Econometrica,Vol. 72, No. 1 (January,2004), 77-92

PRECAUTIONARYBIDDING IN AUCTIONS BY PETER Eso AND LUCY WHITE1 We analyze bidding behavior in auctions when risk-aversebuyers bid for a good whose value is risky.We show that when the risk in the valuations increases, DARA bidders will reduce their bids by more than the appropriateincrease in the risk premium. Ceteris paribus,buyerswill be better off biddingfor a more riskyobject in first price, second price, and Englishauctionswith affiliatedcommon (interdependent)values. This "precautionarybidding"effect arisesbecause the expected marginalutilityof income increaseswith risk, so buyers are reluctantto bid so highly.We also show that precautionarybiddingbehaviorcan make DARA biddersprefer biddingin a common values setting to bidding in a private values one when risk-neutralor CARA bidders would be indifferent.Thus the potential for a "winner'scurse" can be a blessing for rationalDARA bidders. KEYWORDS: Risk, risk aversion, prudence, first price auctions, second price auctions, Englishauctions,winner'scurse. 1. INTRODUCTION IN MANY REAL WORLD AUCTIONS the value of the goods for sale is subject

to ex post risk. At the time of the sale, buyers can only estimate the value of the good and they are well aware that the true value to them will be revealed only some time after the sale. Partof this risk is what might be called "winner's curse"risk:uncertaintyabout other buyers'(or the seller's) informationwhich is not revealed in the course of the auction. However, there is also almost invariablypure risk in the valuations, arising from informationthat none of the buyers (nor the seller) can obtain, that will be resolved after the good has been allocated. The sale of oil tracts,art, antiques,wine, and procurementcontracts provide obvious examples that exhibit both types of risk. In each case, there is something about the future resale price, authenticity,quality,and so on, of these goods that cannot be perfectlyforeseen, and that from the buyers'point of view is purelyrandom. Despite the ubiquityof pure ex post risk,there has to date been no analysisof its effect on the biddingbehaviorof risk-averseagents.2The core of this paper is devoted to providingsuch an analysisin a frameworksimilarto the general symmetricinterdependent-valuesmodel of Milgromand Weber (1982). 'We thankLiamBrunt,Drew Fudenberg,Paul Klemperer,Eric Maskin,Min Shi, Jean Tirole, and seminaraudiences at HarvardUniversity,NorthwesternUniversity,the Congressof the European Economic Association (Bolzano, 2000), the WorldCongressof the EconometricSociety (Seattle, 2000), and the InfonomicsWorkshopon Electronic Market Design (Maastricht,2001) for helpful comments, and Ron Borkovskyfor research assistance.We are also grateful to an editor and three referees for constructivecommentsthat substantiallyimprovedthe paper. 2Buyersdisplayriskaversionin a varietyof auctionscenarios.For a surveyof the experimental evidence,see Kagel (1995). Econometricevidencebased on datafromtimberauctionsis provided by Paarsch(1992) and Athey and Levin (2001). 77

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Our main result is that in common auction forms (first price, second price, and English auctions), symmetricDARA bidders (buyerswhose utility functions are the same and exhibit decreasingabsolute risk aversion) reduce their bids by more than the correspondingincrease in the risk premiumwhen pure risk is added to their values. Therefore, holding the number and the ex ante characteristicsof the participantsfixed, buyerswill be better off bidding for a more riskyobject. In the first price auction, this result follows from an effect we may call precautionarybidding.As with precautionarysaving,when agents face a risk,their marginalutilityof income rises.3This causes buyersto bid less aggressivelybecause they value more highly each extra dollar of income, as compared to the increased probabilityof winning the good. In the case of DARA preferences, this effect is so strong that the buyers end up with higher expected utilities when the noise is present in their valuations. This result is surprisingfor the following reason. Under general conditions, DARA individualsbecome more risk-aversein facing one risk (i.e., losing the object) when they are forced to face an independent risk (i.e., object value).4 Since increasing the degree of risk aversionleads to more aggressivebiddingin a firstprice auction,we might therefore expect that increasingthe riskinessof the good would make buyers more risk-averseand so raise bids and make them worse off.5 However, this latter effect turns out to be smallerthan the precautionaryeffect. In the second price and English auctions, as valuations become noisy, the buyers also reduce their bids by more than the correspondingincrease in the risk premium.The necessary and sufficient conditions for the effect to occur are the same (i.e., DARA), although the intuition behind the result is slightly different. Recall that in these auctions, the buyers submit bids assumingthat they will receive zero surplusfrom winningwhenever they win. Therefore, in the presence of noise, DARA bidders will reduce their bids by the large risk premiumthat would be requiredif their surpluswere zero. But when they actuallywin, the buyerswill have a positive surpluson averageand their expected payment will have been reduced by a risk premium that was "too large." So overall they will be better off. Our finding that the seller would like to reduce the pure risk faced by buyers is distinct from the linkage principle (due to Milgrom and Weber (1982)). This principleimplies that the seller should commit to reveal any information affiliatedwith the buyers'signalsbecause the commitmentreduces the potential winner'scurse that the buyersface. Note, however,that the winner'scurse arises because winning provides informationabout the value of the good, not 3See Kimball(1990) and the literaturedating back to Leland (1968), Dreze and Modigliani (1972), and Sandmo (1970). 4See Eeckhoudt,Gollier, and Schlesinger(1996) and the referencescited therein. 'Auctions with risk-aversebuyers have been studied by Holt (1980), Riley and Samuelson (1981), Maskinand Riley (1984), and others.

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because of the pure risk in the good's value. Conversely,it is completely possible for the private value of a good to an individualto be riskywithout any winner's curse implications for bidding. An obvious distinction between the linkage principle and the effects of white noise is that precautionarybidding will arise only when buyersare decreasinglyrisk-averse(DARA), whereas the linkage principlewill hold even when buyers are risk-neutral,but have affiliated common values. The behavior of risk-aversebuyers in an environmentwith affiliated common values has to our knowledgebeen hardlystudied at all. We use our analysis of precautionarybidding to throw more light on this topic. We show that DARA buyers engage in precautionarybidding in response to the risk inherent in other buyers'signals not revealed in the course of the auction. Because of this, DARA buyersmay prefer an interdependentvalues auctionto a private values setting that would be equally attractivefor risk-neutralbuyers. The paper is laid out as follows. In Section 2, we consider the consequences of adding pure noise to the prize in a symmetricauction model with affiliated signals and interdependentvaluations.We prove that DARA buyersengage in precautionarybiddingin firstprice, second price, and Englishauctions,andwill benefit from more risk in the good's value. We also discuss some implications of our result, including for the revenue ranking of auctions by the seller. In Section 3, we show that even in the absence of additionalnoise, pure commonvalue components alone suffice to generate the precautionarybidding effect. In Section 4, we offer concludingremarks. 2. PRECAUTIONARYBIDDING

2.1. GeneralSymmetricModelwithNoisy Valuations We assume that there are n potential buyers for a given good. The seller's reservation value for the good is zero. Buyer i receives a private signal (type), si e [s, s]. The joint distributionof the signals has a positive, twicedifferentiable density, which is symmetricand affiliated.6We will denote the vector of signals of buyersother than i by s_i, the highest among the signals of buyers other than i by smx= maxji{sj}, and the joint density of maxand si by f(smaX, si). Since all signals are affiliated, smaxand si are also affiliated, that is,

d2 n f (y, x)/dxdy > O.

The ex post monetaryvalue of the good for buyer i is Vi = V(Si, S-i) + Zi,

where v: [s, s]" -> t is a continuous, weakly increasing function, which is

strictlyincreasingin its first argumentand invariantto permutationsof its last

n - 1 arguments (i.e., for all s_i permutations of s_i, v(si, s_i) = v(si, S-i)); the 6Forthe definitionand propertiesof affiliation,see Milgromand Weber(1982).

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additionalterm, Zi,is the realizationof a zero-mean randomvariable zi. Note that the specificationof the valuationfunctionsis symmetric,and a buyer'svaluation depends only on the collection of signals of the other buyers (besides his own), not on the identities of the other buyers.7 We assume that the zi's come from a symmetricjoint distributionand that each zi is independent of (s, ..., s,,).8 The noise is interim unobservable and

uninsurable.When the 2z'sare degenerate, zi - 0, we say that the buyershave deterministicvaluations,and when the zi's are nondegenerate,we say that they have noisy valuations.We can interpretthis "noise term" affecting the buyers' values in either of two ways.First, it could be a result of common shocks (such as a change in the oil price or the amount of oil underground);or second, it could be buyer-specificsymmetricallydistributedshocks (such as unforeseen productioncosts). The buyersevaluate their monetarysurplus(consistingof their initialwealth minus the transfer paid to the seller, plus the good's value when they win) accordingto a strictlyconcave and thrice differentiableutilityfunction, u, and they are expected utilitymaximizers.We normalizetheir initialwealth and u(0) to zero, and assume that the good is valuable to them for all realizations of the signals, that is, E[u(vi + 2z) IVj, sj = s] > 0. We will use the notions of decreasingand constantabsolute riskaversion(DARA and CARA, respectively), defined the standardway as -(2 u/lx2)/(du/dx) being strictlydecreasingand constant, respectively.From now on we will assume that u belongs to either the DARA or CARA family. 2.2. Main Results We now analyze how ex ante symmetricDARA or CARA buyers' behavior and indirect utility changes as a result of more noise being added to their valuations.In particular,we will comparetwo situations,one where zi = 0 (deterministicvaluations), and another where zi is an independent randomvariable with zero mean and finite variance(noisy valuations).Holding everything else the same, we show that DARA buyershave higher indirectutilities (while CARA buyers are indifferent)when noise is present in their valuationsin the English (button-), the first price, and the second price auctions.9 7Analternativenotationwouldbe to writethe valuationfunctionas v(X1, Yi,..., Y,_ ), where X1 standsfor i's own signal(si), and Ykstandsfor the kth highestamongthe otherbuyers'signals. Hence, the deterministicpartof the valuationin our model is equivalentto the buyer's(expected) valuationin the general symmetricaffiliatedmodel of Milgromand Weber(1982). 8Note that we allow the 2i's to be correlated,or even zi = z for all i. 9The results are confined to comparingnoisy and deterministicvaluations,but immediately extend to situationswhere another independentnoise is added to make alreadynoisy valuations still riskier.This is so because the DARA propertyis preservedunderthe additionof independent mean-zeronoise (see Kihlstrom,Romer, and Williams(1981)).

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As a preparationfor the proofs, define (1)

u(w; x, y) = E[u(v(si,

S_i) + w) | Si = x, smax =y].

This is the (expected) utilityof buyer i when he gets the noise-free good (whose value is still risky due to interdependentvalues), given that his wealth level is w, his own signal is x, and the highest of the other buyers' signals is y. We will use three key properties of u; a short explanationof each property (with references) is providedbelow. PROPERTY1: The function u is strictly increasing in x, weakly increasing

in y; and for all x and y, u is a concave, strictlyincreasingutilityfunction in w. This propertyfollows because v is weaklyincreasing(strictlyincreasingin si), and the monotonicityand concavityof u in w are preservedunder expectation. PROPERTY2: If u is DARA, then for x' > x and all y and w, u(w; x', y) is

strictlyless risk-aversein w than u(w; x, y) is; similarly,for y' > y and all x and w, u(w; x, y') is weakly less risk-aversein w than u(w; x, y) is. On the other hand, if u is CARA, then for all x, y, and w, u is also CARA in w with the same degree of absolute risk aversion. This property follows because, by affiliation,the random variable v(si, s_i)

given si and smaxincreases in si in the monotone likelihood ratio sense, and

therefore, when u is DARA, the resultingexpected utility function, u, will exhibit a lower level of riskaversionin w for a highersi (this result is due to Jewitt (1987); see also Eeckhoudt,Gollier, and Schlesinger(1996) and Athey (2000)). The same holds for an increase in smax, except that v(Si, s_i) increases in smax weakly, and hence the decrease in the level of risk aversionwill also be weak. The CARA property (and the level of absolute risk aversion) are preserved when a backgroundrisk is added. PROPERTY 3: The functions -hu/lx and -du/ly are increasing and concave functions of w as well. If u is DARA then, holding x and y fixed, -du/dx exhibits a strictlyhigher and -du/dy a weakly higher level of risk aversion in w than u does. If u is CARA, then all three functions exhibitthe same degree of absolute risk aversionin w. This last propertyfollows easily from Property2 combined with the observations that (i) u is strictlyconcave with a positive third derivativeand (ii) decreasingabsolute risk aversionof a utilityfunction is equivalentto the negative of the marginalutilitybeing more risk-aversethan the utilityfunction (Kimball (1990)).

We are now readyto prove Theorem 1.

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1: Considerthegeneralsymmetricaffiliatedmodel withDARAbuyTHEOREM ers. Thebuyers'utilitiesare strictlyhigherwithnoisyratherthan deterministicvaluationsin the symmetricequilibriaof the secondpriceand the Englishauctions. PROOF: First, assume that valuationsare deterministic.Let v(x, y) _ E[v(si, s-i) I Si = x, smta= y].

Define rT(si)solving (2)

u(-V(si, Si) + Ir(Si);Si,Si) = 0,

which means, by (1), that 7(Si) compensates buyer si for the common value = si at zero expected surplus. risk conditional on smaX

We claim that a symmetricincreasingequilibriumexists,with bid functions

(3)

b(s) = v(si, si) - 7(Si).

First, note that b(si) is strictlyincreasing.By differentiatingidentity(2) in si, _-u(-b(si); si, si)(-b'(si))

dw

+-u(-b(si);

8y

+ -

dx

(-b(Si); si, si)

s,, S) = 0,

where cu/ldw > 0, du/dx > 0, and du/dy > 0, therefore b' > 0. In order to establishthat biddingb(si) by buyeri of type si is a best response to b playedby all j 4 i, suppose towardscontradictionthat i bids b > v(si, Si) - rr(Si) instead of (3) while all others play accordingto b. This makes a difference only if, for smax= y, b > b(y) > v(si, si) u(-v(y,

r(si). Then i will receive, instead of 0,

y) + 7(y); si, y) < u(-v(y,

y) + 7T(y); y, y) = 0,

where the inequalityfollows from Property 1 of u and si < y. Hence bidding b > v(si, si) - 7r(Si) is not profitable.An analogousargumentrules out bidding b < v(Si, Si) -

r(Si). Therefore (3) is a symmetric increasing equilibrium bid

function.1 If valuationsare risky,then in the symmetricequilibriumbuyersi bids (4)

f3(si) = V(Si, Si) - 7T(si) - 7r(si),

1?Underriskneutralitythe equilibriumbid is v(si, si). We have shownthat in the case of DARA bidders it is reduced by the risk premium 7T(si)that compensates the buyer for the risk of the others' signalsat zero expected surplus.

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where 7Tz(si) solves (5)

Ezu(-v(si,

Si) + 7r(si) + 2i + 'fz(Si); Si, Si) = 0.

That is, type si further reduces his bid by the compensatingrisk premiumfor Zi at the risky initial wealth (risky due to the common value risk) that gives him zero surplus.The derivationis identical to that of the equilibriumunder deterministicvaluations,(3), and is therefore omitted. By (2) and (5), u(w; y, y) = Ez,(w+

i + 7rz(y); y, y) at w = -v(y, y) + r7(y).

Therefore, by Property2 of u, for all si > y, u(-v(y,

y) + 7r(y); Si, y) < Ez,u(-(y,

y) + 7r(y) + zi + T,z(y); Si, y).

Taking expectations over smax y < si, and using the definition of u, we obtain E[(u(v(si,
s-i)-

V(Smax, mx) + i)- v(sma

Tr(sma))){smax
smax) + 7(Smax

+ Z + 7z(mx)))l{smax
This means that buyer i with type si is strictlybetter off in the equilibriumwith noisy valuationsthan in the equilibriumwith deterministicvaluations. The argumentis similar,althoughsomewhatsimpler,in the case of the English auction. In the efficient symmetricequilibriumof this auction with deterministicvalues, buyer i plans to quit at v(s, s_i), such that for all active j = i, i, sj equals j's true type. By strictmonotonicityof sj = si, and for all inactivejI vi in si, lower types plan to quit earlier.When a buyerdrops out, the other buyers infer his type and repeat the above calculationuntil only one buyerremains which is (weakly)less active.The winnerwill therefore pay v(x, s_i) at x = smax, than his actualvaluationbecause si > smax.With noisy valuations,buyer i plans to quit at v(Si, s_i) - 7T such that for all active]j i, sj = si, for all inactivej = i, If i is the winner,then solves u(0) = Eu(zi + TrO). sj equals j's true type, and Tro he pays v(x, s_i) - 7roat x = smax.Since the deterministicpart of his surplus is nonnegative, the compensating risk premium for noise zi is less than Tr0; therefore the buyerprefers the equilibriumunder noisy valuations. Q.E.D. The same result holds for the firstprice auction as well. THEOREM2: Considerthegeneralsymmetricaffiliatedmodel withDARAbuyers. Thebuyers'utilitiesare strictlyhigherwithnoisyratherthan deterministicvaluationsin the symmetricequilibriumof thefirstprice auction.

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PROOF:Let the symmetric equilibriumbid functions under deterministic and noisy values be b(si) and 3(si), respectively.l1Under deterministicvalues, in equilibrium,the utilityof buyer i with signal si from pretendingto have type Si is

U(si, si) =

u(-b(si);

i, y)f(ylsi) dy,

where f(ylsi) is the probability density function of y = smaxconditional on si.

Let V(si) = U(si, si). By incentive compatibilityand the Envelope Theorem, for all si E [s, s), the derivativeof V (from the right,in case si = s) must be (6)

V' (s) =

-u(- b(sS); si, y)f(Y si) dy

dx

+

u(-b(s); si, y)

f

f(ysi) dy.

Similarly,if V(Si) denotes si's indirectexpected utilityunder noisyvalues in the equilibriumof the FPA, then we have (7)

- E u(-

V'(si)= +

E, u(-(si)

(si)+ z;si,y)f(y

si)dy

+ Zi;Si, y)

f(yS,)dy.

Note that buyer i with type Siwins with probability0 in either setup; therefore V( s) = V( s) = 0. Moreover,both V and V are continuous (by incentive compatibility).We will now show that if V(si) = V(si) for some si e [s, s), then V'(si) < V'(si), which implies that V(si) < V(si) for all si E (s, s], as claimed.

Suppose that V(si) = V(si) for some s, E [s, s). Then si

(8)

j

[E u(-f(s,)

+ Zi;si, y) - u(-b(s); si, y)]f(y, s,) dy - 0.

The expression in square bracketsis weakly increasingand continuous in y = wheneverit is nonnegativeby Property2 of u, while its expectedvalue with smaX respect to smaX (given s,) is 0; thereforethe integrandswitchessign at most once, from negative to positive. By affiliation, (df(y, si)/(9si)/f(y, si) is increasing "The existence of a symmetricequilibriumin our symmetric,affiliatedenvironmentwith riskaverse buyers follows from standardarguments(see Milgrom and Weber (1982, Section 6) for risk-neutralbuyers).

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in y. Therefore the product of the integrandin (8) and (df (y, i)/lsi)/lf(y, si) has a nonnegative integral (a simple proof can be given along the lines of Lemma 1 in Persico (2000)). That is, ' [Ezt(-3(si)

+ z; s, y) - u(-b(i);

si, y)] -f (y, si) dy > 0,

and so the second term in (7) is not less than the second term in (6). By Property3 of u, which is preservedunder integration, (9)

Xt

dx U(-b(si); si, y)f(ylsi) dy


(si) + Zi;Si,y)f(ylsi) dy,

and therefore V'(si) < V'(si). This completes the proof.

Q.E.D.

REMARK:It is clear from the proofs of Theorem 1 and Theorem 2 that in the general symmetricmodel with CARA preferences, all types of every buyer are indifferentbetween playingthe equilibriumwith deterministicor noisyvaluations in the first price, second price, and English auctions. The intuition underlyingprecautionarybidding in the first price auction is the following. For DARA (CARA) individuals,the precautionarypremiumthe amount required to compensate the marginalutility of income for an increase in risk-exceeds (equals) the risk premium (Kimball (1990)). Thus, if a DARA buyer reduces his bid by the amount of the compensatingrisk premium, his marginalutilityof income is still higherthan in the absence of noise, so he will reduce his bid still further to save income in case he wins and must bear the risk of the good.12(For CARA buyers, marginaland total utility are both exactlycompensatedby the risk premium.) This intuition applies to some other auction formatsas well, so the phenomenon of precautionarybidding is more general than what is demonstratedin Theorems 1 and 2. In the workingpaper version of this article (Eso and White (2001)), we show that DARA bidderswill be better off if a riskis added to their values in an all-pay auction with interdependentvalues and independent signals, or with privatevalues and affiliatedsignals (providedthe all-payauction has an equilibriumin the latter case). White (2003) shows that the same type of precautionarybehaviormakes a risk-aversebargainerbetter off bargaining 12Evidently,the same feature of the utilityfunctioncauses precautionarysavingin response to income risk,hence the name for our phenomenon:"precautionarybidding."

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over a risky cake than a certain one. In Section 3 below, we will examine another incidenceof precautionarybiddingwhere the added noise is not independent of buyers'signals.In particular,we show that common value components in the buyers'valuationssuffice to generate the precautionarybiddingeffect. 2.3. Implicationsof the Results Up until now, the auction literature has abstractedfrom the issue of pure ex post risk. Our results imply that this abstractionis fullyjustified only in the case of CARA preferences as these buyerswould reduce their bids by exactly the risk premium.Since DARA buyersbenefit from the introductionof risk in their valuations,they may attempt collectivelyto commit not to acquireinformation about shocks that affect their payoffs (e.g., not doing test drillingon a tract of oil for sale), if it is possible to make such actions publiclyobservable. Further,it is not clear whether even an individualbuyer would wish to learn his valuation more precisely. One may conjecture that while more information about his valuation should benefit the buyer, it might draw forth a more aggressiveresponse from rivalbiddersas well. The results also imply that a seller who is as risk-averseas the buyers will wish to provide insuranceagainstthe noise. This is not simplybecause the risk reduction directlyincreases the buyers'willingnessto pay, but also because it limits precautionarybidding and thus intensifies competition. By contrast, if the auction-designer'sprofit depends not only on the expected revenue from the auction itself, but also on the numberof bidders attendingthe auction (as is for examplethe case for competinginternetauctionwebsites), then it maybe a good idea for him to auction goods whose value is uncertain.Auctions with risky objects (such as online auctions, where unseen goods are bought from complete strangers)may be very popular with DARA buyers who anticipate a large surplus to be made. It is importantto see that this effect arises with risk-aversebuyersratherthan risk-lovingones, and that the popularityof risky auctions may be completely rational. Finally,let us remarkon the empiricaltestabilityof our model of precautionarybidding.In order to run a direct empiricaltest, one would need an accurate estimate of both the degree of the buyers'risk aversionand the riskinessof the good, and estimatingthese parametersis a difficulteconometric exercise (see Campo, Perrigne,and Vuong (2000)). An indirectbut simplertest would be to see whether auctions of riskieritems attractmore buyers (controllingfor the type of the auction, the ex ante distributionof the expected value of the good, and the characteristicsof the buyers). One way of measuringthe "riskiness" of the good's value might be the level of detail in the seller's description of the good, or the extent to which potential buyerscan examine the good before the auction (for example, the ease of access to a forest or oil tract for pre-bid investigation).

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2.4. Rankingof AuctionswithNoisy PrivateValuations In this subsection,we explore some issues that arise when the buyers'baseline valuations, to which the noise is added, are private (rather than interdependent). The proof of Theorem 1 reveals that in the second price or English auction, as the risk increases,bids are reduced by the amount of the risk premiumthat would obtain if buyershad zero surplus.In the firstprice auction (Theorem 2), we do not have such a simple interpretation.Our result thus appears to raise a question regardingthe rankingof auctions: does the differential reduction in the bids alter the preference orderingof the seller (or the buyers) over the different auction forms, relative to the situationwhen noise is not present? The answer to this question turns out to be straightforward.Under independent private values and with risk-averse(not necessarily DARA) buyers, Maskinand Riley (1984) show that the seller prefersthe firstprice auctionover the second price auction. Under independent private values but with DARA buyers,Matthews(1987) shows that the buyersprefer the second price auction over the first price auction.13When an independent noise is added to the independent private valuations in these models, both results remain true. This is so because both risk aversion and the DARA property are preserved after the introductionof noise and taking expectations (Pratt (1964)). In Eso and White (2001), we complete the preferenceorderingof auctionsby DARA buyers (with independent, private, and possibly noisy) valuationsby showing that they prefer the first price auction to the all-payauction. Our result does call into question the well-knownresult that the seller's revenue in a first price auction (with independent signals and privatevalues) will increase as the buyers become more risk-averse.14It remains true that if the good's value is risky,then the fear of losing it will motivate more risk-averse buyers to bid more highly. But against this, they will bid less highly because they dislike the riskinessof the good, and even less highlybecause of the precautionaryeffect. So it is an open question under exactlywhich circumstances the seller will earn higher expected revenue from more risk-aversebuyers. 3. COMPARISONOF COMMON-AND PRIVATE-VALUEAUCTIONS: WHEN THE "WINNER'SCURSE" IS A BLESSING

In this section, we show that the precautionarybidding effect is not limited to exogenous, mean-zero risks, and that other forms of risk are likely to cause the same type of behavior.In particular,in interdependent-valuecontexts, the good's value is riskyfor the buyerbecause when he wins, he does not necessarily know the other buyers' signals affecting his valuation. As a result of this, 13Underindependent private values the English and second price auctions are outcomeequivalent.Thereforeboth the seller and the buyersare indifferentbetween these two formats. 14Forreferences,see footnote 5.

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DARA buyers will prefer certain common-value environments to "comparable" private-valueenvironments.'sIn other words, the common-valuerisk (associated with the potential "winner'scurse") may in fact be a blessing for DARA buyers. In order to simplifythe analysis,let us assume that the buyers' signals are independentdrawsfrom the same distribution,and that the valuations,though interdependent,can be written as (10)

vi = si + h(s_i),

where h: 'R"- -> R is weakly increasing and invariant to permutations of its arguments (i.e., for all s_i permutations of s_i, h(s_i) = h(s_i)). The advan-

tage of this specificationis that it will simplifythe task of finding comparable interdependent-and private-valueenvironments. In Proposition 1, we compare an environmentwhere the buyers'valuations are private and equal to their signal with another one where their valuations are interdependentas in (10). The two environmentsare the same in all other respects, includingthe numberof buyersand the (i.i.d.) ex ante distributionof their signals. Interestingly,if the selling mechanismis the English (button-) auction, then the buyersobtain the same indirectutilities in the two environmentsno matter what their riskpreferencesare (as long as the model is symmetric).'6However, if the selling mechanism is either a first price or a second price auction, then only risk-neutralor CARA buyers will be indifferent between these privateand interdependent-valuesenvironments;DARA buyerswill strictlyprefer the latter. This, as we establish,is a consequence of precautionarybidding.

1: Assume that the buyers'typesare independentlyand identiPROPOSITION distributed. cally If the buyersare risk-neutralor have the same CARApreferences and the mechanismis eithera firstprice or a secondprice auction, then theyreceivethesame utilitiesin environmentsin whichvi = sifor all i, or vi = si + h(s_i) for all i, whereh is weaklyincreasingand invariantto permutationsof its arguments.If the buyershave the same DARApreferencesand the mechanismis either a firstprice or a secondprice auction, then theypreferthe environmentwherevi has a common component as in (10) to the environment where vi = si.

'5Wewill define the private-valueenvironments"comparable"to certaincommon-valueenvironmentssuch that risk-neutralbuyerswill be exactlyindifferentbetween participatingin either of the two auctions. 6Whenonly two buyersremainin the English auction, say, i and j with signalssi and sj, they both know all the other buyers'signals (the true sk, for all k 0 i, j) from the drop-outprices. In the interdependent-valuescase, the buyers build this informationinto their valuations and bid si + h(si, (Sk)ki,j) and sj + h(sj, (Sk)k,ij), respectively. Suppose the winner is i; then his utility is u(si + h(sj, (Sk)kij) - sJ - h(Sj, (Sk)ki,j) = U(Si - Sj). In the private-values case, if si wins,

then he pays sj, yielding a utilityof u(si - sj). Hence if i wins, then his utility is the same in the common- and private-valuessetups for all realizationsof s,.

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PROOF: First, consider the indirectutility of risk-neutralbuyersobtained in an efficient mechanismin an ex ante symmetricenvironmentwith independent signals and interdependentvaluations. By incentive compatibility,using wellknown arguments(the Envelope Theorem or following Myerson (1981)), the payoff of a buyerwith type si is VRN(s,) = VRN(s) +

Si Pr(sma

< x)E

-

8

v(x, s i) Smax< x dx.

In this environment,both the firstprice and second price auctions are efficient and yield zero surplus to the lowest type, VRN(S) = 0. Moreover, when (10) holds, the second term of the above expression (the integral) is the same no matter whether or not h = 0. Therefore, given that the selling mechanism is either a first price or a second price auction, risk-neutralbuyers receive the same payoff in the two environmentsunder consideration. Second, suppose that the buyers have the same risk-averseutility functions (either CARA or DARA). The additivelyseparableinterdependent-valuesenvironment specified in (10) can be thought of as one with noisy privatevalues where the "noise" comes from the signals of the other buyers. While the hypotheses of Theorems 1 and 2 do not hold (the "noise" is not independent of s_i, nor is it independent of si conditional on winning), their proofs go throughwith some modifications. In the second price auction, under private values, buyer i with type si bids b(si) = si and his equilibrium expected utility is E[u(si- smax)l{smax
interdependentvalues of the form (10), buyer i with signal si bids /3(s) = si + h(si) - rTh(Si),where h(y) = E[h(s_i) sma= y] and Th(y) is defined implicitly by E[u(h(s_ ) - h(y) + Trh(Y)) | smax=

y]

- 0. Note that the random variable

h(s_i) - h(y) given smax= y is independent of si. The equilibriumpayoff of

buyer i with type si conditional on winning against a given smx = y < si, is E[u(si + h(si)

(11)

- y - h(y)+ 7Th(Y)) I Smx =y]

> u(i

- y),

where the inequality holds as equality for CARA, and strict inequality for DARA preferences. The inequality follows because the two sides are equal at Si = y, that is, u is exactly compensated by the risk premium rTh(y) for the added risk, h(s_i) - h(y) given smax= y. When si > y, by the CARA (DARA) property,this risk premium, 7Th(y),exactly (strictlymore than) compensates the buyer's utility for the same risk. By taking expectations of (11) over

smax

= y < si, we complete the proof for the second price auction.

In the firstprice auction, let the symmetricequilibriumbid functionbe b under privatevalues, and 3 under interdependentvalues as in (10). The utilities of type si pretending si under privatevalues and, respectively,interdependent values as in (10), can be written as UPV(si, Si) = Pr(smax < Si)u(si-

b(Si)),

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UCV(si, si) = Pr(smx < Si)E[u(si + h(s-i) - 3(Si)) Imx < s]. Denote V(si) = UPV(si, si) and V(si) = UCV(si, si). Using standard arguments

(the Envelope Theorem), for all si c [s, s), the (right-side)derivativesare, V'(si) = Pr(sm,x< si)u'(si - b(si)), V'(si) =Pr(smrn < si)E[u'(s + h(s_) - /3(s)) | smax< i]. If V(si) = V(si) for some si c [s, s), that is, u(

- b(Si)) = E[u(si + h(si)

- 3(si))

smax< Si],

then, by the CARA or DARA propertyof u, u(i - b(si)) < E[u'(si + h(si)

- /3(s)) | Smx s],

where the inequalityholds as equality under CARA, and strict inequalityunder DARA preferences. Therefore, for all si E [s, s), V(si) = J(si) implies V'(si) < V'(si). Since both functions are continuous and V(s) = V(s), we obQ.E.D. tain V(si) < V(si) for all si E (s, s]. This concludes the proof. The result of Proposition 1 is of practicalimportancebecause in some settings biddersmay be able to choose between entering auctionswhere they will face significantcommon-valuesrisks and those where they will not. In other cases, whether the auction of a given good is largely a privateor common values affair may be determined by prior moves taken by the bidders. For example, consider two firms that will later compete in a procurement auction. If-prior to the auction-these two firmschoose similarproductiontechnologies, then the subsequentauctionwill have a strongcommon-valuecomponent: one firm'sestimate of the likely cost of fulfillingthe contractis likely to be importantinformationfor the other firm.But if the two firmschoose production technologies that are very different from one another, then informationabout one firm's production costs may not be at all useful in estimating the other firm's likely cost, and the auction will take place in a private-valuesenvironment. The results of this section suggest that if the firms are DARA,17they may be better off choosing technologies that are "too correlated"(from the seller's and perhaps the social point of view) in order to benefit from the softened (precautionary)bidding.18 '7For a model of why firms in imperfect capital markets will tend to display decreasing absolute risk aversion, see Froot, Scharfstein, and Stein (1993). '8Similar remarks could apply to the choice of customer base by car, art, wine, and antique retailers who buy their product in wholesale auctions: if they choose to serve customers with sim-

PRECAUTIONARYBIDDING

91

4. CONCLUSIONS

We have shown that in a general symmetricmodel with affiliatedsignalsand interdependentvalues, risk-averseDARA buyersare better off when the value of the good auctioned becomes more risky. This research can be thought of as extending Matthews'(1987) comparisonof auction environmentsfrom the buyers'perspective in a new direction. Instead of comparingdifferent auction formatswith the same informationstructure,we have compared different information structuresholding the auction format fixed. We have shown that when the object's value is subject to an additional independent risk, buyers behave less aggressivelyin the first price, second price, and English auctions, reducingtheir bids by more than the amountof the appropriateincrease in the risk premium.We call this effectprecautionarybidding. We have shown that the same effect occurswhen the risk associatedwith the value of the good arises because buyers care about each others' signals (i.e., "winner'scurse risk").Thus in first and second price auctions, DARA buyers are better off bidding in a common-valuessetting than a private-valuesone, where risk-neutralbuyerswould be indifferent.Thus environmentsthat allow for a potential winner'scurse may in fact be a "blessing"to DARA buyers. KelloggSchool of Management,NorthwesternUniversity,2001 SheridanRd., Evanston,IL 60208, U.S.A.;[email protected], and HarvardBusiness School and CEPR, SoldiersField Rd., Boston MA 02163, U.S.A.;[email protected]. ManuscriptreceivedMay,2002;final revisionreceivedApril,2003. REFERENCES Journalof EcoATHEY,S. (2000): "MonotoneComparativeStaticsunderUncertainty,"Quarterly nomics, 97, 187-223. ATHEY,S., ANDJ. LEVIN(2001): "Informationand Competition in US Forest Service Timber Auctions,"Journalof PoliticalEconomy,109 ,375-417. ANDQ. VUONG(2000): "Semi-ParametricEstimation of First Price CAMPO,S., I. PERRIGNE, Auctions with Risk-AverseBidders,"Mimeo, Universityof SouthernCalifornia;presented at the 2000 WorldCongressof the EconometricSociety. DREZE, J., AND F MODIGLIANI(1972): "Consumption Decisions under Uncertainty," Journal of

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CMS-EMSDiscussion Paper #1331.

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