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Optimal Auctions

John G. Riley; William F. Samuelson

The American Economic Review, Vol. 71, No. 3. (Jun., 1981), pp. 381-392.

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Optimal Auctions

In the two decades since the seminal paperby William Vickrey, literature on the theoryof auctions has developed at a rapid thoughuneven pace.' Much of this literature is frag-mentary, varies widely in scope, and is noteasily accessible to economists. As a result,the implications of different auction rules invarious settings remain relatively unknown.This paper provides a systematic examina-tion of alternative forms of auctions. In sodoing it presents a general characterizationof the imp lications for resource allocation ofdifferent auction designs withn the modeloriginally proposed by Vickrey.

The auction model is a useful descriptionof "thin markets" characterized by a funda-mental asymmetry of market position. Whilethe standard model of perfect competitionposits buyers and sellers sufficiently numer-ous that no economic agent has any degreeof market power, the bare bones of the auc-tion model involves competition on only one

side of the market. In this setting a singleseller of an indivisible good faces a number(n) of potential buyers. Competition amongthe (possibly small number of) buyers takesplace according to a well-defined set of auc-tion rules calling for the submission of priceoffers from the buyers. Most commonly, thechoice of a uction m ethod employed rests withthe monopolistic seller.

These brief observations suggest two natu-ral questions for analysis: First, what formdoes the competition among the few buyers

take under the most common auction proce-

'Professor of economics. University of California-L osAngeles, and assistant professor, School of Manage-ment. Boston University, respectively. Riley's researchwas supported by National Science Foundation grantSOC79-07573. Helpful discussions with H owa rd R aiffa,Jerry Green, Eric Maskin, and Paul Samuelson, and thevaluable comments of a referee are gratefully acknowl-edged.

'A current bibliography by Ro bert Sta rk and MichaelRothkopf lists nearly 500 papers w ritten over this period.For a recent survey of this literature, see RichardEngelbrecht-Wiggans.

dures? In turn, how is a sale price de-termined? Second, by what means can theseller best exploit his monopoly position?For example, would it be more pro fitable forthe seller to require payment not only by theh g h bidder, but also by those with lowerranked bids?2

As one might expect, any change in therules of the auction results in different bid-ding strategies on the part of the buyers. Inparticular, if the auction rules posit a mini-mum payment by on e or more of the bidders(determined by rank), those with sufficientlylow reservation values will be discouragedfrom entering a bid. Our analysis will dem-onstrate that, in a risk-neutral setting, it isthe reservation value below which a buyeropts to remain out of the auction which iscruc ial. To be precise, if the lowest reserva-tion value for whch it is worthwhile biddingis the same for two different auction rules.then the expected return to the seller is also

the same.Throughout the paper we shall retain thefollowing basic assum ption.

a) A single seller with reservation valuev, faces n poten tial buyers, where buyer iholds reservation value v,, i = 1,. . .,n.

b) The reservation values of the partiesare independent and identically distributed,drawn from the common distribution F(v)w i t h F (v )= O , F (6 )= 1 and F(v) strictly in-creasingand differentiable over the interval[ g ,61. We will refer to this as the I I D as -

sumption.The I ID assum ption was first presented by

Vickrey, and has been frequently employed

'Vickrey's com pariso n of the open "ascending bid"auction and the sealed "high bid" auction has beengeneralized in unpublished dissertations by ArmandoOrtega-Reichert . Gerard R. Butters, and WilliamSamuelson. Mdton Harris and Artur Raviv (1979) pro-vide the first discussion of optimal auction design. Em-ploying very different me thods they consider the specialcase in which each buyer has flat (unifo rm) prior prob -abilistic beliefs about the amount others are willing to

Pay.



382 THE AMERIC AN ECO.VOMIC REVIE W J UhrE1 981

in the bidding literature. In practical terms,each party is uncertain about the others'reservation values, believing that each indi-vidual decides the maximum amount he is

willing to pay inde penden tly of the others. Inaddition, the parties share common priorswith respect to the possible reservation val-ues of eac h individual. With the IID assump-tion, the bidding procedures we outline be-low belong to the class of games of incom-plete information first formulated by JohnHarsanyi.

Th e paper is organized as follows: First wepresent our central result dem onstrating thatexpected seller revenue from quite different

auctions can be very easily com pared. As animm edia te implication, the equivalence of the"English" or "ascending bid" auction andthe "Dutch" or "high bid" auc tion is estab-lished. More important, it is shown that, fora broad family of auction rules, expectedseller revenue is maximized using either ofthe two common auctions if the seller an-nounces that he will not accept bids belowsome appropriately chosen minimum or "re-serve" price. Surprisingly, this reserve priceis independent of the number of buyers and

is always strictly greater than the seller'spersonal value of the object. In Section I1several alternative auction rules are de-scribed in detail and their implications forthe seller are compared. Finally, in Section111, the two commonly used auctions areonce again compared under the assumptionthat the buyers are risk averse rather thanrisk neutral. It is shown in this setting thatthe English auction is dominated by thesealed high bid auction, and that the optimalreserve price in the latter is a declining func-

tio n of the deg ree of buy er risk aversion.

I. Comparison of Alternative Auction Rules

Before characterizing a broad family ofalternative auction rules, a few rem arks abo utthe English or ascending bid auction will behelpful.

In auc tions of a ntiques, estate objects, an dworks of art, the good is awarded to thebuyer who makes the final and highest bid.The buyer placing the hlghest valuation onthe good therefore pays approximately the

maximum of the reservation values of theother n - 1 buyers. As Vickrey noted, thls isequivalent to a sealed bid auction in whicheach buyer submits a bid a nd the high bidder

pays the second highest rather than thehlghest bid.3 To see this, suppose the ithbuyer considers shading his bid, b , , below hlsreservation value v,. If the largest of all theother bids, b,, exceeds v , , another buyer isthe high bidder so that buyer 1's gain re-mains zero. If b,<b,, buyer i remains thehigh bidder and continues to gain v, -b,.

However, if b, <b,<v,, the shading yields azero gain, whereas without shading the gainis u, -b,. A parallel argument establishes

that there is no advantage in making a bid,b , , greater than 0 , . The optimal strategy ofeach buyer is therefore to submit his reserva-tion value. It follows that, just as in theEnglish auction, the high bidder ends uppaying the second highest reservation value.This equivalence greatly simplifies the com-parison between the English and sealed highbid auctions4 since it implies that we needonly com pare sealed bid auctions.

In the high and second bid auctions, onlythe winner makes a payment to the seller.

How ever, there is an infinity of auction rulesinvolving payment by more than one bidder.For example, all buyers might be charged afixed entry fee. Alternatively, losers might berequired to pay some fraction of their bids.A third possibility, discussed in Section 11, isthat the seller might attempt to encouragehigher bids by offering to return some of themoney paid by the winner to each of thelosers, the size of the rebat e depe ndin g on aloser's bid.

Each of these alternatives is an ex amp le of

an auction with the following properties.First, a buyer can make any bid above someminimum "reserve" price announced by theseller. Second, the buyer malung the highestbid is awarded the object. Third, the auctionrules are anonymous: each buyer is treated

' ~ h sype of auction is sometimes referred to as a

Vickrey auction.

4 ~ h eealed h igh b id auc t ion a lso has i ts open auc -tion equivalent. In t h s Dutch auction, the sale price is

initially set at a hlgh level and is then lowered until a bidis made.



383O L . 71 NO . 3 R I L E Y A N D S A M U E L S O N : O P T I M A L A U CT I O N S

alike. Fourth, there is a common equilibriumbidding strategy in whlch each buyer makesa bid b , , whlch is a strictly increasing func-t ion of h ~ seservation value u , , i.e.,

Thro ugh out this section we shall consider thefamily @ of auc tion rules for which these fou rassumptions a re ~a t i s f ie d . ~e be g n with themain result.

PROPOSITION 1: Suppose the I ID assump-tion holds and all buyers are risk neutral. Thecommon equilibrium bidding strategy for any

mem ber of the familyW

of auction rules yieldsan expected revenue to the seller of

where v , is the reservation value below whichit is unprofitable to submit a bid.

Proposition 1 is important because it tellsus that the expected revenue of the sellerfrom quite different auctions can be com-

pared simply by determining the lowestreservation value, v , , for which it isworthwhile bidding. We begin the proof byexamining the behavior of a single buyer.The expected return to malung a bid can beexpressed as follows.

(2)

expected probability

[I = {res:~::On} ( wi:ling 1

Below we obtain simple expressions forboth the probability of winning and the ex-

' ~ f t e r er iving Proposit ion 1, we became aware of apaper by Roger Myerson which uses a much moretechnically demanding approach to examine expectedseller revenue in an even broader class of auctions.

Generalizing our approach, Eric Maslun and Riley(1980a) have shown that Myerson's results imply there

there are circumstances i n which expected seller revenuecan b e increased by prohibiting bids over certain ranges.

pected buyer gain. Then, from ( 2 ) , we areable to derive the expected payment of atypical buyer. Given the symmetry of theauc tion rule, exp ected seller revenue is just ntimes this expected payment.

With buyers behaving noncooperatively, acommon strategy, b, = b ( v , ) , is an equi-librium strategy if, when adopted by allbuyers but one, the latter 's best response isto adopt it also. Without loss of generality,we may suppose that the buyer consideringan alternative bidding strategy is buyer 1.With all other buyers bidding according tob ( v ) , buyer 1, if he bids at all, will wish tobid in the range of this function. Hence, we

can write any bid as b , = b ( x ) and viewbuyer 1 as choosing x . It follows that b ( v ) san equilibrium bidding strategy if buyer 1can do no better than choose x = v , , and sobids b ( v ,).

T o examine the optima l choice of buyer 1,

we begn by assuming his bid is b ( x ) , andthen ask what restrictions are implied by therequirement that his optimal bid is b ( v , ) .Any auction rule must specify the am oun t hemust pay, p , given his own bid b , = b ( x ) andthose by the other n- 1 buyers, i.e.,

We may therefore write the expected pay-ment by buyer 1, given a bid of b , = b ( x ) as

Also, the bid of b ( x ) is the winning bid ifan d only if all other buyers have m ade lowerbids. By assumption the equilibrium bidfunction is strictly increasing in v , thereforebuyer 1 wins if all other valuations are lessthan x . Since the probability buyer j has areservation value less than x is F ( x ) , buyer 1wins with probability F n - ' ( x ) . Combiningthis last result with ( 2 ) and ( 3 ) , the expectedgain to buyer 1, if he chooses to enter theauction, can be expressed as



384 T H E A M E R IC A N E C O N O MI C R E V I E W J U N E 1981

For b(u) to be the equilibrium bidding

strategy, buyer 1's optimal choice must be to

select x= u , and bid b(v,). Then buyer 1's

maximized expected gain is IT(u,, u,) and

the following first-order condition must be

sati~fied.~

This must hold for all reservation values

exceeding v,, the reservation value for which

a buyer is indifferent between submitting the

bid, b(v,), and not entering the auction.

That is, (5) holds for all v, a v , , where u,,

satisfies

Setting x=v, in (9,t follows that the equi-

librium expected payment by buyer 1 must

satisfy the differential equation

Integrating and making use of the boundary

condition, (6), buyer 1's expected payment is

therefore

Integrating the second term by parts, this

can be rewritten more conveniently as

V ' a v *

6 ~ r o m4) an d (5) we also have

an,(x, o l ) = (o l - x )( n - 1 ) ~ " - ' ( X )F ' ( X )U A

Therefore n ( x , o , ) is increasing in x fo r x < o l anddecreasing for x > o l and the first-order condition yields

the global maximum for buyer 1.

The final step is to consider the auction

from the seller's viewpoint. As far as the

seller is concerned, u, and hence the ex-

pected payment, P(u ,), is a random variable.

The seller's expected revenue from buyer 1 is

therefore the expectation of P(v,). Since the

seller knows that u, has distribution F(v,)

his expected revenue is

Substituting for P(v,) from (8b) and in-tegrating by parts, the expected revenue from

buyer 1 can be rewritten as follows:

Given the equal treatment of all n buyers,

expected seller revenue is just n times the

expected revenue from buyer 1 and the pro-

position is proved.

One of the striking features of our deriva-

tion is that nowhere is there explicit refer-

ence to the equilibrium bidding strategy b(u).However, under any particular auction rule

this is readily derived. The key to such der-

ivation is (8), the expression for expected

payment, P(v) , of a buyer with reservation

value v.

For example, in the high bid auction, sup-

pose the seller announces a reserve price b,.

Any buyer with a reservation value v>b, has

an incentive to enter, i.e., v, =b,. Since a

buyer pays if and only if he is the high

bidder, his expected payment is

(10) P(u ) =Prob {b( v) is high bid) b(u )

But b(u) is the high bid if and only if all

other buyers have lower reservation values.

Then Prob {b(v) is high bid) =Fn-'(v) and,

from (lo), b(u )=P (u) /Fn-'( v). Substitut-

ing for P(u) from (8b), we therefore have the

following additional result.

PROPOSITION 2: Suppose the IID assump-

tion aN are risk and theseller announces a reserve price b,. In the high



385O L . 71 NO . 3 RILEY AND SAMU ELSON: OPT IMAL AUCTIONS

bid auction, the equilibrium bidding strategy ofa typical buyer with reservation value v 2 b, is

Proposition 2 indicates the degree to whlcha buyer will "shade" his bid, b ( v ) , below hisreservation value v in the high bid auction. Itis a straightforward matter to confirm thatb ( v ) s strictly increasing in v . Therefore thehigh bid auctio n is a mem ber of the family ofauctions described by Proposition 1 . Cer-tainly the second bid auction is in this familysince anyone entering w ill bid his reservationvalue. In both auctions, a buyer will enter if

and only if his reservation value exceeds b,.Then in both cases u , = b, and, from Prop-osition 1 , expected seller revenue is the same.

We now demonstrate that these two com-mon auction rules are optimal for the seller,gven the appropriate choice of a reserveprice. For any auction rule there is someimplied m inimum reservation value v , belowwhich buyers will choose not to bid. Thenthere is a probability of F n ( v , ) that all nbuyers will decide not to submit a bid. Inthis case the seller's gain is his own personalvaluation v , . Then, from (9), the total ex-pected return to the seller is

It follows that any two auction s in the family&, for which v , is the same, yield the sameexpected gain to the seller.' Moreover, dif-ferentiating with respect to v , the expected

gain of the seller is maximized for some v ,satisfying the condition8

oreo over, any two auctions for which o, is thesame yield the same expected gain to buyer i conditionalon o,. This result follows directly from equations (4) and

'8bj.Expression (12) w ill, in general, have m ultiple roots.

If this is the case, it is necessary to evaluate the expectedreturn, ( 1 I), at each root to determine the global maxi-

mum.

We therefore have the following further re-sult.

PROPOSITlON 3: If the I I D assumption

holds and buyers are risk neutral, the membersof the family & of auction rules, which maxi-mize the expected gain of the seller are thosefor which the reservation value v , , below whichit is not worthwhile bidding, satisfies

independent of the number of buyers.

An immediate implication of Proposition3 is that the high and second bid auctions

with reserve price b , = v , are both optimal inthe family of auctions @.Note also that v ,exceeds v , : the seller announces a reserveprice strictly greater than his personal valua-tion.

To gain an understanding of this strongresult, it is helpful to consider a second bidauction in which there are two buyers and toexamine the implications of introducing areserve price slightly higher than u,; i.e.,v , = v , + 6 where 6 is small. Since eachbuyer's dominant strategy is to bid his res-ervation value, the expected gain to the selleris affected (i) if b oth valua tions lie betweenu, and v , , and (ii) if one valuation liesbetween v , and v , and the other exceeds 0 , .

In the first case the seller retains the item,which he values at u,, rather than selling it atsome price between v , and v , . His loss istherefore of order 6 . Since this outcome oc-

curs with probability ( F ( v , )- ~ ( 0 , ) ) ~ ( F ' ( v , ) ) ~ ~ ~ ,he expected loss is of orde r a 3 .In the second case the seller receives a pay-ment of v , rather than some price betweenv , and v , hence has a gain of order 6 . Sincethis outcome occurs with probability 2 ( F ( v , )-F(u , ) ) ( l -F ( v , ) = 2 F ' ( v , ) ( l -F ( v 0 ) 6 , heexpected gain is of order S 2 . Therefore, forsufficiently small 6 , the gain to raising thereserve price above the seller's personal val-uation outweighs the cost.

While we have focused on the reserve pricebecause of its com mon usage, there are manydifferent ways in which to discourage theappropriate subset of buyers from participat-

ing in the bidding. Suppose, for example,



386 T H E A M E R I C A N E C O N O M I C R E V I E W J U N E 1981

that the seller annou nces a fixed entry fee c .For all buyers with valuations less than somenumber v , , it will be optimal to remain outof the auction. Consider a buyer with the

borderline reservation value v , . In the secondbid auction he enters, and, since the entry feeis now sunk, bids h s rue value v , . H e wins ifand only if there are no other bidders, inwhch case there is no addit ional payment.Since this occurs with probability F ( v , ) " - 'h s expected profit is

But for v , to be the borderline reservationvalue, the expected profit must be zero. Theseller then chooses an entry fee c , satisfying

A similar argument holds for the high bidauction. If a buyer has the borderline res-ervation value v , , he wins if and only if thereare no bidders. The optimal bid in suchcircumstances is zero, therefore the expectedprofit is again given by ( 1 3 ) and the optimal

entry fee by ( 1 4 ) .Our general results are also helpful inanalyzing the expected payoff to multiplerounds of bidding. Suppose, for concrete-ness, that the seller is using a high bid auc-tion. If he can co nvince buyers that there willonly be a single round with optimal reserveprice v , , there will be a chance that no bidswill be su bm itted. Since v , exceeds the seller'sreservation value v , , the seller, after the fact,has an incentive to lower his reserve priceand to call for a second round of bids.

However, if buyers are not fooled, they willadopt first-round strategies in the expec-tation of a possible second round. In particu-lar, all those with reservation values v, >boo,the reserve price in the second round, willplan to enter the two round auction. Then,from Proposition 3 the seller's expected gainis lower since the entry value is no longeroptimal.

A final point concerns the decision of theseller whether or not to announce a reserveprice. In the second bid auction, the strategy

of bidding one's reservation value is a domi-

nant strategy. Therefore the seller cannotinfluence bids by concealing his reserve price.It follows that the optim al silent reserve priceis the same as the optimal an noun ced reserve

price and that expected seller revenue isidentical.

The argument is more complex in the caseof the high bid auction, but on ce again it canbe shown that there is no adv antage in usinga silent reserve price. The proof, whchinvolves a straightforward extension of Prop-osition 1 , is provided in ou r earlier paper.

11. Alternative Au ctions

To illustrate the results of Section I, wenow compare some specific auctions underthe simplifying assumptions that there areonly two buyers, and that reservation valuesare uniformly distributed o n the unit interval( F ( v ) = v , fo r v E [O , 1 ) . We assume also thatthe object for sale has no value to the seller,v , = O . First we indicate the gains to employ-ing an optimal reserve price in the high bidauction. We then present an unusual pair ofauction designs which happen to belong tothe class of optima l auc tion s9 In con trast, a

t h r d example shows that a seemingly natural(and commonly employed) auction proce-dur e is suboptimal.

Under our simplifying assumptions, it fol-lows from Proposition 3 that it is optimal forthe seller to design the auction so that onlythose with reservation values exceeding v , =1 / 2 find it worthwhile bidding. Then, in thehigh bid auction it is optimal for the seller toannounce a minimum or reserve price b, =1 / 2 . Appealing to Proposition 2, this, inturn, implies that the equilibrium bid of buyer

i, with reservation value v ,2

1 / 2 , is b ( v , ) =v , 2 + 1/ 8 v , . By contrast, if the seller alwayssells the good (by setting the reserve price6 , = 0 ) buyer i 's bid becomes b ( v , )= u,/ 2 .Either by direct computation o r by appealingto Proposition 1 , it can be confirmed thatexpected seller revenue is 5 / 1 2 with b, = 1 / 2

and 1 / 3 with 6 , = O . Thus the optimal re-

h he interested reader is referred to our earlier paper.where it is shown that the auct ions of examples 1 and 2

belong to the class of optimal auctions for arbitrary

F ( c ) and n .



V O L . 71 NO . 3 RILEY AND SAMUELSON: OPTIMAL A UCTIONS 387

serve price strategy results in a 25 percentincrease in expected revenue.

We now consider three quite different auc-

tion rules.

Example 1 S ad Loser Auction. Suppose thereare just two buyers a nd the seller announcesthe following auctio n rules.

(i) Each buyer paying an entry fee c iseligible to subm it as his bid an y positive realnumber."

(ii) T he h g h bidder receives the good butretains his bid.

(iii) Th e lower bidd er (if there is one)loses h s bid.

It is tempting to conjecture that there is noequilibrium bidding strategy for t h s set ofrules. However, not only is t h s incorrect, butthe equilibrium bidding strategy is readilyderived. Un der rules (i)-(iii), the expectedgain of the typical buyer is

For b(v) to be the equilibrium biddingstrategy, thls gain is maximized by setting

x = v , so that the buyer's expected payment isb(u , )( l - F(v ,)) . Then if F (v )= v and c= 1/4,it can be confirmed from equation (14) thatv, = 1/2, and from equation (8b) that

1for v,>-

2

Thus, as in the optimal high bid auc tion, anybuyer with reservation value less than 1/2rem ains out of the auction . Th e bids of thosewith higher reservation values are strictlyincreasing in v and increase without boundas v approaches l! Nevertheless, it is easy toconfirm that expected seller revenue underthis scheme matches th at of the high bid a ndsecond bid auction, cum optimal reserveprice.

Under the high bid and second bid auc-tions, only the recipient of the good gains. Incontrast, the following auction distributes apositive return to all participants.

his rules out bids such as "infinity" or "one morethan my opponent."

Example 2: Santa Claus Auction. Supposethere are just two buyers, and the sellerannounces the following auction rules.

(i) A buyer who submits a bid b>v,receives from the seller an amount S(b)=

l,b*F (v )d v .(ii) T he h g h bidder obtains the good for

h s b id p rice so tha t h s net payment isb- S (b ) .

O ne can confirm that the equilibrium strategyof each buyer is to bid his reservation value.Suppose tha t the second buyer bids b2 =v ?.Th en if buye r 1 bids b ,, his expected profit isgiven by

P r { b , is h g h b i d ) ( v , - b , ) + S ( b , )

It is straightforward to check that thisexpression is maximized at b, = v ,.

In t h s auction the seller's expected netrevenue is the expected value of the hg h e r ofthe two bids less the seller's expected pay-ments. With v, = 1/2 , S ( b )= b2/2- 1 /8.Moreover, each buyer bids h s reservation

value; therefore the seller's expected grossreceipts and payments are easily computed.On ce again it can b e confirmed that expectednet reven ue is 5/12, exactly the sum theseller can expect from the high bid auction.

Since the implication of Proposition 1 isthat many seemingly different auction tech-niques lead to the same ultimate results, it isimportant to illustrate the range of excep-tions.

Example 3: Matching Auction. Supp ose thereare just two buyers an d the seller employsthe following auctio n rules.

(i) The re is a single round of bidding.Buyer 1 is given the opportunity to quo te aprice b, >v,.

(ii) If buy er 1 makes a bid, buyer 2 canmatch it, if he chooses, obtaining the goodfor this price. If buy er 1 makes n o bid, buyer2 can obtain the good at price v, if hechooses.'

" F o r an analysis of the matching auction when mrounds of bidding are permitted, see our earlier paper.



388 T H E A M E R I C A N E C O N O M I C R E V I E W J U N E 1981

Though t h s auction procedure is quite com-mon (for example, in house sales, a renteroccupant is frequently given the right tomatch the offer of any potential buyer), it is

inefficient from the point of view of theseller. In fact, in some circumstances it per-mits a buyer who values the item less highlythan his opponent to obtain the good. Thus,it may produce an allocation of the goodthat is inefficient ex post .

Suppose that F ( v ) = v and v , = 1/2. Thestrategy of bu yer 2 is straigh tforw ard. H ematches b , if and only if v , 2 b . If buyer 1does not open the bidding, buyer 2 bids 1/2for the good if v , 2 1/2. Anticipating thebehavior of buye r 2, buyer 1 bids b , 2 1/2 tomaximize

( v ,- 6 , )Prob{bu yer 2 chooses not to ma tch)

Buyer 2 will not match if his reservationvalue is less than b , , that is, he will notmatch with probability b , . Then buyer 1chooses b ,2 1/2 to maximize his expectedgain ( 0 ,- b , b , . Since this expression is de-creasing in b , for all b ,> 1/2, buyer 1'soptimal strategy is to bid

Consequently, whenever 1/2 t v , < v ,, theobject is awarded to buyer 2 who values itless highly than buyer 1. The expected reve-nue of the seller for t h s example is 3/8, aredu ction of 10 percent relative to the highand second bid auctions.

111. Buyer Risk Aversion

When potential buyers are risk averse, thefundamental equivalence result outlined inSection I is no longer valid.', Re taining the

'*other authors have also considered the effects of

risk aversion on bidding. Butters derives Propositions 4and 5 for the special case in which buyers exhibitconstant relative risk aversion. Charles Holt examinesthe effects of risk aversion in the closely related problemof bidding on incentive contracts. Steven Matthe wscompares high bid an d second bid au ction s when seller

and buyers are risk averse.

assumption of buyer symmetry, it is shownin the Appendix that the high bid auctiondominates the second bid auction underbuyer risk aversion.

PROPOSITION 4: Suppose assumption IIDholds and all buyers share a common utilityfunction displaying risk aversion. Then (i) Inthe second bid auction, bidders continue to bidtheir reservation ualues, that is, b , =u ,. (ii) Inthe high bid auction, as bidders become morerisk averse, they make uniformly higher bids.(iii) Consequently, the seller enjoys a greaterexpected profit under the high bid auction thanunder the second bid auction.

It is evident that the introduction of riskaversion does not affect the strategy domi-nance of bidding one's true reservation valuein a second bid auction, hence part (i). Part(iii) follows directly from part (ii) which isproved in the Appendix.

The intuition behind these results is thatwith risk aversion the marginal increment inwealth associated with a successful, slightlylower bid is weighted less heavily than thepossible loss ( v ,- b , ) if, as a result of lower-

ing the bid, the buyer is no longer the h g hbidder. This leads risk-averse bidders alwaysto shade their bids less than risk-neutralbidders.

Under risk aversion, the general equiva-lence result obtained in Proposition 1 nolonger holds. For instance, an auction em-ploying a seller reserve price will not, ingeneral, be equivalent to one that specifies abuyer entry fee-even when the samereservation value v,, below whch it is notworth bidding, is implied. Still it is natu ral to

explore the effect that buyer risk aversionhas on the optimal seller reserve price in thehlgh bid auction. The following result isderived in the Appendix.

PROPOSITION 5 : Suppose assumption IIDholds and all buyers share a common cardinalutility function. Th en , in the high bid auction,the optimal seller reserve price is a decliningfunction of the degree of risk aversion.

The proposition is intuitively plausible in

view of the fact that as buyers become risk



389O L . 71 N O. 3 R I L E Y A N D S A M U E L S O N : O P T I M A L A U C T I O N S

averse in the extreme, the amount by whichthey will shade their reservation values ap-proaches zero, b(v,)+ v,. Naturally, the sellercan do no better than to announce his per-

sonal valuation as his reserve price, b, =v,.To quote a hgher price cannot "push up"buyer offers and risks the loss of beneficialsales. Of course when b, =v, and b, =v,, thehigh bid auction is also efficient ex post.

IV. ConcludingRemarks

Whle a general result concerning the de-sign of optimal auctions under uncertaintyhas been presented, it is important to pointout the limitations and special assumptionsof the present model. We have assumed that:

(a) A single indivisible good is to be soldto the highest bidder.

(b) The greater a bidder's reservation valuethe more he will bid for the good.

(c) Buyer roles are symmetrical (i.e., buyervalues are drawn from a common distribu-tion) and each buyer is risk neutral.

(d) Buyer values are independent.Additional difficulties are raised when

multiple goods are auctioned or when a di-

visible good must be allocated. Unless buyervaluations are additive and income indepen-dent, auctioning the goods in sequence willbe inefficient (ex post and ex ante). Whenmultiple goods are auctioned, each buyershould logically submit a bid for each subsetof goods. Roughly speaking, the seller willallocate goods to maximize revenue underone of a number of auction schemes. In thecase of a divisible good, each buyer willsubmit a "demand schedule" indicating theprice he is willing to pay for any given

quantity of the good. The seller must for-mulate an auction rule whlch specifies theallocation of the good and appropriate pay-ment of buyers. In either instance the de-termination of optimal auctions for thesemore general environments lies beyond thebounds of the present analysis.I3

Given assumption (b) it follows that thefamily of auctions considered are those in

I3Harris and Raviv (198 1) and M askin and Riley

(1980b) analyze optimal auctions for different classes ofdemand curves.

whch the good is sold to the buyer with thehighest reservation value v, if the good issold at all. Under only moderate restrictionson the form of the distribution F(v), it can

be shown that it is never optimal to utilize arule in whch the winner might be someoneother than the buyer with the highest v.However, when these restrictions are notsatisfied, a stochastic auction is optimal. Insuch an auction a lottery is employed toallocate the good when buyer reservationvalues fall in specified ranges.I4

Dropping the assumption of buyer sym-metry also causes complications in the analy-sis. The derivation of the class of optimalauctions relied explicitly on the existence ofa common equilibrium bidding strategy.Without this, these propositions no longerhold. The asymmetric model, though far morecomplex, is nevertheless amenable to thebasic approach developed herein. Supposethe reservation prices of the buyers are drawnfrom the independent distributions, F, , F,,. ,F,. Some partial results from this settingsuggest a basic conclusion. An optimal auc-tion extends the asymmetry of the buyerroles to the allocation rule itself. The assign-

ment of the good and the appropriate buyerpayment will depend not only on the list ofoffers, but also on the identities of the buyerswho submit the bids. In short, an optimalauction under asymmetric conditions violatesthe principle of buyer anonymity.

As pointed out earlier, the assumption ofrisk neutrality is crucial to our general equiv-alence result. Given risk neutrality, the sellercan do no better than to employ the secondbid auction with an optimal reserve price. Inthis auction, buyers will have no difficulty

formulating an optimal bidding strategy. Norneed they know the form of the distributionfunction F(u). Against any distribution ofopponents' bids, each buyer's dominantstrategy is to bid his reservation value. Theclear advantage of the second bid auction isthat it economizes on the information eachbuyer requires to bid optimally. Further-more, Proposition 3 indicates that the seller

I 4 ~ o r presentation of the more general framework

from which the optimal stochastic auction can be de-rived see Myerson or Maslun and Riley (1980a).



390 T H E A M E R I C A N E C O NO M IC R E V I E W J U N E 1981

can formulate an optimal reserve price policywithout knowledge of the number of buyerswho might enter the auction."

Final ly, one must consider the appropria-teness of the model's most basic assumption,value independence. The analysis has as-sumed that each buyer is informed of hisown reservation price and, more important,tha t t h s pr ice conveys no informat ion aboutany other buyer's value. A different auctionmodel has been applied to bidding for off-shore oil leases.I6 Here, a tract being auc-tioned is assumed to have a common valuefor all parties. The tract value is unknown,though buyers may possess (differing) sam-

ple information allowing inferences aboutthis value. In this setting, each buyer mustdetermine a strategy for acquiring informa-tion concerning the value of the tract an d forsubmitting a bid based on a correct estimateof this value. These features have a directinfluence on th e determination of a n optimalauction and raise additional policy issues.(Should the seller maintain a stake in anawarded tract for the purpose of risk shar-ing? Shou ld the seller undertake measures tofacilitate inform ation acquisition or to allow

information pooling?)"

I 5 ~ h eargest auction houses ( for example. SothebyPark Bernet. Inc. and Christie 's) employ the Englishauction (combining its open bid and sealed bid forms)to sell rare and valuable items (art , antiques, and jewelry).A buyer can bid personally for an item on the da y of theauction or can submit a prior written offer, designatinga representative from the auction house to bid on hisbehalf. This same procedure establishes a silent sellerreserve price, since a house representative is instructedto buv back th e good if the sale price is insufficient. It isa common observation that the competitive features ofthe op en ascending auction serve to elevate buy er offers(above their prior values). This implies that the openascending auction enjoys a practical advantage over thesealed bid version. The "mixed" auction allows writtenbids in order to promote the greatest possible participa-tion while maintainin g the "uplifting" features of theopen ascending auction.

I6see. for example. Robert Wilson (1975) and Mat-thew Oren and Albert Williams. In this model buyersbegin with common prior beliefs about the value of aresource but have different posterior beliefs as a result ofindependent sampling. For discussion of auctions inwhich buyers have different prior beliefs, see Wilson(1967) .

or a discussion of the incentives for the seller tomake information public. see Paul Milgrom and Robert

Weber.

In most real world settings, we would ex-pect tha t a good's econom ic value to a poten-tial buy er consists of two parts-a valueelement whlch is common to all market par-ticipants and one which is buyer specific.Shell's recent $3.6 billion purchase ofBelridge Oil- the most expen sive in U . S .hs to ry - is a d rama t ic example . Belridge wassold by closed sealed bid auction in whichtwenty-odd prospective buyers participated.Differences in bids presumably reflected (i)differences in beliefs about the value ofBelridge's oil holdings (differences that m ighthave been dissipated through pooling of in-formation) and (ii) differences in the extent

to which Belridge's operations comple-mented bidders' other activities. It is easy toimagine, though not to solve, a hybrid modelspecifying both dependent and independentcomponents of buyer reservation values. Aformal analysis of optimal auction design inthls more general environment remains to beundertaken.

PROPOSITION 4 (ii): Suppose assumption

IID holds and all buyers share a commonutility function displaying risk aversion. Thenin the high bid auction, as bidders becomemore risk a verse, they ma ke uniformly higherbids.

P R O O F :Let b ( v ) be the common equilibrium

strategy of n risk averse buyers, each ofw h o m h a s t h e s a me v o n N e u ma n n -Morgenstern utility function u ( x ) . We as-sume that u ( x ) is a strictly increasing, con-

cave function of x and normalize so thatu(O)=O. W ith all other buyers using the equi-librium bidding strategy and buyer j biddingb ( x ) ,j 's expected utility is

F o r b ( x ) o be the equilibrium strategy, ( A l )must have i ts maximum at x=v, . Differenti-ating with respect to x an d s etting the deriva-

tive equal to zero at x=v, , we have the



392 T H E A M E R I C AN E C O N O M I C R E V I E W J U N E 1981

tial equation for ab/av,

From (A4) and (A5) the bracket in (All)is larger for th e utility function u 2( x) ex-hibiting greater risk aversion. The n if we canestablish that ab2 /av ,=8 bl /a v ,> 0 at u=u,, it will follow fro m (A1 1) tha t

for v>v , and hence that ab , /av ,>ab ,/av ,for v>v,.

From (A10) we have,

Since b(v,, v,)= u, an d u(O)=O, it followsfrom (A2) that for any concave utility func-

tion and an y v, > 0, the first term in (A12) iszero. Then the second term in (A12) is equalto unity for both u, (x ) and u2(x) .

REFERENCES

G . R. Butters, "Equilibrium Price Distribu-tions and the Economics of Information,"unpublished doctoral dissertation, Univ.Chicago 1975.

R. Engelbrecht-Wiggans, "Auctions and Bi-dding Models: A Survey," Management

Sci., Feb. 1980, 26, 119-42.J. C. Harsanyi, "Games with Incomplete In-

formation Played by Bayesian Players,"Parts I; 11; 111, Management Sci., Nov.1967; Jan. 1968; Mar. 1968, 14, 159-82;321-34; 486-502.

M . Harris and A. Raviv, "Allocation Mecha-nisms and the Design of Auctions," work-ing paper, Grad. School Ind. Admin.,Carnegie-Mellon Univ. 1979.

and , "A Theory of Monopoly

Pricing Schemes with Demand Uncer-tainty," Amer. Econ. Rev., June 1981, 71,347-65.

C. A. Holt, Jr., "Competitive Bidding for Con-tracts under Alternative Auction Proce-dures," J. Polit. Econ., June 1980, 88 ,43 3-445.

E. S . Maskin and J. G . Riley, (1980a) "Auction-ing an Indivisible Object," working paper,Kennedy School Government, HarvardUniv. 1980.

and -, (1980b) "Price Dis-crimination and Bundling, Monopoly Sell-ing Strategies when Information is Incom-plete," mimeo., MIT, 1980.

S. Matthews, ' 'R~sk Aversion and the Ef-ficiency of First and Second Price Auc-tions," mimeo., Univ. Illinois, 1979.

P. R. Milgrom and R. J. Weber, "A Theory ofAuctions an d Co mpetitive Bidding," work-ing paper, Grad. School Ma,~agement,Northwestern Univ. 1980.

R. B. Myerson, "Optimal Auction Design,"Math. Operations Res., 1981, forthcoming.

M. E. Oren and A. C. Williams, "On Competi-tive Bidding," Operations Research, Nov.-Dec. 1975, 23, 1072-79.

A. Ortega-Reichert, "Models for CompetitiveBidding Under Uncertainty," unpublisheddoctoral dissertation, Stanford Univ. 1968.

J. G . Riley and W. F. Samuelson, "OptimalAuctions," working paper, Univ. Cali-fornia-Los Angeles 1979.

W. F. Samuelson, "Models of CompetitiveBidding," unpublished doctoral disserta-tion, Harvard Univ. 1978.

R. M. Staik and M. H. Rothkopf, "CompetitiveBidding: A comprehensive Bibliography,"Operations Res., Mar.-Apr. 1979, 27, 364-

91.W. Vickrey, "Counterspeculation, Auctions

and Competitive Sealed Tenders," J. Fi-nance, M ar. 1961, 16, 8-37.

R. B. Wilson, "Competitive Bidding withAsymmetrical Information," ManagementSci., July 1967, 13, A8 16-20.

, "On the Incentive for InformationAcquisition in Competitive Bidding withAsymmetrical Information," report, dept.econ., Stanford Univ. 1975.



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Optimal Auctions

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[Footnotes]

13 A Theory of Monopoly Pricing Schemes with Demand Uncertainty

Milton Harris; Artur Raviv


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References

A Theory of Monopoly Pricing Schemes with Demand Uncertainty

Milton Harris; Artur Raviv


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Competitive Bidding for Contracts under Alternative Auction Procedures

Charles A. Holt, Jr.

The Journal of Political Economy, Vol. 88, No. 3. (Jun., 1980), pp. 433-445.

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Counterspeculation, Auctions, and Competitive Sealed Tenders

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