IN MANY REAL WORLD AUCTIONS the value of the goods for sale is subject to ex post ... Econometric evidence based on data
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
78
P.ESO AND L. WHITE
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.
PRECAUTIONARYBIDDING
79
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).
80
P.ESO AND L. WHITE
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)).
PRECAUTIONARYBIDDING
81
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.
82
P.ESO AND L. WHITE
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.
PRECAUTIONARYBIDDING
83
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,
