ISSN (Print) 0473-453X

Discussion Paper No. 1109

ISSN (Online) 2435-0982

THE NET EFFECT OF ADVICEON STRATEGY-PROOF MECHANISMS:
AN EXPERIMENTFOR THE VICKREY AUCTION
Takehito Masuda
Ryo Mikami
Toyotaka Sakai
Shigehiro Serizawa
Takuma Wakayama

December 2020

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

The net eﬀect of advice on strategy-proof
mechanisms: An experiment for the Vickrey auction∗
Takehito Masuda†

Ryo Mikami‡

Shigehiro Serizawa¶

Toyotaka Sakai§

Takuma Wakayamak

December 3, 2020

Abstract
We conducted laboratory experiments for the multi-unit Vickrey auction withand without advice to subjects on strategy-proofness. The rate of truth-telling
among the subjects without advice was 20%, whereas the rate increased to 47%
among those who received advice. By conducting similar experiments for the
pay-your-bid auction, which is not strategy-proof, we conﬁrm that the increasein truth-telling owes signiﬁcantly to the net advice eﬀect, i.e., the eﬀect beyond
the so-called experimenter demand eﬀect. Moreover, providing advice improveseﬃciency in the Vickrey auction, particularly in the early periods, when subjects
are less experienced. It is well known that subjects tend to overbid in Vickrey
auction experiments. Our results indicate the possibility that providing a pieceof simple advice decreases such overbidding by promoting a better understandingof the strategy-proofness property in the Vickrey auction. Strategy-proof mechanisms are sometimes criticized because players often fail to ﬁnd the beneﬁt intelling the truth; however, our observations show that introducing advice on theproperty of strategy-proofness helps them behave “correctly.”
Keywords: advice eﬀect; strategy-proofness; Vickrey auction; pay-your-bid auction; market design
JEL codes: D44; D71; D61; D82∗

A version of this paper was previously circulated under the title “A strategy-proof mechanism
should be announced strategy-proof: An experiment for the Vickrey auction.” We are grateful toJohn Duﬀy, the editor of this journal, and two anonymous referees for their useful comments andsuggestions that improved the paper. We thank Yoshitaka Okano for providing the z-Tree programof the Ausubel auction and very helpful discussion. We also thank Pablo Guillen, Nobuyuki Hanaki,
Naoko Nishimura, Hans-Theo Normann, Alvin Roth, Tayfun Sönmez, Utku Ünver, Takashi Yamagata,and Lianming Zhu for their helpful comments. Yuki Hamada provided excellent research assistance.
We gratefully acknowledge ﬁnancial support from the Joint Usage/Research Center at ISER, Osaka University; Osaka University’s International Joint Research Promotion Program (Type A); andGrant-in-aid for Research Activity, Japan Society for the Promotion of Science (15H05728, 20H05631,
20KK0027).
†
Faculty of Economics and Law, Shinshu University. E-mail: tmasuda@shinshu-u.ac.jp‡
Graduate School of Economics, Osaka University. E-mail: ryomyself@gmail.com§
Department of Economics, Keio University. E-mail: tsakai@z5.keio.jp¶
ISER, Osaka University. E-mail: serizawa@iser.osaka-u.ac.jpkFaculty of Economics, Ryukoku University. 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)), andkidney exchange (Roth et al. (2004)), among many others. A preferred property ofsuch mechanisms is strategy-proofness, which ensures that truthfully reporting theirown type is always a dominant strategy for every participant. As long as a strategyproof mechanism is employed, everyone has an incentive to report his/her true type,so that the social goal is realized through the collection of accurate information.
However, it has often been observed in laboratory experiments involving strategyproof mechanisms that telling the truth is far from universal (e.g., Attiyeh et al. (2000),
Kawagoe and Mori (2001), Cason et al. (2006), and Chen and Sönmez (2006)).1 Theseobservations have been considered an anomaly to the theory, encouraging skepticism
of the practical performance of strategy-proof mechanisms. A prominent example isthe Vickrey auction (Vickrey (1961)), under which many experimental studies haveobserved overbidding.2 However, this does not imply that the strategy-proofness of
the Vickrey auction is useless in reality. Simply, it suggests the necessity of developing
some device that puts strategy-proofness to work, making the Vickrey auction usefulin real-life applications.
It is not easy for an average person to realize that the Vickrey auction is strategyproof.3 That is, the Vickrey auction does not exhibit the expected performance only
by explaining the auction rule to participants. A simple idea to resolve this issue isto announce the strategy-proofness of the Vickrey auction to users, which is what weshall attempt in this experimental research.
In our experiment, we provide advice on the strategy-proofness of the Vickrey
1

Velez and Brown (2020) 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 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
are familiar 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
experiment tend to overbid, compared with subjects in the Ausubel auction (Ausubel (2004)), which
is a dynamic counterpart of the Vickrey auction. See Kagel and Levin (2016) for a survey on theexperimental 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 explanations.

2

auction to the subjects, and examine its eﬀect on subjects’ behaviors and the eﬃciencyof outcomes. Our experiment deals with a case where multiple units of an item areto be sold under the Vickrey auction. We observe that although the rate of sincere
bidding is approximately 20% without advice, it increases to 46.9% once subjects areadvised as to the strategy-proofness of the Vickrey auction.
One might suspect that this increase stems from a so-called experimenter demand
eﬀect (Zizzo (2009)). Thus, we attempt to detect whether an eﬀect of advice exists
beyond the experimenter demand eﬀect, which we call the net advice eﬀect. In doing this, we conduct similar experiments for the pay-your-bid auction, which is not
strategy-proof.4 We used the same advice across auction mechanisms to cause similar
experimenter demand eﬀects in both auctions.5 The comparison with the pay-your-bid
auction reveals that the Vickrey auction has a large net advice eﬀect. Moreover, for
the Vickrey auction, the eﬀect of the advice is stronger for subjects with a perfect scorein a quiz on the auction rules than for other subjects.6
In the literature on 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)), the underestimation of possible losses (Georganas etal. (2017)), and so on. Our results, however, show the possibility that merely providing
a piece of simple advice decreases overbidding by promoting a better understandingof the strategy-proofness property in the Vickrey auction. Our experimental results
suggest that before operating strategy-proof mechanisms, explaining the property ofstrategy-proofness 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 Schotter (2017b)). These studies use two-sided matching models to test whether providing
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 matchingmechanisms increases the rate of truth-telling, while providing a detailed explanation

4

The pay-your-bid auction is used by many countries in practical Treasury bond allocations. See
Brenner et al. (2009) and Marszalec (2017) for more details.
5
To avoid deception, we added a careful notice to the advice. See Section Section 2.2 for moredetails.
6
Guillen and Hakimov (2017) analyze the eﬀect of the quiz score on the rate of truth-telling ina strategy-proof matching mechanism experiment. Similarly, Hanaki et al. (2016) ﬁnd that subjects
with higher cognitive skill are more likely to perform one step of elimination of dominated strategy.

3

of the mechanism has the opposite eﬀect.7 These results suggest that advice helps
participants conﬁrm individual optimality of truth-telling. To the best of our knowledge, our study is the ﬁrst one that tests the role of advice in the literature of auction
studies.8
To enhance how individuals understand incentive properties, Saijo et al. (2007)
and Li (2017) attempt to strengthen strategy-proofness. Saijo et al. (2007) emphasize that, even if a mechanism is strategy-proof, it often admits the presence of anineﬃcient Nash equilibrium composed of mis-reporting. Saijo et al. (2007) strengthen
strategy-proofness to “secure implementability” by additionally requiring that no Nashequilibrium realizes an outcome that is not the “true” outcome. An experimental study
by Cason et al. (2006) compares bidding behavior under a securely implementable (andso strategy-proof) mechanism and that of other strategy-proof mechanisms. They observe that the rate of truth-telling in the former is higher than that in the latter.
Li (2017) proposes a stronger version of strategy-proofness called “obvious strategyproofness” that can be applied to dynamic mechanisms such as ascending auctions ormatching algorithms. In a problem where a single item is auctioned, the ascending
auction is obviously strategy-proof, but the Vickrey auction is not. 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 admissibleclass of mechanisms. On the other hand, our approach of providing advice can beapplied to any strategy-proof mechanism in any environment. The Vickrey auction isneither securely implementable nor obviously strategy-proof, but giving advice drastically improves its performance on truth-telling.

7

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 eﬀectof the advice in that they do not conduct a Vickrey auction experiment without advice. In addition,
in some experimental studies that aim to elicit real valuations 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 do not test whether or not providing such information will aﬀect the rate of truth-telling.

4Organization. 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 discusses our results and provides additional analyses. Section 5 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 mi ∈ R units of money is:9

U (k, mi ; vi ) ≡
v 1 + vi2 − mi ivi1 − mi



−m
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 m : V 3 → R3 . A
rule is a pair of assignment and payment functions, ϕ = (d, m). Given any v ∈ V 3 , theprojection of ϕ(v) = (d(v), m(v)) on i ∈ {1, 2, 3} is denoted by ϕi (v) = (di (v), mi (v)),
that is, di (v) ∈ {0, 1, 2} is the number of the objects i obtains, and mi (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 vi0 ∈ V , U (ϕi (v); vi ) ≥U (ϕi (vi0 , v−i ); vi ).
The following rules are central in the literature on auction theory.

9

In standard auctions such as the Vickrey auction, the pay-your-bid auction, introduced later,mi = 0 if k = 0.

5• 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 highest to the lowest bid and allocates the two units to the two highest bids. If bidder
i wins one (respectively, two) unit, then the bidder has to pay the highest bids
(respectively, the sum of the highest and the second-highest bids) 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 themfrom the highest to the lowest bid and allocates the two units to the two highest
bids. If bidder i wins one (respectively, two) unit, then the bidder has to pay b1i(respectively, b1i + b2i ).
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 valuations are drawn from the uniform distribution with the interval 0 to
1,000 with increments of 10. Then, the larger (respectively, smaller) integer is assigned
the valuation for the ﬁrst (respectively, second) unit. All valuations are displayed inJapanese yen (JPY).

6

Table 1: Summary of treatments.

Treatment Auction rule Advice
Date# of Sessions # of Subjects (Groups)
VA
Vickrey
Yes
Mar-153
63 (21)
VN
VickreyNo
Mar-152
48 (16)PA
Pay-your-bid
Yes
Jul-163
69 (23)PN
Pay-your-bid
No
Jul-16
3
72 (24)

VA, PA, and PN each had three experimental sessions, whereas VN had two sessions.
We conducted these 11 sessions at Osaka University in March 2015 and July 2016. We
recruited student subjects from Osaka University through campus-wide advertisements.
None of the students was experienced in this particular type of experiment. No subject attended more than one session. Our experiment was computerized using theexperimental software z-Tree (Fischbacher (2007)). Twenty-one or twenty-four subjects participated in each session. Table 1 summarizes the number of 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.10 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 $0.3 ($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. Pleaseconsider 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 valuations 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 beginning ofeach period, all subjects were separated into groups of three. At the bidding stage,
10

The full set of experimental instructions (including screen shots, the quiz, and the questionnaire)is provided in Online Appendix F–J and O–P.

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• Valuations are uniformly distributed on {0, 10, . . . , 1000}.
?

Questionnaire
Figure 1: Timeline of one session.

each subject was asked to enter his/her two bids as nonnegative integers—such thatthe 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 $5.9 to $70.3.

3

Experimental results

3.1

Level of understanding for auction rules8Figure 2: Distribution of quiz scores.

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. Givenan auction rule, we found no statistical diﬀerence in the score distributions with andwithout advice.11 Similarly, we found no statistical evidence to support the diﬀerent
levels of understanding between the two auctions regardless of the presence of advice.12
In summary, we have the following result.
Result 1 (Level of understanding for auction rules). The majority of subjects
had a thorough understanding of the auction rules prior to playing regardless of whether
they participated in the Vickrey or the pay-your-bid auction.

11

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 also hold under a chi-square
test.
12
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 valuation drawn
for the unit. Similarly, we say that a bid is over (respectively, under ) if the bid ismore (respectively, less) than the valuation. For example, if valuations and bids are
v1 = (800, 300), v2 = (600, 400), v3 = (900, 500), b1 = (850, 240), b2 = (600, 400), andb3 = (900, 0), then the sincere bids are the ﬁrst and second units of bidder 2 and theﬁrst unit of bidder 3. The unique overbid 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 of bidder 1, 2, and 3 is 0, 1, and 0.5, respectively.
Figure 3 shows the scatter plots for valuations and bids in each treatment. Initially,
the Vickrey treatments graphs suggest that, for unit 1, bidders with higher valuations
more likely to overbid. When valuations are close to the maximum, 1,000, bids jump
as high as 2,000. For unit 2, bidding behavior is polarized with an increase in underbidding: overbidding is prevalent regardless of the realized valuations. At the sametime, we also observe a cluster of zero bidding when the valuation for unit 2 are no
more than 500. Table 2 summarizes the classiﬁcation of bids into three categories bytreatments and periods.13
As 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. In particular, we observed from Panels (b) and (c) of Table 2 that the rate of sincere bidding increased morein the former 13 periods than in the latter 12 periods. Speciﬁcally, the rate of sincere
bidding increased by 28.2% in the former 13 periods, while the rate increased by 24.3%
in the latter 12 periods. Notice that overbidding is prevalent in the Vickrey auction
without advice (VN), which amounts to 63.4% of all bids. This observation is consistent
with ﬁndings in experiments of the multi-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 18.1%. Moreover, we found that underbidding also decreases when theadvice is given. Speciﬁcally, our advice decreased underbidding by 8.3%. To compare
the eﬀect 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 underbidding in VN. As a result, we ﬁnd that the13

See Online Appendix Q for individual bidding data.

10

Figure 3: Scatter plots of bids. Green × and purple • indicate bids for the former 13 periods
and latter 12 periods, respectively.

11

Table 2: Bid category by treatments and periods.
(a) All

Bid category
Over
Sincere
Under
Total

Vickrey
Pay-your-bid
VA
VN
VA−VN
PA
PN
PA−PN
1,427
1,521
19
33
−18.07%−0.37%(45.30%) (63.38%)(0.55%) (0.92%)
1,478
494
287
209
26.34%
2.51%(46.92%) (20.58%)(8.32%) (5.81%)
245
385
3,144
3,358
−8.26%−2.15%(7.78%) (16.04%)(91.13%) (93.28%)
3,150
2,400
3,450
3,600
(b) Periods 1–13

Bid category
Over
Sincere
Under
Total

Vickrey
Pay-your-bid
VA
VN
VA−VN
PA
PN
PA−PN
742
785
8
28
−17.60%−1.05%(45.30%) (62.90%)(0.45%) (1.50%)
765
231
166
109
28.19%
3.43%(46.70%) (18.51%)(9.25%) (5.82%)
131
232
1,620
1,735
−10.59%−2.38%(8.00%) (18.59%)(90.30%) (92.68%)
1,638
1,248
1,794
1,872
(c) Periods 14–25

Vickrey
Bid category
Over
Sincere
Under
Total

VA
VN
VA−VN
685
736
−18.58%(45.30%) (63.89%)
713
263
24.33%(47.16%) (22.83%)
114
153−5.74%(7.54%) (13.28%)
1,512
1,152

12

Pay-your-bid
PA
PN
PA−PN
11
5
0.37%
(0.66%) (0.29%)
121
100
1.52%
(7.31%) (5.79%)
1,524
1,623
−1.89%(92.03%) (93.92%)
1,656
1,728

advice has more eﬀect on underbids (8.26/16.04) than overbids (18.07/63.38). Even
when we break down the data into the former 13 periods and the latter 12 periods,(Panels (b) and (c) of Table 2), the same tendency holds. In contrast, approximately
90% of the subjects in PA and PN are likely to underbid. We also observe a 3%-increaseof sincere bidding although the advice is not true in this case.14

3.3

Net eﬀect of advice

We now apply a normal approximation to examine whether the following three factors
aﬀect sincere bidding behavior: auction rule, advice, and unit.15 Suppose that sincere
bidding behavior in treatment j ∈ {VA, VN, PA, PN} is a realization of a randomvariable Xj with the Bernoulli distribution that takes 1 (= sincere bidding) with thesuccess rate of pj . 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 frequencies of sincere bidding by treatments, whether thequiz score is perfect or not, and by units. In this table, superscripts “all,” “per,”and “im” correspond to all subjects, the subjects whose quiz scores are perfect, andthe subjects whose quiz scores are imperfect, respectively. Hereafter, capital letters
indicate random variables while small letters do realized values. Table 4 summarizes
the frequency of sincere bidding in the former periods (periods 1–13) and in the latter
periods (periods 14–25).16
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 (Advice eﬀect).
14

A simple regression of all bids to valuations 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).
15
See page 255 of Jacquemet and L’Haridon (2018) for the construction of our test statistics.
16
We will examine whether the subject’ bidding behaviors change over time in Section 4.1.

13

Table 3: Frequency of sincere bidding by treatments, quiz scores, and units.
(a) Both units
DataAll
Perfect
Imperfect

xall
j
sall
j
xper
j
sper
j
xim
j
sim
j

Vickrey
VA
VN
0.469
0.206
(7.91 × 10−5 ) (6.81×10−5 )
0.523
0.211
(1.19×10−4 ) (1.11×10−4 )
0.361
0.198
(2.20×10−4 ) (1.76×10−4 )

(1)
VA−VN
0.263∗∗∗

Pay-your-bid
(2)
PA
PN
PA−PN
0.083
0.058
0.025∗∗∗(2.21×10−5 ) (1.52×10−5 )
0.079
0.046
0.034∗∗∗−5−5(2.52×10 ) (1.53×10 )
0.104
0.105
−0.002(1.69×10−4 ) (1.26×10−4 )0.312∗∗∗
0.162∗∗∗(1) − (2)0.238∗∗∗
0.279∗∗∗
0.165∗∗∗(b) Unit 1
Vickrey
Data
All
Perfect
Imperfect

xall
j
sall
j
xper
j
sper
j
xim
j
sim
j

VA
VN
0.416
0.175
(1.54×10−4 ) (1.20×10−4 )
0.463
0.176
(2.37×10−4 ) (1.93×10−4 )
0.322
0.173
(4.16×10−4 ) (3.18×10−4 )

(1)
VA−VN
0.241∗∗∗
0.287∗∗∗
0.149∗∗∗

Pay-your-bid
PA
PN
0.020
0.006
(1.15×10−5 ) (3.37×10−6 )
0.018
0.004
(1.21×10−5 ) (2.94×10−6 )
0.033
0.013
(1.15×10−4 ) (3.51×10−5 )

(2)
PA−PN
0.014∗∗∗(1) − (2)0.227∗∗∗

0.014∗∗∗

0.273∗∗∗

0.019∗∗

0.129∗∗∗(2)
PA−PN
0.036∗∗∗(1) − (2)0.250∗∗∗

0.054∗

0.285∗∗∗−0.0230.201∗∗∗(c) Unit 2
Vickrey
Data
All
Perfect
Imperfect

xall
j
sall
j
xper
j
sper
j
xim
j
sim
j

VA
VN
0.523
0.237
(1.58×10−4 ) (1.51×10−4 )
0.584
0.245
(2.31×10−4 ) (2.47×10−4 )
0.400
0.222
(4.57×10−4 ) (3.84×10−4 )

(1)
VA−VN
0.286∗∗∗
0.339∗∗∗
0.176∗∗∗

Pay-your-bid
PA
PN
0.146
0.110
(7.23×10−5 ) (5.44×10−5 )
0.141
0.087
(8.34×10−5 ) (5.58×10−5 )
0.175
0.197
(5.24×10−4 ) (4.22×10−4 )Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level.

Table 4: Frequency of sincere bidding by treatments, quiz scores, and periods.
(a) Former 13 periods
DataAll
Perfect
Imperfect

xall
j
sall
j
xper
j
sper
j
xim
j
sim
j

Vickrey
VA
VN
0.467
0.185
(1.52 × 10−4 ) (1.21 × 10−4 )
0.510
0.182
(2.29 × 10−4 ) (1.91 × 10−4 )
0.379
0.190
(4.31 ×10−4 ) (3.29×10−4 )

(1)
VA−VN
0.282∗∗∗
0.329∗∗∗
0.189∗∗∗

Pay-your-bid
(2)
PA
PN
PA−PN (1) − (2)
0.093
0.058
0.034∗∗∗ 0.248∗∗∗(4.68 × 10−5 ) (2.93 × 10−5 )
0.088
0.045
0.043∗∗∗ 0.286∗∗∗−5−5(5.34 ×10 ) (2.91 × 10 )
0.115
0.108
0.008
0.181∗∗∗(3.57 ×10−4 ) (2.46 ×10−4 )(b) Latter 12 periods
DataAll
Perfect
Imperfect

xall
j
sall
j
xper
j
sper
j
xim
j
sim
j

Vickrey
VA
VN
0.472
0.228
(1.65 × 10−4 ) (1.53 × 10−4 )
0.537
0.242
(2.47 × 10−4 ) (2.55 × 10−4 )
0.341
0.206
(4.46 ×10−4 ) (3.79 × 10−4 )

(1)
VA−VN
0.243∗∗∗
0.295∗∗∗
0.135∗∗∗

Pay-your-bid
(2)
PA
PN
PA−PN (1) − (2)
0.073
0.058
0.015∗∗ 0.228∗∗∗−5−5(4.09×10 ) (3.16×10 )
0.070
0.046
0.024∗∗∗ 0.271∗∗∗−5−5(4.66 × 10 ) (3.21 × 10 )
0.091
0.103
−0.0120.147∗∗∗(3.13 × 10−4 ) (2.56×10−4 )Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level.

14

(i) Providing advice increases sincere bidding in each auction rule when using all
data. The mean increases are 26.3% (95% conﬁdence interval, 24.0% to 28.7%)
and 2.5% (95% conﬁdence interval, 1.3% to 3.7%) in the Vickrey auction and in
the pay-your-bid auction, respectively.
(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.
³ all
´
all
Support. We ﬁrst establish (i). Let YVall ≡ X VA − X VN . The test statistic is
YVall − E(YVall )alle,
YV ≡ p all
all
SVA + SVN
where E is the expectation operator. By the central limit theorem, YVall approximately
follows a standardized normal distribution. Hence, in what follows, we evaluate probabilities by a standardized normal distribution unless noted otherwise. The following
null hypothesis states that providing advice does not increase sincere bidding in the
Vickrey auction.
Null hypothesis (H01 ): E(YVall ) = 0.
By a direct calculation, yeVall = 21.71. This yields Prob(YeVall ≥ yeVall ) < 0.0001. Hence,advice signiﬁcantly increased sincere bidding in the Vickrey auction. Moreover, the
95% conﬁdence interval of E(YVall ) is [0.240, 0.287].
³ all
´
all
all
We next consider the pay-your-bid auction. Let YP ≡ X PA − X PN . The following null hypothesis states that providing advice does not increase sincere bidding in
the pay-your-bid auction.
Null hypothesis (H02 ): E(YPall ) = 0.
By a direct calculation, yePall = 4.115. This yields Prob(YePall ≥ yePall ) < 0.0001. Hence,
providing 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(YPall ) is
[0.013, 0.037].

15

To establish (ii), we break the data of the Vickrey auction into two types: thesubjects with perfect quiz scores or those with imperfect quiz scores. Given h ∈{per, im}, YVh and YeVh are deﬁned similarly. We then formulate two null hypotheses:
Null hypothesis (H01p): E(YVper ) = 0.
Null hypothesis (H01i): E(YVim ) = 0.
By a direct calculation, we obtain yeVper = 20.63 and yeVim = 8.200, yielding Prob(YeVper ≥yeVper ) < 0.0001 and Prob(YeVim ≥ yeVim ) < 0.0001, respectively. Hence, both E(YVper ) = 0
and E(YVim ) = 0 are rejected.
We ﬁnally establish (iii). Given h ∈ {per, im}, YPh and YePh are deﬁned similarly.
We also break the data of the pay-your-bid auction into the subjects with perfect quizscores and those with imperfect quiz scores. We then formulate two null hypotheses:
Null hypothesis (H02p): E(YPper ) = 0.
Null hypothesis (H02i): E(YPim ) = 0.
Then, by a direct calculation, yePper = 5.298 and yePim = −0.099, yielding Prob(YePper ≥yePper ) < 0.0001 and Prob(YePim ≥ yePim ) = 0.539, respectively. Hence, E(YPper ) = 0 is
rejected, while E(YPim ) = 0 is accepted. Thus, we establish (iii).
The advice eﬀect on the pay-your-bid auction reported in Result 2 should be considered 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 next test whether the advice inthe Vickrey auction has a net eﬀect. We call its eﬀect the net advice eﬀect.
Since the advice provided in both auctions is identical, the experimenter demand
eﬀects would also be similar in both auctions. Moreover, since the advice is false in
the pay-your-bid auction, its advice eﬀect is an all experimenter demand eﬀect. Thus,
we assume that the net advice eﬀect in the Vickrey auction can be measured by thediﬀerence in advice eﬀects on the two auctions. We test the net advice eﬀect in the
Vickrey auction under this assumption.
Result 3 (Net advice eﬀect). Providing advice increases the rate of sincere bidding
much more in the Vickrey auction compared with the pay-your-bid auction, with themean increase of 23.8% (95% conﬁdence interval, 21.2% to 26.5%).

16

¡
¢
Support. We follow a diﬀerence-in-diﬀerence type argument. Let Z all = YVall − YPall .
The test statistic is
Z all − E(Z all )Zeall ≡ p all
.
all
all
all
SVA + SVN
+ SPA
+ SPN
We formulate the null hypothesis as follows:
Null hypothesis (H03 ): E(Z all ) = 0.
By a direct calculation, zeall = 17.54. This yields Prob(Zeall ≥ zeall ) < 0.0001. Moreover,the 95% conﬁdence interval of E(Z all ) is [0.212, 0.265].
The diﬀerence, 23.8%, of the advice eﬀects in the two auctions is considered as the
net advice eﬀect 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.
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 (Relationship between quiz scores and advice eﬀects). 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 interms of an increase in the rate of sincere bidding is 15.0% (95% conﬁdence interval,
10.1% to 20.0%).
Support. Let WV = (YVper − YVim ). The test statistic is
WV − E(WV )fV ≡ p
.
W
per
per
im
im
+ SVA
+ SVN
+ SVN
SVA
We formulate the null hypothesis as follows:
Null hypothesis (H04 ): E(WV ) = 0.
fV ≥ w
By a direct calculation, w
eV = 5.977. This yields Prob(WeV ) < 0.0001. Moreover,the 95% conﬁdence interval of E(WV ) is [0.101, 0.200].
Result 4 suggests that subjects who properly understand the rule of the Vickrey
auction tend to bid sincerely under the advice than others who do not so.

17

Table 5: Frequency of sincere bidding in period one by treatments and quiz scores.
DataAll
Perfect
Imperfect

xall
j
sall
j
xper
j
sper
j
xim
j
sim
j

Vickrey
VA
VN
0.468
0.188
(1.98 × 10−3 ) (1.59 × 10−3 )
0.512
0.233
(2.97 × 10−3 ) (2.98 × 10−3 )
0.381
0.111
(5.61 × 10−3 ) (2.75 × 10−3 )

(1)
VA−VN
0.280∗∗∗
0.279∗∗∗
0.270∗∗∗

Pay-your-bid
PA
PN
0.123
0.021
(7.83 × 10−4 ) (1.42 × 10−4 )
0.138
0.026
(1.02 × 10−3 ) (2.25 × 10−4 )0.046b)0(1.97 × 10−3 )(0)(2)
PA−PN (1) − (2)0.102∗∗∗ 0.178∗∗∗
0.112∗∗∗

0.167∗∗

0.046

0.225∗∗

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.

Remark 1. When we restrict our attention to the ﬁrst period only and use both units’
data, as shown in Table 5, H01 , H02 , and H03 are again rejected at the 10% signiﬁcancelevel 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.
♦

3.4

Eﬃciency

Our eﬃciency measure is due to Kagel and Levine (2009).17 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 6= 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 3.3, we again normally approximated to examine the eﬀect of auction
rules, advice, and experience on eﬃciency. Suppose the eﬃciency ratio r in treatmentj ∈ {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 nj0 realizations of Rj . Let
Pnj
Tj ≡

2
0
k=1 (Rjk − Rj ) /(nj − 1).
n0jSuperscripts “ap,” “fp,” and “lp” correspond to all periods, the former periods (periods
1–13), and the latter periods (periods 14–25), respectively. Table 6 summarizes theresults on eﬃciency with a focus on experience.18 In Table 6, the upper (respectively,
17

See also Kwasnica and Sherstyuk (2013).
Englemann and Grimm (2009) compare the Vickrey auction with the pay-your-bid auction by
conducting an experiment, where two units of an item are auctioned to two bidders with ﬂat demand.
Their experimental results are similar to ours in regard to the comparison of the eﬃciency ratios in
the two auctions.
18

18

Table 6: Eﬃciency by treatments and periods.
Data
rap
j
tap
j
rfp
j
Periods 1–13
tfp
j
rlp
j
Periods 14–25
tlp
j
All periods

Vickrey
(1)
VA
VN
VA−VN
0.977
0.967
0.010∗∗−6−5(8.95×10 ) (1.62×10 )
0.970
0.956
0.015∗∗−5−5(2.32×10 ) (4.59×10 )
0.984
0.978
0.006
(1.13×10−5 ) (1.53×10−5 )Pay-your-bid
(2)
PA
PN
PA−PN (1) − (2)
0.990
0.988
0.002
0.009∗−6−6(1.36×10 ) (1.36×10 )
0.991
0.989
0.002
0.013∗−6−6(2.27×10 ) (2.67×10 )
0.989
0.988
0.001
0.004
(3.26×10−6 ) (2.76×10−6 )Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level.

middle, bottom) panel collects the results obtained for all periods (respectively, periods
1–13, periods 14–25).
Result 5 (Advice eﬀects on eﬃciency). Providing advice improves eﬃciency inthe Vickrey auction, particularly in the former 13 periods.
³ h´hSupport. Given h ∈ {ap, fp, lp}, let UVh = RVA − RVN . The test statistic is
UVh − E(UVh )eh ≡ p.
UVh
TVN+ T hVN
The following hypothesis states that providing advice does not increase eﬃciency within
the Vickrey auction.
Null hypothesis (H05 ): E(UVap ) = 0.
e ap eap ) = 0.020. Therefore,
By a direct calculation, ueap
V = 2.054. This yields Prob(UV ≥ uVadvice signiﬁcantly increased eﬃciency in the Vickrey auction.
To obtain further insights, we break the data into two types: periods 1–13 and
14–25. We then formulate two null hypotheses:
Null hypothesis (H05f ): E(UVfp ) = 0.
Null hypothesis (H05l): E(UVlp ) = 0.
e fp
Then, we obtain u
efp
elp
efp
V = 1.769 and u
V = 1.066, yielding Prob(UV ≥ uV ) = 0.038 and
lp
lp
fp
e
Prob(UV ≥ ueV ) = 0.143, respectively. Hence, E(UV ) = 0 is rejected, while E(UVlp ) = 0
is not rejected.
Result 5 suggests that improvements in the eﬃciency of the Vickrey auction occurs
mainly when subjects are less experienced.

19

Result 6 (Relative eﬀect of advice on eﬃciency). The relative eﬀect of the adviceof strategy-proofness on the eﬃciency of the Vickrey auction compared with the payyour-bid auction exists, particularly in early periods.
Support. We again apply the diﬀerence-in-diﬀerence technique analogously to Result 4.
Let Qap = (UVap − UPap ). The test statistic is
Qap − E(Qap )eap ≡ p
Q
.
ap
ap
ap
ap
TVA
+ TVN
+ TPA
+ TPN
We formulate the null hypothesis as follows:
Null hypothesis (H06 ): E(Qap ) = 0.
eap ≥ qeap ) = 0.052. Again, we
By a direct calculation, qeap = 1.629. This yields Prob(Qeh arebreak the data into two: periods 1–13 and 14–25. Given h ∈ {fp, lp}, Qh and Q
deﬁned similarly. We then formulate two null hypotheses:
Null hypothesis (H06f ): E(Qfp ) = 0.
Null hypothesis (H06l): E(Qlp ) = 0.
efp ≥ qefp ) = 0.073 andThen, we obtain qefp = 1.454 and qelp = 0.770, yielding Prob(Qelp ≥ qelp ) = 0.221, respectively. Hence, E(Qfp ) = 0 is rejected at the 10%
Prob(Qsigniﬁcance level, while E(Qlp ) = 0 is not rejected.
Result 6 suggests that a relative improvement in the eﬃciency of the Vickrey auction compared with the pay-your-bid auction occurs mainly when subjects are lessexperienced.
Remark 2. 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. We found a signiﬁcant increase in the eﬃciency of the Vickrey auction for all
periods, the former periods, and the latter periods. That is, Result 6 is replicated. See
Table 9 in Appendix A.
♦

4Discussion

4.1

Regression analysis of sincere bidding

We perform a regression analysis to check the robustness of the results. Almost all
results are conﬁrmed. The dependent variable is a dummy variable of sincere bidding,20

which takes 1 if subject i in period t bids sincerely. The three independent variables
are the same variables we examined independently in the previous section:• Advice is a dummy for treatments with advice;• Latter is a dummy for periods more than 13; and• Perfect is a dummy for subjects with a perfect score in the quiz.
We also included the interaction terms of these three variables. Table 7 summarizes
the results from linear probability models of sincere bidding by auction rules and units.
For each auction rule and unit, we run three regressions with diﬀerent speciﬁcations.
The highlights of the linear probability model regression results are as follows.19• Vickrey: Both Advice and Advice × Perfect in speciﬁcations from (1) to (6) havethe coeﬃcients of 15% or more for both units, supporting Result 2 and Result 4.
Moreover, the insigniﬁcant eﬀect in Latter and Advice × Latter in speciﬁcations(3) and (6) suggests that the rate of sincere bidding is stable in time, and theadvice quickly improves the rate of sincere bidding in earlier periods.20 This quickeﬀect of advice is consistent with Table 4. Finally, the comparison of Perfect andAdvice × Perfect in speciﬁcations (3) and (6) suggests that advice has quite asimilar eﬀect on sincere bidding across units, given the level of understanding of
the auction rule.
• Pay-your-bid: Speciﬁcations (9) and (12) with negative coeﬃcients of Perfect
conﬁrm that the subjects become less likely to bid sincerely as they understandthe auction rule correctly. This is consistent with Table 3.
The literature reports mixed results on whether the second price auction subjects
learn to bid sincerely from experiencing losses owing to overbidding. For example,
Cooper and Fang (2008) observe learning, while Harstad (2000) do not. Using an
additional regression, we do not ﬁnd such evidence in any of two auction rules, andthis is consistent with Harstad (2000).

19

See Angrist and Pischke (2008) for the wide applicability of the linear probability model.
Together with the insigniﬁcant eﬀect in Latter × Perfect and Advice × Latter × Perfect, thisresult suggests that the learning process is similar between groups with diﬀerent levels of understandingof auction rules.
20

21

Table 7: Regression analysis of sincere bidding.
(a) Vickrey

AdvicePerfect
Advice×Perfect(1)
0.149∗∗∗(0.029)0.003(0.027)0.138∗∗∗(0.036)

Unit 1(2)0.270∗∗∗(0.024)0.044∗(0.026)−0.060∗(0.034)LatterAdvice×Latter
Latter ×Perfect
Advice × Latter ×Perfect
Constant
Observations
Adjusted R2

0.173∗∗∗(0.021)
2,775
0.077

0.154∗∗∗(0.018)
2,775
0.067

(3)
0.178∗∗∗(0.040)−0.021(0.037)0.140∗∗∗(0.049)0.014(0.042)−0.061(0.057)0.048(0.053)−0.003(0.071)0.167∗∗∗(0.029)
2,775
0.077

(4)
0.178∗∗∗(0.030)0.023(0.028)0.161∗∗∗(0.037)

Unit 2(5)0.294∗∗∗(0.025)0.042(0.027)−0.017(0.036)

0.222∗∗∗(0.022)
2,775
0.101

0.216∗∗∗(0.019)
2,775
0.084

(6)
0.200∗∗∗(0.041)
0.004(0.038)0.140∗∗∗(0.052)0.018(0.044)−0.047(0.060)0.039(0.055)0.042(0.074)0.214∗∗∗(0.030)
2,775
0.102

(b) Pay-your-bid

AdvicePerfect
Advice×Perfect(7)
0.019∗∗(0.009)−0.009(0.007)−0.006(0.010)

Unit 1(8)0.010∗(0.005)−0.003(0.005)0.008(0.008)LatterAdvice×Latter
Latter ×Perfect
Advice × Latter ×Perfect
Constant
Observations
Adjusted R2

0.013∗∗(0.006)
3,525
0.005

0.007∗∗(0.004)
3,525
0.003

(9)
0.000
(0.012)−0.016∗(0.009)0.013(0.014)−0.015(0.012)0.039∗∗(0.018)0.015(0.013)−0.038∗(0.020)0.021∗∗(0.008)
3,525
0.005

(10)
−0.023(0.026)−0.110∗∗∗(0.019)0.076∗∗∗(0.029)0.002(0.016)−0.046∗∗(0.022)0.197∗∗∗(0.017)
3,525
0.012

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

22

Unit 2(11)
0.058∗∗∗(0.016)

0.109∗∗∗(0.011)
3,525
0.004

(12)
0.015
(0.037)−0.109∗∗∗(0.027)0.058(0.040)0.005(0.034)−0.079(0.053)−0.004(0.039)0.039(0.058)0.195∗∗∗(0.024)
3,525
0.013

4.2

How did subjects perceive advice?

We examined how many subjects believed the advice in VA by employing a post experimental questionnaire. The questionnaire asks subjects whether they believe theadvice, and if yes, since when they started believing it. Subjects chose one from fouranswers:(a) Believed the advice before practice periods;(b) Believed the advice after practice periods but before real periods;(c) Believed the advice during real periods; and(d) Never believed.
In VA, the number of subjects who chose (a), (b), (c), and (d) is, respectively, 13, 14,
14, and 22. Thus, in VA, 41 subjects believed the advice, and 22 did not.
We categorized the subjects in VA into two groups—agents who believed (a, b, c)and agents who never believed (d)—and ran normal approximation parametric tests
for group comparison. We obtained two reasonable results:
1. The VA subjects who believed the advice bid signiﬁcantly more sincerely compared with the VA subjects who never believed the advice (60.2% and 22.2%,
respectively; z = 22.97, p < 0.001); and2. The rate of sincere bidding among the VA subjects who never believed the advice
is not signiﬁcantly diﬀerent from that of all VN subjects (22.2% and 20.6%,
respectively; z = 1.238, p = 0.215).
These results suggest that providing advice is likely to increase sincere bids whensubjects believe them, and that providing advice makes subjects bid more sincerely—
not because subjects are inﬂuenced by the experimenter demand eﬀect, but because
subjects believe the advice.
Remark 3. Incidentally, in PA, the number of subjects who chose (a), (b), (c), and(d) was, respectively, 3, 1, 0, and 65, that is, 4 subjects believed the advice, and 65
never did. We also ﬁnd that there is no statistical diﬀerence in the rate of sincere
bidding between two groups in PA. Moreover, the rate of sincere bidding among those
who never believed the advice was signiﬁcantly higher than the overall rate of sincere
bidding in PN.
♦

23

Table 8: Average bidder’s payoﬀ and average seller’s revenue per auction.
(a) Bidder’s payoﬀ

Data
π ap
j
All periods
ξjap
π fp
j
Period 1–13
ξjfp
π lp
j
Period 14–25
ξjlp

Vickrey
VA
VN
VA−VN
136.844 122.292 14.553∗(32.174) (40.912)
132.247 115.000 17.247(67.765) (86.076)
141.825 130.191 11.634(60.139) (76.487)

Pay-your-bid
PA
PN
PA−PN
82.82989.928−7.099∗(6.448) (6.901)
78.99783.301−4.305(11.478) (12.388)
86.98197.106 −10.126∗(14.493) (15.317)(b) Seller’s revenue

Vickrey

Data
π ap
j
ξjap
π fp
j
Periods 1–13
ξjfp
π lp
j
Periods 14–25
ξjlp
All periods

Pay-your-bid
VA
VN
VA−VN
PA
PN
PA−PN
1139.410 1125.875
13.535 1303.704 1288.700
15.004(226.595) (465.283)(56.924) (64.599)
1144.689 1142.933
1.756 1313.211 1303.397
9.813(604.420) (882.481)(120.829) (143.281)
1133.690 1107.396
26.295 1293.406 1272.778
20.628(599.881) (985.408)(104.943) (110.987)

Notes: a) ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level; and b) Givenj ∈ {VA, VN, PA, PN}, let Πj denote denote the sample mean for nj0 realizations of an observation of thebidder’s payoﬀ (or the seller’s revenue) Πj . Ξj is deﬁned similarly to Tj introduced in Section 3.4.

4.3

Bidders’ payoﬀs and seller’s revenue

In this section, we ask whether the advice aﬀects bidders’ payoﬀs and the seller’s
revenue in each auction rule. Table 8 summarizes the results on bidders’ payoﬀs andthe seller’s revenue. We then make the following observations.
Result 7 (Bidders’ payoﬀs and seller’s revenue).
(i) The advice marginally increases (respectively, decreases) the bidder’s payoﬀ per
auction in the Vickrey (respectively, pay-your-bid) auction using all data.
(ii) The advice does not aﬀect the seller’s revenue.
The increase in bidders’ payoﬀs in the Vickrey auction is consistent with the eﬃciency gain we reported in Section 3.4. See, also, Online Appendix C for the comparisonof diﬀerent auction rules in regard to bidders’ payoﬀs and seller’s revenue.

24

4.4

Ausubel auction

We further conducted an experiment to test the eﬀect of advice for the Ausubel auction(Ausubel (2004)).21 The Ausubel auction, which is an ascending bid auction, is considered a dynamic version of the Vickrey auction. That is, the equilibrium outcomes ofthe former and the latter are equivalent.22 It is well known that, in the Ausubel auction, sincere bidding, where bidders may reduce their demand for units at prices equal
to their valuations for these units, is not a weakly dominant strategy. In this sense,
the Ausubel auction is not strategy-proof. However, in the Ausubel auction, sincere
bidding is an ex post equilibrium (Ausubel (2004); Okamoto (2018)). Therefore, one
might think that providing advice would improve the performance of the Ausubel auction, as with the Vickrey auction. We then provided the following advice to subjects
in our experiment:“The following advice is about the auction in which you are participating. Pleaseconsider carefully whether this advice is true or false. It is completely up to you
whether you follow the advice.
You can maximize your earnings by reducing your demands at your valuations
as they are, regardless of the prices at which others reduce their demands.”
We have two types of treatments: the Ausubel auction with advice and without
advice.23 In our experiment, almost half of the subjects had a perfect score in the quizof the Ausubel auction. We then conﬁrmed that the advice eﬀect exists in the Ausubel
auction, that is, providing advice increases sincere bidding in the Ausubel auction.
In addition, as with the Vickrey auction, we observed that subjects who understandthe rule of the Ausubel auction well are likely to bid sincerely on the basis of advice
than others who do not. However, providing advice does not increase (value-based)eﬃciency in the Ausubel auction.24
We compare our Ausubel auction data with our Vickrey auction data. On doing
so, we ﬁnd that, when the advice is provided, the rate of sincere bidding for nonwinning bids is signiﬁcantly higher in the Vickrey auction compared with the Ausubel
21

We thank an anonymous referee for suggesting this experiment.
Online Appendix B provides the more formal deﬁnition of the Ausubel auction.
23
To compare the results of the Vickrey auction and those of the Ausubel auction, the settings foreach treatment are identical to the setting for the Vickrey experiment. See Online Appendix B for
the details of our experimental procedures. The full set of experimental instructions (including screen
shots and quiz) for these two treatments can be found in Online Appendix K–P.
24
If we evaluate eﬃciency using the binary eﬃciency, shown in Online Appendix B, the improvement
is marginally signiﬁcant in the former periods.
2225

auction.25 Although Kagel and Levin (2009) observe that the dynamic format brings
sincere bidding more in line with theory than the static format, our experimental result
suggests that providing advice can exceed such a “dynamic” eﬀect and reverse the order.
The logic behind this is as follows: Providing advice improves sincere bidding in the
Ausubel auction mainly through the bids for the ﬁrst unit and does so for both units
in the Vickrey auction. Thus, the total improvement in sincere biding is much larger
in the Vickrey auction.
We formally state these results and provide the statistical analysis of these resultsin Online Appendix B.

5Conclusion

We experimentally showed that introducing advice on strategy-proofness leads to ahigher rate of sincere bidding in a multi-unit static Vickrey auction. Without thisadvice, the rate of sincere bidding in our experiment is similar to rates observed inprevious studies. The same piece of advice has eﬀects not only in the Vickrey auction,
but also in the pay-your-bid auction. The latter eﬀect is attributed to the so-called
experimenter demand eﬀect (Zizzo (2009)). To detect whether the advice has an eﬀect
beyond the experimenter demand eﬀect, which we call the net advice eﬀect, we compared the eﬀects in the two auctions. We found that a large net advice eﬀect exists inthe 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
and Vragov (2006)), researchers have attempted to identify factors that drive overbidding, such as the anticipation of regret, the joy of winning (Cooper and Fang (2008)),
cognitive limits on contingent reasoning (Li (2017)), and the underestimation of possible losses (Georganas et al. (2017)). Our results suggest that one of the key driversof overbidding is the subjects’ failure to grasp the strategy-proofness property of theVickrey 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 can beused in the two auctions. We emphasize that even such a piece of simple advice has
a considerable net advice eﬀect in the Vickrey auction. More detailed advice would
25

According to the rule of the Ausubel auction, winners’ demand reductions are censored in the
Ausubel auction. For this reason, and for a fair comparison, we focus on non-winning bids in the
Vickrey treatments.

26

make the subjects bid more sincerely.
Our results show that a simple advice improves eﬃciency in the Vickrey auction, andthis improvement is higher in early periods, when subjects are less experienced. Thus,
our results show that advice improves eﬃciency in the Vickrey auction by promoting
a better understanding of the strategy-proofness property of the Vickrey auction.
To focus on the advice eﬀect on bidding behavior, we conducted experiments in a“symmetric” environment, that is, an environment in which each bidder in the sameauction has the same distribution function of valuations. As discussed in Section 3.4,
the pay-your-bid auction as well as the Vickrey auction generated almost perfectly efﬁcient outcomes in such a symmetric environment. Thus, even without advice, there islittle room to improve eﬃciency in both auctions. That is, in our experiment, the eﬀects
of advice on eﬃciency are limited in both auctions. We conjecture that experiments inan “asymmetric” environment would clarify the advice eﬀect on eﬃciency.

27

A

Appendix: Binary eﬃciency

Table 9 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 theachievement of eﬃcient allocation in treatment j ∈ {VA, VN, PA, PN} is a realizationof a random variable Ej with a Bernoulli distribution that takes 1 (= eﬃcient) with asuccess rate of qj . That is,1 with prob. q ,jEj ≡0 with prob. 1 − q .
j

Let E j denote the sample mean for nj0 realizations of Ej and let
Γj ≡E j (1 − E j ).
n0jTable 9: Binary eﬃciency by treatments and periods.
Data
eap
j
γjap
efp
j
Periods 1–13
γjfp
elp
j
Periods 14–25
γjlp
All periods

Vickrey(1)
Pay-your-bid
(2)
VA
VN
VA−VN
PA
PN
PA−PN (1) − (2)
0.779
0.705
0.074∗∗∗
0.823
0.783
0.039∗∗ 0.035∗∗(3.28×10−4 ) (5.20×10−4 )(2.54×10−4 ) (2.83×10−4 )
0.744
0.644
0.099∗∗∗
0.823
0.782
0.041∗
0.059∗∗−4−3−4−4(6.98×10 ) (1.10×10 )(4.88×10 ) (5.46×10 )
0.817
0.771
0.047∗
0.822
0.785
0.038∗
0.009
−4
−4
−4
−4
(5.92×10 ) (9.20×10 )(5.29×10 ) (5.87×10 )Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level.

28

References
Abdulkadiroğlu, A. and T. Sönmez (2003) “School choice: A mechanism design approach,” American Economic Review, 93, 729–747.
Angrist, J. D. and J. S. Pischke (2008) Mostly Harmless Econometrics: An Empiricist’sCompanion, Princeton University Press.
Attiyeh, G., R. Franciosi, and R. M. Isaac (2000) “Experiments with the pivotal process
for providing public goods,” Public Choice, 102, 95–114.
Ausubel, L. (2004) “An eﬃcient ascending-bid auction for multiple objects,” American
Economic Review, 94, 1452–1475.
Brenner, M., D. Galai, and O. Sade (2009) “Sovereign debt auctions: Uniform or
discriminatory?” Journal of Monetary Economics, 56, 267–274.
Cason, T., T. Saijo, T. Sjöström, and T. Yamato (2006) “Secure implementation experiments: Do strategy-proof mechanisms really work?” Games and Economic
Behavior, 57, 206–235.
Chen, Y. and T. Sönmez (2006) “School choice: An experimental study,” Journal of
Economic Theory, 127, 202–231.
Cooper, D. J. and H. Fang (2008) “Understanding overbidding in second price auctions:
An experimental study,” Economic Journal, 118, 1572–1595.
Ding, T. and A. Schotter (2017a) “Matching and chatting: An experimental study ofthe impact of network communication on school-matching mechanisms,” Gamesand Economic Behavior, 103, 94–115.
Ding, T. and A. Schotter (2017b) “Learning and mechanism design: An experimental
test of school matching mechanisms with intergenerational advice,” mimeo.
Engelmann, D. and V. Grimm (2009) “Bidding behavior in multi-unit auctions: An
experimental investigation and some theoretical insights,” Economic Journal, 119,
855–882.
Fischbacher, U. (2007) “z-Tree: Zurich toolbox for readymade economic experiments,”
Experimental Economics, 10, 171–178.

29

Garratt, R. J., M. Walker, and J. Wooders (2012) “Behavior in second-price auctions byhighly experienced eBay buyers and sellers,” Experimental Economics, 15, 44–57.
Georganas, S., D. Levin, and P. McGee (2017) “Optimistic irrationality and overbidding
in private value auctions,” Experimental Economics, 20, 772–792.
Grether, D. and C. Plott (1979) “Economic theory of choice and the preference reversal
phenomenon,” American Economic Review, 69, 623–638.
Guillen, P. and R. Hakimov (2017) “Not quite the best response: Truth-telling,strategy-proof matching, and the manipulation of others,” Experimental Economics, 20(3), 670–686.
Guillen, P. and R. Hakimov (2018) “The eﬀectiveness of top-down advice in strategyproof mechanisms: A ﬁeld experiment,” European Economic Review, 101, 505–
511.
Guillen, P. and A. Hing (2014) “Lying through their teeth: Third party advice andtruth telling in a strategy proof mechanism,” European Economic Review, 70,
178–185.
Hanaki, N., N. Jacquemet, S. Luchini, and A. Zylbersztejn (2016) “ Cognitive ability
and the eﬀect of strategic uncertainty,” Theory and Decision, 81, 101–121.
Hassidim, A., D. Marciano, A. Romm, and R. I. Shorrer (2017) “The mechanism istruthful, why aren’t you?” American Economic Review, 107, 220–224.
Harstad, R. M. (2000) “Dominant strategy adoption and bidders’ experience withpricing rules,” Experimental economics, 3, 261–280.
Holmström, B. (1979) “Groves’ scheme on restricted domains,” Econometrica, 47,
1137–1144.
Jacquemet, N. and O. L’Haridon (2018) Experimental Economics: Method and Applications, Cambridge University Press.
Kagel, J. H. and D. Levin (1993) “Independent private value auctions: Bidder behaviour in ﬁrst-, second- and third-price auctions with varying numbers of bidders,” Economic Journal, 103, 868–879.

30

Kagel, J. H. and D. Levin (2001) “Behavior in multi-unit demand auctions: Experiments with uniform price and dynamic Vickrey auctions,” Econometrica, 69,
413–454.
Kagel, J. H. and D. Levin (2009) “Implementing eﬃcient multi-object auction institutions: An experimental study of the performance of boundedly rational agents,”
Games and Economic Behavior, 66, 221–237.
Kagel, J. H. and D. Levin (2016) “Auctions: A survey of experimental research,” in
Handbook of Experimental Economics, Vol. 2, (J. H. Kagel and A. E. Roth eds.),
Princeton University Press.
Kawagoe, T. and T. Mori (2001) “Can the pivotal mechanism induce truth-telling? Anexperimental study,” Public Choice, 108, 331–354.
Kwasnica, A. M. and K. Sherstyuk (2013) “Multiunit auctions,” Journal of Economic
Surveys, 27, 461–490.
Lebrun, B. and M.-C. Tremblay (2003) “Multiunit pay-your-bid auction with onedimensional multiunit demands,” International Economic Review, 44, 1135–1172.
Li, S. (2017) “Obviously strategy-proof mechanisms,” American Economic Review,
107, 3257–3287.
List, J. A. (2001) “Do explicit warnings eliminate the hypothetical bias in elicitation
procedures? Evidence from ﬁeld auctions for sportscards,” American Economic
Review, 91, 1498–1507.
Manelli, A. M., M. Sefton, and B. Wilner (2006) “Multi-unit auctions: A comparisonof static and dynamic mechanisms,” Journal of Economic Behavior and Organization, 61, 304–323.
Marszalec, D. (2017) “The impact of auction choice on revenue in treasury bill auctions– An empirical evaluation,” International Journal of Industrial Organization, 53,
215–239.
Milgrom, P. (2000) “Putting auction theory to work: The simultaneous ascending
auction,” Journal of Political Economy, 108, 245–272.
Okamoto, N. (2018) “An eﬃcient ascending-bid auction for multiple objects: Comment,” American Economic Review, 108, 555–560.

31

Porter, D. and R. Vragov (2006) “An experimental examination of demand reduction in multi-unit versions of the Uniform-price, Vickrey, and English auctions,”
Managerial and Decision Economics, 27, 445–458.
Roth, A. E., T. Sömez, and M. Ünver (2004) “Kidney exchange,” Quarterly Journalof Economics, 119, 457–488.
Saijo, T., T. Sjöström, and T. Yamato (2007) “Secure implementation,” Theoretical
Economics, 2, 203–229.
Shogren, J. F., M. Margolis, C. Koo, and J. A. List (2001) “A random nth-priceauction,” Journal of Economic Behavior and Organization, 46, 409–421.
Velez, R. A. and A. L. Brown (2020) “Empirical strategy-proofness,” mimeo.
Vickrey, W. (1961) “Counterspeculation, auctions, and competitive sealed tenders,”
Journal of Finance, 16, 8–37.
Zizzo, D. J. (2009) “Experimenter demand eﬀects in economic experiments,” Experimental Economics, 13, 75–98.

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