation about the distribution of buyers’valuations (for example, from a history of sell-ing similar products), there is no reason not toutilize it in designing the mechanism. At thesame time, it is useful to have mechanisms thatcan be used even when the seller has “no idea”of the distribution from which the buyers’ val-uations are drawn. Both kinds of mechanismsare suggested in the present paper. While theBayes optimal mechanism utilizes the seller’sprior, this prior become irrelevant with a largenumber of buyers, and asymptotically the seller

does just as well using classical statistical esti-mation of demand. Non-Bayesian knowledgeabout the distribution, such as that of itssmoothness or functional form, can also be usedto accelerate convergence to optimal monopolyprofits. Thus, the paper provides a flexibleframework allowing the utilization of differentkinds of prior knowledge, Bayesian or non-Bayesian, in designing the optimal mechanism.

An important concern for the practical imple-mentation of any novel economic mechanism iswhether its participants can understand how themechanism works. However, a buyer participatingin a mechanism proposed in this paper does notneed to understand the intricacies of the pricingformula: as long as he believes that his own bidwill not affect the price he ends up facing, he willfind it optimal to bid his valuation. Of course, it iscrucial that the seller’s commitment not to basethe price offered to a buyer on his own bid becredible. The same commitment issue arises insecond-price auctions, and usually it is success-fully resolved in real life. For example, eBay.comallows buyers to submit proxy bids, effectivelyconverting the English auction into a second-priceauction. The web site explains to buyers that sincea buyer’s proxy bid will not be used to raise theprice above the minimum necessary for him towin the auction, he should bid his true valuationfor the object. There is no reason for the sameexplanation not to be effective for the mechanismsproposed in this paper.

The proposed mechanisms could be mademore transparent by allowing buyers to submitand raise their bids over time, while observingthe current price they face. Many Internet pric-ing mechanisms are realized in this dynamicfashion.31 An interesting distinguishing featureof our mechanism is that in its dynamic realiza-

31 A downside of such transparency is that it facilitates tacitcollusion among buyers (the same is true of other dynamicmechanisms, such as the English auction). For example, at abid profile at which each buyer receives a unit, no buyer has astrict incentive to raise his bid, even if it is below his valuation.He may even strictly prefer not to raise his bid to avoidretaliation by other bidders. The rules may have to be modifiedto reward buyers for breaking such collusive equilibria (see,e.g., the suggestions in McAdams, 2002). Note that collusiveequilibria are unlikely in the one-shot mechanism, since abuyer with sufficient uncertainty about others’ behavior willfind it strictly optimal to bid truthfully.


tion, the price facing a buyer can either rise orfall as demand grows. This contrasts with stan-dard auctions, in which price can only rise asdemand grows, and with “demand aggregation”mechanisms, in which it can only fall.

The present analysis can be extended in sev-eral directions. One extension is to allow buyersto demand more than one unit. The optimalmechanism would in general involve second-degree price discrimination, charging eachbuyer different prices for different units, as inthe model of Maskin and John Riley (1984).When the seller’s marginal cost is constant, herproblem is again additively separable acrossbuyers, and she should offer each buyer theoptimal nonlinear tariff using the informationinferred from other buyers’ reported prefer-ences. It should be kept in mind, however, thatunless the buyers’ preferences are seriously re-stricted a priori (say, to a one-dimensionaldomain with a single-crossing property), com-puting the optimal tariff may be quite difficult.

Another possible extension is the addition ofvalue interdependence (i.e., a common-valuecomponent) among buyers, which can be ana-lyzed using Chung and Ely’s (2001) concept ofex post implementation. The buyers in this set-ting may need to submit more complex bids.For example, with unit demands, they couldreport functions describing how their valuationsdepend on those of others, as in the mechanismof Dasgupta and Maskin (2000).

Finally, note that the mechanisms proposed inthis paper typically charge different buyers differ-ent prices for identical units. This occurs becausethe price to each buyer is calculated upon exclud-ing this buyer’s bid. However, I conjecture that auniform-pricing mechanism in which the price iscalculated using all buyers’ bids would work justas well when the number of buyers is large. In-deed, each buyer will then realize that his bid’seffect on the price is very small, and therefore willbid close to his true valuation.

APPENDIX: PROOFS OF PROPOSITIONS 6 AND 7

PROOF OF PROPOSITION 6:Let �i( p�v�i) denote the objective function

in (8). If the seller uses the pricing rule p(v�i)solving (8), her expected loss relative to �* onbuyer i given v�i is bounded above as follows:

(A1) �* � ��p�v�i ��

� �� p*� � �i �p*�v�i ��

� ��i �p*�v�i � � �i �p�v�i ��v�i ��

� ��i �p�v�i ��v�i � � �� p�v�i ��

� 2 supp � �0,v� �

�� p� � �i �p�v�i ��.

In words, the loss is bounded above by twice thesupremum absolute difference between the ob-jective functions in (5) and (8). This supremumabsolute difference can in turn be boundedabove as follows:

supp � �0,v� �

�� p� � �i �p�v�i ��

� M supp � �0,v� �

�F� p�v�i � � F� p��

� supp � M

�p�1 � F� p�v�i ��

� supp � M

�p�1 � F� p��

for any M � 0. A simple extension of Lemma2.11 in van der Vaart (1998) shows that, by theconsistency of F(vi�v�i), the first term abovegoes to zero in probability as n 3 � for anyfixed M. The other assumption on F(vi�v�i)implies that the second term goes to zero inprobability as M, n3 �. Finally, the third termgoes to zero as M 3 � due to the assumptionthat E[v�F] � �. Putting together, we see thatfor all �, � � 0 we can find M � M(�, �) anda corresponding n(�, �) such that for all n �n(�, �), each term is less than �/3 with proba-bility at least 1 � �/3. This implies thatPr{supp � [0,v� )��( p) � �i( p�v�i)� � �} � 1 �� for n � n(�, �), which by (A1) implies that�� p�v�i �� ¡

p�* as n 3 �. Since �( p(v�i)) is

bounded, it follows that the expected profitEv�i�F[�( p(v�i))] 3 �* as n 3 �.

PROOF OF PROPOSITION 7:Note that the proposition’s assumptions ver-

ify those of Proposition 3, which implies thatthe maximum expected profit with F known


converges to �* as n 3 �. Therefore, it suf-fices to show that the expected loss from notknowing F goes to zero as n 3 �.

The allocation rule described in Proposition 5maximizes the virtual surplus (7) in each state,and therefore maximizes its expectationEv�FJ(x(v), v). The seller’s expected profit un-der the true distribution F is instead

Ev�F� �i

mF �vi �xi �v� � C� �i

xi �v�� ,

where mF(vi) � vi � (1 � F(vi))/f(vi). By anargument similar to that in the beginning ofthe proof of Proposition 6, the seller’s expectedloss from not knowing F is bounded above bytwice the supremum absolute difference be-tween the two expectations over all allocationrules x(�). Using symmetry, this supremum ab-solute difference is in turn bounded above asfollows:

supx:�0,v� �n3 �0,1/n n

Ev�F� �i

�mi �v� � mF �vi ��xi �v�� Ev�F�mi �v� � mF �vi ��.

The consistency of F(vi�v�i) and f(vi�v�i) im-plies that for each vi , mi �v� � mF �vi � ¡

p0 as

n 3 �. Since the absolute value of the differ-ence is asymptotically uniformly integrable byassumption, Theorem 2.20 in van der Vaart(1998) implies that Ev�i�F�mi(v) � mF(vi)� 30 as n 3 �. Take expectation Evi�F usingLebesgue’s Dominated Convergence Theoremto see that the right-hand side of the aboveinequality goes to zero.

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