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http://www.econometricsociety.org/

Econometrica, Vol. 78, No. 4 (July, 2010), 1435–1452

CAN RELAXATION OF BELIEFS RATIONALIZE THE WINNER’S

CURSE?: AN EXPERIMENTAL STUDY

ASEN IVANOV

Virginia Commonwealth University, Richmond, VA 23284, U.S.A.

DAN LEVIN

The Ohio State University, Columbus, OH 43210, U.S.A.

MURIEL NIEDERLE

Stanford University, Stanford, CA 94305-6072, U.S.A.

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Econometrica, Vol. 78, No. 4 (July, 2010), 1435–1452

NOTES AND COMMENTS

CAN RELAXATION OF BELIEFS RATIONALIZE THE WINNER’S

CURSE?: AN EXPERIMENTAL STUDY

BY ASEN IVANOV, DAN LEVIN, AND MURIEL NIEDERLE1

We use a second-price common-value auction, called the maximal game, to exper-

imentally study whether the winner’s curse (WC) can be explained by models which

retain best-response behavior but allow for inconsistent beliefs. We compare behavior

in a regular version of the maximal game, where the WC can be explained by inconsis-

tent beliefs, to behavior in versions where such explanations are less plausible. We find

little evidence of differences in behavior. Overall, our study casts a serious doubt on

theories that posit the WC is driven by beliefs.

KEYWORDS: Common-value auctions, winner’s curse, beliefs, cursed equilibrium,

level-k model.

1. INTRODUCTION

A WELL DOCUMENTED PHENOMENON in common-value auctions is the win-

ner’s curse (WC)—a systematic overbidding relative to Bayesian Nash equi-

librium (BNE) which results in massive losses in laboratory experiments.2Two

recent papers, Eyster and Rabin (2005) and Crawford and Iriberri (2007), ra-

tionalize the WC within theories that retain the BNE assumption that players

best respond to beliefs (hence, we refer to these theories as belief-based), but

relax the requirement of consistency of beliefs. Eyster and Rabin introduced

the concept of cursed equilibrium (CE) in which players’ beliefs do not fully

take into account the connection between others’ types and bids. Crawford

and Iriberry used the level-k model which was introduced by Stahl and Wilson

(1995) and Nagel (1995). In this model, level-0 (L0) players bid in some pre-

specified way and level-k (Lk) players (k = 1?2????) best respond to a belief

that others are Lk−1.3

In response to Eyster and Rabin (2005) and Crawford and Iriberri (2007),

we investigate experimentally whether the WC in common-value auctions is

indeed driven by beliefs.4We use a second-price common-value auction, called

themaximal(ormaximum)game,thatwasfirstintroducedinBulowandKlem-

1We would like to thank David Harless and Oleg Korenok for useful discussions. We would

also like to thank a co-editor and three anonymous referees for their comments and suggestions.

An extended working-paper version is available at http://www.people.vcu.edu/~aivanov/.

2See Bazerman and Samuelson (1983), Kagel and Levin (1986), Kagel, Harstad, and Levin

(1987), Dyer, Kagel, and Levin (1989), Lind and Plott (1991), and the papers surveyed in Kagel

(1995, Section II) and Kagel and Levin (2002).

3CE and the level-k model can be applied to environments other than common-value auctions.

4Our study applies to any belief-based explanation of the WC. This includes, for example,

analogy-based expectation equilibrium (Jehiel (2005), Jehiel and Koessler (2008)). We focus on

CE and the level-k model because they are the two most prominent belief-based explanations.

© 2010 The Econometric SocietyDOI: 10.3982/ECTA8112

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A. IVANOV, D. LEVIN, AND M. NIEDERLE

perer (2002) and Campbell and Levin (2006). This game has the special prop-

erty of being two-step dominance-solvable and our experimental design ex-

ploits this property. We focus on initial periods of play as this seems like a

natural starting point for evaluating belief-based theories.

The paper most closely related to ours is Charness and Levin (2009). This

study finds that the WC is alive and well in an individual-choice variant of the

“acquiring a company” game, that is, in an environment where the WC cannot

be rationalized by inconsistent beliefs about other players’ behavior.

Three concerns arise in interpreting the results in Charness and Levin

(2009). First, one cannot reasonably expect CE or the level-k model to ex-

plain every aspect of behavior.5Thus, even if Charness and Levin’s setup rules

out belief-based explanations, we should still expect some anomalies. The key

question is, “Is the WC more pronounced in environments where it can be

rationalized by belief-based explanations than in environments where such ex-

planations are less plausible?” If the answer is “yes,” the difference could be

attributed to the level-k model or CE, whereas a negative answer casts doubt

on the validity of such models. However, Charness and Levin (2009) cannot

answer this question because it does not include a regular acquiring a com-

pany game against human opponents, that is, an environment of the former

type. In contrast, our paper studies and compares behavior in both types of

environments.

Second, the acquiring a company game represents a lemons market (see Ak-

erlof (1970)) and is not a common-value auction. Although both types of en-

vironments admit a WC, they are quite different and it is not obvious that

Charness and Levin’s conclusions readily extend to common-value auctions.6

Finally, Charness and Levin (2009) studied behavior in individual-choice

settings. However, it is possible that subjects employ very different cognitive

mechanisms in interactions with other players; such interactions may trigger

all sorts of thought processes about others’ reasoning, beliefs, and intentions.

Thus, the conclusions from Charness and Levin (2009) do not necessarily ex-

tend to games against human opponents. In our study, subjects play against

other people.7

Another related study is Costa-Gomes and Weizsäcker (2008), which finds a

systematic inconsistency between chosen actions and stated beliefs in normal-

form games. This study differs from ours in two important ways. First, it con-

cerns an environment which is very different from common-value auctions.

Second, it is based on eliciting subjects’ beliefs. In addition, the study cannot

distinguish between two possible interpretations: (i) that subjects do not best

5For example, they do not explain bidding above values in private-value second-price auctions.

6It is plausible that the WC in both types of environments is driven by the same forces. How-

ever, given that these are quite different environments, this cannot be taken for granted.

7In one of our environments, each subject plays against the computer which, however, mimics

the strategy of a person (actually, the subject’s own past strategy).

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RATIONALIZING THE WINNER’S CURSE

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respond to beliefs when choosing actions and (ii) that subjects form different

beliefs when choosing actions and stating beliefs.8

We proceed as follows. In Section 2, we describe the maximal game and de-

rive the relevant theoretical predictions. In Section 3, we describe our exper-

imental design and in Section 4 we examine the experimental data. Section 5

concludes.

2. THEORETICAL CONSIDERATIONS

We begin by describing the maximal game. There are n bidders, each of

whom privately observes a signal Xithat is independent and identically dis-

tributed (i.i.d.) from a cumulative distribution function F(·) on [0?10]. Let

Xmax= max({Xi}n

note particular realizations of Xiand Xmax, respectively. Given (x1?????xn),

the ex post common value to the bidders is v(x1?????xn) = xmax. Bidders bid

in a sealed-bid second-price auction where the highest bidder wins, earns the

common value, xmax, and pays the second highest bid. In case of a tie, each

tying bidder gets the object with equal probability.

We will say that, given signal xi, a player bidding b underbids, bids her signal,

overbids, or bids above 10 if b < xi, b = xi, xi< b ≤ 10, or b > 10, respectively.

We now state our first result.

i=1) be the highest of the n signals, and let xiand xmaxde-

PROPOSITION 1: b(xi) = xi is the unique bid function remaining after two

rounds of iterated deletion of weakly dominated bid functions.9In the first round,

all bid functions bi(·) with bi(xi) < xior bi(xi) > 10 for some xiare deleted. In

the second round, all bid functions bi(·) with xi< bi(xi) ≤ 10 for some xiare

deleted.

The proof is given in the Appendix A. Here, we give the intuition. It is ob-

vious that bidding above 10 is weakly dominated. Underbidding is also weakly

dominated since, under the second-price rule, one could lose the auction at a

price below one’s signal even though the value of the object is greater than or

equal to one’s signal. Given that no one underbids, bi(xi) > xiis weakly dom-

inated for any xi, because, in case the highest bid among others is between xi

and bi(xi), i makes nonpositive (and possibly negative) profits.

That bidding one’s signal is a BNE follows directly from Proposition 1. In

fact, we can say more than that (the proof is given in Appendix A):

8Another related study is Pevnitskaya (2008). This study investigates whether deviations from

the risk-neutral BNE in first-price private-value auctions are caused by inconsistent beliefs, risk

aversion, or probability misperception. All components seem to be at work.

9A bid function is weakly dominated if, for some signal, it prescribes a weakly dominated bid.

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A. IVANOV, D. LEVIN, AND M. NIEDERLE

PROPOSITION 2: The bid function b(xi) = xiis the unique symmetric BNE

(including mixed strategies).10

We now show that overbidding can arise within the level-k model and CE.

First, let us consider the level-k model. In this model, level-0 (L0) players bid

in some prespecified way and level-k (Lk) players (k = 1?2????) best respond

to a belief that others are Lk−1. For auction settings, Crawford and Iriberry

(2007) considered a version of L0, called random L0(RL0), which, regardless

of its signal, bids uniformly over all bids between the minimal and maximal

value of the object. RLk(k ≥ 1) best responds to RLk−1. The next proposition

shows that RL1can overbid.11

PROPOSITION 3: The bid function of RL1is bRL1(xi) = E(Xmax|Xi= xi) ≥ xi.

If F(xi) < 1, the inequality is strict.12

The proof is given in Appendix A. It hinges on the fact that, because RL0’s

bid is uninformative about its signal, RL1cannot draw any inference about

Xmaxfrom winning the auction.13

Let us turn to CE. In a χ-CE (χ ∈ [0?1]),players best respond to a belief that

each other player j, with probability χ, chooses a bid that is type-independent

and is distributed according to the ex ante distribution of j’s bids and, with

probability 1 − χ, chooses a bid according to j’s actual type-dependent bid

function. Thus, χ captures players’ level of “cursedness”: if χ = 0, we have a

standard BNE, and if χ = 1, players are fully cursed and draw no inferences

about other players’ types. Based on Proposition 5 in Eyster and Rabin (2005),

we can state the following proposition.

PROPOSITION4: Assuming Xihasastrictlypositiveprobabilitydensityfunction

(p.d.f.),14the following bid function constitutes a symmetric χ-CE: bCE(xi) = (1−

χ)xi+χE(Xmax|Xi= xi) ≥ xi. If χ > 0 and F(xi) < 1, the inequality is strict.15

10In our experiment, matching of subjects is anonymous and there is no feedback, so it seems

implausible that subjects should coordinate on an asymmetric BNE. For more on asymmetric

equilibria, see our working paper at http://www.people.vcu.edu/~aivanov/

11Crawford and Iriberry (2007) also considered a version of the level-k model based on a so-

called truthful L0(TL0). In our settings, the behavior of TLkand RLk+1coincides for k ≥ 0.

12If signals have the discrete uniform distribution on the set {0?1?2?????10} and there

are two bidders (this is relevant for our experiment), then bRL1(xi) = E(Xmax|Xi= xi) =

(x2

13The behavior of RLkfor k ≥ 2 is not uniquely determined. The point, however, is that a RL1

can rationalize overbidding.

14Although the assumption of a strictly positive p.d.f. is not satisfied for the discrete distribu-

tion in our experiment, we suspect that the proposition nevertheless holds.

15If signals have the discrete uniform distribution on the set {0?1?2?????10} and there are two

bidders, then bCE(xi) = (1−χ)xi+χE(Xmax|Xi= xi) = (1−χ)xi+χ(x2

i+xi+110)/22.

i+xi+110)/22.

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RATIONALIZING THE WINNER’S CURSE

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3. EXPERIMENTAL DESIGN

3.1. Treatments and Procedures

TheexperimentconsistsoftheBaseline,ShowBidFn,andMinBid treatments.

The Baseline treatment consists of two parts. In part I, subjects play the max-

imal game for 11 periods. In each period, subjects are randomly and anony-

mously rematched in separate two-player auctions. Each subject’s signals for

the 11auctions aredrawnwithequalprobabilityandwithout replacementfrom

the set {0?1?2?????10}.16Signals are independent across subjects. Subjects can

bid anything between 0 and 1?000?000 experimental currency units (ECU). To

minimize the effect of learning, we provide no feedback whatsoever during the

experiment. This also ensures that, in any auction, each bidder’s prior over the

other bidder’s signal is the discrete uniform distribution on {0?1?2?????10}.

Part II is similar to part I. The only difference is that each subject i bids

againstthe computer ratherthan againstanother subject. The computer, which

“receives” a uniformly distributed signal, mimics i’s behavior from part I by

using the same bid function that i used in part I. For example, if the computer

receives signal y, it makes the same bid that i made in part I when she received

signal y. Effectively, in part II each subject is playing against herself from part I

(and knows that this is the case).17

The ShowBidFn treatment is identical to the Baseline treatment except that

in part II we explicitly show subjects their bid functions from part I. The Min-

Bid treatment is identical to the Baseline treatment except that subjects are

explicitly not allowed to underbid.

We conducted three sessions of the Baseline (62 subjects), two sessions of

the ShowBidFn (46 subjects), and one session of the MinBid treatment (26 sub-

jects). Subjects were students at The Ohio State University (OSU) who were

enrolled in undergraduate economics classes. The sessions were held at the

Experimental Economics Lab at OSU and lasted around 45 minutes. At the

start of each session, the experimenter read the instructions for part I aloud

as subjects read along. After that, subjects did a practice quiz. Experimenters

walked around checking subjects’ quizzes, answering questions, and explain-

ing mistakes. After part I of the relevant treatment, the instructions for part II

were read. After part II, subjects were paid. Subjects’ earnings consisted of a

$5 show-up fee, plus 10 ECU starting balances, plus their cumulative earnings

from the 22 auctions,18converted at a rate of $0.50 per ECU. Average earnings

16Our design for part I ensures that each subject receives each signal from the set

{0?1?2?????10} exactly once. In effect, we are eliciting subjects’ bid functions. This simplifies

the design of part II.

17Note that although in part II a subject bids against the computer, the bidding strategy of the

opponent is that of a person. The fact that this person is herself from part I should only make the

cognitive processes of the opponent all the more salient.

18In case a subject incurred losses which could not be covered by the 10 ECU starting balances,

she was paid just her $5 show-up fee.

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A. IVANOV, D. LEVIN, AND M. NIEDERLE

were $18.53/$18.03/$15.53 in the Baseline/ShowBidFn/MinBid treatment. The

instructions for the Baseline treatment are given in the Supplemental Material

(Ivanov, Levin, and Niederle (2010)).19The experiment was programmed and

conducted with the software z-Tree (Fischbacher (2007)).

3.2. Possible Implications for Belief-Based Theories

In part I of the Baseline and ShowBidFn treatments, underbidding and bid-

ding above 10 are weakly dominated and can hardly be explained by any belief-

based theory. The most interesting behavior is overbidding because it leads to

a WC (as long as others are also appropriately overbidding) and because it

could potentially be explained by belief-based theories. Notice that, to explain

overbidding, both the level-k model and CE require that beliefs place a posi-

tive weight on underbidding, that is, on weakly dominated bids. Although not

implausible, this requirement puts some strain on belief-based explanations of

overbidding.

However, the real test of belief-based theories comes from part II of each

treatment and part I of the MinBid treatment. In particular, we argue below

that if behavior is driven by beliefs, we should observe a reduction in overbid-

ding (i) in part II of each treatment relative to part I and (ii) in part I of the

MinBid treatment relative to part I of the Baseline and ShowBidFn treatments.

The absence of any such reduction would cast a serious doubt on belief-based

theories.

Ourargumentisbasedontheassumptionthatifbehaviorisdrivenbybeliefs,

these beliefs are at least consistent with the objectively known features of the

environment. That is, we assume that a subject’s belief in part II is consistent

with the fact that the computer uses her own bid function from part I20and that

a subject’s belief in the MinBid treatment is consistent with the fact that the

opponent cannot underbid. Later, we will consider alternative interpretations

of belief-based theories under which beliefs can be at odds with the objectively

known features of the environment.

Consider a subject i who overbids (for all signals) in part I of one of the

three treatments.21From Proposition 1, it follows that bidding her signal is a

best response in part II. Although underbidding may not be a best response,

it is at least a response in the right direction.22If i continues to overbid but

19The instructions in the other two treatments are very similar and are available upon request.

20In the Baseline treatment, this assumption entails that subjects are able to recall their bidding

behavior from part I (which was just a few minutes ago) or perhaps, at least, whether they tended

to underbid, bid their signal, overbid, or bid above 10. In the ShowBidFn treatment, subjects do

not need to recall anything because they are explicitly shown their bid functions.

21To be precise, overbidding is not possible for signal 10: a subject can underbid (except in the

MinBid treatment), bid her signal, or bid above 10. Therefore, the correct statement is “a subject

i who overbids for all signals 0–9 and bids above 10 for signal 10.”

22Of course, underbidding is not possible in the MinBid treatment.

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RATIONALIZING THE WINNER’S CURSE

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corrects her overbidding downward, this may or may not be a best response,23

but again it is a response in the right direction. On the other hand, if i contin-

ues overbidding without a downward correction or even starts bidding above

10 in part II, she is clearly not best responding to her behavior from part I.

The bottom line is that if i’s behavior is driven by beliefs, we should observe a

downward correction of bids in part II relative to part I.

In part I of the MinBid treatment, anything other than bidding one’s signal

is weakly dominated. Thus, if behavior is driven by beliefs, we would expect a

reduction in the frequency and (average) magnitude of overbidding relative to

part I of the Baseline and ShowBidFn treatments.24

Let us turn to three interpretations of belief-based theories under which be-

liefs can be at odds with the objectively known features of the environment.

The first interpretation is that subjects are using some simple rule of thumb

which leads them to behave “as if” they were best responding to beliefs. For

example, a player using a rule like “bid based on the expected value conditional

on my signal and ignore everything else” would behave just like a fully cursed

or a RL1player. Because subjects do not deliberately form beliefs, the beliefs

describing their behavior could be at odds with objectively known features of

the environment. Thus, subjects in any of our environments could behave as if

they had cursed or RL1beliefs.

Note that this interpretation requires that the rule of thumb be rigid across

environments. For example, a subject using the above rule of thumb needs to

ignore the opponent’s bidding strategy just as much in part II as in part I, even

though in part II it is her own past bidding strategy (which is even explicitly

shown to her in the ShowBidFn treatment).

The second and third interpretations pertain to CE. According to the sec-

ond interpretation, cursed players do not fully think through the connection

between others’ types and bids. As a result, they come up with cursed beliefs

to which, however, they best respond by appropriately conditioning the ex-

pected value of the object on winning the auction. Under this interpretation,

CE would explain behavior in our experiment only if players equally fail to re-

alize the connection between others’ types and bids when bidding against (i)

other people whose bids are unrestricted, (ii) their own bidding strategy (even

when it is shown to them), and (iii) other people who are explicitly not allowed

to underbid.

23For overbidding in part II to be a best response, i would need to shift her bid function in

part II, bII

(xi?bII

24In the MinBid treatment, a subject’s available bids depend on her type so that CE is not

formally defined. Nevertheless, our point remains valid: if subjects’ behavior is driven by beliefs

(whether these beliefs are appropriately redefined cursed beliefs or other beliefs), we should

observe a reduction in overbidding.

i(·), downward in a way that, for all signals xi, none of the bids she made in part I lies in

i(xi)]. Otherwise, there is a positive probability that she wins the auction and loses money.

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A. IVANOV, D. LEVIN, AND M. NIEDERLE

According to the third interpretation, players are aware of others’ type-

contingent strategies but underappreciate the information content of winning

the auction. Under this interpretation, CE could explain overbidding in any of

the environments of our experiment.25The problem with this interpretation is

that rather than being about inconsistent beliefs, it is about a failure to prop-

erly update the expected value of the object conditional on winning. Such a

failure is not at all part of the formal definition of CE according to which play-

ers perfectly update given their (albeit cursed) beliefs. Although theoretically

awkward, this interpretation could have validity if CE can accurately capture

behavior which is actually driven by improper updating. For example, in the

special case of bidders who have correct beliefs but completely fail to condi-

tion on winning, fully CE perfectly describes behavior. However, it is unclear

to what extent CE can accurately capture the behavior of bidders who update,

albeit incompletely. Thus, it is unclear to what extent this interpretation is gen-

erally valid.26

4. RESULTS

We start by studying and comparing behavior in parts I and II within each

treatment. After that, we compare the part I behavior of the Baseline and

ShowBidFn treatments with that of the MinBid treatment.

4.1. Behavior in Part I and Part II

We start by placing each bid b, given signal x, in one of the following cat-

egories: (i) b < x − 0?25, (ii) x − 0?25 ≤ b ≤ x + 0?25, (iii) x + 0?25 < b ≤ 10,

and (iv) b > 10.27That is, we count all bids within 0.25 ECU of one’s signal as

if they were precisely equal to the signal.28Based on this, we classify subjects

(separately for each part of each treatment) in the following way: Underbid-

ders/Signal Bidders/Overbidders/Above-10 Bidders are those who make at least

6 (out of 11) bids in category (i)/(ii)/(iii)/(iv); subjects who fall in none of these

four classes are classified as Indeterminate.29

We start the analysis with the Baseline treatment. Table I shows how many

subjects were in each class in part I (last column) and part II (last row). The

25The same holds for Charness and Levin’s variant of the acquiring a company game.

26To shed light on the issue, one would formally have to define an equilibrium concept in which

players have correct beliefs but incompletely update their beliefs (conditional on winning).

27Actually, for signal x = 10, a bid needs to be above 10.25 to fall into category (iv); a bid

9?75 ≤ b ≤ 10?25 falls into category (ii). We ignore this into our notation.

28Counting only bids which are precisely equal to the signal in category (ii) (and adjusting the

other categories appropriately) does not change any of our results.

29Using 7 or 8 (instead of 6) class-consistent decisions as the cutoff for a player to be assigned

to a class does not affect the analysis much (apart from increasing the number of Indeterminate

subjects).

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RATIONALIZING THE WINNER’S CURSE

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TABLE I

SUBJECT CLASSIFICATION IN PARTS I AND II OF THE BASELINE TREATMENT

Part I / II Underbidders Signal Bidders Overbidders Above-10 Bidders Indeterminate

Underbidders

Signal Bidders

Overbidders

Above-10 Bidders

Indeterminate

2

0

1

2

2

7

0

5

5

1

2

2

3

1

1

1

6

5

0

0

4

0

1

5

5

9

14

1

3

23

25

10

13

13 14

table also shows how subjects switched between classes from part I to part II.

For example, the entry in the first row and third column shows that 2 subjects

who were Underbidders in part I became Overbidders in part II. Based on the

table, we can make the following statements:

RESULT 1:

(a) In part I, a large percentage of subjects make a weakly dominated bid

(b < x−0?25 or b > 10) in at least 6 (out of 11) auctions (30.7%).30

(b) In part I, Overbidders are the largest class (40.3%).

(c) Only a minority of Overbidders from part I become Signal Bidders or

Underbidders in part II (24%).

(d) The majority of Overbidders from part I remain Overbidders in part II

(56%).

For a large proportion of subjects, behavior can hardly be explained by

belief-based theories (point (a)). However, the largest proportion of subjects

in part I are Overbidders. These subjects’ behavior is potentially driven by be-

liefs. However, only a minority of them (best) respond in part II by becoming

Signal Bidders or Underbidders. The key question is whether those who remain

Overbidders in part II are best responding in part II or are at least responding

in the right direction by correcting their bids downward.31

For subjects who are Overbidders in parts I and II, we find that only 23% of

bids in part II are best responses to part I behavior. These subjects are fore-

going, on average, 5.62 ECU (median is 4.07 ECU) in expected profits by not

behaving optimally in part II. Figure 1 plots, for each signal, the median bid in

part I (circles) and part II (stars).32Based on the figure, we see no downward

correction of bids in part II. We can state the following result:

30ThispercentageincludesallUnderbiddersandOverbidders,aswellas4Indeterminatesubjects.

31The one Overbidder from part I who becomes an Above-10 Bidder in part II is clearly not

(best) responding to her behavior from part I.

32We plot median, rather than average, bids because averages are distorted by bids above 10.

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A. IVANOV, D. LEVIN, AND M. NIEDERLE

FIGURE 1.—Median bids in parts I (circles) and II (stars) for subjects who are Overbidders in

parts I and II of the Baseline treatment.

RESULT 2: For subjects who are Overbidders in parts I and II, we make the

following observations:

(a) In part II, they forego substantial expected profits.

(b) In part II, there is no evidence of a downward correction of bids.

Result 1 extends to the ShowBidFn and MinBid treatments; see Tables II

and III in Appendix B which are the analogs of Table I.33Result 2 also extends

to the ShowBidFn and MinBid treatments. In the ShowBidFn/MinBid treat-

ment, for subjects who are Overbidders in parts I and II, 15%/19% of bids in

part II are best responses to part I behavior; these subjects are foregoing, on

average, 6.60 ECU34/5.40 ECU (median is 7.19 ECU/3.84 ECU) in expected

profits in part II. For analogs of Figure 1, see Figures 4 and 5 in Appendix B.

4.2. Baseline and ShowBidFn versus MinBid

If behavior is driven by beliefs, we would expect a reduction in the frequency

and (average) magnitude of overbidding in part I of the MinBid treatment rel-

33In the ShowBidFn/MinBid treatment, the percentage of subjects who make a weakly domi-

nated bid in at least 6 (out of 11) auctions is 28.3%/7.8%. In the MinBid treatment, the percentage

is smaller largely because subjects cannot underbid.

34This average excludes one subject who bid very high both in parts I and II so that she incurred

huge losses in part II.

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FIGURE 2.—Average bids in part I of Baseline and ShowBidFn (circles) and MinBid (stars)

(based on bids of form x+0?25 < b ≤ 10).

ative to part I of the Baseline and ShowBidFn treatments. The frequency of

overbidding is 42.8% in the Baseline and ShowBidFn treatments35and 60.5%

in the MinBid treatment. Overbidding is probably more frequent in the Min-

Bid treatment because underbidding is impossible, so all bids are distributed in

three, rather than four, categories. Given this, the frequencies of overbidding

seem quite comparable.

What about the magnitude of overbidding? Figure 2 shows, for each sig-

nal, the average bid of the form x + 0?25 < b ≤ 10 in part I of the Baseline

and ShowBidFn treatments (circles) and in part I of the MinBid treatment

(stars). Average bids are astonishingly close. We can now state our final re-

sult:

RESULT 3: Relative to part I of the Baseline and ShowBidFn treatments, we

find no evidence in part I of the MinBid treatment of (a) a lower frequency of

bids of the form x+0?25 < b ≤ 10 or (b) a reduction in the average size of bids

of the form x+0?25 < b ≤ 10.

35We pool the data from part I of the Baseline and ShowBidFn treatments because part I is the

same in both treatments.

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A. IVANOV, D. LEVIN, AND M. NIEDERLE

5. CONCLUDING REMARKS

We investigate experimentally whether belief-based theories can explain the

WC in common-value auctions in initial periods of play.36The main idea of

our approach is to compare behavior in an environment where overbidding can

be rationalized by belief-based theories with behavior in environments where

belief-based explanations are less plausible. We observe no reduction in over-

bidding in the latter environments. We conclude that, our results cast serious

doubt on belief-based explanations of the WC in initial periods of play unless

one is willing to accept one of the following statements:

(i) Subjects use a rule of thumb which leads them to behave as if they were

best responding to beliefs and which is fixed across the environments in our

study.

(ii) Subjects equally fail to realize the connection between others’ types and

bids in all environments in our study.

(iii) CE, contrary to its formal definition, can be interpreted as being about

improper updating rather than about inconsistent beliefs.

APPENDIX A: PROOFS

PROOF OF PROPOSITION 1: First round of deletion of weakly dominated bid

functions. It is obvious that bidding above 10 is weakly dominated. Under the

second-price rule, for any xi, any bid strictly below xiis also weakly dominated

(by bidding xi) since one could lose the auction at a price below xieven though

xmax≥ xi. Therefore, we can delete all bid functions, such that bi(xi) < xior

bi(xi) > 10 for some xi.

Second round of deletion of weakly dominated strategies. Suppose that bidder i

with signal xi considers bidding b+> xi. In the event that bidding xi wins,

bidding b+rather than xidoes not matter. In the event that bidding b+does

not win, bidding b+rather than xialso does not matter.

Now consider the third possible event: that bidding xidoes not win but bid-

ding b+does. Then bidder i pays the highest bid among the other n − 1 play-

ers,?b, where?b ≥ xmax. The inequality holds because?b ≥ xi(otherwise xiwould

round of deletion of weakly dominated bid functions). But then i would make

nonpositive profits by bidding b+, whereas she would make zero profits by bid-

ding xi. Moreover, if?b is strictly above xmax, then b+makes strictly negative

theories of learning, such as fictitious play (Brown (1951) and Robinson (1951)), in which players

best respond to others’ past actions. (We thank an anonymous referee for this point.) We do not

emphasize this point because learning models are usually about multiple repetitions. Therefore,

even if players in part II in our experiment do not best respond to their own past behavior, with

experience, they may very well learn to best respond to others’ past behavior.

have won) and because none of the other bidders ever underbid (by the first

36Our study, and particularly behavior in part II of our treatments, may have implications for

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RATIONALIZING THE WINNER’S CURSE

1447

profit. Therefore, b+is weakly dominated and we can delete all bid functions

such that bi(xi) > xifor some xi. Q.E.D.

PROOF OF PROPOSITION 237: A strategy for player i is a probability mea-

sure H on [0?10] × [0?∞) with marginal cumulative distribution function

(c.d.f.) on the first coordinate equal to F(·). A pure strategy is a bid func-

tion b:[0?10] ?→ [0?∞) such that H({x?b(x)}x∈[0?10]) = 1. That b(x) = x is a

BNE follows directly from Proposition 1. Here, we prove uniqueness among

all symmetric BNE.38

Assume that H is a symmetric BNE. Let L = {(x?b)|x ∈ [0?10]?b < x} and

U = {(x?b)|x ∈ [0?10]?b > x}. That is, L and U are the sets in [0?10]×[0?10]

strictly below and strictly above the 45◦line, respectively. We need to show that

H(L∪U) = 0 or, equivalently, that H(L) = 0 and H(U) = 0.

First, assume H(L) > 0. Let sk(·) be the step function, defined by sk(x) =

10

kint(kx

in the left graph in Figure 3). Let Ak= {(x?b)|b ≤ sk(x)} ∩ L, that is, Akis

the area in L below the sk(·) function. Note that k?< k??implies A2k? ⊂ A2k??

and that L =?

10), where int(·) gives the integer part of a real number (s3(·) is depicted

k≥1A2k. Therefore, H(L) = limk→∞H(A2k) > 0.39Therefore,

FIGURE 3.—s3(·) and S3(·).

37Under standard assumptions on F(·), we could simply invoke Proposition 1 in Pesendorfer

and Swinkels (1997) so that no proof would be necessary. However, these assumptions do not

hold in the case of the discrete distribution in our experiment.

38Of course, any bid function which differs from b(x) = x only on a set of measure zero will

also be a symmetric BNE.

39To see this, let B2= A2 and Bl = Al/Al−1 for l ≥ 3. Then H(L) = H(?

and fifth equalities follow from the (countable) additivity of probability measures.

l≥2Al) =

H(?

l≥2Bl) =?

l≥2H(Bl) = limk→∞

?k

l=2H(Bl) = limk→∞H(Ak) = limk→∞H(A2k). The third

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A. IVANOV, D. LEVIN, AND M. NIEDERLE

for some k, H(A2k) > 0. Because A2kconsists of finitely many rectangles like

ABCD in Figure 3 (ABCD includes its boundaries, except for point D), it fol-

lowsthatatleastoneoftheserectangleshaspositivemeasure.Assume,without

loss of generality, H(ABCD) > 0.

We will show that, for a positive measure (with respect to (w.r.t.) H) of

points (x?b) ∈ ABCD, bidding b given signal x is strictly worse than bidding

x because there is a positive probability that one will lose the auction to a

bid strictly below x. Let g(?b) = H({(x?b)|(x?b) ∈ ABCD?b ≤?b}). Note that

b = min({b|(x?b) ∈ ABCD}) and b = max({b|(x?b) ∈ ABCD}).

If g(b) > 0, then {(x?b)|(x?b) ∈ ABCD?b = b} has positive measure. For

anypoint (x?b) inthisset,bidding b givensignal x isstrictlyworsethanbidding

x since there is a positive probability of a tie at b.

Assume g(b) = 0. If g(·) is continuous, choose b∗∈ (b?b), such that

0 < g(b∗) < g(b).40Then {(x?b)|(x?b) ∈ ABCD?b ≤ b∗} and {(x?b)|(x?b) ∈

ABCD?b > b∗} each have positive measure. But then for a positive measure of

points (x?b) (the points in the former set), bidding b given signal x is strictly

worse than bidding x since there is a positive probability of losing the auction

to a bid b, such that b∗< b < x.

If g(·) is not continuous, then it has a jump point41at, say, b∗∗. Therefore,

{(x?b)|(x?b) ∈ ABCD?b = b∗∗} has positive measure. For any point (x?b) in

this set, bidding b∗∗given signal x is strictly worse than bidding x since there is

a positive probability of a tie at b∗∗. This proves that we cannot have H(L) > 0.

The proofthat we cannot have H(U) > 0 is analogous,so only a brief outline

is provided. Assume that H(U) > 0. Let Sk(·) be the step function, defined by

Sk(x) = sk(x +10

show analogously to the above discussion that a rectangle of the sort EFGK

in Figure 3 has positive measure. Then defining h(?b) = H({(x?b)|(x?b) ∈

(x?b) ∈ EFGH, bidding b given signal x is strictly worse than bidding x be-

cause there is a positive probability that one will win the auction at a price

strictly above xmax.

g(·) is a nondecreasing function and that g(b) ≥ 0 and g(b) > g(b), where

k) (S3(·) is depicted in the right graph in Figure 3). Then we

EFGH?b ≤?b}), we show that for a positive measure (w.r.t. H) of points

Q.E.D.

PROOF OF PROPOSITION 3: Let? B denote the highest bid among the n − 1

E(payoff|Xi= xi)

= prob(? B < b|Xi= xi)E(Xmax−? B|Xi= xi?? B < b)

40This can clearly be done by the intermediate value theorem.

41Any nondecreasing function is either continuous or has countably many jump points.

42Ties are ignored because they occur with zero probability.

subjects other than i. Given Xi= xi, subject i chooses her bid, b, to maximize42