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

Secondly Revised October 2021
Revised May 2021
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

October 5, 2021

Abstract
We conduct 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 stays at 20%, whereas the rate increases to
47% among those who have received advice. By conducting similar experiments
for the pay-your-bid auction, which is not strategy-proof, we conﬁrm that theincrease in truth-telling is due signiﬁcantly to the net advice eﬀect (i.e., theeﬀect beyond the so-called experimenter demand eﬀect). Moreover, we ﬁnd thatproviding advice improves eﬃciency in the Vickrey auction, particularly in theearly periods, when the subjects are less experienced. In general, subjects tend
to overbid in Vickrey auction experiments. Our results indicate the possibility
that providing simple advice decreases such overbidding by promoting a better
understanding of the strategy-proofness of the Vickrey auction. Strategy-proof
mechanisms are sometimes criticized because players often fail to recognize thebeneﬁt of telling the truth. However, our observations show that introducing
advice on the property of strategy-proofness helps them behave “correctly.”
Keywords: advice eﬀect; strategy-proofness; Vickrey auction; pay-your-bid auction; Ausubel 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.”†
Faculty of Economics and Law, Shinshu University, Nagano 390-8621, Japan.
E-mail:tmasuda@shinshu-u.ac.jp‡
ISER, Osaka University, Osaka 567-0047, Japan. E-mail: ryomyself@gmail.com§
Department of Economics, Keio University, Tokyo 108-8345, Japan. E-mail: tsakai@z5.keio.jp¶
ISER, Osaka University, Osaka 567-0047, Japan E-mail: serizawa@iser.osaka-u.ac.jpkCorresponding author. Faculty of Economics, Ryukoku University, Kyoto 612-8577, Japan. Email: wakayama@econ.ryukoku.ac.jp1

1

Introduction

1.1

Motivation and outline

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 his/her
own type is always a weakly dominant strategy for every participant. Provided thata strategy-proof mechanism is employed, everyone has an incentive to report his/hertrue type to ensure that a social goal is realized through the collection of accurate
information.
However, in the experimental literature on strategy-proof mechanisms, it has often
been observed 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 theVickrey auction is useless in reality. Simply put, it suggests the necessity of developing
some device that enables strategy-proofness to work, making the Vickrey auction usefulin real-life applications.
For an average person, it is not easy to realize that the Vickrey auction is strategyproof.3 That is, the Vickrey auction does not exhibit the expected performance bymerely explaining the auction rule to participants. A simple method to address this1

Velez and Brown (2020) study the observations in the experimental literature on strategy-proof
mechanisms from a diﬀerent perspective. They identify the conditions 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 isapproximately 27% of all bids. Garratt et al. (2012) report that despite the fact that the subjects arefamiliar with bidding in eBay auctions, approximately four out of ﬁve subjects fail to bid sincerely.
Manelli et al. (2006) and Kagel and Levin (2009) observe that subjects in a Vickrey auction experimenttend to overbid, compared with those in the Ausubel auction (Ausubel (2004)), which is a dynamic
counterpart of the Vickrey auction. See Kagel and Levin (2016) for a survey on the 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 explanations.

2issue is to announce the strategy-proofness of the Vickrey auction to users, which iswhat we shall attempt in this experimental research.
In our experiment, we provide a piece of simple advice on the strategy-proofness of
the Vickrey auction to the subjects and examine its eﬀect on the subjects’ behavior andthe eﬃciency of outcomes. Our experiment focuses on a case where multiple units ofidentical objects are to be sold under the Vickrey auction. We observe that, although
the rate of sincere bidding is 20.6% without advice, it increases to 46.9% once subjects
are advised regarding the strategy-proofness of the Vickrey auction.
One might suspect that this increase is due to the 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. To this end,
we conducted similar experiments for the pay-your-bid auction, which is not strategyproof.4 We used the same advice across auction rules to cause similar experimenter
demand eﬀects in both auctions.5 The comparison results for the pay-your-bid auction
reveal that the Vickrey auction has a large net advice eﬀect. Moreover, for the Vickrey
auction, the eﬀect of the advice is stronger for the subjects with a perfect score in aquiz on the auction rules than for the other subjects.6
In the literature on auction theory, overbidding in the Vickrey/second price auction has been attributed to various factors such as the joy of winning (Cooper andFang (2008)), the presence of a cognitive limit on contingent reasoning (Li (2017)),
the underestimation of possible losses (Georganas et al. (2017)), and so on. However,
our results show the possibility that merely providing a piece of simple advice decreases overbidding by promoting a better understanding of the strategy-proofness of
the Vickrey auction. Our experimental results suggest that, before operating strategyproof mechanisms, explaining the property of strategy-proofness helps those who areunaware of its property to behave “correctly.”
Finally, we conducted an experiment to test the eﬀect of advice for the Ausubel
auction (Ausubel (2004)), which is known as a dynamic counterpart of the Vickrey
auction.7 This auction rule is not strategy-proof because in it, sincere bidding, wherebidders may reduce their demand for units at prices equal to their valuations forthese units, is not a weakly dominant strategy. Meanwhile, it is well known that
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 note to the advice. See Section 2.2 for more details.
6
Guillen and Hakimov (2017) analyze the eﬀect of quiz scores on the rate of truth-telling in a
strategy-proof matching mechanism experiment. Similarly, Hanaki et al. (2016) ﬁnd that subjects
with higher cognitive ability are more likely to perform one step of elimination of dominated strategies.
7
We thank an anonymous referee for suggesting this experiment.

3

in the Ausubel auction, sincere bidding is an ex-post equilibrium (Ausubel (2004);
Okamoto (2018)). Thus, based on our experimental results above, one would expectthat providing simple advice on this property would increase the rate of sincere bidding in the Ausubel auction, similar to that in the Vickrey auction. Although the
Ausubel auction is a dynamic mechanism, we set the instruction and advice in the
Ausubel auction treatment as similar to the main experiments to the extent possible,
but without making deception. We ﬁnd that the advice eﬀect also exists in the Ausubel
auction. This ﬁnding suggests that providing “appropriate” advice could serve as atool to improve the performance of even non-strategy-proof mechanisms.

1.2

Related literature

The eﬀect of providing advice on strategy-proof mechanisms has recently been investigated by several authors (Guillen and Hing (2014); Ding and Schotter (2019)). In thesestudies, two-sided matching models are used to verify whether providing advice willaﬀect 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
increases the rate of truth-telling, while providing a detailed explanation of the mechanism has the opposite eﬀect.8 These results suggest that advice helps participants
conﬁrm individual optimality of truth-telling. To the best of our knowledge, our study
is the ﬁrst to test the role of advice in the literature of auction studies.9
To improve individuals’ understanding on incentive properties, Saijo et al. (2007)
and Li (2017) attempt to strengthen the notion of strategy-proofness. Saijo et al.
(2007) emphasize that, even if a mechanism is strategy-proof, it often admits the presence of an ineﬃcient Nash equilibrium composed of mis-reporting. Saijo et al. (2007)
strengthen strategy-proofness to “secure implementability” by additionally requiringthat no Nash equilibrium achieves an outcome that is not the “true” outcome. An
experimental study by Cason et al. (2006) compares bidding behavior under a securely
8

Guillen and Hakimov (2018) consider a 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 (2019)), peer information sharing in networks (Ding and Schotter (2017)), and third-party advice such as websites (Guillen and Hing (2014)).
9
Shogren et al. (2001) conduct a Vickrey auction experiment under the condition where subjects
are informed of the strategy-proofness of the Vickrey auction. However, they do not test the eﬀect of
the advice, that is, 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 providing such information aﬀects the rate of truth-telling.

4implementable (and so strategy-proof) mechanism and that of other strategy-proof
mechanisms. They observe that the rate of truth-telling in the former is higher thanin the latter.
Li (2017) proposes a stronger version of strategy-proofness called “obvious strategyproofness” that could be applied to dynamic mechanisms, such as ascending auctions
or matching algorithms. In a problem where a single object is auctioned, the ascending
auction is obviously strategy-proof, but the Vickrey auction is not. In an experiment
where a single object is auctioned, Li observes that the ratio of dominant strategies
played by subjects is signiﬁcantly higher in the ascending clock auction than in 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. Conversely, our approach of providing advice can be applied toany strategy-proof mechanism in any environment. The Vickrey auction is neither
securely implementable nor obviously strategy-proof, but providing advice drastically
improves its truth-telling performance.

1.3

Organization

The remainder of the paper is organized as follows. Section 2 describes the theoreticalbackground and our experimental design and proposes several hypotheses. Section 3
presents our experimental results. Section 4 describes the Ausubel auction treatment indetail and discusses our experimental results on the Ausubel auction. Finally, Section 5
provides the 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 vi1 ≥ vi2 ≥ 0 and vij denotes the value that bidder iassigns to the j-th unit. Given any vi , bidder i’s utility of obtaining k units of objects
and paying mi units of money is:10
10

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

5

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 ) is a valuation proﬁle. A bid submitted by bidder i is denoted bybi = (b1i , b2i ), where b1i ≥ b2i ≥ 0 and bji denotes the bid of bidder i for the j-th unit. Let
b = (b1 , b2 , b3 ) be a bid proﬁle.
An assignment function is a function d = (d1 , d2 , d3 ) that speciﬁes, for each bid
proﬁle b, the number of the objects bidder i obtains di (b) ∈ {0, 1, 2} under the following
resource constraint: d1 (b) + d2 (b) + d3 (b) = 2. A payment function is a function m =(m1 , m2 , m3 ) that speciﬁes, for each bid proﬁle b, the amount of money bidder i paysmi (b). An rule is a pair of assignment and payment functions (d, m).
We are interested in rules in which sincere bidding (i.e., bidding the true valuations)is a weakly dominant strategy for everyone. Formally:
Strategy-proofness: For each bidder i, each bidder i’s valuation vi , each bidder i’s
bid bi , and each other bidders’ bids b−i ,U (di (vi , b−i ), mi (vi , b−i ); vi ) ≥ U (di (bi , b−i ), mi (bi , b−i )); vi ).
The following rules are central in the literature on auction theory.
• Vickrey auction: Each bidder i simultaneously submits his/her bid bi = (b1i , b2i ).
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 i simultaneously submits hies/her bid bi =(b1i , b2i ). 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 (respectively, two) unit, then the bidder has to pay b1i(respectively, b1i + b2i ).
The most important feature of the Vickrey auction is that it is strategy-proof. However,many previous experimental studies report that bidders tend to overbid in Vickrey

6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 is not
strategy-proof.

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:• Treatment VA: the Vickrey auction with advice• Treatment VN: the Vickrey auction without advice• Treatment PA: the pay-your-bid auction with advice• Treatment PN: the pay-your-bid auction without advice
In each treatment, three bidders compete for two units of identical objects. For eachbidder, two integer valuations are drawn from the uniform distribution with the interval 0 to 1000 with increments of 10. Then, the larger (respectively, smaller) integer
is assigned the valuation for the ﬁrst (respectively, second) unit. All valuations are
displayed in Japanese yen (JPY).
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.11 The experimenterread aloud the instructions. Then, subjects answered a 17-question quiz that tested
11

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

7

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)

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,
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 2000.
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. Individual payments including (or excluding) participation feesand quiz points ranged from $19.1 to $85.4 (or from $5.9 to $70.3).

2.3

Hypotheses

We examine whether the following three factors aﬀect sincere bidding behavior: auction
rule, advice, and unit. Thereafter, we apply a normal approximation, unless noted
otherwise.12
12

See Online Appendix A for the detail of statistics.

8

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.

Although it is well known in the literature that the Vickrey auction is strategy-proof,
several experimental results demonstrate that this fact is non-trivial for an average
person. Thus, we predict that the subjects may become aware of the beneﬁt of truthtelling if we provide advice on the strategy-proofness of the Vickrey auction to thesubjects. To examine whether this conjecture is true, we formulate the following null
hypothesis:
Hypothesis 1 (Advice eﬀect in the Vickrey auction). Providing advice does notincrease the rate of sincere bidding in the Vickrey auction.
Even if we conﬁrm that the advice eﬀect exists in the Vickrey auction, it is very
probable that its eﬀect involves the so-called experimenter demand eﬀect (Zizzo (2009)).

9

An excess of eﬀect over the experimenter demand eﬀect is considered a “net eﬀect” ofthe advice to promote sincere bidding in the Vickrey auction. We call this eﬀect the
net advice eﬀect.
As the advice provided in both the Vickrey auction and the pay-your-bid auction is
identical, the experimenter demand eﬀects would be similar in both auctions. Moreover,as the advice is false in the pay-your-bid auction, the advice eﬀect is an all experimenter
demand eﬀect. Thus, we assume that the net advice eﬀect in the Vickrey auction canbe measured by the diﬀerence in the advice eﬀects in the two auctions. To detect
the net advice eﬀect in the Vickrey auction under this assumption, we formulate thefollowing null hypotheses:
Hypothesis 2 (Advice eﬀect in the pay-your-bid auction). Providing advice
does not increase the rate of sincere bidding in the pay-your-bid auction.
Hypothesis 3 (Net advice eﬀect in the Vickrey auction). The net advice eﬀect
on sincere bidding does not exist in the Vickrey auction.
Subjects who fully understand the procedure of the Vickrey auction are expected
to quickly recognize the strategy-proofness of the Vickrey auction based on the advice,even if they are not initially aware. The subjects with perfect scores could be consideredthose who understand the procedure of the Vickrey auction well, whereas the subjects
with imperfect scores could be considered those who do not. Considering this case,
we predict that subjects with perfect scores would be more likely to follow the advice
than those with imperfect scores. To test this prediction, we formulate the following
null hypothesis:
Hypothesis 4 (Relationship between quiz scores and advice eﬀects). The
advice eﬀect on sincere bidding does not depend on the pre-play understanding of theVickrey auction.
As mentioned in the introduction, several experimental studies have observed overbidding in the Vickrey auction. This could cause a severe loss of eﬃciency. Therefore,
we assume that, in the Vickrey auction, providing advice would improve eﬃciency by
reducing such overbidding behavior. To examine whether this assumption is true, we
formulate the following null hypothesis:
Hypothesis 5 (Advice eﬀect on eﬃciency in the Vickrey auction). Providing
advice does not improve eﬃciency within the Vickrey auction.

10

Figure 2: Distribution of quiz scores.

As in Hypothesis 3, we consider that the net advice eﬀect on eﬃciency in the
Vickrey auction can be measured by the diﬀerence in the advice eﬀects on eﬃciency
across the two auctions. To test whether such a net advice eﬀect on eﬃciency exists inthe Vickrey auction, we formulate the following null hypothesis:
Hypothesis 6 (Net advice eﬀect on eﬃciency in the Vickrey auction). The
net advice eﬀect on eﬃciency does not exist in the Vickrey auction.

3

Experimental results

3.1

Level of understanding of the auction rules

Figure 2 displays the distributions of quiz scores. The scores range from 0 to 17. A
ﬁrst look at Figure 2 clearly shows that over 60% of the subjects received a perfectscore in the Vickrey auction, while over 80% did so in the pay-your-bid auction. For
each auction rule, we ﬁnd no statistical diﬀerence in the score distributions with and11without advice.13 Similarly, we ﬁnd no statistical evidence to support the diﬀerent
levels of understanding between the two auctions, regardless of the presence of advice.14
In summary, we obtain the following result.
Result 1 (Level of understanding of the auction rules). The majority of subjects
had a thorough understanding of the auction rule prior to playing, regardless of whether
they participated in the Vickrey or the pay-your-bid auction.

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 (or under ) if the bid is more(or 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), and b3 = (900, 0),then the sincere bids are the ﬁrst and second units of bidder 2 and the ﬁrst unit ofbidder 3, respectively. 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 bidders 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 graphs for the Vickrey treatments suggest that, for unit 1, bidders with higher
valuations are more likely to overbid. When valuations are close to the maximum,
1000, bids jump as high as 2000. For unit 2, bidding behavior is polarized with anincrease in underbidding: overbidding is prevalent regardless of the realized valuations.
Further, we observe a cluster of zero biddings when the valuation for unit 2 is no
more than 500. Table 2 summarizes the classiﬁcation of bids into three categories bytreatments and periods.15
As presented in Panel (a) of Table 2 (VA−VN column), sincere bidding increases by
26.3% with the statement of strategy-proofness in the Vickrey auction. In particular,
we observe, from Panels (b) and (c) of Table 2, that the rate of sincere bidding increases
further in the former 13 periods than in the latter 12 periods. Speciﬁcally, the rate ofsincere bidding increases by 28.2% in the former 13 periods, while the rate increases
13

A two-sample Kolmogorov-Smirnov (K-S) test for the equality of two score distributions has ap-value > 0.90 for both VA vs. VN and PA vs. PN. The same conclusions also hold under a chi-square
test.
14
The K-S test has 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.
15
See Online Appendix S for the individual bidding data.

12

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

13

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

Vickrey
Pay-your-bid
Bid category
VA
VN
VA − VN
PA
PN
PA − PN
0.469
0.206
0.263∗∗∗ 0.083
0.058
0.025∗∗∗
Sincere(0.499) (0.404)(0.276) (0.234)
0.453
0.634 −0.181∗∗∗ 0.006
0.009 −0.004∗∗
Over(0.498) (0.482)(0.074) (0.095)
∗∗∗
0.078
0.160 −0.083
0.911
0.933 −0.021∗∗∗Under(0.268) (0.367)(0.284) (0.250)(b) Former 13 periods

Bid category
SincereOver
Under

Vickrey
Pay-your-bid
VA
VN
VA − VN
PA
PN
PA − PN
∗∗∗
0.467
0.185
0.282
0.093
0.058
0.034∗∗∗(0.499) (0.389)(0.290) (0.234)
∗∗∗
0.453
0.629 −0.176
0.004
0.015 −0.010∗∗∗(0.498) (0.483)(0.067) (0.121)
0.080
0.186 −0.106∗∗∗ 0.903
0.927 −0.024∗∗∗(0.271) (0.389)(0.296) (0.261)(c) Latter 12 periods

Vickrey
Pay-your-bid
Bid category
VA
VN
VA − VN
PA
PN
PA − PN
∗∗∗
0.472
0.228
0.243
0.073
0.058
0.015∗∗
Sincere(0.499) (0.420)(0.260) (0.234)
∗∗∗
0.453
0.639 −0.186
0.007
0.003
0.004∗
Over(0.498) (0.481)(0.081) (0.054)
∗∗∗
0.075
0.133 −0.057
0.920
0.939 −0.019∗∗
Under(0.264) (0.340)(0.271) (0.239)
Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level. Numbers inparentheses are standard deviations.

14

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 observationis consistent with ﬁndings in the 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 is overbidding; this implies that our advice decreases overbidding by
18.1%. Moreover, we ﬁnd that underbidding also decreases when advice is given to
the subjects. Speciﬁcally, our advice decreases underbidding by 8.3%. To compare theeﬀect of the 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. Thus, we ﬁnd that the advice
has more eﬀect on underbids (8.26/16.04) than overbids (18.07/63.38). Even when webreak down the data into the former 13 periods and the latter 12 periods, (Panels (b)and (c) of Table 2), the same tendency holds. Conversely, approximately 90% of thesubjects in PA and PN are likely to underbid. We also observe a 3%-increase of sincere
bidding although the advice is not true in this case.16

3.3

Advice eﬀect

Table 3 summarizes the frequencies of sincere bidding by treatments, whether the quizscore is perfect, and by units. Table 4 summarizes the frequency of sincere bidding in
the former period (periods 1–13) and in the latter period (periods 14–25).17
Using mainly both units’ data shown in Panel (a) of Table 3, we ﬁrst test whether
the advice increases sincere bidding in each auction rule. We summarize our ﬁndings
as follows:
Result 2 (Advice eﬀect).
(i) Providing advice increases the rate of sincere bidding in each auction rule whenusing all data. The mean increases are 26.3% (95% conﬁdence interval, 24.0% to28.7%) 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.
16

A simple regression of all bids to valuations strongly shows that the coeﬃcients are 0.8032 for PA
and 0.7927 for PN (for the multi-unit pay-your-bid auction, see Lebrun and Tremblay (2003)).
17
We will examine whether the subject’s bidding behavior changes over time in Section 3.4.

15

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

Data
All
Perfect Score
Imperfect Score

Vickrey(1)
Pay-your-bid
(2)
VA
VN
VA − VN
PA
PN
PA − PN
0.469
0.206
0.263∗∗∗
0.083
0.058
0.025∗∗∗(0.499) (0.404)(0.276) (0.234)
0.523
0.211
0.313∗∗∗
0.079
0.046
0.034∗∗∗(0.500) (0.408)(0.270) (0.209)
0.361
0.198
0.163∗∗∗
0.104
0.105 −0.002(0.481) (0.399)(0.305) (0.307)(1) − (2)0.238∗∗∗
0.279∗∗∗
0.165∗∗∗(b) Unit 1

Data
All
Perfect Score
Imperfect Score

Vickrey(1)
VA
VN
VA − VN
0.416
0.175
0.241∗∗∗(0.493) (0.380)
0.463
0.176
0.287∗∗∗(0.499) (0.381)
0.322
0.173
0.149∗∗∗(0.468) (0.379)

Pay-your-bid
(2)
PA
PN
PA − PN (1) − (2)
0.020
0.006
0.014∗∗∗
0.227∗∗∗(0.141) (0.078)
0.018
0.004
0.014∗∗∗
0.273∗∗∗(0.133) (0.065)
0.033
0.013
0.019∗
0.129∗∗∗(0.178) (0.115)(c) Unit 2

Data
All
Perfect Score
Imperfect Score

Vickrey(1)
Pay-your-bid
(2)
VA
VN
VA − VN
PA
PN
PA − PN
0.523
0.237
0.286∗∗∗
0.146
0.110
0.036∗∗∗(0.500) (0.425)(0.353) (0.313)
∗∗∗
0.584
0.245
0.338
0.141
0.087
0.054∗∗∗(0.493) (0.431)(0.348) (0.282)
0.400
0.222
0.178∗∗∗
0.175
0.197 −0.023(0.490) (0.416)(0.380) (0.399)(1) − (2)0.250∗∗∗
0.285∗∗∗
0.201∗∗∗

Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level. Numbers inparentheses are standard deviations.

(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. We ﬁrst establish (i). Hypothesis 1 is rejected at p < 0.001. Similarly,
Hypothesis 2 is rejected at p < 0.001. The conﬁdence intervals are calculated using
the standard normal distribution. To establish (ii) and (iii), we divide the data into
two types: the subjects with perfect scores and those with imperfect scores. Regarding(ii), Hypothesis 1 is rejected at p < 0.001 in both the data of the subjects with perfectscores and that of those with imperfect scores. We ﬁnally establish (iii). Hypothesis 2 isrejected in the data of the subjects with perfect scores (p < 0.001), while Hypothesis 2
is not rejected in the data of the subjects with imperfect scores (p = 0.461).

16

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

Data
All
Perfect Score
Imperfect Score

Vickrey(1)
VA
VN
VA − VN
0.467
0.185
0.282∗∗∗(0.499) (0.389)
0.511
0.182
0.329∗∗∗(0.500) (0.386)
0.379
0.190
0.189∗∗∗(0.486) (0.393)

Pay-your-bid
(2)
PA
PN
PA − PN (1) − (2)
0.093
0.058
0.034∗∗∗ 0.248∗∗∗(0.290) (0.234)
0.088
0.045
0.043∗∗∗ 0.286∗∗∗(0.284) (0.208)
0.115
0.108
0.008
0.181∗∗∗(0.320) (0.310)(b) Latter 12 periods

Data
All
Perfect Score
Imperfect Score

Vickrey(1)
Pay-your-bid
(2)
VA
VN
VA − VN
PA
PN
PA − PN (1) − (2)
0.472
0.228
0.243∗∗∗
0.073
0.058
0.015∗∗ 0.228∗∗∗(0.499) (0.420)(0.260) (0.234)
0.537
0.242
0.295∗∗∗
0.070
0.046
0.024∗∗∗ 0.271∗∗∗(0.499) (0.428)(0.255) (0.210)
0.341
0.206
0.135∗∗∗
0.091
0.103 −0.012
0.147∗∗∗(0.475) (0.405)(0.288) (0.304)

Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level. Numbers inparentheses are standard deviations.

The advice eﬀect in the pay-your-bid auction reported in Result 2 should be considered an all experimenter demand eﬀect as the advice is false in that auction. Thus,
one might think that the advice eﬀect in the Vickrey auction might also be an experimenter demand eﬀect. However, Result 2 also reports that the advice eﬀect inthe Vickrey auction is much larger than that in the pay-your-bid auction. Moreover,the experimenter demand eﬀects would be similar in both auctions because the advice
provided in both auctions is identical. These observations suggest that the net advice
eﬀect could exist in the Vickrey auction. The following result supports this.
Result 3 (Net advice eﬀect). Providing advice increases the rate of sincere bidding
further in the Vickrey auction compared with the pay-your-bid auction, with the mean
increase of 23.8% (95% conﬁdence interval, 21.2% to 26.5%).
Support. Hypothesis 3 is rejected at p < 0.001. The conﬁdence interval is calculatedusing the standard normal distribution.
The diﬀerence, 23.8%, of the advice eﬀects in the two auctions is considered the
net advice eﬀect in the Vickrey auction. Many authors investigate why subjects do not
bid sincerely in the Vickrey auction by proposing alternative explanations. Result 317

suggests that a considerable amount of such behavior might be mitigated by providing
advice that helps understand the strategy-proofness of the Vickrey auction.
We ﬁnally test whether quiz scores aﬀect the responsiveness to our advice in the
Vickrey auction. The following result demonstrates that the responsiveness to ouradvice 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 thanthose with imperfect quiz scores. The mean diﬀerence between the two groups in termsof an increase in the rate of sincere bidding is 15.0% (95% conﬁdence interval, 10.1%
to 20.0%).
Support. Hypothesis 4 is rejected at p < 0.001. The conﬁdence interval is calculatedusing the standard normal distribution.
Result 4 suggests that under the advice, subjects who properly understand theprocedure of the Vickrey auction tend to bid more sincerely than those who do not.

3.4

Regression analysis of sincere bidding

We perform a regression analysis to check the robustness of the results, and almost all
results are conﬁrmed. We employ a logit model with a random eﬀect at the individuallevel. The dependent variable is a dummy variable of sincere bidding, which takes thevalue of 1 if subject i in period t bids sincerely. The three independent variables are thesame ones we examined independently above: Advice is a dummy for treatments withadvice; Latter is a dummy for periods more than 13; and Perfect Score is a dummy
for subjects with a perfect score in the quiz. We also include the interaction terms ofthese three variables. Table 5 summarizes the results by auction rules and units. For
each auction rule and each unit, we run three regressions with diﬀerent speciﬁcations.
The highlights of the logit model regression results are as follows.
• Vickrey: Advice in speciﬁcations from (1) to (6) has positive coeﬃcients forboth units, supporting Result 2. Meanwhile, Advice × Perfect Score has an
insigniﬁcant impact. This is seemingly inconsistent with Result 4, but seems anatural consequence of allowing a random eﬀect, where the impact is regarded as
individual characteristics. Moreover, the insigniﬁcant eﬀect in Latter and Advice
× Latter in speciﬁcations (3) and (6) suggests that the rate of sincere bidding

18

Table 5: Regression analysis of sincere bidding.
(a) Vickrey

AdvicePerfect Score
Advice × Perfect Score(1)
1.986∗(1.103)−0.135(1.059)1.308(1.375)

Unit 1(2)
3.289∗∗∗(0.709)0.652∗∗∗(0.225)−0.802∗∗∗(0.273)LatterAdvice × Latter
Latter × Perfect Score
Advice × Latter × Perfect Score
Constant
Observations
Akaike Inf. Crit.

−3.253∗∗∗ −3.707∗∗∗(0.845)(0.565)2,775
2,775
1,909.857 1,902.138(3)
2.291∗∗(1.135)−0.583(1.100)1.563(1.420)0.187(0.347)−0.622(0.438)0.798∗(0.459)−0.373(0.564)−3.360∗∗∗(0.872)
2,775
1,903.642(4)
1.829∗∗(0.888)0.613(0.831)0.732(1.105)

Unit 2(5)
2.466∗∗∗(0.556)(6)2.044∗∗(0.916)0.424(0.861)0.592(1.141)0.452∗∗
0.211(0.182)(0.309)−0.236−0.433(0.231)(0.389)0.363(0.382)0.337(0.485)−2.323∗∗∗ −2.173∗∗∗ −2.432∗∗∗(0.662)(0.422)(0.685)
2,775
2,775
2,775
2,193.692 2,189.629 2,187.752(b) Pay-your-bid

AdvicePerfect Score
Advice × Perfect Score
LatterAdvice × Latter

(7)
1.177
(1.111)−1.091(0.970)0.447(1.290)

Unit 1(8)
1.100∗(0.660)−0.506(0.643)0.783(0.735)Latter × Perfect Score
Advice × Latter × Perfect Score
Constant
Observations
Akaike Inf. Crit.

−5.841∗∗∗ −6.549∗∗∗(0.949)(0.761)3,525
3,525
445.609446.079(9)0.215(1.240)−1.594(1.079)1.405(1.476)−1.443(1.160)2.334∗(1.381)
1.541(1.425)−2.343(1.660)−5.379∗∗∗(0.974)3,525
450.302(10)−0.005(0.664)−0.936∗(0.496)0.609(0.740)

Unit 2(11)
0.673∗∗(0.317)(12)0.331(0.699)−0.924∗(0.529)0.469(0.782)0.028
0.046(0.172)(0.308)−0.499∗∗ −0.778(0.234)(0.497)−0.025(0.371)0.341(0.566)−2.144∗∗∗ −2.901∗∗∗ −2.169∗∗∗(0.439)(0.237)(0.465)
3,525
3,525
3,525
2,231.654 2,226.571 2,230.219

Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level. Numbers inparentheses are standard errors.

19

does not change over time and the advice quickly improves the rate of sincere
bidding in earlier periods.18 This quick eﬀect of advice is consistent with Table 4.
• Pay-your-bid: Speciﬁcation (9) suggests that, with advice, the sincere bidding
rate marginally increases over time. Speciﬁcation (12) with the negative coeﬃcient of Perfect Score conﬁrms that the subjects become less likely to bid sincerely
as they understand the auction rule correctly. These are consistent with Table 2
and 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) does not. Using an
additional regression, we do not ﬁnd such evidence in any of the two auction rules, andthis is consistent with Harstad (2000).

3.5

Eﬃciency, bidders’ payoﬀs, and seller’s revenue

Our eﬃciency measure is taken from Kagel and Levin (2009).19 In one game, if bidder
i is the winner with the highest bid and if bidder j is the winner with the secondvi1 + vj1
highest bid, then the observed eﬃciency ratio r is given by
if i 6= j and byv[1] + v[2]vi1 + vi2
if i = j, where v[1], v[2] denote the two highest units among six valuationsv[1] + v[2](v11 , v12 , v21 , v22 , v31 , v32 ). The eﬃciency ratio of the full or part of a treatment is the averageof the eﬃciency ratios in the games in the full treatment or part of the treatment.
Table 6 summarizes the results on eﬃciency with a focus on experience.20 In Table 6,
the upper (or middle, bottom) panel collects the results obtained for all periods (periods
1–13 and periods 14–25, respectively).
Result 5 (Advice eﬀect on eﬃciency). Providing advice improves eﬃciency in the
Vickrey auction, particularly in the former 13 periods.
Support. Hypothesis 5 is rejected at the 5% signiﬁcance level (p = 0.020). To obtainfurther insights, we divide the data into two types: periods 1–13 and 14–25. Then,
Together with the insigniﬁcant eﬀect in Advice × Latter × Perfect Score, this result suggests thatthe learning process is similar between groups with diﬀerent levels of understanding of auction rules
whenever advice is provided.
19
See also Kwasnica and Sherstyuk (2013).
20
Englemann and Grimm (2009) compare the Vickrey auction with the pay-your-bid auction by
conducting an experiment, where two units of an identical object are auctioned to two bidders withﬂat demand. Their experimental results are similar to ours in regard to the comparison of the eﬃciency
ratios in the two auctions.
18

20

Table 6: Eﬃciency by treatments and periods.

Data
All periods
Periods 1–13
Periods 14–25

Vickrey(1)
VA
VN
VA − VN
0.977
0.967
0.010∗∗(0.069) (0.081)
0.970
0.956
0.015∗∗(0.080) (0.098)
0.984
0.978
0.006
(0.053) (0.054)

Pay-your-bid
(2)
PA
PN
PA − PN (1) − (2)
0.990
0.988
0.002
0.009∗(0.028) (0.029)
0.991
0.989
0.002
0.012∗(0.026) (0.029)
0.989
0.988
0.001
0.004
(0.030) (0.028)

Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level. Numbers inparentheses are standard deviations.

Hypothesis 5 is rejected at the 5% signiﬁcance level in the data of periods 1–13 (p =0.039), while Hypothesis 5 is not rejected in the data of periods 14–25 (p = 0.143).
Result 5 suggests that improvements in the eﬃciency of the Vickrey auction occur
mainly when the subjects are less experienced.
Result 6 (Net advice eﬀect on eﬃciency). The net advice eﬀect on eﬃciency
exists in the Vickrey auction, particularly in the former 13 periods.
Support. Hypothesis 6 is rejected at the 10% signiﬁcance level (p = 0.051). Again, we
divide the data into two: periods 1–13 and 14–25. Then, Hypothesis 6 is rejected at
the 10% signiﬁcance level in the data of periods 1–13 (p = 0.074), while Hypothesis 6
is not rejected in the data of periods 14–25 (p = 0.218).
Result 6 suggests that a relative improvement in the eﬃciency of the Vickrey auction
compared with the pay-your-bid auction occurs mainly when the subjects are lessexperienced.21
We ﬁnally ask whether the advice aﬀects bidders’ payoﬀs and the seller’s revenue
in each auction rule. Table 7 summarizes the results on the bidders’ payoﬀs and theseller’s revenue. We then make the following observations:
Result 7 (Bidders’ payoﬀs and seller’s revenue).
(i) The advice marginally increases (or decreases) the bidder’s payoﬀ per auction in
the Vickrey (or pay-your-bid) auction using all data.
21

We also test the same hypotheses using an alternative way of deﬁning eﬃciency, which we callthe “binary eﬃciency.” In this eﬃciency measure, we focus only on whether an eﬃcient allocation isrealized in a group observation. Then, Result 5 holds even if we use the binary eﬃciency as a measureof eﬃciency (see Online Appendix B for more details).

21

Table 7: Average bidder’s payoﬀ and average seller’s revenue per auction.
(a) Bidder’s payoﬀ

VickreyData
All periods
Periods 1–13
Periods 14–25

VA
VN
VA − VN
136.844122.292
14.553∗∗(225.109) (221.572)
132.247
115.000
17.247∗(235.584) (231.757)
141.825130.191
11.634(213.225) (209.896)

Pay-your-bid
PAPN
PA − PN
82.82989.928−7.099∗∗(105.467) (111.454)
78.99783.301−4.305(101.469) (107.679)
86.98197.106−10.126∗∗(109.545) (115.037)(b) Seller’s revenue

Pay-your-bid
VA
VN
VA − VN
PA
PN
PA − PN
1139.410 1125.875
13.5351303.704 1288.700
15.004∗(397.611) (431.408)(180.917) (196.875)
1144.689 1142.933
1.756
1313.211 1303.397
9.813(406.210) (428.434)(190.073) (211.432)
1133.690 1107.396
26.2951293.406 1272.778
20.628∗(388.806) (434.969)(170.189) (178.786)
VickreyData
All periods
Periods 1–13
Periods 14–25

Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level. Numbers inparentheses are standard deviations.

(ii) Providing advice does not aﬀect the seller’s revenue in the Vickrey auction, while
the advice marginally increases the seller’s revenue in the pay-your-bid auction
using all data.
The increase in the bidders’ payoﬀs in the Vickrey auction is consistent with theeﬃciency gain that we reported previously.

4

Ausubel auction

We further conducted an experiment to test the eﬀect of advice for the Ausubel auction(Ausubel (2004)), which is considered a dynamic version of the Vickrey auction. In
the Ausubel auction (Ausubel (2004)), rather than reporting their valuations for theobjects to the auctioneer, all bidders gradually reveal their demands at given prices (orequivalently, their dropout prices), until the auction ends. The Ausubel auction canbe described as follows:• Ausubel auction: In this auction, a price is gradually increased from 0. At
each price p, each bidder i reports their demand Di (p) ∈ {0, 1, 2}. Note that, for22each bidder i, Di (0) = 2 and Di are non-increasing in p. For each price p andeach bidder i, let D(p) ≡ D1 (p) + D2 (p) + D3 (p) and D−i (p) ≡ D(p) − Di (p)denote the aggregate demand at p and the aggregate demand of all other bidders
except i, respectively. As the price p rises, the aggregate demand D(p) will drop,and, at a certain p∗ , there is a bidder, say bidder i∗ , such that D−i∗ (p∗ ) = 1. At
this price, the competitors of bidder i∗ demand one unit less than the available
number of units (two units); thus, bidder i∗ surely obtains one unit. Then, we
say that bidder i∗ has clinched one unit. Bidder i∗ ’s payment for it is the priceat which they have clinched it, p∗ . Therefore, both the available number of units
and the bidder i∗ ’s demand are reduced by one unit, and the auction continues
until the available number of units becomes 0.
We focus on the Ausubel auction with dropout information, where all dropout
prices are publicly revealed as they occur. In this auction rule, bidder i’s sincere
bidding, which consists of both reporting two demands until the price equals vi2 andreporting one demand until the price equals vi1 , is not a weakly dominant strategy(Kagel and Levin (2009)).22 However, it is well known that sincere bidding is an expost equilibrium (Ausubel (2004); Okamoto (2018)).23 Therefore, one can expect thatproviding advice on this property improves the performance of the Ausubel auction,
similar to that of the Vickrey auction.

4.1

Experimental procedures

To compare the results of the Ausubel auction with those of the Vickrey auction, theexperimental settings for the Ausubel auction are the same as the experimental settings
for the Vickrey auction. This follow-up experiment consists of two treatments:• Treatment AA: the Ausubel auction with advice• Treatment AN: the Ausubel auction without advice
We conducted three experimental sessions in each of AA and AN at Osaka University in February 2020. A total of 9, 21, or 24 subjects participated in each session.
22

If dropout information is not publicly announced (i.e., the Ausubel auction without dropout
information), sincere bidding is a weakly dominant strategy for everyone (Ausubel (2004); Kagel andLevin (2009)).
23
This is equivalent to saying that the Ausubel auction is ex-post incentive compatible. Moreover,In the Ausubel auction, sincere bidding is an ex-post equilibrium even oﬀ the equilibrium paths; thatis, it is an ex-post perfect equilibrium (Ausubel (2004); Okamoto (2018)).

23

Table 8: Summary of treatments.

Treatment
AA
AN

Auction rule Advice Date
AusubelYes
Feb-20
AusubelNo
Feb-20# of Sessions # of Subjects (Groups)
3
54 (18)3
57 (19)We recruited student subjects from Osaka University through campus-wide advertisements. None of the students were experienced in this particular type of experiment,
and no subject attended more than one session. Table 8 summarizes the number ofobservations.
The experimental procedure is exactly the same as that described in Section 2.2
except for the content of advice.24 Speciﬁcally, the text of the advice is as follows:“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 reducing your demands at your values as
they are, regardless of the prices at which others reduce their demands.”
Note that the advice involves no deception problem for AA. One period of the Ausubel
auction proceeds as follows. The price is gradually increased from 0 to 2000, whichis equal to the maximum possible bid in our Vickrey treatments. The price rises inincrements of 10 JPY every two seconds.25 Each subject clicks a button on the screen
to reduce his/her demand by one unit. The auction ends when the aggregate demand
becomes two (i.e., the button is clicked four times), unless the price reaches 2000.
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. Individual payments including (or excluding) participation feesand quiz points ranged from $32.6 to $91.5 (or from $18.1 to $76.4).

4.2

Hypotheses

Similarly to Section 2.3, we apply a normal approximation to examine whether thefollowing three factors aﬀect sincere bidding behavior: auction rule, advice, and unit.
24

The full set of experimental instructions (including screen shots, quiz, and questionnaire) is provided in Online Appendix M–R.
25
The price increased by 10 JPY every 3 seconds in the ﬁrst sessions in each of AA and AN. However,to complete one session within three hours, we subsequently modiﬁed the clock speed to 2 seconds
from the second session onward.

24

As mentioned previously, sincere bidding is an ex-post equilibrium in the Ausubel
auction. Based on this theoretical result, we expect that providing advice could increase
the rate of sincere bidding in the Ausubel auction similar to the Vickrey auction. To
examine whether this conjecture is true, we formulate the following null hypothesis:
Hypothesis 1’ (Advice eﬀect in the Ausubel auction). Providing advice does
not increase the rate of sincere bidding in the Ausubel auction.
If an advice eﬀect exists in the Ausubel auction, then not all of the eﬀect would be
the experimenter demand eﬀect because the advice is correct in the Ausubel auction.
Thus, the Ausubel auction would have a net advice eﬀect. We are concerned withthe diﬀerence in the net advice eﬀect between strategy-proof and non-strategy-proof
mechanisms (i.e., the Vickrey and Ausubel auctions) at this time. Since winners’
demand reductions are censored in the Ausubel auction, we focus on non-winning bids
in the Vickrey treatments for a fair comparison of these two auction rules. To examine
this comparison, we consider the following null hypothesis:
Hypothesis 3’ (Relative eﬀect of advice on sincere bidding). When we focus
on non-winning bids, no diﬀerence exists in the advice eﬀect on sincere bidding between
Ausubel and Vickrey auctions.
Similar to the Vickrey auction, we predict that subjects with perfect scores would
be more likely to follow the advice than those with imperfect scores even in the Ausubel
auction. We also expect that providing advice would improve eﬃciency even in the
Ausubel auction. To test these predictions, we formulate the following null hypotheses:
Hypothesis 4’ (Relationship between quiz scores and advice eﬀects). The
advice eﬀect on sincere bidding does not depend on the pre-play understanding of the
Ausubel auction.
Hypothesis 5’ (Advice eﬀect on eﬃciency in the Ausubel auction). Providing
advice does not improve eﬃciency within the Ausubel auction.

4.3

Level of understanding and bidding behavior

Figure 4 displays the distributions of quiz scores. The scores range from 0 to 17.
In both treatments, almost half of the subjects (50.0% and 49.3% in AA and AN,
respectively) had a perfect score in the quiz conducted immediately after instructions
had been provided. Then, we obtain the following result:25

Figure 4: Distribution of quiz scores for each treatment prior to the auctions.

Result 8 (Level of understanding of the Ausubel auction). In the Ausubel
auction, 49.6% of the subjects had a thorough understanding of the auction rule prior
to playing.
Figure 5 illustrates how bidding behavior contrasts between units in the Ausubel
auction. The subjects typically bid sincerely for the ﬁrst unit, while they overbid forthe second unit. This behavior explains 59.0% and 52.9%, respectively, of the ﬁrst andsecond unit bids, when combining the data of the AA and AN. A full categorizationof each bid into “Sincere,” “Over,” and “Under” is summarized in Table 9. Bids are
more likely to be sincere over time when the advice is provided.26,27

26Interestingly, the dynamic nature of the Ausubel auction induces a unique type of overbidding,which does not prevail in the Vickrey auction treatments: The Ausubel auction subjects tend to
reduce demand for the ﬁrst time when the price reaches the average of two drawn values, (v1 + v2 )/2.
We now regard the average bidding as the demand reduction within $0.1 of the average of the twovaluations. It appears that 16.9% (= 213/1263) and 11.2% (= 143/1276) of the overall ﬁrst demand
reductions in AA and AN, respectively, are caused by such average bidding behavior. A bid observation
can simultaneously be interpreted as both the average bid and a sincere one. For example, consider
the case where a subject with (v1 , v2 ) = (500, 490) reduced demand to one at p = 490.
27
Sincere bidding for the ﬁrst unit has a justiﬁcation both from the theoretical and experimental
perspectives. Baisa (2020) theoretically shows that two types of bids dominate in the multi-unit
Vickrey auction: non-sincere bidding for the ﬁrst unit and underbidding for the second unit. Moreover,
we can regard the situation for the Ausubel subject after his/her ﬁrst demand reduction as the singleunit English auction. Recall that subjects in English auction experiments have revealed to recognize
easily that it is beneﬁcial to remain in the auction up to their own value (Li (2017); McGree andLevin (2019); and Breitmoser and Schweighofer-Kodritsc (2021)). Accordingly, it is natural that mostsubjects bid sincerely for the ﬁrst unit in our Ausubel auction treatments.

26

Figure 5: Scatter plots of bids. Green × and purple • indicate bids for the former 13 periods
and latter 12 periods, respectively.

27

Table 9: Bid category by treatments and periods in the Ausubel auction.
(a) All

Bid category
Sincere
Over
Under

AA
AN
AA − AN
0.389
0.305
0.084∗∗∗(0.488) (0.460)
0.415
0.427 −0.012(0.493) (0.495)
0.196
0.268 −0.072∗∗∗(0.397) (0.443)(b) Former 13 periods

Bid category
Sincere
Over
Under

AA
AN
AA − AN
0.339
0.295
0.044∗∗(0.474) (0.456)
0.451
0.418
0.033∗(0.498) (0.493)
0.210
0.287 −0.077∗∗∗(0.408) (0.453)(c) Latter 12 periods

Bid category
Sincere
Over
Under

AA
AN
AA − AN
0.443
0.317
0.126∗∗∗(0.497) (0.466)
0.376
0.437 −0.060∗∗∗(0.485) (0.496)
0.181
0.246 −0.066∗∗∗(0.385) (0.431)

Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level. Numbers inparentheses are standard deviations.

28

Table 10: Frequency of sincere bidding, by treatments, quiz scores, and units.
(a) Both units

Data
All
Perfect Score
Imperfect Score

Ausubel(1)
Vickrey(2)
AA
ANAA − AN
VA
VN
VA − VN (1) − (2)
0.389
0.305
0.084∗∗∗ 0.550
0.240
0.310∗∗∗ −0.226∗∗∗(0.488) (0.460)(0.498) (0.427)
0.427
0.295
0.132∗∗∗ 0.602
0.255
0.347∗∗∗ −0.215∗∗∗(0.495) (0.456)(0.490) (0.436)
0.351
0.314
0.037∗∗
0.434
0.216
0.218∗∗∗ −0.181∗∗∗(0.477) (0.464)(0.496) (0.412)(b) Unit 1

Data
All
Perfect Score
Imperfect Score

Ausubel(1)
Vickrey(2)
AA
ANAA − AN
VA
VN
VA − VN (1) − (2)
0.672
0.512
0.160∗∗∗ 0.573
0.226
0.347∗∗∗ −0.187∗∗∗(0.470) (0.500)(0.495) (0.419)
0.685
0.541
0.144∗∗∗ 0.616
0.252
0.365∗∗∗ −0.220∗∗∗(0.465) (0.499)(0.487) (0.435)
0.659
0.486
0.173∗∗∗ 0.457
0.189
0.268∗∗∗ −0.095∗(0.475) (0.501)(0.500) (0.393)(c) Unit 2

Data
All
Perfect Score
Imperfect Score

Ausubel(1)
Vickrey(2)
AA
ANAA − AN
VA
VN
VA − VN (1) − (2)
0.268
0.213
0.055∗∗∗ 0.540
0.246
0.294∗∗∗ −0.239∗∗∗(0.443) (0.410)(0.499) (0.431)
∗∗∗
0.316
0.189
0.127
0.595
0.257
0.339∗∗∗ −0.212∗∗∗(0.465) (0.392)(0.491) (0.437)
0.220
0.236 −0.016
0.425
0.229
0.197∗∗∗ −0.212∗∗∗(0.415) (0.425)(0.495) (0.421)

Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level. Numbers
in parentheses are standard deviations. In the Vickrey treatments, this table focuses on non-winning
bids.

4.4

Advice eﬀect

We now examine whether the advice eﬀect exists in the Ausubel auction. We also
compare the Ausubel auction with the Vickrey auction. As mentioned previously, thissection focuses on non-winning bids in the Vickrey treatments to fairly compare these
two auction rules.
Table 10 summarizes the frequencies of sincere bidding by treatments, whether
the quiz score is perfect, and by units. Table 11 summarizes the frequency of sincere
bidding in the former period (periods 1–13) and in the latter period (periods 14–25).28
28

We will examine whether the subject’s bidding behavior changes over time in Online Appendix S.

29

Table 11: Frequency of sincere bidding by quiz scores and periods.
(a) Former 13 periods

Data
All
Perfect Score
Imperfect Score

Ausubel(1)
Vickrey(2)
AA
ANAA − AN
VA
VN
VA − VN (1) − (2)
0.339
0.295
0.044∗∗
0.560
0.218
0.342∗∗∗ −0.298∗∗∗(0.474) (0.456)(0.497) (0.413)
0.390
0.289
0.101∗∗∗ 0.603
0.214
0.389∗∗∗ −0.288∗∗∗(0.488) (0.454)(0.490) (0.411)
0.284
0.300 −0.016
0.459
0.223
0.237∗∗∗ −0.253∗∗∗(0.451) (0.459)(0.499) (0.417)(b) Latter 12 periods

Data
All
Perfect Score
Imperfect Score

Ausubel(1)
Vickrey(2)
AA
ANAA − AN
VA
VN
VA − VN (1) − (2)
0.443
0.317
0.126∗∗∗ 0.540
0.264
0.275∗∗∗ −0.149∗∗∗(0.497) (0.466)(0.499) (0.441)
0.470
0.302
0.168∗∗∗ 0.601
0.299
0.302∗∗∗ −0.134∗∗∗(0.500) (0.460)(0.490) (0.458)
0.418
0.331
0.087∗∗∗ 0.407
0.208
0.199∗∗∗ −0.111∗∗(0.494) (0.471)(0.492) (0.407)

Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level. Numbers
in parentheses are standard deviations. In the Vickrey treatments, this table focuses on non-winningbids.

Using mainly both units’ data shown in Panel (a) of Table 10, we ﬁrst test whether
the advice increases sincere bidding in the Ausubel auction. We summarize our ﬁndings
as follows:
Result 9 (Advice eﬀect in the Ausubel auction).
(i) Providing advice increases the rate of sincere bidding in the Ausubel auction whenusing all data. The mean increase is 8.4% (95% conﬁdence interval, 5.3% to11.5%).
(ii) The advice is eﬀective even when only focusing on the data of the subjects withperfect quiz scores only or of those with imperfect quiz scores.
Support. We ﬁrst establish (i). Hypothesis 1’ is rejected at p < 0.001. To establish(ii), we divide the data into two types: the subjects with perfect scores and thosewith imperfect scores. Then, Hypothesis 1’ is rejected in the data of the subjects withperfect scores (p < 0.001), and it is also rejected in the data of the subjects withimperfect scores (p < 0.05).
We observe from Result 9 that the advice eﬀect exists in the Ausubel auction. Thus,
we compare our Ausubel auction data with our Vickrey auction data to examine the30

diﬀerences in the advice eﬀects between these two auction rules. Moreover, we compare
the two auction rules according to their rates of sincere bidding when advice is provided,
given that some experimental studies report that, without advice, the rate of sincere
bidding in the Ausubel auction is higher than that in the Vickrey auction (e.g., Kageland Levin (2009)). We then obtain the following observations from comparing Ausubel
and Vickrey auctions.
Result 10 (Comparison of non-winning bids with the Vickrey treatments).
(i) Without advice, the rate of sincere bidding for non-winning bids is signiﬁcantly
lower in the Vickrey auction than that in the Ausubel auction. Conversely, when
advice is provided, the rate of sincere bidding for non-winning bids is signiﬁcantly
higher in the Vickrey auction than that in the Ausubel auction.
(ii) The advice has a negligible eﬀect in mitigating overbidding for the second unit in
the Ausubel auction, while it decreases overbidding for the second unit by about
18% in the Vickrey auction.
(iii) The eﬀect of advice on the rate of sincere bidding for non-winning bids is significantly higher in the Vickrey auction than that in the Ausubel auction.
Support. Table 12 summarizes the non-winning bids in the Vickrey and Ausubel auctions. Support for (i) is provided in Panel (a) of Table 12. We observe that the ratesof sincere bidding in AA and VA are 38.9% and 55%, respectively. Support for (ii) is
given by ﬁnding the majority bid categories in Table 12 (shown in bold). In Panel(c) for the second unit, the rate of overbidding is nearly equal between AA and AN,
while it decreased by about 18% in the Vickrey treatments. Support for (iii) is given
by testing Hypothesis 3’, and we have p < 0.001.
Result 10 shows that the rate of sincere bidding in the Ausubel auction is signiﬁcantly lower than that in the Vickrey auction when advice is provided, unlike the casewithout advice. Moreover, Result 10 suggests that the net advice eﬀect of the Ausubel
auction is signiﬁcantly smaller than that of the Vickrey auction. This may be because the Vickrey auction is strategy-proof, but the Ausubel auction is not. However,considering that our instructions and the statement of advice in the Ausubel auction
treatments were designed to be in a similar format as those in our main experiment,
this might lead to poor performance of the Ausubel auction. This point is discussed
in detail in Section 4.7.

31

Table 12: Non-winning bid category by treatments and units.
(a) Both units

Bid category
SincereOver
Under

AusubelVickreyAA
ANAA − AN
VA
VN
VA − VN
∗∗∗
0.389
0.305
0.084
0.550
0.240
0.310∗∗∗(0.488) (0.460)(0.498) (0.427)
0.415
0.427 −0.012
0.340
0.536 −0.196∗∗∗(0.493) (0.495)(0.474) (0.499)
0.196
0.268 −0.072∗∗∗ 0.110
0.224 −0.114∗∗∗(0.397) (0.443)(0.312) (0.417)(b) Unit 1

AusubelVickrey
Bid categoryAA
ANAA − AN
VA
VN
VA − VN∗∗∗
0.672 0.512
0.160
0.573
0.226
0.347∗∗∗
Sincere(0.470) (0.500)(0.495) (0.419)∗∗
0.145
0.195 −0.050
0.375
0.599 −0.224∗∗∗Over(0.353) (0.397)(0.485) (0.491)
∗∗∗
0.182
0.293 −0.110
0.052
0.174 −0.123∗∗∗
Under(0.387) (0.455)(0.222) (0.380)(c) Unit 2

AusubelVickrey
Bid categoryAA
ANAA − AN
VA
VN
VA − VN
0.268
0.213
0.055∗∗∗ 0.540
0.246
0.294∗∗∗
Sincere(0.443) (0.410)(0.499) (0.431)
0.530 0.529
0.001
0.325
0.508 −0.182∗∗∗Over(0.499) (0.499)(0.469) (0.500)
∗∗∗
0.202
0.258 −0.056
0.135
0.246 −0.111∗∗∗Under(0.402) (0.438)(0.341) (0.431)
Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level. Numbers
in parentheses are standard deviations. In the Vickrey treatments, this table focuses on non-winning
bids.

32

Table 13: Regression analysis of sincere bidding in the Ausubel auction.

AdvicePerfect Score
Advice × Perfect Score
LatterAdvice × Latter(13)
0.778∗(0.406)0.219(0.399)0.058(0.578)

Unit 1(14)
0.436(0.323)0.395∗(0.202)0.934∗∗∗(0.306)

Latter × Perfect Score
Advice × Latter × Perfect Score
Constant
Observations
Akaike Inf. Crit.

0.029−0.039(0.281)(0.224)
1,101
1,101
1,324.765 1,286.744

(15)
0.113
(0.463)0.196(0.452)0.598(0.653)0.371(0.285)1.456∗∗∗(0.441)
0.051(0.404)−1.020∗(0.616)−0.136(0.318)1,101
1,289.184

(16)
0.338
(0.580)0.052(0.576)0.467(0.820)

Unit 2(17)
0.341(0.431)(18)
0.121(0.607)0.018(0.602)0.433(0.859)0.075
0.037(0.171)(0.253)0.443∗
0.411(0.237)(0.338)0.070(0.343)0.085(0.475)−2.060∗∗∗ −2.074∗∗∗ −2.079∗∗∗(0.411)(0.304)(0.430)
2,539
2,539
2,539
2,012.186 2,003.285 2,010.236

Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level. Numbers inparentheses are standard errors.

We ﬁnally test whether quiz scores aﬀect the responsiveness to our advice in the
Ausubel auction. The following result demonstrates that a better understanding of theauction rule increases the eﬀect of advice in the Ausubel auction.
Result 11 (Relationship between quiz scores and advice eﬀects). Within the
Ausubel auction, the subjects with perfect quiz scores are more responsive to advice thanthose with imperfect quiz scores. The mean diﬀerence between the two groups in termsof an increase in the sincere bidding rate is 9.5% (95% conﬁdence interval, 3.4% to15.7%).
Support. Hypothesis 4’ is rejected at p < 0.001.

4.5

Regression analysis of sincere bidding

We perform a regression analysis to check the robustness of the results by utilizing
a logit model with a random eﬀect at the individual level. Almost all results are
conﬁrmed.
Table 13 summarizes the results. In this table, speciﬁcations from (13) to (18)
jointly clarify how the advice changes sincere bidding behavior for each unit. For theﬁrst unit, the advice eﬀect appears as the subjects play repeatedly, regardless of theirunderstanding level of the Ausubel auction. Conversely, speciﬁcation (18) shows that33

Table 14: Eﬃciency by treatments and periods in the Ausubel auction.

Data
All periods
Periods 1–13
Periods 14–25

AA
AN
0.986
0.982
(0.050) (0.054)
0.981
0.975
(0.062) (0.061)
0.992
0.989
(0.032) (0.043)(1)
AA − AN0.005∗(1) − (VA − VN)−0.0050.006−0.0090.003−0.003

Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level. Numbers
in parentheses are standard deviations. In the Vickrey treatments, this table focuses on non-winningbids.

the advice does not mitigate overbids and underbids for the second unit, as illustrated
in Figure 5.
More interestingly, in the Ausubel auction, the advice eﬀect appears as the subjects
play repeatedly only for the ﬁrst unit (signiﬁcance and non-signiﬁcance of Advice ×
Latter in speciﬁcations (15) and (18) of Table 13, respectively). Meanwhile, in theother auction rules, the impact of Advice × Latter is insigniﬁcant in (3), (6), (9), and(12) of Table 5. This implies that the advice has less impact on sincere bidding forboth units in the latter 12 periods. These observations indicate that the advice has adiﬀerent impact on sincere bidding between units in the latter 12 periods only underthe Ausubel auction; that is, the eﬀect of advice on sincere bidding in the Ausubel
auction is higher in the latter 12 periods than in the former 13 periods.29

4.6

Eﬃciency, bidders’ payoﬀs, and seller’s revenue

We ﬁrst examine the eﬀect of auction rules, advice, and experience on eﬃciency. We
here use the eﬃciency measure introduced in Section 3.5.
Result 12 (Advice eﬀect on eﬃciency). Providing advice improves eﬃciency in
the Ausubel auction.
Support. Table 14 summarizes the results on eﬃciency with a focus on experience. In
this table, the upper (or middle, bottom) panel collects the results obtained for all
periods (periods 1–13 and periods 14–25, respectively). Hypothesis 5’ is rejected at the
10% signiﬁcant level in all data, while Hypothesis 5’ is not in the data of periods 1–13
(p = 0.154) and periods 14–25 (p = 0.224).
29

These results hold under logit models without random eﬀects.

34

Table 15: Average bidder’s payoﬀ and average seller’s revenue in the Ausubel auction.
(a) Bidder’s payoﬀ

Data

AA
165.711All periods(218.738)
158.376
Period 1–13
(216.427)
173.657
Period 14–25(221.107)AN
AA − AN158.0077.704(212.836)
154.399
3.977(213.467)
162.191
11.466(212.192)(b) Seller’s revenue

Data

AA
1075.111
All periods(354.358)
1095.897
Periods 1–13
(370.447)
1052.593
Periods 14–25(335.465)AN
AA − AN1060.935
14.176(356.833)
1090.283
5.614(343.964)
1026.901
25.691(369.092)

Notes: ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level. Numbers inparentheses are standard deviations.

Through Result 12, we ﬁnd that, as opposed to the Vickrey auction, the advice
eﬀect on eﬃciency does not appear in the Ausubel auction in early periods.
Thereafter, we ask whether providing advice aﬀects the bidder’s payoﬀ in the
Ausubel auction. Table 15 summarizes the results on the bidders’ payoﬀs and seller’s
revenue. We observe that, in the Ausubel auction, the advice does not aﬀect both thebidders’ payoﬀs and the seller’s revenue.
Finally, we compare the three auction rules in terms of seller’s revenue and bidders’payoﬀs. Several existing literature shows that the static Vickrey auction yields a higher
revenue than the English/Ausubel auction (Kagel et al. (1987) for a single unit soldand Manelli et al. (2006) for multiple units sold). Thus, we determine whether thetransparency of dynamic format mitigates overbidding and increases (or decreases) the
bidders’ payoﬀs (or the seller’s revenue). Table 16 shows the comparison of the seller’s
revenue and bidders’ payoﬀs across diﬀerent auction rules.30
Given that the valuations of unit 1 are higher than those of unit 2, the overbids
in unit 1, when rejected, often determine the prices in the Vickrey auction. Thus, the30

It is well known that the revenue equivalence result does not generally hold when multiple units are
sold (Krishna (2009)). In this table, we do not evaluate how close the observed payoﬀs and revenues
are to the predicted ones.

35

more overbids in unit 1 could make the prices higher and bidders’ payoﬀs smaller inthe Vickrey auction. This is also true in the Ausubel auction. Panel (b) of Table 12
reports that the overbids in the Vickrey auction are much higher than those in the
Ausubel auction in unit 1. This would be the factor of the observations in Table 16
that, when the two auctions are compared, bidders’ payoﬀs in the Vickrey auction aresmaller than in the Ausubel auction regardless of whether advice is provided.
Table 6 and Table 14 report that the Vickrey and Ausubel auctions are both almost
eﬃcient, implying that both auctions generate almost equal surpluses. Thus, the sellers’revenues in the two auctions are reversed from the bidders’ payoﬀs; that is, the sellers’revenues in the Vickrey auction are larger than those in the Ausubel auction.
We also observe from Table 16 that the bidders’ payoﬀs in the pay-your-bid auction
are less than those in the Vickrey and Ausubel auctions, regardless of whether advice
is provided. It is a well-known theoretical result that, in single-object auctions, thebidders’ payoﬀs in the pay-your-bid auction (ﬁrst-price auction) are less than thosein the Vickrey auction when bidders are risk-averse (Holt (1980); Krishna (2009),
Ch. 4.2). Table 16 reports similar phenomena in the bidders’ payoﬀs, and the reversed
relationships in the sellers’ revenues for multi-unit auctions.

4.7

Comparison with Kagel and Levin (2009)

In this section, we compare the Ausubel auction experimental data of Kagel andLevin (2009) (hereinafter, KL) and the data of this paper because KL include the
Ausubel treatments that resemble our treatments. Among several settings KL used,
we focus on the periods wherein subjects were given information feedback and competed for two units, similar to our design.31
We carefully deal with the increment of valuation and bid/price to allow a reasonable comparison. Each valuation is drawn from the uniform distribution over $0 to$7.50 with one cent increment. The price increment is $0.25. KL categorize a demand
reduction as sincere if the reduction occurred within $0.125 of the true valuation.

31

KL vary the number of bidders in one auction and the number of units sold, within one session.

36

37

VA
VN
PA
PN
PA
<∗∗∗
<∗∗∗

PA
<∗∗∗
<∗∗∗

AN
>
>∗∗∗ (∗∗)
>∗∗∗
>∗∗∗
AN
∗∗ (∗)>>∗ (N.Sig.)
>∗∗∗
>∗∗∗
AN
∗∗∗
>
>∗∗
>∗∗∗
>∗∗∗

∗∗∗

Bidders’ payoﬀsPA
PNAA∗∗∗ (∗)∗∗∗ (N.Sig.)
∗∗∗
VA >
>
<
VN >∗∗∗ (∗∗)
>∗∗∗
<∗∗∗
PA
<∗∗∗
PN
<∗∗∗
PA
PN
AA
∗∗∗ (N.Sig.)∗∗∗ (N.Sig.)∗∗ (∗∗∗)VA >><VN >∗∗∗ (∗∗)>∗∗∗ (∗∗)
<∗∗∗
PA
<∗∗∗
PN
<∗∗∗
PA
PN
AA
∗∗∗ (∗∗)∗∗∗ (N.Sig.)
∗∗∗
VA >
>
<
∗∗∗ (N.Sig.)∗∗∗ (N.Sig.)
VN >
>
<∗∗∗
PA
<∗∗∗
PN
<∗∗∗

<
<∗∗∗
<∗∗∗
<∗∗∗
AN
∗∗ (∗∗∗)
<
<∗∗∗
<∗∗∗
<∗∗∗
AN
∗∗ (∗∗∗)
<
<∗∗∗
<∗∗∗
<∗∗∗

AN
∗∗∗

Notes: a) ∗ denotes signiﬁcant at the 10% level, ∗∗ at the 5% level, and ∗∗∗ at the 1% level. b) “>” (or “<”) indicates the treatment in the row(or column) has a larger value. c) We used normal approximation test. d) The star in parenthesis indicates how Nonparametric Wilcoxon rank sum
test results diﬀer.

VA
VN
Periods 14–25
PA
PN

Periods 1-13

All periods

VA
VN
PA
PN

PA
<∗∗∗
<∗∗∗

Seller’s revenue
PN
AA
∗∗∗
∗∗∗
<
>
<∗∗∗ >∗∗ (∗)
>∗∗∗
>∗∗∗
PN
AA
∗∗∗
∗ (N.Sig.)<><∗∗∗ N.Sig.
>∗∗∗
>∗∗∗
PN
AA
∗∗∗
∗∗∗ (∗∗)<>∗∗∗<>∗ (N.Sig.)
>∗∗∗
>∗∗∗

Table 16: Treatment comparison of seller’s revenue and bidders’ payoﬀs.

Table 17: Comparison with KL Ausubel treatments.

KL Ausubel
Bid categoryAA
ANAA − AN Unit 1 Unit 2
KL Sincere
69.4% 69.6%
−0.2%
83%
77%
KL Over
24.1% 22.9%
1.2%
5%
17%
KL Under
6.5% 7.5%
−1.0%
12%
5%
Notes: See Table 2 of Kagel and Levin (2009) for KL Ausubel treatment statistics.

Based on the KL design detail mentioned previously and our $0.1 bid/price increment, we redeﬁne sincere bidding in our Ausubel treatments as the demand reduction
within $1.2 from the true valuation. We denote this category by “KL Sincere.” Further, both “KL Over” and “KL Under” are deﬁned accordingly. The bid categories à la
KL of our Ausubel treatments are summarized in Table 17. The frequencies of sincere
bidding in Table 17 are closer to those observed in the corresponding treatments in KL
than in Table 9. However, those frequencies in Table 17 remain lower than each unit
in KL.
The lower rate of sincere bidding in Table 17 compared to KL might be partiallycaused by the diﬀerence (to KL) in the instructions. As mentioned in the introduction,
our objective in this paper is to examine net advice eﬀects on strategy-proof mechanisms. Therefore, we ﬁrst focused on the Vickrey auction, which is a well-known
strategy-proof auction rule. To achieve our objective, in our Vickrey and pay-yourbid auction treatments, we carefully designed both experimental instructions and thestatement of advice in order to minimize the diﬀerences among the treatments while
avoiding deception. Such design constraints lead to the provision of simple advice. To
ensure fair comparison as much as possible, we designed Ausubel auction instructions
similar to those in the two static auction rules, and then we provided simple advice in
our Ausubel auction treatment. Certainly, the instruction à la KL and the statement ofadvice tailored to the Ausubel auction may have increased the rate of sincere bidding.
However, we leave this topic for future research.

4.8

Nonparametric tests

Almost pairwise comparison results are consistent with normal approximation tests.
Speciﬁcally, our nonparametric tests yield the results consistent with Result 2 andResult 9 (see Panel (a) of Table 18), Result 5 and Result 12 (see Panel (e) of Table 18),
and Result 7 (see Panels (g) and (h) of Table 18).

38

39

AA − AN VA − VN
0.037
0.000
0.001
0.000
0.582
0.000

(e) Eﬀciency

PA − PN
0.000
0.000
0.753
d)

VA − VNd)
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

Data
VA − VN
All periods
0.015
Periods 1–13
0.063
Periods 14–25
0.108

PA − PN
0.192
0.644
0.164

(g) Bidder’s payoﬀ
AA − AN
0.431
0.791
0.404

Data
VA − VN PA − PN AA − ANAll periods
0.007
0.097
0.088
Periods 1–13
0.016
0.203
0.100
Periods 14–25
0.181
0.280
0.592

All
Perfect Score
Imperfect Score

VA − VN
0.000
0.000
0.000

(c) Former 13 periods

PA − PN AA − AN
0.000
0.000
0.000
0.000
0.921
0.096
0.000
0.000
0.000
0.001
0.093
0.000
0.001
0.001
0.000
0.000
0.463
0.500

(a) All periods

VA − VN
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

VA − VN
0.000
0.000
0.000

Data
VA − VN PA − PN AA − ANAll periods
0.621
0.237
0.532
Periods 1–13
0.905
0.881
0.835
Periods 14–25
0.538
0.095
0.405

(h) Seller’s revenue

AA − AN VA − VNd)
0.000
0.000
0.000
0.000
0.008
0.000

AA − AN
0.115
0.131
0.711

PA − PN
0.074
0.008
0.622

PA − PN
0.106
0.246
0.307

(f) Binary eﬀciency

All
Perfect Score
Imperfect Score(b) Period one
VA − VN PA − PN AA − AN VA − VNd)
0.000
0.001
0.273
0.000
0.001
0.002
0.294
0.001
0.018
0.423
0.616
0.002

(d) Latter 12 periods

Data
VA − VN
All periods
0.013
Periods 1–13
0.024
Periods 14–25
0.275

Both

Data

All
Both Perfect Score
Imperfect Score

Data

Notes: a) Each number in the cells in panels (a), (c)–(e), and (g)–(h) is a two-sided p-value for a Wilcoxon rank sum test for the diﬀerence between
two groups; b) Each number in the cells of panel (b) is a Fischer exact p-value; c) Each number in the cells of panel (f) is a p-value for a chi-square
test; and d) We focus on non-winning bids in the Vickrey treatments.

Both

Data

Both

All
Perfect Score
Imperfect Score
All
Unit 1 Perfect Score
Imperfect Score
All
Unit 2 Perfect Score
Imperfect Score

Data

Table 18: Nonparametric test results.

5Conclusion

We experimentally demonstrated that introducing advice on strategy-proofness leads
to a higher rate of sincere bidding in the multi-unit Vickrey auction. Without thisadvice, the rate of sincere bidding in our experiment is similar to the 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 determine whether the advice has aneﬀect beyond the experimenter demand eﬀect, which we call the net advice eﬀect, we
compared the eﬀects in the two auctions. The results indicate that a substantial net
advice eﬀect exists in the Vickrey auction.
Considering that overbidding in the Vickrey auction has been widely observedin 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 underestimation of possible losses (Georganas et al. (2017)). Our results suggest that oneof the key drivers of overbidding is the subjects’ failure to grasp the strategy-proofnessof the Vickrey auction, despite the fact that the subjects perfectly understand theprocedure of the Vickrey auction.
Note that our statement of advice is relatively simple to ensure that the same advice
can be used in the two auctions. This simplicity makes the eﬀect of advice limited.
However, even such simple advice has a considerable net advice eﬀect in the Vickrey
auction. Further, more detailed advice would make the subjects bid more sincerely.
Our results indicate that simple advice improves eﬃciency in the Vickrey auction.
This improvement is higher in early periods, when subjects are less experienced. Thus,
our results demonstrate that advice improves eﬃciency in the Vickrey auction by promoting a better understanding of the strategy-proofness of the Vickrey auction.
To focus on the advice eﬀect on bidding behavior, we conducted experiments in a“symmetric” environment, that is, an environment wherein each bidder in the sameauction has the same distribution function of valuations. As discussed in Section 3.5,
the pay-your-bid auction and the Vickrey auction generated almost perfectly eﬃcient
outcomes in such a symmetric environment. Thus, even without advice, eﬃciency inboth auctions has little room for improvement. 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.

40

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Acknowledgement
We are grateful to John Duﬀy, the editor of this journal, and two anonymous referees
for their useful comments and suggestions that helped us to improve the paper. We
thank Yoshitaka Okano for providing the z-Tree program of the Ausubel auction andthe very helpful discussion. We also thank Kou Fujimori, 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 the ﬁnancial support
from the Joint Usage/Research Center at ISER, Osaka University; Osaka University’s
International Joint Research Promotion Program (Type A); and Grant-in-aid for Research Activity, Japan Society for the Promotion of Science (15H05728, 20H05631,
20K01555, 20KK0027).

Declarations
Funding
We gratefully acknowledge the ﬁnancial support from the Joint Usage/Research Center
at ISER, Osaka University; Osaka University’s International Joint Research Promotion
Program (Type A); and Grant-in-aid for Research Activity, Japan Society for the
Promotion of Science (15H05728, 20H05631, 20K01555, 20KK0027).

Conﬂicts of interest/Competing interests
The authors declare that they have no conﬂict of interest.

Availability of data and material (data transparency)
Data are available at journal’s website.

Code availability
Code used in analysis is available at journal’s website.

45