ISSN (Print) 0473-453X

Discussion Paper No. 1048

ISSN (Online) 2435-0982

A STRATEGY-PROOF MECHANISMSHOULD BE ANNOUNCED
TO BE STRATEGY-PROOF:
AN EXPERIMENTFOR THE VICKREY AUCTION
Takehito Masuda
Toyotaka Sakai
Shigehiro Serizawa
Takuma Wakayama

Revised November 2019
January 2019

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanA strategy-proof mechanism
should be announced to be strategy-proof:
An experiment for the Vickrey auction∗
Takehito Masuda† Toyotaka Sakai‡
Shigehiro Serizawa§ Takuma Wakayama¶
November 13, 2019
Abstract
We conduct laboratory experiments for the multi-unit Vickrey auction withand without providing advice to subjects on strategy-proofness. Although therate of truth-telling among the subjects stays at 20% without advice, the rateincreases to 47% with advice. By conducting similar experiments for the payyour-bid auction, which is not strategy-proof, we conﬁrm that our results are
not due to so-called experimenter demand eﬀects. Moreover, advice improveseﬃciency in the Vickrey auction, particularly in early periods in which subjects are less experienced. It is well known that subjects tend to overbid inseveral Vickrey auction experiments. Our results indicate the possibility that
simple advice decreases such overbidding by promoting better understanding of
the strategy-proofness property in the Vickrey auction. Strategy-proof mechanisms are sometimes criticized because players often fail to ﬁnd the beneﬁt oftruth-telling, but our observations show that introducing advice on the propertyof strategy-proofness helps them to behave “correctly.”
Keywords: advice eﬀect; strategy-proofness; Vickrey auction; pay-your-bid auction; market design
JEL codes: D44; D71; D61; D82∗

We thank Pablo Guillen, Nobuyuki Hanaki, Naoko Nishimura, Hans-Theo Normann, Alvin Roth,Tayfun Sönmez, Utku Ünver, Takashi Yamagata, and Lianming Zhu for their comments. Yuki Hamada
and Ryo Mikami provided excellent research assistance. We gratefully acknowledge ﬁnancial support
from the Joint Usage/Research Center at ISER, Osaka University and Grant-in-aid for Research
Activity, Japan Society for the Promotion of Science (15H03328, 15H05728).
†
Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka
567-0047, Japan. E-mail: tmasuda@iser.osaka-u.ac.jp‡
Department of Economics, Keio University, 2-15-45 Mita, Minato-ku, Tokyo 108-8345, Japan.
E-mail: tsakai@z5.keio.jp§
Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka
567-0047, Japan. E-mail: serizawa@iser.osaka-u.ac.jp¶
Faculty of Economics, Ryukoku University, 67 Tsukamoto-cho, Fukakusa, Fushimi-ku, Kyoto 6128577, Japan. E-mail: wakayama@econ.ryukoku.ac.jp1

1

Introduction

The technology for the design of economic mechanisms has grown rapidly in the lastfew decades. Practical applications of economic mechanisms now include spectrum auctions (Milgrom, 2000), school choice (Abdulkadiroğlu and Sönmez, 2003), and kidneyexchange (Roth et al., 2004), among many others. A preferred property of mechanisms
is strategy-proofness, which ensures that truthfully reporting his own type is always
a dominant strategy for every participant. As long as a strategy-proof mechanism is
employed, no participant is supposed to mis-report his type, so that a social optimum
is realized through the collection of accurate information.
On the other hand, laboratory experiments often exhibit that the percentage oftruth-reporting subjects is not high or is even low (e.g., Attiyeh et al. (2000), Kawagoe and Mori (2001), Cason et al. (2006), and Chen and Sönmez (2006)).1 Theseobservations have been considered as an anomaly to the theory, encouraging skepticism concerning the practical performance of strategy-proof mechanisms. A prominent
example is the Vickrey auction, under which many experimental studies have observed
overbidding behavior.2 However, this does not imply that strategy-proofness of theVickrey auction is not useful in reality. Simply, it shows the necessity of developing
some device that puts strategy-proofness to work, making the Vickrey auction usefulin real-life applications.
The fact that the Vickrey auction is strategy-proof is not so trivial that the average
person would not notice it.3 This means that the Vickrey auction does not exhibit the
expected performance if the explanation on the rule is merely given to participants. A
simple idea to overcome this diﬃculty is to announce strategy-proofness of the Vickrey
auction to users, which is what we shall attempt in this experimental research. In our
experiment, we provide advice on the strategy-proofness of the Vickrey auction to the1

Velez and Brown (2019) study the phenomenon of strategy-proof mechanism experiments froma diﬀerent angle. They identify the condition under which the untruthful behavior is “empirically”
plausible.
2
Several experimental auction studies report that most subjects do not reveal their true type in
the Vickrey auction even in the single-unit case. Kagel and Levin (1993) ﬁnd that the rate of sincere
bidding is approximately 27% of all bids. Garratt et al. (2012) report that even though the subjects arefamiliar with bidding in eBay auctions, approximately four out of ﬁve subjects failed to bid sincerely.
Manelli et al. (2006) and Kagel and Levin (2009) observe that subjects in a Vickrey auction experimenttend to overbid, compared with subjects in the Ausubel auction, which is a dynamic counterpart of
the Vickrey auction (Ausubel, 2004; Okamoto, 2018). See Kagel and Levin (2016) for a survey on
experimental results of the Vickrey auction.
3
Hassidim et al. (2017) provide possible explanations as to why individuals do not report their true
preferences under the deferred acceptance mechanism, which is a well-known strategy-proof mechanismin the context of two-sided matching both in the ﬁeld and in the lab. They cite “failure to identify
the dominant strategy” as one of the explanations2

subjects and examine its eﬀect on subject behavior.
Our experiment deals with a case where multiple units of an item are to be soldunder the Vickrey auction. We investigate the eﬀect of providing advice on bidding
behavior and the eﬃciency of outcomes. Since the Vickrey auction is truly strategyproof, providing such advice involves no ethical problem concerning deception.4,5 In
our experiment, even after subjects were given advice, they were not required to follow
the advice, and some of them in fact did not.
We observe that although the rate of sincere bidding is approximately 20% without
advice, it increases to 46.9% once subjects are advised as to the strategy-proofnessof the Vickrey auction. One might suspect that this increase stems from a so-called
experimenter demand eﬀect (Zizzo, 2009). To detect a “net eﬀect” of advice in the
Vickrey auction, we conduct similar experiments for a pay-your-bid auction, which is
not strategy-proof.6 We used the same advice across auction mechanisms to avoid
deception. Then, we ﬁnd that the advice eﬀect in the Vickrey auction is much greaterthan the experimenter demand eﬀect. Moreover, for the Vickrey auction, the impactof the advice is stronger for subjects who well understand the auction rule than forother subjects.
In the literature of auction theory, overbidding in the Vickrey/second price auction
has been attributed to the joy of winning (Cooper and Fang, 2008), a cognitive limit on
contingent reasoning (Li, 2017), and so on. Our results, however, show the possibility
that simple advice decreases overbidding by promoting better understanding of thestrategy-proofness property in the Vickrey auction. Our experimental results suggestthat before operating strategy-proof mechanisms, explaining the property of strategyproofness helps those who are unaware of its property to behave “correctly.”
Related literature. The eﬀect of providing advice on strategy-proof mechanisms
has recently been studied by several authors (Guillen and Hing, 2014; Ding and Schot4

Our approach sounds an intervention of freedom of choice or free play, but we consider thatsaying so is an overreaction. Suppose a consumer purchases a new electronic device and the makerof the device kindly explains how the consumer should use it. The consumer still has the freedom of
choice to not follow the advice. Then, we do not say that the person giving the advice is intervening
in the consumer’s freedom of choice. What we would like to do for economic mechanisms is exactly
like this. This scenario represents a type of consumer protection rather than an intervention. Or, we
could describe the situation as an opportunity for education rather than an intervention.
5
Even in practical auctions, this type of advice is sometimes given to bidders.
For example, the eBay auction provides bidders with “Tips for winning actions”(https://www.ebay.com/help/buying/bidding/tips-winning-auctions?id=4015).
6
The pay-your-bid auction is used by many countries in practical Treasury bonds allocation. See
Brenner et al. (2009) and Marszalec (2017) for more details.

3ter, 2017b). These studies use a two-sided matching model to test whether or notproviding advice will aﬀect the rate of truth-telling. A closely related work by Guillen
and Hakimov (2018) ﬁnds that announcing the strategy-proofness of strategy-proof
matching mechanisms raises the rate of truth-telling, although giving a detailed explanation of the deﬁnition of a mechanism rather decreases it.7 These results suggestthat advice helps participants to conﬁrm individual optimality of truth-telling. To thebest of our knowledge, this study is the ﬁrst one that tests the role of advice in theliterature of auction studies.8
To enhance that people understand incentive properties, Saijo et al. (2007) andLi (2017) attempt to strengthen strategy-proofness. Saijo et al. (2007) emphasize thateven if a mechanism is strategy-proof, it often admits the presence of an ineﬃcient Nashequilibrium composed of mis-reporting. They strengthen strategy-proofness to “secure
implementability” by additionally requiring that no Nash equilibrium realizes an outcome that is not the “true” outcome. An experimental study by Cason et al. (2006)
compares bidding behavior under a securely implementable (and so strategy-proof)
mechanism and that of other strategy-proof mechanisms. They observe that the rateof truth-telling in the former is higher than that in the latter. Li (2017) proposes astronger version of strategy-proofness called “obvious strategy-proofness” that can beapplied to dynamic mechanisms such as ascending auctions or matching algorithms.
In a problem where a single item is auctioned, the ascending auction is obviously
strategy-proof, but the Vickrey auction is not so. In an experiment where a single
item is auctioned, Li observes that the ratio of dominant strategies played by subjects
is signiﬁcantly higher under the ascending clock auction than under the Vickrey auction. However, given that the class of strategy-proof mechanisms is already narrow(e.g., Holmström (1979)), strengthening strategy-proofness severely restricts the admissible class of mechanisms. On the other hand, our approach of providing advice
can be applied to any strategy-proof mechanism in any environment. The Vickrey auc7

Guillen and Hakimov (2018) consider top-down advice (e.g., advice given by the mechanism
administrator). Several studies examine whether the source of advice aﬀects the truth-telling behavior:advice given to children by their parents (Ding and Schotter, 2017b); peer information sharing in
networks (Ding and Schotter, 2017a); and third-party advice such as websites (Guillen and Hing,2014).
8
Shogren et al. (2001) conduct a Vickrey auction experiment under the condition that subjects
are informed of the strategy-proofness of the Vickrey auction. However, they do not test the eﬀect of
the advice because they do not conduct a Vickrey auction experiment without advice. In addition,
in some experimental studies that aim to elicit real values for items through incentive-compatible
mechanisms such as the Vickrey auction, the experimental instructions explicitly inform subjects thattruth-telling is the best strategy (e.g., Grether and Plott (1979) and List (2001)). However, thesestudies are not interested in testing whether or not providing such information will aﬀect the rate oftruth-telling.

4

tion is neither securely implementable nor obviously strategy-proof, but giving advice
drastically improves its performance on truth-telling.
The remainder of the paper is organized as follows. Section 2 describes the theoretical background and our experimental design. Section 3 presents our experimental
results. Section 4 provides concluding remarks.

2

Experimental design

2.1

Theoretical considerations

There are three bidders, {1, 2, 3}, and two indivisible identical objects to be auctioned.
Each bidder is admitted to demand two units. Bidder i’s valuation for the objects isdenoted by vi = (vi1 , vi2 ) where vij denotes the value that bidder i assigns to the j-thunit. Bidder i’s valuation is drawn independently from the uniform distribution on
V ≡ {(vi1 , vi2 ) ∈ [0, v]2 : vi1 ≥ vi2 } where v > 0. Given any vi ∈ V , bidder i’s utility ofobtaining k units of objects and paying pi ∈ R units of money is

U (k, pi ; vi ) ≡
v 1 + vi2 − pi ivi1 − pi



−p
i

if k = 2
if k = 1
if k = 0.

A list v = (v1 , v2 , v3 ) ∈ V 3 is a valuation proﬁle. An assignment function is afunction d : V 3 → {0, 1, 2}3 that satisﬁes the following resource constraint: for eachv ∈ V 3 , d1 (v) + d2 (v) + d3 (v) = 2. A payment function is a function p : V 3 → R3 . A
rule is a pair of assignment and payment functions, f = (d, p). Given any v ∈ V 3 , theprojection of f (v) = (d(v), p(v)) on i ∈ {1, 2, 3} is denoted by fi (v) = (di (v), pi (v));
that is, di (v) ∈ {0, 1, 2} is the number of the objects i obtains, and pi (v) ∈ R is theamount of money i pays.
We are interested in rules in which sincere bidding is a weakly dominant strategy
for everyone. Formally:
Strategy-proofness: For each v ∈ V 3 , each i ∈ {1, 2, 3}, and each vi′ ∈ V , U (fi (v); vi ) ≥U (fi (vi′ , v−i ); vi ).
The following rules are central in the literature on auction theory.
• Vickrey auction: Each bidder simultaneously submits a bid vector bi = (b1i , b2i ) ∈V . After the seller collects all bidders’ bids, the seller ranks them from the high-5

est to the lowest bid and allocates the two units to the two highest bids. If bidder
i wins one (or two, respectively) unit, then the bidder has to pay the highest bids
(or the sum of the highest and the second-highest bids, respectively) from among
the other bidders’ losing bids.
• Pay-your-bid auction: Each bidder simultaneously submits a bid vector bi =(b1i , b2i ) ∈ V . After the seller collects all bidders’ bids, the seller ranks them fromthe highest to the lowest bid and allocates the two units to the two highest bids.
If bidder i wins one (or two, respectively) unit, then the bidder has to pay b1i (orb1i + b2i , respectively).
The most important feature of the Vickrey auction is strategy-proofness. However,many previous experimental studies report that bidders tend to overbid in Vickrey
auctions. The pay-your-bid auction is widely used in real life, while the Vickrey auction
is not so. However, in contrast to the Vickrey auction, the pay-your-bid auction violates
strategy-proofness.

2.2

Experimental procedures

We conducted an experiment to test the eﬀect of advice for the Vickrey auction. We
used between-subjects design. To distinguish between whether subjects follow theadvice because they understand it or obey the advice without understanding it, we
also conducted corresponding experiments for the pay-your-bid auction. We have fourtypes of treatments in total:
1. Treatment VA: the Vickrey auction with advice
2. Treatment VN: the Vickrey auction without advice
3. Treatment PA: the pay-your-bid auction with advice
4. Treatment PN: the pay-your-bid auction without advice
In each treatment, three bidders compete for two units of an item. For each bidder,
two integer values are drawn from the uniform distribution with the interval 0 to 1,000
with increments of 10. Then, the larger (smaller) integer is assigned the value of theﬁrst (second) unit. All values are displayed in Japanese yen (JPY).
Each of VA, PA, and PN had three experimental sessions, whereas VN had two
sessions. We conducted these 11 sessions at Osaka University in March of 2015 and

6Treatment
VA
VN
PA
PN

Auction rule
Vickrey
Vickrey
Pay-your-bid
Pay-your-bid

Advice
Yes
No
Yes
No

Date
Mar-15Mar-15
Jul-16
Jul-16# of Sessions32
3
3# of Subjects (Groups)
63 (21)48 (16)69 (23)72 (24)

Table 1: Summary of treatments.

July of 2016. We recruited student subjects from Osaka University through campuswide advertisements. None of the students were experienced in this particular typeof experiment. No subject attended more than one session. Our experiment wascomputerized using the experimental software z-Tree (Fischbacher, 2007). Twenty-one
or twenty-four subjects participated in each session. Table 1 summarizes the numberof observations. Figure 1 illustrates the timeline of one session.
Each subject was seated at a computer terminal assigned by a lottery. All terminals
were separated by partitions. No communication among subjects was allowed. Eachsubject had a set of printed instructions and a recording sheet.9 The experimenterread aloud the instructions. Then, subjects answered a 17-question quiz that tested
whether they understood the auction rule that they had been informed of a short timeago. Every correct answer was worth $0.30 ($1 = JPY 100). The experimenter read
aloud the answers to the quiz. Subsequently, only in VA and PA, the experimenter
distributed a paper with written advice and also read it aloud. The text of the advice
is as below:“The following advice is about the auction in which you are participating.
Please consider carefully whether this advice is true or false. It is completely
up to you whether you follow the advice or not.
You can maximize your earnings by bidding your values as they are, regardless of what others bid.”
Note that the advice involves no deception problem for VA and PA. The subjects
were given time to ask questions before proceeding to two practice periods and to the
25 successive payment periods under the random matching protocol. At the beginningof each period, all subjects were separated into groups of three. At the bidding stage,
each subject was asked to enter his/her two bids as nonnegative integers—such that
9

The full set of experimental instructions (including screen shots, the quiz, and the questionnaire)is provided in Appendices C–H.

7

Instructions?

Quiz (on a scale of 17)?

Advice?

?

Q and A?

?

Vickrey

Pay-your-bid• Group of three;• Two units;• Twenty-ﬁve periods, random match; and• Values are uniformly distributed on {0, 10, . . . , 1000}.
?

Questionnaire

Figure 1: Timeline of one session.

the ﬁrst unit bid is weakly greater than that of the second unit bid—into a box on thedisplay screen. We set the maximum feasible ﬁrst unit bid to 2,000.
After the 25 payment periods, the subjects completed a questionnaire and wereimmediately paid in cash. Each subject was privately paid the sum of his/her earnings
over the 25 periods. The value of the individual payments ranged from $5.90 to $70.30.

3

Experimental results

3.1

Level of understanding of auction rules

Figure 2 displays the distributions of quiz scores. The scores range from 0 to 17. A ﬁrst
look at Figure 2 shows clear evidence that over 60% of the subjects received a perfectscore in the Vickrey auction while over 80% did so in the pay-your-bid auction. Given8

Figure 2: Distribution of quiz scores for each treatment prior to the auctions.

an auction rule, we found no statistical diﬀerence in the score distributions with andwithout advice.10 Similarly, we found no statistical evidence to support the diﬀerent
levels of understanding between the two auctions regardless of the presence of advice.11
In summary, we have the following result.
Result 1 (Level of understanding auction rules). The majority of subjects had athorough understanding of the auction rules prior to playing regardless of whether they
participated in the Vickrey or the pay-your-bid auction.

10

A two-sample Kolmogorov-Smirnov (K-S) test for the equality of two score distributions had ap-value > 0.90 for both VA vs. VN and PA vs. PN. The same conclusions hold under a chi-square test
also.
11
The K-S test had a p-value > 0.10 for all pairwise comparisons of (i) VA vs. PA, (ii) VA vs. PN,(iii) VN vs. PA, and (iv) VN vs. PN.

9

3.2

Bidding behavior

For a given unit, we say that a bid is sincere if it matches exactly the value drawn for
the unit. Similarly, we say that a bid is over (resp. under ) if the bid is more (resp. less)than the value. For example, if values and bids are v1 = (800, 300), v2 = (600, 400),v3 = (900, 500), b1 = (850, 240), b2 = (600, 400) and b3 = (900, 0), then the sincere bids
are the ﬁrst and second units of bidder 2 and the ﬁrst unit of bidder 3. The uniqueoverbid is bidder 1’s ﬁrst unit, while the remaining two bids are underbids. Hence,
the overall average sincere bid rate is 0.5 (i.e., we divide the number of sincere bids by
the total number of bids by the three bidders), whereas the average sincere bid rate ofbidder 1, 2, and 3 is 0, 1, and 0.5, respectively.
Figure 3 shows the scatter plots for values and bids in each treatment. Initially, theVickrey treatments graphs suggest that, for unit 1, bidders with higher values more
likely to overbid. When values are close to the maximum, 1,000, bids jump as highas 2,000. For unit 2, bidding behavior is polarized with an increase in underbidding:overbidding is prevalent regardless of the realized values. At the same time, we also
observe a cluster of zero bidding when unit 2 values are no more than 500. Table 2
summarizes the classiﬁcation of bids into three categories by treatment and unit byunit.12As Panel (a) of Table 2 (VA−VN column) shows, sincere bidding increased by
26.3% with the statement of strategy-proofness in the Vickrey auction. Notice thatoverbidding is prevalent in the Vickrey auction without advice (VN), which amounts
to 63.29% of all bids. This observation is consistent with ﬁndings in experiments of themulti-unit Vickrey auction by Manelli et al. (2006), Engelmann and Grimm (2009),
and Kagel and Levin (2009). With an explicit statement on the strategy-proofness of
the Vickrey auction (VA), 45.3% of the case was overbidding; that is, this means that
our advice decreased overbidding by 17.99%. Moreover, we found that underbidding
also decreases when the advice is given. Speciﬁcally, our advice decreased underbidding
by 8.31%. In order to compare the impact of advice on overbids and underbids, we
calculate the ratio of the reduction rate of overbidding to the rate of overbidding in VN
as well as the ratio of the reduction rate of underbidding to the rate of underbiddingin VN. As a result, we ﬁnd the advice has more impact on underbids (8.31/16.08)
than overbids (17.99/63.29). When we break down the data into each unit (Panels (b)and (c) of Table 2), the same tendency holds. In contrast, approximately 90% of thesubjects in PA and PN are likely to underbid. We also observe a 3%-increase of sincere
12

See Appendix B for individual-oriented analysis.

10Figure 3: Scatter plots of bids. Blue and red plots indicate bids for unit one and two,
respectively.

11

bidding although the advice is not true in this case.13
Table 3 summarizes the frequencies of sincere bidding by treatment, whether thequiz score is perfect or not, and by unit. In what follows, we apply a normal approximation to examine whether the following three factors aﬀect sincere bidding behavior:auction rule, advice, and unit.14 Suppose that sincere bidding behavior in treatmentj ∈ {VA, VN, PA, PN} is a realization of a random variable Xj with the Bernoulli
distribution that takes 1 (= sincere bidding) with the success rate of pj .15 That is,1 with prob. p ,jXj =0 with prob. 1 − p .
j

Let X j denote the sample mean for nj realizations of Xj and let
Sj =X j (1 − X j ).
njTable 3 summarizes the observed sincere bidding. Superscripts a, p, and i correspond to all subjects, the subjects whose quiz scores are perfect, and the subjects
whose quiz scores are imperfect, respectively. Hereafter, capital letters indicate randomvariables while small letters do realized values.
3.2.1

Advice eﬀects and experimenter demand eﬀects

Using mainly both units’ data shown in Panel (a) of Table 3, we ﬁrst test whether
or not the advice increases sincere bidding in each auction rule. We summarize our
ﬁndings as follows:
Result 2.
(i) The advice increases sincere bidding in each auction rule when using all data.
The mean increases are 26.3% (95% conﬁdence interval, 23.9% to 28.7%) and2.5% (95% conﬁdence interval, 1.3% to 3.7%) in the Vickrey auction and in the
pay-your-bid auction, respectively.

13

A simple regression of all bids to values strongly shows the coeﬃcients are 0.8032 for PA and
0.7927 for PN. For the theory of multi-unit, pay-your-bid auctions, see Lebrun and Tremblay (2003).
14
See Jacquemet and L’Haridon (2018, p. 255) for the construction of our test statistics.
15
It is plausible to assume independence among subjects and periods since the values are independent and identically distributed draws from the uniform distribution, and we employed random
matching. On the other hand, two bids in a given period and subject can depend on each other.

12

13

1,427
1,478
245
3,150

875
655
45
1,575

552
823
200
1,575

Bid category
Over
Sincere
Under
Total

Bid category
Over
Sincere
Under
Total

Bid category
Over
Sincere
Under
Total

876
210
114
1,200

VA−VN
−17.44%24.09%−6.64%

17
252
1,456
1,725

2
35
1,688
1,725

19
287
3,144
3,450

2
11
1,787
1,800

PN
(0.11%)(0.61%)(99.28%)Pay-your-bid
PAPN(0.99%)
31 (1.72%)(14.61%)198 (11%)(84.41%) 1,571 (87.28%)
1,800

PA
(0.12%)(2.03%)(97.86%)

Pay-your-bid

Pay-your-bid
PA
PN(0.55%)
33 (0.92%)(8.32%)
209 (5.81%)(91.13%) 3,358 (93.28%)
3,600

Table 2: Bid category by treatment and unit.

VA−VN−18.54%28.50%−9.97%(c) Unit 2

VN
(73%)
(17.5%)(9.5%)

Vickrey
VA
VN(35.05%)643 (53.58%)(52.25%)285 (23.75%)(12.7%)
272 (22.67%)
1,200

VA
(55.56%)(41.59%)(2.86%)

Vickrey

VA−VN−17.99%26.30%−8.31%(b) Unit 1

Vickrey
VA
VN(45.3%)
1,519 (63.29%)(46.92%)495 (20.63%)(7.78%)
386 (16.08%)
2,400(a) Both unit

PA−PN
−0.74%
3.61%−2.87%

PA−PN
0.00%
1.42%
−1.42%

PA−PN−0.37%
2.51%−2.15%

14

xaj
saj
xpj
spj
xij
sij

xaj
saj
xpj
spj
xij
sij

xaj
saj
xpj
spj
xij
sij

VN
0.206
(6.82×10−5 )0.211(1.11×10−4 )
0.199(1.77×10−4 )

VN
0.175
(1.20×10−4 )
0.176(1.93×10−4 )0.173(3.18×10−4 )

VA
0.523
(1.58×10−4 )0.584(2.31×10−4 )0.400(4.57×10−4 )

VN
0.237
(1.51×10−4 )
0.245(2.47×10−4 )
0.224(3.86×10−4 )VickreyVA
0.416(1.54×10−4 )0.463(2.37×10−4 )0.322(4.16×10−4 )VickreyVA
0.469(7.91 × 10−5 )0.523(1.19×10−4 )0.361(2.20×10−4 )Pay-your-bid
PA
PN
0.083
0.058
(2.21×10−5 ) (1.52×10−5 )
0.079
0.046
−5
(2.51×10 ) (1.54×10−5 )
0.104
0.105
−4
(1.69×10 ) (1.25×10−4 )0.176∗∗∗

0.339∗∗∗yV•
VA−VN
0.286∗∗∗

Pay-your-bid
PA
PN
0.02
0.006
(1.14×10−5 ) (3.31×10−6 )
0.018
0.004
−5
(1.22×10 ) (2.80×10−6 )
0.033
0.013
−4
(1.16×10 ) (3.42×10−5 )Pay-your-bid
PA
PN
0.146
0.110
(7.23×10−5 ) (5.44×10−5 )
0.141
0.087
(8.35×10−5 ) (5.57×10−5 )
0.175
0.197
(5.25×10−4 ) (4.22×10−4 )(c) Unit 2

0.149∗∗∗

0.287∗∗∗yV•
VA−VN
0.241∗∗∗(b) Unit 1

0.162∗∗∗

0.312∗∗∗yV•
VA−VN
0.263∗∗∗−0.0220.054∗yP•
PA−PN
0.036∗∗∗

0.020∗

0.014∗∗∗yP•
PA−PN
0.014∗∗∗−0.0010.033∗∗∗yP•
PA−PN
0.025∗∗∗

Table 3: Frequency of sincere bidding, by treatment, quiz score and unit.

Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level.

Imperfect

PerfectAll

Data

Imperfect

PerfectAll

Data

Imperfect

Perfect

All

Data

Vickrey(a) Both unit

0.198∗∗∗

0.285∗∗∗

0.250∗∗∗

z•0.129∗∗∗

0.273∗∗∗

0.227∗∗∗

z•0.163∗∗∗

0.279∗∗∗

0.238∗∗∗z•(ii) The advice is eﬀective in the Vickrey auction even when focusing on the data of
the subjects with perfect quiz scores only or of those with imperfect quiz scores
only.
(iii) The same advice is eﬀective in the pay-your-bid auction for the subjects withperfect quiz scores rather than those with imperfect quiz scores.
¡¢
Support. Given • ∈ {V, P}, let Y•a ≡ X •A − X •N . The test statistic isY•a − E(Y•a )
a
e
p
,
Y• ≡
a
a
+ S•N
S•A
where E is the expectation operator. By the central limit theorem and reproducibilityof normal distributions, Y•a follows a standardized normal distribution. Hence, in what
follows, we evaluate probabilities by a standardized normal distribution unless noted
otherwise. The following null hypotheses state that advice does not increase sincere
bidding within an auction rule.
Null hypothesis (H01 ): E(YVa ) = 0.
Null hypothesis (H02 ): E(YPa ) = 0.
By a direct calculation,yeVa = √0.263= 21.68.
7.91 × 10−5 + 6.82 × 10−5This yields Prob(YeVa ≥ yeVa ) < 0.0001. Hence, advice signiﬁcantly increased sincere
bidding in the Vickrey auction. Moreover, the 95% conﬁdence interval of E(YVa ) is
[0.239, 0.287]. To establish (ii), we break the data into two: the subjects with perfectquiz scores or those with imperfect quiz scores. Then, we obtain
0.312= 20.58;
1.19 × 10−4 + 1.11 × 10−4
0.162
yeVi = √= 8.132,−4
2.20 × 10 + 1.77 × 10−4
yeVp = √yielding Prob(YeVp ≥ yeVp ) < 0.0001 and Prob(YeVi ≥ yeVi ) < 0.001 respectively. Hence,both E(YVp ) = 0 and E(YVi ) = 0 are rejected.
By a direct calculation,yePa = √0.025
2.21 × 10−5 + 1.52 × 10−5

15

= 29.50.

This yields Prob(YePa ≥ yePa ) < 0.0001. Hence, advice signiﬁcantly increased sincere
bidding in the pay-your-bid auction although the advice is false. Therefore, this result
supports a so-called experimenter demand eﬀect (Zizzo, 2009). Moreover, the 95%
conﬁdence interval of E(YPa ) is [0.013, 0.037]. Nevertheless, when breaking the data
into the subjects with perfect quiz scores and those with imperfect quiz scores,
yePp = √yePi = √0.033
2.51 × 10−5 + 1.54 × 10−5

= 5.186;−0.001= −0.058,
1.69 × 10−4 + 1.25 × 10−4

yielding Prob(YePp ≥ yePp ) < 0.0001 and Prob(YePi ≥ yePi ) = 0.523 respectively. Hence,
E(YPp ) = 0 is rejected, while E(YPi ) = 0 is accepted. Thus, we establish (iii).
3.2.2

Net eﬀect of advice

Result 2 reports the advice eﬀect on the pay-your-bid auction. The eﬀect should beconsidered as an experimenter demand eﬀect since the advice is false in that auction.
Thus, there is a possibility that the advice eﬀect in the Vickrey auction might also be
an experimenter demand eﬀect. However, Result 2 also reports that the advice eﬀect
in the Vickrey auction is much higher than that in the pay-your-bid auction. An excess
of eﬀect over the experimenter demand eﬀect is considered as a “net eﬀect” of advice
to promote sincere bidding in the Vickrey auction. We test whether the advice in the
Vickrey auction has a net eﬀect.
Since the advice given in both auctions is identical, the experimenter demand eﬀects
would also be similar in both auctions. Moreover, since the advice is false in the payyour-bid auction, its advice eﬀect is an all experimenter demand eﬀect. Thus, we
assume that the net eﬀect of advice in the Vickrey auction can be measured by thediﬀerence in advice eﬀects on the two auctions. We test the net eﬀect of advice in the
Vickrey auction under this assumption.
Result 3. The advice increases sincere bidding much more in the Vickrey auction
compared to the pay-your-bid auction, with the mean increase of 23.8% (95% conﬁdence
interval, 21.1% to 26.5%).
Support. We follow a diﬀerence-in-diﬀerence type argument. Let Z a = (YVa − YPa ). The
test statistic is
Z a − E(Z a )Zea ≡ p a.
aaa
SVA + SVN
+ SPA
+ SPN

16

We formulate the null hypothesis as follows:
Null hypothesis (H03 ): E(Z a ) = 0.
By a direct calculation,zea = √

0.263 − 0.025= 17.52.
7.91 × 10−5 + 6.82 × 10−5 + 2.21 × 10−5 + 1.52 × 10−5This yields Prob(Zea ≥ zea ) < 0.0001. Moreover, the 95% conﬁdence interval of E(Z a )is [0.211, 0.265].
The diﬀerence, 21.1%, of the advice eﬀects in the two auctions is considered as a net
eﬀect of advice on the Vickrey auction. Many authors investigate why subjects do not
bid sincerely in the Vickrey auction by proposing alternative explanations. Result 3
suggests that a considerable amount of such behavior might be mitigated by providing
advice that helps to understand the strategy-proofness property of the Vickrey auction.
3.2.3

Relationship between quiz scores and advice eﬀects

We ﬁnally test whether quiz scores aﬀect the responsiveness to advice in the Vickrey
auction. The following result demonstrates that the responsiveness to advice depends
on the quiz scores in the Vickrey auction.
Result 4. Within the Vickrey auction, the subjects with perfect quiz scores are more
responsive to advice than the subjects with imperfect quiz scores. The mean diﬀerence
between the two groups in terms of an increase in the sincere bidding rate is 15.0%(95% conﬁdence interval, 10.1% to 20.0%).
¡¢
Support. Given • ∈ {p, i}, let YV• = X VA − X VN and let WV = (YVp − YVi ). The test
statistic is
WV − E(WV )fV ≡ pW.
ppii
SVA + SVN
+ SVA
+ SVN
We formulate the null hypothesis as follows:
Null hypothesis (H04 ): E(WV ) = 0.
By a direct calculation,w
eV = √

0.312 − 0.162
1.19 × 10−4 + 1.11 × 10−4 + 2.20 × 10−4 + 1.77 × 10−4

17

= 5.992.

fV ≥ w
This yields Prob(WeVa ) < 0.0001. Moreover, the 95% conﬁdence interval of E(WV )is [0.101, 0.200].
Result 4 suggests that subjects who well understand the rule of the Vickrey auction
tend to bid sincerely under the advice than others who do not so.
3.2.4

Robustness

We comment on the robustness of our results regarding bidding behaviors.
1. When we restrict attention to unit-by-unit data in Table 3, almost all results
carry over: H01 , H02 , and H03 are again rejected at the 1% signiﬁcance level except
for the cases including imperfectly scored subjects in the pay-your-bid auction.
Additionally, H04 is rejected at the 1% signiﬁcance level for both units.
2. When we restrict our attention to the ﬁrst period only and use both units’ data,
as shown in Table 5 in Appendix A, H01 , H02 , and H03 are again rejected at the
10% signiﬁcance level except for the cases including imperfectly scored subjects
in the pay-your-bid auction where the sample size is smaller than 30 while H04 is
accepted.
3. We also examined whether the subjects’ bidding behaviors change over time. All
observations stated in this subsection held even when we focus on the former
periods (periods 1–13) or the latter periods (periods 14–25). See Table 6 inAppendix A for more details.

3.3

Eﬃciency

Our eﬃciency measure is due to Kagel and Levine (2009).16 In one game, if i is thewinner with the highest bid and if j is the winner with the second-highest bid, thenvi1 + vj1
v 1 + vi2
the observed eﬃciency ratio r is given by
if i ̸= j and by i
if i = j,v[1] + v[2]v[1] + v[2]where v[1], v[2] denote the two highest units among six valuations (v11 , v12 , v21 , v22 , v31 , v32 ).
The eﬃciency ratio of the full or part of a treatment is the average of the eﬃciency
ratios in the games in the full treatment or part of the treatment.
As in Section 3.2, we again normally approximated to examine the eﬀect of auction
rules, advice, and experience on eﬃciency. Suppose the eﬃciency ratio r in treatment

16

See also Kwasnica and Sherstyuk (2013).

18

j ∈ {VA, VN, PA, PN} is a realization of a random variable Rj with mean E(Rj ) ∈ [0, 1]
and variance Var(Rj ). Let Rj denote the sample mean for nj′ realizations of Rj . Let
Pnj
Vj ≡

2
k=1 (Rjk − Rj )n′j − 1and Tj ≡Vj.
n′jSuperscripts a, f , and ℓ correspond to all periods, the former periods (periods 1–13),
and the latter periods (periods 14–25), respectively. Table 4 summarizes the results oneﬃciency with a focus on experience. In Table 4, the upper (resp., middle, bottom)
panel collects the results obtained for all periods (resp., periods 1–13, periods 14–25).
Result 5. Advice improves eﬃciency in the Vickrey auction, particularly in the former
13 periods.
³ •´•
Support. Given • ∈ {a, f, ℓ}, let UV• = RVA − RVN . The test statistic is
UV• − E(UV• )eV• ≡ p.
U•
TVN
+ T •VN
The following hypothesis states that advice does not increase eﬃciency within the
Vickrey auction.
Null hypothesis (H05 ): E(UVa ) = 0.
By a direct calculation,ueaV = √

0.010
8.95 × 10−6 + 1.62 × 10−5

= 2.054.

ea ≥ uThis yields Prob(UeaV ) = 0.020. Therefore, advice signiﬁcantly increased eﬃciency
Vin the Vickrey auction.
To obtain further insights, we break the data into two: periods 1–13 and 14–25.
Then, we obtain
0.015= 1.769;
2.32 × 10−5 + 4.59 × 10−50.006ueℓV = √
= 1.066,
1.13 × 10−5 + 1.53 × 10−5uefV = √

ef ≥ ueℓ ≥ uyielding Prob(UefV ) = 0.038 and Prob(UeℓV ) = 0.143, respectively. Hence,
V
V
E(UVf ) = 0 is rejected, while E(UVℓ ) = 0 is not rejected.

19

20

raj
taj
rfj
tfj
rℓj
tℓj

VA
0.977
(8.95×10−6 )0.970(2.32×10−5 )
0.984(1.13×10−5 )

VN
0.967
(1.62×10−5 )
0.956(4.59×10−5 )
0.978(1.53×10−5 )0.006

0.015∗∗u•V
VA−VN
0.010∗∗
Pay-your-bid
PA
PN
0.990
0.988
(1.36×10−6 ) (1.35×10−6 )
0.991
0.989
−6
(2.27×10 ) (2.60×10−6 )
0.989
0.988
(3.26×10−6 ) (2.75×10−6 )Table 4: Eﬃciency by treatment and period.

Notes : ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level.

Periods 14–25

Periods 1–13

All

Data

Vickrey

0.001

0.002

u•P
PA−PN
0.002

0.004

0.013∗

0.009∗q•Result 5 suggests that improvements in the eﬃciency of the Vickrey auction occurs
mainly when subjects are less experienced.
Result 6. The relative impact of the advice of strategy-proofness on the eﬃciency of
the Vickrey auction compared with the pay-your-bid auction exists, particularly in early
periods.
Support. We again apply the diﬀerence-in-diﬀerence technique analogously to Result
4. Let Qa = (UVa − UPa ). The test statistic is
Qa − E(Qa )ea ≡ p
.
Q
a
a
a
a
+ TVN
+ TPA
+ TPN
TVA
We formulate the null hypothesis as follows:
Null hypothesis (H06 ): E(Qa ) = 0.
By a direct calculation,
qea = √

0.009
8.95 × 10−6 + 1.62 × 10−5 + 1.36 × 10−6 + 1.35 × 10−6

= 1.629.

ea ≥ qea ) = 0.052.
This yields Prob(QAgain, we break the data into two: periods 1–13 and 14–25. Then, we obtain
0.013= 1.454;
2.32 × 10−5 + 4.59 × 10−5 + 2.27 × 10−6 + 2.60 × 10−60.0044qeℓ = √
= 0.770,
1.13 × 10−5 + 1.53 × 10−5 + 3.26 × 10−6 + 2.75 × 10−6

qef = √

ef ≥ qef ) = 0.073 and Prob(Qeℓ ≥ qeℓ ) = 0.221, respectively. Hence,yielding Prob(QE(Qf ) = 0 is rejected at the 10% signiﬁcance level while E(Qℓ ) = 0 is not rejected.
Result 6 suggests that a relative improvement in the eﬃciency of the Vickrey auction
compared to the pay-your-bid auction occurs mainly when subjects are less experienced.
Result 6 and Table 4 show that the pay-your-bid auction is not necessarily superior
to the Vickrey auction with our advice.17 In Table 7 in Appendix A, we also tested
the same hypotheses using an alternative way of deﬁning eﬃciency where we care
only whether eﬃcient allocation is realized in a group observation. Then, we found a
17

Englemann and Grimm (2009) compare the Vickrey auction and the pay-your-bid auction in anexperiment where two units of an item are auctioned to two bidders with ﬂat demand. They ﬁnd
observations similar to ours regarding the comparison of the eﬃciency ratios in two auctions.

21

signiﬁcant increase in the eﬃciency of the Vickrey auction but no signiﬁcance in thediﬀerence-in-diﬀerence test.

4

Conclusion

We experimentally showed that introducing advice on strategy-proofness leads to ahigher sincere bidding rate in a multi-unit static Vickrey auction. Without this advice,
the rate of sincere bidding in our experiment is similar to rates observed in previous
studies. The same advice has eﬀects on the Vickrey auction but also on the payyour-bid auction. The latter eﬀect is considered as a so-called experimenter demand
eﬀect (Zizzo, 2009). To test whether the advice has an excess eﬀect over the experimenter demand eﬀect in the Vickrey auction, we compared the eﬀects of the advice
on the two auctions and found that the advice has a large excess eﬀect in the Vickrey
auction.
Since overbidding in the Vickrey auction has been widely observed in the literature(Kagel and Levin, 1993; Garratt et al., 2012; Engelmann and Grimm, 2009; Porter andVragov, 2006), researchers have attempted to identify factors that drive overbidding,such as the anticipation of regret, the joy of winning (Cooper and Fang, 2008), andcognitive limits on contingent reasoning (Li, 2017). Our results suggest that one of the
key drivers of overbidding is the subjects’ failure to be aware of the strategy-proofness
property of the Vickrey auction, even for those who perfectly understand the rule of
the Vickrey auction.
Note that our statement of advice is quite simple so that the same advice canbe used in the two auctions. We emphasize that even such a simple advice has aconsiderable excess eﬀect in the Vickrey auction. Much detailed advice would makethe subjects bid more sincerely.
Our results show that simple advice improves eﬃciency in the Vickrey auction, andthis improvement is higher in early periods in which subjects are less experienced. Thus,
our results show that advice improves eﬃciency in the Vickrey auction by promoting
better understanding of the strategy-proofness property in the Vickrey auction.
To focus on the advice eﬀect on bidding behavior, we conducted experiments ina “symmetric” environment; that is, an environment where each bidder in the sameauction has the same distribution function of valuations. In the pay-your-bid auction insuch an environment, there is a symmetric equilibrium bidding strategy whose outcomesare eﬃcient. As discussed in Section 3.3, the pay-your-bid auction as well as the Vickrey
auction generated almost perfectly eﬃcient outcomes. Thus, even without advice, there22

is little room to improve eﬃciency in both auctions. Thus, in our experiment, the eﬀects
of advice on eﬃciency are limited in both auctions. We conjecture that experiments inan “asymmetric” environment would make the advice eﬀect on eﬃciency clearer.

23

AppendicesA

Appendix: Additional analysis• Table 5 summarizes the sincere bidding rates in the ﬁrst period only in the samemanner as Table 3.
• Table 6 summarizes the sincere bidding rates in the former 13 periods and thelatter 12 period.
• Table 7 summarizes the frequencies of realization of eﬃcient allocation by treatment. In what follows, we apply normal approximation as we do in Section 3.2.
Suppose achievement of eﬃcient allocation in treatment j ∈ {VA, VN, PA, PN}
is a realization of a random variable Ej with the Bernoulli distribution that takes
1 (= eﬃcient) with the success rate of qj .18 That is,1 with prob. q ,jEj =0 with prob. 1 − q .
j

Let E j denote the sample mean for nj′ realizations of Ej and let
Fj =E j (1 − E j )
.
n′j

18

It is plausible to assume independence among subjects and periods since the values are independent and identically distributed draws from the uniform distribution, and we employed random
matching. On the other hand, two bids in a given period and subject can depend on each other.

24

25
VN
0.208
(3.43 × 10−3 )0.267(6.52 × 10−3 )0.111(5.48 × 10−3 )Vickrey0.318∗∗∗

0.281∗∗∗

VA−VN
0.300∗∗∗

yV•

PN
0.042
(5.59 × 10−4 )0.053(8.81 × 10−4 )0(0)b)

Pay-your-bid
PA
0.203(2.34 × 10−3 )0.224(3.00 × 10−3 )0.091(7.52 × 10−3 )b)(c) Unit 2

0.222∗∗

0.276∗∗∗

00.227∗

0.11

0.139∗

0.222∗∗

z•0.225∗∗

0.167∗∗

0.224∗∗

0.171∗∗∗
0.091

z•

z•
0.178∗∗∗

0.219∗∗∗yP•
PA−PN
0.161∗∗∗

0.052∗∗yP•
PA−PN
0.043∗∗

0.045

0.112∗∗∗yP•
PA−PN
0.102∗∗∗

Table 5: Frequency of sincere bidding in period one, by treatment, quiz score and unit.

Notes : a) ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level and b) The number of observations
is no more than 30.

xaj
saj
xpj
spj
xij
sij

xaj
saj
xpj
spj
xij
sij

yV•
VA−VN
0.262∗∗∗
Pay-your-bid
PA
PN
0.043
0
−4
(5.96 × 10 ) (0)
0.052
0
−4
(8.50 × 10 ) (0)
0b)
0b)
(0)
(0)

Pay-your-bid
PA
PN
0.123
0.021
(7.82 × 10−4 ) (1.43 × 10−4 )
0.138
0.026
−3
(1.03 × 10 ) (2.22 × 10−4 )0.045b)0−3(1.95 × 10 )(0)(b) Unit 1

0.270∗∗∗

0.279∗∗∗yV•
VA−VN
0.280∗∗∗

VN
0.167(2.90 × 10−3 )0.2(5.33 × 10−3 )0.111(5.48 × 10−3 )VickreyVN
0.188(1.59 × 10−3 )0.233(2.98 × 10−3 )0.111(2.74 × 10−3 )

VA
0.429
(3.89 × 10−3 )0.476(5.94×10−3 )0.333(1.06 × 10−2 )

VA
0.468
(1.98 × 10−3 )0.512(2.97 × 10−3 )
0.381(5.62 × 10−3 )

VA
0.508
(3.97 × 10−3 )0.548(5.90 × 10−3 )0.429(1.17 × 10−2 )Imperfect

Imperfect

Perfect

All

Data

xaj
saj
xpj
spj
xij
sij

PerfectAll

Data

Imperfect

Perfect

All

Data

Vickrey(a) Both unit

26

xaj
saj
xpj
spj
xij
sij

xaj
saj
xpj
spj
xij
sij

VA
0.472
(1.65 × 10−4 )
0.537(2.47 × 10−4 )
0.341(4.47 ×10−4 )0.187∗∗∗

0.329∗∗∗

Pay-your-bid
PA
PN
0.093
0.058
(4.68 × 10−5 ) (2.93 × 10−5 )
0.088
0.045
(5.34 ×10−5 ) (2.92 × 10−5 )
0.115
0.108
(3.58 ×10−4 ) (2.47 ×10−4 )

VN
0.228
(1.53 × 10−4 )
0.242(2.55 × 10−4 )0.206(3.80 × 10−4 )0.135∗∗∗

0.295∗∗∗yV•
VA−VN
0.243∗∗∗

Pay-your-bid
PA
PN
0.073
0.058
(4.09×10−5 )(3.16×10−5 )
0.070
0.046
−5
(4.66 × 10 ) (3.21 × 10−5 )
0.090
0.103
−4
(3.14 × 10 ) (2.57×10−4 )(b) Latter 12 periods

VN
0.186
(1.21 × 10−4 )0.182(1.91 × 10−4 )0.192(3.33×10−4 )VickreyVA
0.467(1.52 × 10−4 )0.510(2.29 × 10−4 )0.379(4.32 ×10−4 )

yV•
VA−VN
0.281∗∗∗−0.0120.024∗∗∗yP•
PA−PN
0.015∗∗

0.008

0.043∗∗∗yP•
PA−PN
0.034∗∗∗

Table 6: Frequency of sincere bidding in the former 13 periods and the latter 12 periods.

Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level.

Imperfect

PerfectAll

Data

Imperfect

Perfect

All

Data

Vickrey(a) Former 13 periods

0.147∗∗∗

0.271∗∗∗

0.228∗∗∗

z•0.179∗∗∗

0.286∗∗∗

0.246∗∗∗

z•

27

ej
fj
ej
fj
ej
fj

VA
0.779
(3.28×10−4 )0.744(6.98×10−4 )
0.817(5.92×10−4 )

VN
0.705
(5.20×10−4 )0.644(1.10×10−3 )0.771(9.20×10−4 )0.047

0.099∗∗∗(1)
VA−VN
0.074∗∗∗

Pay-your-bid
PA
PN
0.823
0.783
(2.54×10−4 ) (2.83×10−4 )
0.823
0.782
−4
(4.88×10 ) (5.46×10−4 )
0.822
0.785
(5.29×10−4 ) (5.87×10−4 )Table 7: Binary eﬃciency by treatment and period.

Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level.

Periods 14–25

Periods 1–13

All

Data

Vickrey

0.038

0.041

(2)
PA−PN
0.039∗

0.009

0.059

0.035

(1)−(2)

B

Appendix: Individual-oriented analysis

B.1

VA1

Sixteen subjects (ID numbers 1–3, 6, 7, 9–16, 18, 19, 21) received a perfect score on
the quiz.
Unit 1• Sincere bidding. Four subjects (ID numbers 3, 11, 20, 21) bid sincerely in all
periods. Two subjects (ID numbers 1, 10) bid sincerely in most periods (thefrequency that each of these subjects bid sincerely was at least 88% (= 22/25)).
• Overbidding. Three subjects (ID numbers 12, 15, 17) overbid in all periods. Six
subjects (ID numbers 4, 5, 6, 8, 13, 18) overbid in most periods (the frequency
that each of these subjects overbid was at least 80% (= 20/25)).
• Other patterns.
◦ One subject (ID number 2) overbid until period 11 and bid sincerely in mostof the remaining periods.
◦ One subject (ID number 7) bid any one of 0, 1,000, and 2,000.
◦ Four subjects (ID numbers 9, 14, 16, 19) ﬂuctuated between sincere bidding
and overbidding in most or all periods.
Unit 2• Sincere bidding. Four subjects (ID numbers 10, 11, 20, 21) bid sincerely inall periods. Two subjects (ID numbers 3, 19) bid sincerely in most periods (thefrequency that each of these subjects bid sincerely was at least 92% (= 23/25)).
• Overbidding. One subject (ID number 12) overbid in all periods. Five subjects(ID numbers 1, 4, 5, 15, 17) overbid in most periods (the frequency that each ofthese subjects overbid was at least 76% (= 19/25)).
• Underbidding. One subject (ID number 7) bid 0 in all but one period.
• Other patterns.
◦ One subject (ID number 2) overbid until period 11 and bid sincerely in theremaining periods.

28◦ Three subjects (ID numbers 6, 8, 9) ﬂuctuated between sincere bidding andoverbidding in most or all periods.
◦ One subject (ID number 13) ﬂuctuated between underbidding and overbidding in most periods.
◦ One subject (ID number 14) ﬂuctuated between sincere bidding and underbidding in most periods.
◦ One subject (ID number 16) ﬂuctuated between underbidding and overbidding until period 7, switched to sincere bidding for 12 periods, and ﬂuctuatedbetween sincere bidding and overbidding in the last 6 periods.
◦ One subject (ID number 18) ﬂuctuated between sincere bidding and underbidding in most of the ﬁrst 8 periods, switched to overbidding for 8 periods,
and sincere bidding for the next ﬁve periods, then chose underbidding in thelast 4 periods.

29

Figure 4: Time evolution of the gap between bid and value for unit 1 in VA1.

30

Figure 5: Time evolution of the gap between bid and value for unit 2 in VA1.

31

B.2

VA2

Thirteen subjects (ID numbers 2–4, 7, 8, 11, 13, 15–20) received a perfect score on thequiz.
Unit 1• Sincere bidding. Three subjects (ID numbers 8, 10, 15) bid sincerely in all
periods.
• Overbidding. One subject (ID number 7) overbid in all periods. Eleven subjects(ID numbers 1–4, 9, 12, 14, 16–18, 20) overbid in most periods (the frequency
that each of these subjects overbid was at least 76% (= 19/25)).
• Other patterns.
◦ One subject (ID number 5) ﬂuctuated among overbidding, underbidding,and sincere bidding.
◦ One subject (ID number 6) ﬂuctuated between sincere bidding and overbidding in the former 13 periods and bid sincerely in most of the latter 12
periods.
◦ Four subjects (ID numbers 11, 13, 19, 21) ﬂuctuated between sincere bidding
and overbidding in most or all periods.
Unit 2• Sincere bidding. Three subjects (ID numbers 8, 10, 15) bid sincerely in all
periods. Three subjects (ID numbers 6, 11, 13) bid sincerely in most periods (thefrequency that each of these subjects bid sincerely was at least 76% (= 19/25)).
• Overbidding. Six subjects (ID numbers 2, 4, 7, 12, 18, 20) overbid in mostperiods (the frequency that each of these subjects overbid was at least 76% (=19/25)).
• Underbidding. Two subjects (ID numbers 5, 21) underbid in most periods (thefrequency that each of these subjects underbid was at least 76% (= 19/25)).
• Other patterns.
◦ Two subjects (ID numbers 1, 16) ﬂuctuated between underbidding and overbidding in most periods.

32◦ One subject (ID number 3) underbid in the ﬁrst 4 periods and overbid inmost of the remaining periods.
◦ Two subjects (ID numbers 9, 17) ﬂuctuated among overbidding, underbidding, and sincere bidding.
◦ One subject (ID number 14) basically ﬂuctuated between underbidding andoverbidding until period 16 and underbid in the remaining periods.
◦ One subject (ID number 19) ﬂuctuated between sincere bidding and overbidding in most periods.

33

Figure 6: Time evolution of the gap between bid and value for unit 1 in VA2.

34

Figure 7: Time evolution of the gap between bid and value for unit 2 in VA2.

35

B.3

VA3

Thirteen subjects (ID numbers 2, 4–6, 10–12, 14–16, 18, 19, 21) received a perfect scoreon the quiz.
Unit 1• Sincere bidding. Four subjects (ID numbers 4, 13, 15, 18) bid sincerely inall periods. Five subjects (ID numbers 6, 8, 11, 14, 16) bid sincerely in mostperiods (the frequency that each of these subjects bid sincerely was at least 76%(= 19/25)).
• Overbidding. One subject (ID number 17) overbid in all periods. Seven subjects(ID numbers 2, 3, 7, 9, 12, 19, 21) overbid in most periods (the frequency thateach of these subjects overbid was at least 76% (= 23/25)).
• Other patterns.
◦ One subject (ID number 1) overbid in most of the former 13 periods andﬂuctuated between sincere bidding and overbidding in the latter 12 periods.
◦ Two subjects (ID numbers 5, 10) ﬂuctuated between sincere bidding andoverbidding.
◦ One subject (ID number 20) bid sincerely in most of the former 13 periods
and ﬂuctuated between sincere bidding and overbidding in the latter 12
periods.
Unit 2• Sincere bidding. Seven subjects (ID numbers 4, 6, 10, 13, 15, 16, 18) bid
sincerely in all periods. Six subjects (ID numbers 5, 7, 8, 11, 12, 14) bid sincerely
in most periods (the frequency that each of these subjects bid sincerely was atleast 72% (= 18/25)).
• Overbidding. Three subjects (ID numbers 2, 3, 9) overbid in most periods (thefrequency that each of these subjects overbid was at least 80% (= 20/25)).
• Other patterns.
◦ One subject (ID number 1) ﬂuctuated among overbidding, underbidding,and sincere bidding.

36◦ One subject (ID number 17) overbid until period 12 and ﬂuctuated among
overbidding, underbidding, and sincere bidding in the remaining periods.
◦ One subject (ID number 19) bid sincerely until period 7 and ﬂuctuatedbetween sincere bidding and overbidding in the remaining periods.
◦ Two subjects (ID numbers 20, 21) ﬂuctuated between sincere bidding andoverbidding in most periods.

37

Figure 8: Time evolution of the gap between bid and value for unit 1 in VA3.

38

Figure 9: Time evolution of the gap between bid and value for unit 2 in VA3.

39

B.4

VN1

Sixteen subjects (ID numbers 1, 2, 4, 5, 8, 9, 11–15, 17–20, 24) received a perfect scoreon the quiz.
Unit 1• Sincere bidding. One subject (ID number 18) bid sincerely in all periods.
• Overbidding. Six subjects (ID numbers 3, 9, 12, 13, 15, 20) overbid in all
periods. Nine subjects (ID numbers 1, 4, 14, 16, 17, 19, 21, 22, 24) overbid inmost periods (the frequency that each of these subjects overbid was at least 76%(= 19/25)).
• Underbidding. One subject (ID number 10) underbid in most periods (thefrequency that the subject underbid was 72% (= 18/25)).
• Other patterns.
◦ One subject (ID number 2) overbid until period 12 and ﬂuctuated between
sincere bidding and overbidding in the remaining periods.
◦ One subject (ID number 5) overbid in most of the former 13 periods andﬂuctuated between sincere bidding and overbidding in the latter 12 periods.
◦ One subject (ID number 6) bid either 0 or 1,000 in the former 13 periods
and bid sincerely in most of the latter 12 periods.
◦ One subject (ID number 7) bid 1,000 in most of the former 13 periods andbid either 0 or 1,000 in the latter 12 periods.
◦ One subject (ID number 8) ﬂuctuated between underbidding and overbidding in most periods.
◦ One subject (ID number 11) ﬂuctuated between sincere bidding and underbidding in most periods.
◦ One subject (ID number 23) ﬂuctuated between sincere bidding and overbidding in most periods.
Unit 2• Sincere bidding. Two subjects (ID numbers 15, 18) bid sincerely in all periods.
One subject (ID number 23) bid sincerely in most periods (the frequency thatthe subject bid sincerely was 80% (= 20/25)).

40

• Overbidding. One subject (ID number 3) overbid in all periods. Six subjects(ID numbers 9, 12, 13, 17, 19, 22) overbid in most periods (the frequency thateach of these subjects overbid was at least 72% (= 18/25)).
• Underbidding. Three subjects (ID numbers 1, 6, 20) underbid in most periods(the frequency that each of these subjects underbid was at least 76% (= 19/25)).
• Other patterns.
◦ Two subjects (ID numbers 2, 21) ﬂuctuated between sincere bidding andoverbidding in most or all periods.
◦ One subject (ID number 4) overbid in most of the former 13 periods andbid sincerely in most of latter 12 periods.
◦ One subject (ID number 5) ﬂuctuated between sincere bidding and overbidding in most of the ﬁrst 14 periods and bid sincerely in the remaining
periods.
◦ Two subjects (ID numbers 7, 16) ﬂuctuated between underbidding and overbidding in most or all periods.
◦ One subject (ID number 8) ﬂuctuated among overbidding, underbidding,and sincere bidding.
◦ Two subjects (ID numbers 10, 24) ﬂuctuated between overbidding and underbidding in the former 13 periods and overbid in most of the latter 12
periods.
◦ One subject (ID number 11) ﬂuctuated between sincere bidding and underbidding.
◦ One subject (ID number 14) ﬂuctuated between overbidding and underbidding in the former 13 periods and underbid in most of the latter 12 periods.

41

Figure 10: Time evolution of the gap between bid and value for unit 1 in VN1.

42

Figure 11: Time evolution of the gap between bid and value for unit 2 in VN1.

43

B.5

VN2

Fourteen subjects (ID numbers 2–6, 8, 11, 12, 14, 15, 18–21) received a perfect scoreon the quiz.
Unit 1• Sincere bidding. Two subjects (ID numbers 1, 5) bid sincerely in all periods.
One subject (ID number 15) bid sincerely in most periods (the frequency thatthe subject bid sincerely was 72% (= 18/25)).
• Overbidding. Six subjects (ID numbers 2, 4, 6, 9, 21, 22) overbid in all periods.
Ten subjects (ID numbers 3, 7, 10, 11, 12, 14, 16, 19, 20, 23) overbid in mostperiods (the frequency that each of these subjects overbid was at least 72% (=18/25)).
• Other patterns.
◦ One subject (ID number 8) ﬂuctuated between overbidding and underbidding.
◦ One subject (ID number 13) ﬂuctuated between sincere bidding and overbidding until period 19 and bid sincerely in the remaining periods.
◦ One subject (ID number 17) ﬂuctuated between sincere bidding and overbidding.
◦ One subject (ID number 18) ﬂuctuated among overbidding, underbidding,and sincere bidding.
◦ One subject (ID number 24) underbid until period 15 and overbid in theremaining periods.
Unit 2• Sincere bidding. Two subjects (ID numbers 1, 5) bid sincerely in all periods.
• Overbidding. Three subjects (ID numbers 4, 6, 9) chose overbidding in all
periods. Nine subjects (ID numbers 2, 7, 11, 12, 14, 19, 21, 22, 24) overbid inmost periods (the frequency that each of these subjects overbid was at least 72%(= 18/25)).

44

• Underbidding. One subject (ID number 8) underbid in most periods (thefrequency that the subject underbid was 80% (= 20/25)).
• Other patterns.
◦ One subject (ID number 3) ﬂuctuated between overbidding and underbidding in the former 13 periods and overbid in most of the latter 12 periods.
◦ Two subjects (ID numbers 10, 18) ﬂuctuated between overbidding and underbidding in most periods.
◦ One subject (ID number 13) ﬂuctuated between sincere bidding and overbidding until period 18 and bid sincerely in the remaining periods.
◦ Two subjects (ID numbers 15, 17) ﬂuctuated between sincere bidding andunderbidding.
◦ One subject (ID number 16) overbid until period 11 and ﬂuctuated among
overbidding, underbidding, and sincere bidding in the remaining periods.
◦ One subject (ID number 20) ﬂuctuated between sincere bidding and overbidding in most periods.
◦ One subject (ID number 23) underbid in most of the former 13 periods andoverbid in most of the latter 12 periods.

45

Figure 12: Time evolution of the gap between bid and value for unit 1 in VN2.

46

Figure 13: Time evolution of the gap between bid and value for unit 2 in VN2.

47

B.6

PA1

Twenty subjects (ID numbers 1, 3–21) received a perfect score on the quiz.
Unit 1• Underbidding. Fifteen subjects (ID numbers 2, 4, 6, 8, 10–20) underbid in all
periods. Six subjects (ID numbers 1, 3, 5, 7, 9, 21) underbid in most periods (thefrequency that each of these subjects underbid was at least 92% (= 23/25)).
Unit 2• Underbidding. Seven subjects (ID numbers 2, 4, 6, 8, 14, 15, 17) underbid in
all periods. Eight subjects (ID numbers 5, 7, 10, 11, 13, 19–21) underbid in mostperiods (the frequency that each of these subjects underbid was at least 76% (=19/25)). Two subjects (ID numbers 12, 18) bid 0 whenever his/her value for thesecond unit was 0 and underbid in other periods.
• Other patterns.
◦ One subject (ID number 1) ﬂuctuated between underbidding and sincere
bidding in most periods.
◦ One subject (ID number 3) ﬂuctuated between underbidding and sincere
bidding in most of the former 13 periods and underbid in most of the latter
12 periods.
◦ One subject (ID number 9) underbid in most of the former 13 periods andﬂuctuated between underbidding and sincere bidding in the latter 12 periods.
◦ One subject (ID number 16) bid sincerely until period 6 and underbid inmost of the remaining periods.

48

Figure 14: Time evolution of the gap between bid and value for unit 1 in PA1.

49

Figure 15: Time evolution of the gap between bid and value for unit 2 in PA1.

50

B.7

PA2

Eighteen subjects (ID numbers 1– 8, 10, 11, 15, 16, 18–22, 24) received a perfect scoreon the quiz.
Unit 1• Underbidding. Eighteen subjects (ID numbers 2, 3, 5, 6, 8–13, 15–17, 19, 21–24) underbid in all periods. Six subjects (ID numbers 1, 4, 7, 14, 18, 20) underbid
in most periods (the frequency that each of these subjects underbid was at least
92% (= 23/25)).
Unit 2• Underbidding. Seven subjects (ID numbers 2, 3, 5, 8, 11, 15, 16) underbid in
all periods. Thirteen subjects (ID numbers 1, 4, 7, 9, 10, 12–14, 17, 19, 20, 23,24) underbid in most periods (the frequency that each of these subjects underbid
was at least 76% (= 19/25)). Two subjects (ID numbers 6, 21) bid 0 whenever
his/her value for the second unit was 0 and underbid in other periods.
• Sincere bidding. One subject (ID number 18) bid sincerely in most periods(the frequency that the subject bid sincerely was 88% (= 22/25)).
• Overbidding. One subject (ID number 22) overbid in most periods (the frequency that the subject overbid was 72% (= 18/25)).

51

Figure 16: Time evolution of the gap between bid and value for unit 1 in PA2.

52

Figure 17: Time evolution of the gap between bid and value for unit 2 in PA2.

53

B.8

PA3

Twenty-one subjects (ID numbers 1–12, 14–17, 19–22, 24) received a perfect score on
the quiz.
Unit 1• Underbidding. Sixteen subjects (ID numbers 1, 4, 5, 7, 9–13, 15–18, 20–22)
underbid in all periods. Seven subjects (ID numbers 2, 3, 6, 8, 14, 19, 23)
underbid in most periods (the frequency that each of these subjects underbid
was at least 80% (= 20/25)).
• Other pattern. One subject (ID number 24) ﬂuctuated between sincere bidding
and underbidding.
Unit 2• Underbidding. Three subjects (ID numbers 1, 7, 17) underbid in all periods.
Nine subjects (ID numbers 2–4, 6, 9, 10, 12, 14, 20) underbid in most periods(the frequency that each of these subjects underbid was at least 72% (= 18/25)).
Six subjects (ID numbers 5, 15, 18, 19, 21, 22) bid 0 whenever his/her value forthe second unit was 0 and underbid in other periods.
• Other patterns.
◦ One subject (ID number 8) ﬂuctuated between sincere bidding and underbidding in most periods.
◦ One subject (ID number 11) ﬂuctuated between sincere bidding and underbidding in the former 13 periods and underbid in most of the latter 12
periods.
◦ Four subjects (ID numbers 13, 16, 23, 24) ﬂuctuated between sincere bidding
and underbidding.

54

Figure 18: Time evolution of the gap between bid and value for unit 1 in PA3.

55

Figure 19: Time evolution of the gap between bid and value for unit 2 in PA3.

56

B.9

PN1

Twenty-one subjects (ID numbers 1–4, 6–8, 10–15, 17–24) received a perfect score on
the quiz.
Unit 1• Underbidding. Twenty subjects (ID numbers 3, 5–15, 17–24) underbid in all
periods. Four subjects (ID numbers 1, 2, 4, 16) underbid in most periods (thefrequency that each of these subjects underbid was at least 84% (= 21/25)).
Unit 2• Underbidding. Twelve subjects (ID numbers 3, 5–10, 14, 17–19, 21) underbid
in all periods. Six subjects (ID numbers 4, 11, 13, 15, 20, 23) underbid in mostperiods (the frequency that each of these subjects underbid was at least 80% (=20/25)). Three subjects (ID numbers 12, 22, 24) bid 0 whenever his/her value
for the second unit was 0 and underbid in other periods.
• Other patterns.
◦ One subject (ID number 1) ﬂuctuated between overbidding and other strategies in the former 13 periods and underbid in most of the latter 12 periods.
◦ One subject (ID number 2) underbid in most of the former 13 periods andbid sincerely in most of the latter 12 periods.
◦ One subject (ID number 16) ﬂuctuated between sincere bidding and underbidding in the former 13 periods and underbid in most of the latter 12
periods.

57

Figure 20: Time evolution of the gap between bid and value for unit 1 in PN1.

58

Figure 21: Time evolution of the gap between bid and value for unit 2 in PN1.

59

B.10

PN2

Eighteen subjects (ID numbers 2, 4, 5, 7–10, 12–14, 16–20, 22–24) received a perfectscore on the quiz.
Unit 1• Underbidding. All subjects underbid in all periods.
Unit 2• Underbidding. Ten subjects (ID numbers 3, 6, 7, 9–12, 18, 19, 22) underbid
in all periods. Six subjects (ID numbers 1, 2, 4, 13, 17, 23) underbid in mostperiods (the frequency that each of these subjects underbid was at least 84% (=21/25)). Five subjects (ID numbers 5, 8, 14, 16, 20) bid 0 whenever his/her value
for the second unit was 0 and underbid in other periods.
• Other patterns.
◦ One subject (ID number 15) ﬂuctuated between sincere bidding and underbidding.
◦ One subject (ID number 21) underbid in most of the former 13 periods
and ﬂuctuated between sincere bidding and underbidding in the latter 12
periods.
◦ One subject (ID number 24) ﬂuctuated between underbidding and otherstrategies in the former 13 periods and underbid in most of the latter 12
periods.

60

Figure 22: Time evolution of the gap between bid and value for unit 1 in PN2.

61

Figure 23: Time evolution of the gap between bid and value for unit 2 in PN2.

62

B.11

PN3

Eighteen subjects (ID numbers 1, 3–5, 7–14, 16–18, 20, 22, 23) received a perfect scoreon the quiz.
Unit 1• Underbidding. Twenty subjects (ID numbers 1–7, 9, 11, 13, 14, 16–24) underbid in all periods. Four subjects (ID numbers 8, 10, 12, 15) underbid in mostperiods (the frequency that each of these subjects underbid was at least 88% (=22/25)).
Unit 2• Underbidding. Five subjects (ID numbers 4, 9, 14, 18, 19) underbid in all
periods. Nine subjects (ID numbers 1, 6–8, 13, 16, 17, 21, 24) underbid in mostperiods (the frequency that each of these subjects underbid was at least 80% (=20/25)). One subject (ID number 20) bid sincerely in the ﬁrst 6 periods and thenswitched to underbidding in the remaining periods. Four subjects (ID numbers 5,
11, 22, 23) bid 0 whenever his/her value for the second unit was 0 and underbid
in other periods.
• Other patterns.
◦ One subject (ID number 2) ﬂuctuated between sincere bidding and otherstrategies in the former 13 periods and bid sincerely in most of the latter 12
periods.
◦ Two subjects (ID numbers 3, 15) ﬂuctuated between sincere bidding andunderbidding.
◦ Two subjects (ID numbers 10, 12) ﬂuctuated between sincere bidding andunderbidding in all but one period.

63

Figure 24: Time evolution of the gap between bid and value for unit 1 in PN3.

64

Figure 25: Time evolution of the gap between bid and value for unit 2 in PN3.

65

C

Appendix: Experimental instructions (VA)

Note: In this experiment, please remember that you cannot communicate with othersubjects. If there is communication, this experiment will be stopped at that point.
First, please conﬁrm the following items. If any of the items are missing, pleasecontact the experimenter.
• Instructions (this handout)• Record Sheet• Ballpoint pen, Pencil and Eraser• Calculator

C.1

Overview

In this experiment, you will act as bidders in a sequence of auctions. Three bidders
will participate in each auction. At the beginning of each period of auction, the experimenter randomly match you with two persons from the other subjects. The threeof you will form a group. This experiment consists of 25 periods. The individuals you
are matched with will change each period. You will not know who you are matchedwith either during or after the experiment.
The rewards you receive after the experiment are determined based on the decisions
you and others make in the experiment. Your rewards will be paid to you in cash at
the end of the experiment. A detailed explanation of the rewards you will receive will
be provided later in C.3. Rewards.

C.2

Auction

C.2.1

Procedure in each period

1. Two units of an identical object will be auctioned oﬀ in every period. Threebidders, including yourself, will participate in an auction.
2. Each bidder will be assigned a value for the ﬁrst unit (“1st unit’s value”) andanother value for the second unit (“2nd unit’s value”). For each bidder, values
will be randomly drawn from the interval JPY 0 to JPY 1,000 with incrementsof 10 yen. Any value within this interval has an equally likely chance of being66

drawn and being assigned as a value. The higher of these two values will be the“1st unit’s value” while the lower will be the “2nd unit’s value.” Each of theother two bidders will be assigned values for two units in the same way that yourvalues were assigned. The particular values assigned to the other two bidders will
typically be diﬀerent from yours. Please note that for each bidder, the second
unit’s value will always be lower than the ﬁrst unit’s value. In addition, your
“1st unit’s value” and “2nd unit’s value” are only for your private information.
The other bidders will not know your values.
3. You will submit your “bid for the 1st unit” and “bid for the 2nd unit” to theexperimenter. Then, please note the following three points:• “Your bid for the 1st unit” must be higher than “your bid for the 2nd unit”.
• Your bid must exceed 0 yen.
• Your bid must be increments of 10 yen.
4. Each of the three bidders will submit two bids. Therefore, there will be a total of
six bids. The two highest bidders will each earn an item. In case of a tie among
the high bids, the experimenter will randomly determine who earns an item. This
described procedure determines the number of units earned by each bidder.
5. The “earnings” for a bidder who earns an item are equal to his/her value of theitem less his/her payment. The “earnings” for a bidder who has not earned any
units are zero.
C.2.2

Earnings calculation

We ﬁrst explain how to calculate a bidder’s earnings if the bidder earns an item using
examples. There are three bidders, A, B, and C. Here, we focus on bidder A. In thefollowing examples, the numbers are displayed in Japanese yen.
(1) The case where a bidder earns one unit A bidder who earns one unit will pay theamount of the highest bid from among the other bidders’ losing bids.
Example C.1. Suppose that A’s value for the ﬁrst unit is 680 and 480 for the second
unit. The bids from the three bidders are shown below.

67

Bidder

Bid for 1st Unit

Bid for 2nd Unit

A

600

450

B

500

300

C

400

250

Here, the winning bids are A’s 600 and B’s 500. The losing bids are A’s 450, B’s
300, C’s 400, and C’s 250. When A’s 450 is excluded from the losing bids, B’s 300,
C’s 400, and C’s 250 are left. Bidder A pays the highest amount among these, which
is 400. Please note that the payment diﬀers from the winning bid. Then, A’s earnings
are calculated as follows:
680 (1st item’s value) − 400 (payment) = 280
Example C.2. Suppose that A’s value for the ﬁrst unit is 680 and 480 for the second
unit. The bids from the three bidders are shown below.
Bidder

Bid for 1st Unit

Bid for 2nd Unit

A

800

350

B

750

300

C

700

250

Here, the winning bids are A’s 800 and B’s 750. Thus, bidder A earns one unit.
Excluding A’s other bid, the losing bids were B’s 300, C’s 700, and C’s 250. Bidder A
pays 700, the highest bid. Please note that the payment diﬀers from the winning bid.
Then, A’s earnings are calculated as follows:
680 (1st item’s value) − 700 (payment) = −20(2) The case where a bidder earns two units A bidder who earns two units will pay
the sum of the highest and second-highest bids from among the other bidders’ losing
bids.
Example C.3. Suppose that A’s value for the ﬁrst unit is 680 and 480 for the second
unit. The bids from the three bidders are shown below.

68Bidder

Bid for 1st Unit

Bid for 2nd Unit

A

600

550

B

500

300

C

400

250

Here, the winning bids are A’s 600 and A’s 550. The losing bids are B’s 500, B’s
300, C’s 400, and C’s 250. Thus, bidder A pays 900, the sum of the highest bid of 500
and the second-highest bid of 400 from the losing bids. Here too, A’s payment diﬀers
from A’s own bids. A’s earnings are calculated as follows:
680 (1st item’s value) + 480 (2nd item’s value) − 900 (payment) = 260
Example C.4. Suppose that A’s value for the ﬁrst unit is 680 and 480 for the second
unit. The bids from the three bidders are shown below.
Bidder

Bid for 1st Unit

Bid for 2nd Unit

A

900

850

B

800

600

C

700

550

Here, the winning bids are A’s 900 and A’s 850. The losing bids are B’s 800, B’s
600, C’s 700, and C’s 550. Thus, bidder A pays 1,500, the sum of the highest bid of
800 and the second-highest bid of 700 from the losing bids. Here too, A’s payment
diﬀers from A’s own bids. A’s earnings are calculated as follows:
680 (1st item’s value) + 480 (2nd item’s value) − 150 (payment) = −340
The earnings calculation method may be summarized as follows:• When you earn one unit and the third-highest bid is yours:
Earnings = 1st item’s value − 4th highest bid• When you earn one unit and the third-highest bid is not yours:
Earnings = 1st item’s value − 3rd highest bid

69When your payment is higher than your ﬁrst item’s value, please note that yourearnings will be negative.
• When you earn two units:
Earnings = (1st unit’s value + 2nd unit’s value)− (3rd highest bid + 4th highest bid)
When your payment is higher than the sum of your ﬁrst and second units’ values,
please note that your earnings will be negative.
• When you earn an item, your earnings will be zero.

C.3

Rewards

We will explain the rewards you receive after the experiment. Your rewards are thesum of your earnings over all 25 periods plus a participation fee of 1,000 yen.
For example, if the sum of your earnings is 2,580 yen, your rewards will be 3,580
yen. In other words, the more you earn from each period, the higher your rewards will
be.

70

71

12

11

10

9

8

7

6

5

4

3

2

1

Round

Value for1st item

Value for2nd item

Bid for
1st item

Bid for
2nd item

Payment(1st item)

Recording Sheet
Payment(2nd item)

Payment(Total)Seat Number

Earnings

72

24

23

22

21

20

19

18

17

16

15

14

13

Round

Value for1st item

Value for2nd item

Bid for
1st item

Bid for
2nd item

Payment(1st item)

Recording Sheet
Payment(2nd item)

Payment(Total)Seat Number

Earnings

D

Appendix: Experimental instructions (PA)

Note: In this experiment, please remember that you cannot communicate with othersubjects. If there is communication, this experiment will be stopped at that point.
First, please conﬁrm the following items. If any of the items are missing, pleasecontact the experimenter.
• Instructions (this handout)• Record Sheet• Ballpoint pen, Pencil and Eraser• Calculator

D.1

Overview

In this experiment, you will act as bidders in a sequence of auctions. Three bidders
will participate in each auction. At the beginning of each period of auction, the experimenter will randomly match you with two persons from the other subjects. The threeof you will form a group. This experiment consists of 25 periods. The persons you arematched with will change each period. You will not know who you are matched witheither during or after the experiment.
The rewards you receive after the experiment are determined based on the decisions
you and others make in the experiment. Your rewards will be paid to you in cash at
the end of the experiment. A detailed explanation of the rewards you receive will be
provided later in D.3. Rewards.

D.2

Auction

D.2.1

Procedure in each period

1. Two units of an identical object will be auctioned oﬀ in every period. Threebidders, including yourself, will participate in an auction.
2. Each bidder will be assigned a value for the ﬁrst unit (“1st unit’s value”) andanother value for the second unit (“2nd unit’s value”). For each bidder, values
will be randomly drawn from the interval JPY 0 to JPY 1,000 with incrementsof 10 yen. Any value within this interval has an equally likely chance of being

73

drawn and being assigned as a value. The higher of these two values will be the“1st unit’s value” while the lower will be the “2nd unit’s value.” Each of theother two bidders will be assigned values for two units in the same way that yourvalues were assigned. The particular values assigned to the other two bidders will
typically be diﬀerent from yours. Please note that for each bidder, the second
unit’s value will always be lower than the ﬁrst unit’s value. In addition, your
“1st unit’s value” and “2nd unit’s value” are only for your private information.
The other bidders will not know your values.
3. You will submit your “bid for the 1st unit” and “bid for the 2nd unit” to theexperimenter. Then, please note the following three points:• “Your bid for the 1st unit” must be higher than “your bid for the 2nd unit”.
• Your bid must exceed 0 yen.
• Your bid must be increments of 10 yen.
4. Each of the three bidders will submit two bids. Therefore, there will be a total of
six bids. The two highest bidders will each earn an item. In case of a tie among
the high bids, the experimenter will randomly determine who earns an item. The
described procedure determines the number of units earned by each bidder.
5. The “earnings” for a bidder who earns an item are equal to his/her value of theitem less his/her payment. The “earnings” for a bidder who has not earned any
units are zero.
D.2.2

Earnings calculation

We ﬁrst explain how to calculate a bidder’s earnings if the bidder earns an item using
examples. There are three bidders, A, B, and C. Here, we focus on bidder A. In thefollowing examples, the numbers are displayed in Japanese yen.
(1) The case where a bidder earns one unit A bidder who earns one unit will
pay the amount of his/her bid for ﬁrst unit.
Example D.1. Suppose that A’s value for the ﬁrst unit is 680 and 480 for the second
unit. The bids from the three bidders are shown below.

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Bidder

Bid for 1st Unit

Bid for 2nd Unit

A

600

450

B

500

300

C

400

250

Here, the winning bids are A’s 600 and B’s 500. Bidder A pays the amount of A’s
bid for the ﬁrst unit, which is 600. Then, A’s earnings are calculated as follows:
680 (1st item’s value) − 400 (payment) = 280
Example D.2. Suppose that A’s value for the ﬁrst unit is 680 and 480 for the second
unit. The bids from the three bidders are shown below.
Bidder

Bid for 1st Unit

Bid for 2nd Unit

A

800

350

B

750

300

C

700

250

Here, the winning bids are A’s 800 and B’s 750. Thus, bidder A earns one unit. A
pays 800, which is A’s bid for the ﬁrst unit. A’s earnings are calculated as follows:
680 (1st item’s value) − 800 (payment) = −120
(2) The case where a bidder earns two units A bidder who earns two units will
pay the sum of his/her bid for the ﬁrst unit and second unit.
Example D.3. Suppose that A’s value for the ﬁrst unit is 680 and 480 for the 2ndunit. The bids from the three bidders are shown below.
Bidder

Bid for 1st Unit

Bid for 2nd Unit

A

600

550

B

500

300

C

400

250

Here, the winning bids are A’s 600 and A’s 550. Bidder A pays 1,150, the sum of
the bid of 600 for the ﬁrst unit and 550 for the second unit. A’s earnings are calculated
as follows:
680 (1st item’s value) + 480 (2nd item’s value) − 1150 (payment) = 10

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Example D.4. Suppose that A’s value for the ﬁrst unit is 680 and 480 for the second
unit. The bids from the three bidders are shown below.
Bidder

Bid for 1st Unit

Bid for 2nd Unit

A

900

850

B

800

600

C

700

550

Here, the winning bids are A’s 900 and A’s 850. Bidder A pays 1,750, the sum of
the bid of 900 for the ﬁrst unit and 850 for the second unit. A’s earnings are calculated
as follows:
680 (1st item’s value) + 480 (2nd item’s value) − 1750 (payment) = −590
The earnings calculation method may be summarized as follows:• When you earn one unit:
Earnings = 1st item’s value − bid of 1st unit
When your payment is higher than your ﬁrst item’s value, please note that yourearnings will be negative.
• When you earn two units:
Earnings = (1st unit’s value + 2nd unit’s value)− (bid for 1st unit + bid for 2nd unit)
When your payment is higher than the sum of your ﬁrst and second unit’s values,
please note that your earnings will be negative.
• When you earn an item, your earnings will be zero.

D.3

Rewards

We will explain the rewards you receive after the experiment. Your rewards are thesum of your earnings over all 25 periods and a participation fee of 1,000 yen plus youscore in the quiz to check your understanding of each instruction.
For example, if the sum of your earnings is 2,580 yen and your score on the quiz is420, your rewards will be 4,000 yen.

76

77

12

11

10

9

8

7

6

5

4

3

2

1

Round

Value for1st item

Value for2nd item

Bid for
1st item

Bid for
2nd item

Payment(1st item)

Recording Sheet
Payment(2nd item)

Payment(Total)Seat Number

Earnings

78

24

23

22

21

20

19

18

17

16

15

14

13

Round

Value for1st item

Value for2nd item

Bid for
1st item

Bid for
2nd item

Payment(1st item)

Recording Sheet
Payment(2nd item)

Payment(Total)Seat Number

Earnings

E

Appendix: The text of advice

Advice on decision making
The following advice is about the auction in which you are participating. Please consider carefully whether this advice is true or not. It is completely up to you whether
you follow the advice or not.
You can maximize your earnings by bidding at your values as they are, regardless ofwhat others bid.

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F

Appendix: Auction screens and procedures

F.1

Bidding screen

When you are ready to submit your bid, the following screen will be displayed. The
upper-left corner of the screen displays the experiment period. The following screen
shows the ﬁrst period in a total of 25 periods. In the center of the screen, “Your 1st
Unit’s Value” and “Your 2nd Unit’s Value” are displayed. First, please transcribe thisinformation into the corresponding columns on the record sheet.
Next, you submit “Your Bid for the 1st Unit” and “Your Bid for the 2nd Unit”,
both in increments of 10 yen. Please input your bids into the corresponding cells on
the screen. Note that your bid for the ﬁrst unit must be higher than your bid for thesecond unit. After that, please transcribe your bids into the corresponding columns on
the record sheet. After the transcription, click the “OK” button.
You will have 60 seconds to ﬁnalize your bid on this screen.

F.2

Auction results screen

After all the subjects have clicked the “OK” button, the following screen will be displayed. On the left side of the screen, all the bids will be ranked from highest to
lowest. The right side will be divided into three sections. The top section will show

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“Your Bid for the 1st Unit,” “Your Bid for the 2nd Unit,” and the number of units
you have earned in this auction. In the middle section, your payment will be displayedin the following order: the amount you paid for the ﬁrst unit, the amount you paid
for the second unit, and the total amount you paid in this auction. Please transcribe
this information into the corresponding columns on your record sheet. Finally, thebottom section will show the amount of “Your Earnings from this Auction.” Pleasetranscribe this information into the corresponding column on your record sheet. Afterthe transcription, click the “Next” button.

After all the subjects have clicked the “Next” button, the next auction will start.
This marks the end of one auction round. This experiment includes a series of 25
auctions.

81

G

Appendix: Quiz

Please answer all questions below.
The bid from three bidders are shown in the following table. Suppose that B’s value
for the ﬁrst unit is 900, and B’s value for the second unit is 700. Suppose that C’svalue for the 1st unit is 600.
Bidder

Bid for 1st Unit

Bid for 2nd Unit

A

800

550

B

800

700

C

600

500

(1) Find (two) winning bids.
(2) Calculate B’s payment.
(3) Calculate B’s earnings.
(4) Calculate C’s payment.
(5) Calculate C’s earnings.
Suppose that B’s bid for the ﬁrst unit is 900; all other bids are shown in the table
above.
(6) Find (two) winning bids.
(7) Calculate B’s payment.
(8) Calculate B’s earnings.
Suppose that B’s bid for the ﬁrst unit is 700 and 300 for the second unit; all otherbids are shown in the table above.
(9) Find (two) winning bids.
(10) Calculate B’s payment.
(11) Calculate B’s earnings.

82

Suppose that C’s bid for the ﬁrst unit is 1,000; all other bids are shown in the table
above.
(12) Find (two) winning bids.
(13) Calculate C’s payment.
(14) Calculate C’s earnings.
Suppose that C’s bid for the 1st unit is 500; all other bids are shown in the table
above.
(15) Find (two) winning bids.
(16) Calculate C’s payment.
(17) Calculate C’s earnings.

83

H

Appendix: Post-experiment survey
Seat Number
Faculty / Department

Age

Gender: Male / Female

1. Did you understand the auction rules (that is, how to calculate a bidder’s earnings)?
(a) I understood it after receiving the instructions.
(b) I understood it after conﬁrming the rules.
(c) I understood it by following the practice, and before the ﬁrst real round.
(d) I understood it during the real rounds（from round）(e) I did not understand it.
2. Did you trust the decision-making advice?
(a) I trusted it before the practice.
(b) I trusted it after the practice, and before the ﬁrst real round.
(c) I began trusting it during the real rounds.（from round）(d) I did not trust it.
If you answer is (c) or (d), please write your reason(s) below.

3. How did you decide to bid? Please write speciﬁcally.
The number of times you bid your valuations: ﬁrst unit (times), second unit (times).

4. What did you think about others’ bids? Please circle your selection and write
speciﬁc details.

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(Thought deeply, Didn’t think much, Didn’t think at all)

5. What do you think is the optimal bidding strategy? Please write your ideas andtheir reasoning.

6. If you had been aware of the others’ valuations, would you have changed the way
you decided to bid? If so, how would it have changed? Please write your ideas
and reasons.

Thank you for your answers.

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