ISSN (Print) 0473-453X

Discussion Paper No. 1229

ISSN (Online) 2435-0982

TOWARD AN UNDERSTANDING
OF INSINCERE BIDDING
IN A VICKREY AUCTION EXPERIMENT

Shigehiro Serizawa
Natsumi Shimada
Tiffany Tsz Kwan Tse

February 2024

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

Toward an Understanding of Insincere Bidding in a
Vickrey Auction Experiment∗
Shigehiro Serizawa† Natsumi Shimada‡ . Tiffany Tsz Kwan Tse§
February 16, 2024

Abstract
This study explores two key factors influencing subjects’ deviation from sincere
bidding in Vickrey auction experiments. The first factor examines subjects’ understanding of strategy-proofness (SP), while the second focuses on “human interaction” which includes social preferences (spite and altruism), responses to strategic
uncertainty, and tacit collusion. To analyze the effect of understanding SP, we quiz
subjects before an experimental Vickrey auction and examine whether their bidding behavior changes if one of the quizzes includes hints about SP. We design thequiz carefully, incorporating implicit hints about SP and ensuring the avoidanceof explicit demands or advice to mitigate experimenter demand effects. However,completing the quiz enables the subjects to understand SP themselves. To analyze
the effects of human interaction, we examine whether subjects’ bidding behavior
changes if they compete against robots instead of human rivals in the auctions. We
design 2×2 treatments by varying the type of quiz (with or without hints about SP)
and the nature of the rivals (humans or robots). We found that the quiz with hints
about SP increases sincere bidding. The nature of rivals also influences the bidding
behavior; nonetheless, its impact is not as robust as that of SP hints. Thus, themain factor causing insincere bidding in Vickrey auction experiments is not humaninteraction but a lack of understanding of SP.

Keywords: Market design; Strategy-proofness; Vickrey auction; Hints; Human interaction; Human vs robot rivals
JEL Classification: D44, D82, D61

∗

†
‡
§

We are grateful to John Duffy for his practical and insightful feedback, which helped us improve
this paper. We thank Takehito Masuda for providing the z-Tree program of the Vickrey auction and helpful comments. We also thank Soo Hong Chew, Clayton Thomas, Morimitsu Kurino,
Nobuyuki Hanaki, and Toshiji Kawagoe for their helpful comments. Further, we thank Ryo Mikami
for assisting with coding the z-Tree program. We are also deeply grateful to Yuki Hamada, Yuya
Wakabayashi, and other staff for their assistance with conducting the experiment at Osaka University. We gratefully acknowledge financial support from the Joint Usage/Research Center at the
ISER, Osaka University, and Japan Society for the Promotion of Science KAKENHI Grant Numbers JP20H05631, JP20KK0027, JP21H00696, JP22H00062 and JP23K01307. The experimentreported in this paper was approved by the Research Ethics Committee at the Institute of Socialand Economic Research, Osaka University.
Corresponding author: Faculty of Economics, Osaka University of Economics.
E-mail: serizawa.8558@gmail.com
Organization for IT Initiatives, Shinshu University. E-mail: n shimada@shinshu-u.ac.jp
Institute of Social and Economic Research, Osaka University. E-mail: tiffany.econ@gmail.com

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1

Introduction

The Vickrey auction (Vickrey, 1961) is strategy-proof (SP). It is a dominant strategy
for bidders to bid sincerely—to bid according to their true valuation. However, many
experimental studies have reported that most subjects deviate from sincere bidding in
Vickrey auctions with single units (Garratt et al., 2012; Kagel & Levin, 1993; Shogren
et al., 2006) and multiple units (Kagel & Levin 2009; Manelli et al., 2006; Masuda etal., 2022).
In experiments, many researchers have proposed hypotheses to explain insincere
bidding, such as the joy of winning (Cooper & Fang, 2008), spiteful behaviors (Andreoni
et al., 2007; Nishimura et al., 2011), underestimation of possible losses (Georganas etal., 2017), and cognitive limit of reasoning (Li, 2017). These studies suggest that several
factors may cause insincere bidding in Vickrey auctions.
Subjects’ cognitive reasoning limitations may make it difficult for them to understandthe concept of SP in Vickrey auction rules. Conversely, most subjects find it easy tounderstand that sincere bidding represents the dominant strategy in English auctions. Li(2007) demonstrated that this difference leads to divergent performances in Vickrey andEnglish auctions despite both auction rules being SP. In matching models, Hassidim et al.
(2017) proposed a potential explanation for preference misrepresentation—subjects inthe experiments may have failed to understand the SP nature of the deferred acceptancemechanism, a widely recognized SP mechanism.
Several researchers guide subjects to address the difficulty of understanding SP.
They examine whether this guidance contributes to an increase in truth-telling rates
across various SP mechanisms, such as pivotal mechanism (Kawagoe & Mori, 2001), top
trading cycle matching algorithms (Guillen & Hing, 2014; Guillen & Hakimov, 2018),
school matching mechanisms (Ding & Schotter, 2019), and multi-unit Vickrey auctions(Masuda et al., 2022). However, it remains unclear whether subjects express theirgenuine preferences because they understand SP or if they simply follow the guidanceas a result of experimenter demand effects (Zizzo, 2010).
To promote subjects’ understanding of SP without inducing demand effects, we administer quizzes containing hints about SP. These quizzes refrain from offering explicit
2

advice; however, completing them enables subjects to understand the concept of SP independently. In essence, by solving the quizzes, subjects experience an “Aha!” moment,
realizing that the Vickrey auction is SP. Thus, the effect of these quizzes on subjects’
bidding behavior can be attributed to the factor of understanding SP.
Specifically, we designed three distinct types of quizzes: Quiz 1, Quiz 2 with hints
about strategy-proofness (hereafter, Quiz 2 w/ hints), and Quiz 2 without hints
about strategy-proofness (hereafter, Quiz 2 w/o hints). All subjects receive thesequizzes before the auction begins. Quiz 1 assesses whether subjects comprehend theauction rules and simultaneously enhances their comprehension of the auction rules. It
is administered to all subjects after receiving initial instructions. Quiz 2 w/ hints assesses whether subjects understand the concept of SP and simultaneously enhances theirunderstanding of SP. This is administered to subjects in two out of the four treatments
immediately following Quiz 1. Quiz 2 w/o hints closely replicates Quiz 2 w/ hints but
omits SP hints. It focuses solely on the computations of payoffs in the Vickrey auction.
Its purpose is to adjust the subjects’ earnings and comprehension levels of auction rules
without the influence of SP hints before the auction. It is administered to subjects inthe other two treatments immediately following Quiz 1. Our analytical focus centers onunderstanding SP, which involves comparing the effects of Quiz 2 w/ hints and Quiz 2
w/o hints on subjects’ bidding behaviors.
As mentioned earlier, insincere bidding in Vickrey auctions can be influenced bymultiple factors. In addition to examining the impact of understanding SP, we focus
on another significant factor, which we term “human interaction.” This factor includes
various elements, such as social preferences (spite and altruism), responses to strategic
uncertainty regarding other bidders’ behavior, and tacit collusion. All these aspects
stem from interactions among human subjects.
These human interaction factors do not affect bidding behavior when each subject
competes solely against robot rivals, whose bids are automatically generated using aknown distribution. In this controlled environment, the effects of understanding SP
on subjects’ bidding behavior emerge in a purer form. Thus, to analyze the factor ofunderstanding SP within two of the four treatments, we conduct an auction experiment3

in which each human subject competes against robot rivals. Simultaneously, we analyze
the effects of the human interaction factor by contrasting the bidding behavior of subjects
in the two treatments, where each subject competes against robot rivals, with those in
the other two treatments, where only human subjects compete.
Several researchers have recently investigated the use of experimental designs inwhich human subjects play against robot rivals in auction experiments (Chen & Takeuchi
2010; Kagel & Levin, 2001, 2009).5 They analyze subjects’ learning of the robot bidders’strategies and their responses to them. However, we attempt to minimize these effects sothat the effects of understanding SP and human interaction appear in purer forms. Thus,
we inform subjects that the bids of the robot bidders are automatically generated using
a known distribution. We intentionally avoid mentioning the robot bidders’ strategies oreven their valuations to prevent the subjects from consciously thinking about the robot
bidders’ strategies.
In summary, our study focuses on two factors that influence sincere bid rates inVickrey auctions: understanding SP and human interaction. To analyze the effects ofthese two factors, we designed 2×2 treatments varying whether subjects took Quiz 2 w/
hints or Quiz 2 w/o hints and whether they competed against human or robot rivals.
In our experiment, we auction two homogeneous object units. There are two reasonsfor choosing a multi-unit Vickrey auction. First, the multi-unit Vickrey auction is amore complicated design than single-unit Vickrey auctions (or second-price auctions).
Therefore, understanding the concept of SP proves to be more difficult for subjects ina multi-unit Vickrey auction than for those in a single-unit Vickrey auction. Indeed,extremely low sincere bidding rates of approximately 20% were observed in multi-unit
Vickrey auction experiments, such as Kagel and Levin (2009) and Masuda et al. (2022).
Thus, in multi-unit Vickrey auction experiments, the effects of understanding SP andhuman interactions are more salient and easier to observe. In pursuit of a broader
spectrum of understanding levels regarding SP, we opt for a multi-unit Vickrey auction
rather than a single-unit auction. Second, multi-unit auctions are widely used in reality,
such as in spectrum and treasury bill auctions, although Vickrey-type auctions are not5

In addition to auction experiments, the interactions between human subjects and robots have been
investigated in public goods game experiments (Yamakawa et al., 2016).

4

commonly used.
In our experimental results, most subjects obtain maximum or near maximum scoresin Quiz 1. This implies that most subjects have a thorough comprehension of the Vickrey auction rules. Therefore, it is improbable that insincere bidding can be attributed
to insufficient comprehension of the auction rules. Even after eliminating human interactions in treatments where subjects compete against robot rivals, insincere bidding
persists. This suggests that the human interaction factor can only partially explain theoccurrence of insincere bids in Vickrey auctions. We find that hints about SP (Quiz
2 w/ hints) increase sincere bids in the Vickrey auction. This is observed irrespectiveof whether the subjects compete against human or robot rivals. Subjects who scorehigher on Quiz 2 w/ hints tend to make sincere bids more frequently. Thus, subjects
who understand SP bid sincerely more frequently. These experimental data imply thatthe major factor causing insincere bidding in Vickrey auctions is the subject’s inability to comprehend SP. Although present, the effect of human interactions appears to besmaller. Additionally, our auction outcomes are efficient, with efficiency rates of approximately 98% across all treatments. Nevertheless, bidders’ payoffs and seller’s revenues
exhibit mixed results.
The remainder of this paper is organized as follows. Section 2 describes the experimental design and presents several proposed hypotheses. Subsequently, Section 3
presents our experimental results, and Section 4 discusses efficiency, the bidder’s payoff,and the seller’s revenue. Finally, Section 5 concludes the paper.

2

Experimental Design

The purpose of this experiment is to investigate two factors that influence the sincere
bid rates in a multi-unit Vickrey auction: (i) the understanding of SP and (ii) humaninteraction. Our experimental design follows that of Masuda et al. (2022). For thetheoretical background of the multi-unit Vickrey auction, please refer to Masuda et al.
(2022).

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2.1

Theoretical considerations

There are three bidders, {1, 2, 3}, and two indivisible, identical objects to be auctioned.
Each bidder is allowed to bid for two units. Bidder i’s valuation of objects is vi = (vi1 , vi2 ),where vi1 ≥ vi2 ≥ 0 and vij denote the valuations that bidder i assigns to the j-th unit.
Given any vi , bidder i’s utility of obtaining k units of objects and paying mi units ofmoney is as follows:6

U (k, mi ; vi ) ≡
v 1 + vi2 − mi


 i

vi1 − mi






−mi

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

A list v = (v1 , v2 , v3 ) is a valuation profile. The bid submitted by bidder i is denoted asbi = (b1i , b2i ), where b1i ≥ b2i ≥ 0 and bji denote the bidder’s bid i for the j-th unit. Let
b = (b1 , b2 , b3 ) be the bid profile.
An assignment function is a function d = (d1 , d2 , d3 ) that specifies that for eachbid profile b, the number di (b) ∈ {0, 1, 2} of the objects bidder i is obtained under
the resource constraint: d1 (b) + d2 (b) + d3 (b) = 2. A payment function is a functionm = (m1 , m2 , m3 ) that specifies, for each bid profile b, the amount mi (b) of money
bidder i pays. A rule is a pair of assignment and payment functions (d, m).
We are interested in rules in which sincere bidding (i.e., bidding true valuations) isa 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 all other bids b−i ,U (di (vi , b−i ), mi (vi , b−i ); vi ) ≥ U (di (bi , b−i ), mi (bi , b−i )); vi ).

Vickrey auction rules are central to the auction theory literature.
Vickrey auction: Each bidder i simultaneously submits their bid bi = (b1i , b2i ). After
6

In standard auctions, such as the Vickrey auction, mi = 0 if k = 0.

6the seller collects a bid profile, the seller ranks the bids {bki : i = 1, 2, 3 & k = 1, 2}
from highest to lowest and allocates the two units to the two highest bids. Ties arebroken by equal probabilities. If a bidder wins one (or two) unit, then they haveto pay the highest bid (the sum of the highest and second-highest bids) among thelosing bids. If a bidder wins nothing, they pay nothing.
The most important feature of the Vickrey auction is that it is SP. However, several
experimental studies report that bidders tend to overbid in Vickrey auctions.

2.2

Procedure

In each treatment, three bidders compete for two units of identical objects. For eachbidder, two integer valuations are drawn from a uniform distribution with intervals of
10 to 1,000 in increments of 10. The larger (smaller) integer is then assigned a valuation
for the first (second) unit. All values are presented in Japanese yen (JPY).
We designed four treatments, varying the type of Quiz 2 (with/without SP hints)
and the nature of the rivals (human/ robot). Employing a between-subject design, thefour treatments are as follows:• Treatment HHnohint: only human subjects compete, and each subject completes
Quiz 2 without SP hints• Treatment HHhint: only human subjects compete, and each subject completes
Quiz 2 with SP hints• Treatment HRnohint: each subject competes with two robot rivals and completes
Quiz 2 without SP hints• Treatment HRhint: each subject competes with two robot rivals and completes
Quiz 2 with SP hints
Table 1 summarizes the basic data for the four treatments.

7

Table 1: Summary of treatments
Treatments

HHnohint

HHhintHRnohint

HRhintType of Quiz 2
Nature of rivals
No. of sessions
Duration (min)No. of rounds
No. of subjects
Avg. payment (JPY)7
Avg. score of Quiz 1
Avg. score of Quiz 2 w/ hints
Avg. score of Quiz 2 w/o hints

w/o hints
human
2
150
20
48
4,780
9.604
/
9.500

w/ hints
human
2
150
20
45
4,941
9.800
9.200
/

w/o hints
robot
2
150
20
43
5,020
9.488
/
9.628

w/ hints
robot
2
150
20
48
5,026
9.521
9.146
/

The experiments were conducted at Osaka University in eight sessions between
February and March 2022. We recruited 184 students at Osaka University registeredin the ORSEE (Greiner, 2015) database of the Institute of Social and Economic Research. Each subject took part in a single session. Subjects provided their consent
online by clicking a button before registering for the experiment. Each session lasted
approximately 150 min and was conducted by the same experimenter. Our experiment
was computerized using the experimental software z-Tree (Fischbacher, 2007). Figure 1
illustrates the experimental flow.
Subjects were randomly assigned to seats. Each subject received printed handouts
containing the instructions and then listened to audio instructions. Subsequently, subjects were given 3 minutes to read the instructions. Throughout the entire experiment,
subjects were not allowed to ask questions. However, they were allowed to use a calculator during the experiment.
First, subjects engaged in Quiz 1, the details of which are explained in Subsection
2.3.1. Each subject received a Quiz 1 sheet and promptly began answering the questions.
Subjects received 100 JPY for each correct answer, with a total of ten questions andthe maximum potential payoff of 1,000 JPY. The time limit for Quiz 1 was set at 10
minutes. Upon completing Quiz 1, subjects received the answer key and listened to an7

We compare payment across treatments using an OLS regression model, regressing payment onfour treatment dummies while incorporating robust standard errors. Subsequently, we compare
the estimated dummy coefficients using an F test, with the results presented through Bonferroniadjusted p values. There is no statistically significant difference in payments across treatments (p= 0.604).

8

Figure 1: Flow of the experimentaudio explanation. Subsequently, subjects were then given three minutes to review theanswer key for Quiz 1.
Second, the subjects engaged in Quiz 2. Each subject received Quiz 2, the details of
which are provided in Subsections 2.3.2 and 2.3.3. In HHnohint and HRnohint, subjects
completed Quiz 2 w/o hints. In HHhint and HRhint, subjects completed Quiz 2 w/
hints. For Quiz 2 w/o hints and Quiz 2 w/ hints, they received 100 JPY for each correct
answer, with a total of ten questions and a maximum potential payoff of 1,000 JPY.
Subjects promptly began answering the Quiz 2 questions. The time limit for Quiz 2
was set at 20 minutes. Upon completing Quiz 2, subjects in HHnohint and HRnohint
received the answer key for Quiz 2 w/o hints, and those in HHhint and HRhint received
the answer key for Quiz 2 w/ hints. They listened to the audio explanations and were
given five minutes to review the answer keys.
Third, the subjects participated in a Vickrey auction experiment. In HHnohint and
HHhint, the subjects competed against human rivals. In HRnohint and HRhint, eachhuman subject competed against robot rivals. The subjects proceeded to two practiceperiods and 20 successive payment periods under a random matching protocol. At the
beginning of each period, each subject was randomly matched with two other subjects
into groups of three in HHnohint and HHhint and matched with two robot rivals into
groups of three in HRnohint and HRhint. In the bidding stage, each subject received

9

two integer valuations for two units of identical objects. Subjects were then instructedto enter two bids, ranging from 0 to 2,000, as integers in increments of 10. A crucial
requirement was that the first-unit bid must be weakly greater than the second-unitbid. In HRnohint and HRhint, subjects were informed that the bids of their robot rivals
are generated by a uniform distribution over {10, 20, . . . , 1, 000}, and the first-unit bidof the robot rivals is weakly greater than the second-unit bid.
After 20 successive payment periods, the subjects completed a questionnaire andwere immediately paid in cash. Each subject received a private payment equal to thesum of their earnings over the 20 periods, along with earnings from Quiz 1 and Quiz 2.
The average payment amounted to 4,780 JPY in HHnohint, 4,941 JPY in HHhint, 5,020
JPY in HRnohint, and 5,026 JPY in HRhint.

2.3

Design of Quizzes

Before the auction experiment, we provided three types of quizzes to subjects: Quiz 1
for all the subjects, Quiz 2 w/ hints for subjects in HHhint and HRhint, and Quiz 2
w/o hints for subjects in HHnohint and HRnohint. Quiz 1 assesses and enhances subject
comprehension of the Vickrey auction rules. Quiz 2 w/ hints assesses and enhances the
subjects’ understanding of SP in Vickrey auctions. Quiz 2 w/o hints adjusts levels of
earnings and auction rule comprehension for the subjects in HHnohint and HRnohint
before the auction, similar to those of HHhint and HRhint.

2.3.1

Quiz 1

Quiz 1 is administered to the subjects of all treatments and assesses their comprehension of Vickrey auction rules while enhancing their comprehension. Quiz 1 includes 10questions, and the maximum score is 10 points. The subjects receive 100 JPY for eachcorrect answer.
Figure 2 shows sample questions used in Quiz 1. As illustrated in the figure, the quiz
gives numerical examples of the bid profiles and valuations, and the questions ask about
the winning bids, payments, and payoffs of the examples. Three out of 10 questions
are about winning bids, four are about payments, and the remaining three are about

10

payoffs.
The payoffs for two questions are positive, and the payoff for one question is negative.
Thus, if a subject correctly answers these questions, they can realize that their payoff
can be negative because of an overbid. However, to avoid giving hints regarding SP, we
designed the quiz such that bids are equal to valuations in no numerical example.
The time limit for Quiz 1 is 10 minutes. After the subjects complete Quiz 1, they
are provided with a handout containing the correct answers, and they listen to the
explanation of the correct answers using reading software. Subsequently, subjects are
given 3 minutes to review the handout before moving to Quiz 2.

Figure 2: Quiz 1 sample questions

2.3.2

Quiz 2 w/ hint (Quiz 2 w/ hint of strategy-proofness)

Quiz 2 w/ hints is exclusively administered to subjects inHHhint and HRhint. It aims
to assess the subjects’ understanding of the concept of SP in Vickrey auctions while
enhancing their understanding of it. Quiz 2 w/ hints comprises 10 questions, and the
maximum score is 10 points. Subjects receive 100 JPY for each correct answer.
Figure 3 shows sample questions used in Quiz 2 w/ hints. As indicated at the11beginning of the quiz, each subject plays the role of Bidder B, with valuations of 400 and
700 for the first and second units in the auction, respectively. Quiz 2 w/ hints provides
six bidding choices (Choices a - f) for the first and second units, and Choice c coincides
with Bidder B’s valuations (400, 700). Each of the 10 questions in the quiz requires
subjects to select the correct answers from the provided six choices—the questions are
multiple-choice in nature. Five of the ten questions ask about the choices that make
Bidder B’s earnings positive, zero, or negative for given bid profiles of other bidders(A and C). Five of the ten questions ask about the choices that maximize Bidder B’s
payoff. Note that the bidding choices that maximize Bidder B’s payoff for bid profilesof the other bidders are typically not unique. Nevertheless, as Vickrey auctions are SP,
Choice C (Bidder B’s valuations) is always one of the payoff maximizers. Moreover,
Question 10 specifically addresses choices that maximize the payoff for three different
bid profiles of other bidders, and Choice C is the unique answer to this question. Thus,
if subjects correctly answer these questions, they may observe that bidding valuations
maximize one’s payoff regardless of the other bidders’ bids, demonstrating the conceptof SP. Accordingly, the quiz provides subjects with hints regarding SP. Nonetheless,we deliberately refrain from stipulating or advising subjects to bid according to theprescribed valuations to mitigate, or at least minimize, potential demand effects (Zizzo,2009).
Calculation sheets are provided to the subjects to compute payoffs efficiently. The
time limit for answering Quiz 2 w/ hints is 20 minutes. After subjects finish the quiz,
they are provided with a handout of the correct answers, and they listen to an explanation of the correct answers using reading software. The subjects are then given 5
minutes to review the handout before the auction begins.

2.3.3

Quiz 2 w/o hint (Quiz 2 w/o hint of strategy-proofness)

Quiz 2 w/o hints is administered to subjects inHHnohint and HRnohint, aiming to adjust
their earnings levels and comprehension levels of the auction rules so that these aspectsalign with subjects in HHhint and HRhint, except their understanding of SP before theauction. While the primary purpose of Quiz 2 w/ hints is to provide hints regarding SP,
12

Figure 3: Quiz 2 w/ hint

13

Figure 4: Quiz 2 w/o hint14

it inevitably induces other effects.
First, the correct answer to each of the 10 questions in Quiz 2 w/ hints earns subjects
in HHhint and HRhint 100 JPY, providing them with a substantial amount from the quiz.
Thus, unless subjects in HHnohint and HRnohint are compensated with amounts similar
to those in HHhint and HRhint, the subjects in HHnohint and HRnohint might behave
differently, regardless of their understanding of SP.
However, if we simply provide subjects in HHnohint and HRnohint amounts similar
to those in HHhint and HRhint, this may induce the “house money effect” (Corgnet etal., 2015). To avoid such an effect, it is necessary to offer subjects the opportunity
to earn similar amounts to those available from Quiz 2 w/hints through similar effort
rather than simply providing them with similar amounts.
Second, Quiz 2 w/ hints does not specifically enhance the understanding of SP;
instead, it further improves the comprehension of the auction rules. Therefore, to accurately analyze the effects of understanding SP, it becomes crucial to adjust the comprehension levels of subjects inHHnohint and HRnohint, making them similar to those in
HHhint and HRhint.
To summarize, the purposes of Quiz 2 w/o hints are as follows: (i) to give subjects
in HHnohint and HRnohint the opportunity to earn amounts similar to those from Quiz
2 w/ hints through similar efforts and (ii) to align their earnings and comprehension
levels of the auction rules with those of subjects in HHhint and HRhint before the auction.
However, Quiz 2 w/o hints should not provide hints regarding SP. Thus, we designed
Quiz 2 w/o hints as follows.
Figure 4 shows sample questions used in Quiz 2 w/o hints. Similar to Quiz 2 w/
hints, Quiz 2 w/o hints includes 10 questions, the maximum score is 10 points, and eachsubject receives 100 JPY for each correct answer. Each subject plays the role of Bidder
B. The quiz provides six bidding choices (Choices a – f) for the first and second units
at the beginning, and each question in the quiz asks the subjects to select the correct
answer from the six choices. Calculation sheets are also provided to subjects to compute
payoffs efficiently.
Nevertheless, the six choices in Quiz 2 w/o hints do not include Bidder B’s valuations,15

and all the questions in the quiz ask only about bidding choices making positive, zero, or
negative payoffs; nonetheless, no question asks about choices that maximize Bidder B’s
payoffs. These questions do not provide hints regarding SP, similar to Quiz 2 w/hints.
However, they enhance the subjects’ further comprehension of the auction rules. Thisis because three questions ask about bidding choices that result in negative payoffs. By
answering these questions, the subjects can realize that their payoff can be negative
because of overbidding. Notably, Quiz 2 w/o hints varies Bidder B’s valuations suchthat its difficulty level is similar to that of Quiz 2 w/ hints. Otherwise, Quiz 2 w/o hints
would be simpler than Quiz 2 w/ hints.
As reported in Subsection 3.2, Quiz 2 w/o hints is devised with its average scores
similar to that of Quiz 2 w/ hints. This implies that, as intended, subjects’ comprehension levels of the auction rules in HHnohint and HRnohint are similar to those in HHhint
and HRhint, and the subjects of HHnohint and HRnohint can earn amounts similar to
those of HHhint and HRhint.

2.4

Hypotheses

Although the literature has firmly established the SP nature of the Vickrey auction,
various experimental results have shown that many subjects overbid or underbid relative
to their valuations. We hypothesize that this discrepancy may be due to the complexityof understanding the concept of SP as opposed to merely comprehending the Vickrey
auction rules. To examine this, we first ensure that most of the subjects comprehend
the Vickrey auction rules by administering a quiz on the rules (Quiz 1) to all subjects.
Subsequently, we administer two different types of a second quiz: one includes hints
about SP (Quiz 2 w/ hints) and is given to subjects in HHhint and HRhint, whereas
the other lacks these hints (Quiz 2 w/o hint) and is given to subjects in HHnohint andHRnohint. We predict that Quiz 2 w/ hints may make subjects more aware of thebenefits of sincere bidding. To statistically evaluate the effect of the hints about SP, we
propose the following null hypothesis:
Hypothesis 1 (Effect of Hints on Strategy-Proofness ): The rate of sincere bidding
among subjects who take Quiz 2 w/ hints in HHhint is statistically similar to that16

among those who take Quiz 2 w/o hints in HHnohint.
Some experimental results suggest that other factors, such as strategic interaction
and social preferences, may also play a significant role in insincere bidding. These
factors are closely related to interactions among human subjects. When each subject
competes against robot rivals in an auction, the human interaction factor is eliminated.
Nevertheless, we predict that hints regarding SP in Quiz 2 make the subjects more awareof the advantages of sincere bidding, even when their rivals are robots. To statisticallyevaluate the effect of hints about SP, we propose the following null hypothesis:
Hypothesis 2 (Effect of Hints on Strategy-Proofness with robot rivals): The rate ofsincere bidding among subjects who take Quiz 2 w/ hints in HRhint is statisticallysimilar to that among those who take Quiz 2 w/o hints in HRnohint.
To investigate the impact of factors related to human interaction, we compare therate of sincere bidding in treatments in which each subject competes against robot rivals
with the rate in treatments in which only human subjects compete. To examine the effectof human interactions on sincere bidding, we propose the following null hypotheses:
Hypothesis 3a (Effect of human interaction with Quiz w/o hints) The sincere bidding rate is similar regardless of whether subjects compete against human subjects in
HHnohint or solely against robot rivals in HRnohint.
Hypothesis 3b (Effect of human interaction with Quiz w/ hint) The sincere bidding
rate is similar regardless of whether subjects compete against human subjects in HHhintor solely against robot rivals in HRhint.

3

Experimental results

3.1

Level of comprehension of the Vickrey auction rules.

Figure 5 illustrates Quiz 1 score distributions across the four treatments. As shown,a significant majority of subjects in each treatment achieved perfect scores (77% for
HHnohint, 91% for HHhint, 84% for HRnohint, and 75% for HRhint). Other subjects in
17

Figure 5: Distribution of Quiz 1 scoreseach treatment group attained nearly perfect scores (8 or 9 points). To compare Quiz 1
scores across treatments, we employ an OLS regression model, regressing Quiz 1 scores onfour treatment dummies. Robust standard errors are utilized, and the estimated dummycoefficients are compared using an F test, with results assessed through Bonferroniadjusted p values. There is no statistical difference in the Quiz 1 scores across thetreatments (p = 0.328).8 Most subjects in all treatments have a good comprehensionof the Vickrey auction rules. Moreover, their comprehension levels do not differ acrosstreatments.
After Quiz 1, the correct answers are distributed to each subject and explainedusing voice software. As this step further enhances the comprehension, most subjects
exhibit a thorough understanding of the Vickrey auction rules prior to the auction.
This is confirmed by the post-experiment survey (See Online Appendix F): 79.167% of
HHnohint subjects, 91.111% of HHhint subjects, 74.419% of HRnohint subjects, and 75%
of HRhint subjects indicated that they understood the Vickrey auction rules thoroughly
before the auction commenced.
8

We also compare Quiz 1 scores across treatments using a one-way analysis of variance (ANOVA).
There is no statistical difference in Quiz 1 scores across the treatments (p = 0.481).

18

3.2

Level of understanding of the strategy-proofness
Figure 6: Distribution of Quiz 2 scores

Figure 6 displays the distributions of the scores of Quiz 2 w/ hints for HHhint and
HRhint, as well as those of Quiz 2 w/o hints for HHnohint and HRnohint. The figure
highlights that most subjects in HHhint and HRhint (69% in HHhint and 77% in HRhint)
attained perfect scores (10 points) in Quiz 2 w/ hints. Additionally, most of the othersubjects in these treatments scored near the maximum (8 or 9 points). Thus, mostsubjects in HHhint and HRhint received hints, which facilitated their understanding of
SP. After Quiz 2 w/ hints, we distributed the correct answers to each subject in HHhint
and HRhint and explained them using voice software. As this step further enhances theunderstanding of SP, most subjects in HHhint and HRhint received hints to understandSP prior to the auction.
As shown in the figure, most subjects in HHnohint and HRnohint (85% in HHnohint
and 93% in HRnohint ) attained perfect scores (10 points) in Quiz 2 w/o hints. Additionally, most of the other subjects in these treatments scored near the maximum (7, 8, or
9 points). As outlined in Subsection 2.3.3, Quiz 2 w/o hints focuses solely on questions
that ask whether the payoffs are positive, negative, or zero. Although this quiz does19

not explicitly provide hints about SP, its emphasis on a thorough comprehension of theVickrey auction rules could indirectly make subjects recognize the concept of SP.
In comparing Quiz 2 scores across treatments, we employed an OLS regression model,
regressing Quiz 2 scores on four treatment dummies while incorporating robust standard
errors. Subsequently, we compared the estimated dummy coefficients using an F test,with results presented through Bonferroni-adjusted p-values. There are no statisticallysignificant differences in Quiz 2 scores across treatments (p = 0.454).9 This implies that
Quiz 2 w/o hints adjusts the subjects’ earnings levels and comprehension levels of theauction rules before the auction in HHnohint and HRnohint similar to those in HHhint
and HRhint.
In summary, most subjects in HHhint and HRhint had a good understanding of SP
in Vickrey auctions by obtaining perfect scores in Quiz 2 w/ hints. Additionally, theearnings levels and comprehension levels of the auction rules before the auction in alltreatments are well controlled.

3.3

Bidding behavior

We classify a subject’s bids as follows: given subject i’s valuation, vi1 (vi2 ) for the first(second) unit, their bid b1i (b2i ) for the first (second) unit is sincere if b1i = vi1 (b2i = vi2 ),over if b1i > vi1 (b2i > vi2 ), and under if b1i < vi1 (b2i < vi2 ). For example, assume
that the valuations vi = (vi1 , vi2 ), i = 1, 2, 3, of the three subjects are v1 = (800, 200),v2 = (600, 400), v3 = (910, 500) and that their bids bi = (b1i , b2i ), i = 1, 2, 3, are b1 =
(850, 240), b2 = (600, 450), and b3 = (900, 500). Subsequently, the sincere bids are
as follows: bidder 2’s bids for the first unit (b12 = 600 = v21 ) and the second unit ofbidder 3 (b23 = 500 = v32 ). Overbids are bidder 1’s bids for the first and second units(b11 = 850 > v11 = 800 and b21 = 240 > v12 = 200) and the second unit of bidder 2(b22 = 450 > v22 = 400). In this example, an underbid is unique and is the first unitof bidder 3 (b13 = 900 < v31 = 910). The sincere bidding rate is the number of sincere
bids divided by the total number of bids. Thus, the sincere bid rates of the subjects
9

We also compare Quiz 2 scores across the four treatments by using one-way ANOVA. There are
no statistically significant differences in Quiz 2 scores across treatments (p = 0.444). However, theratio of the subjects with perfect scores is higher in HHnohint and HRnohint (0.890) than in HHhint
and HRhint (0.731) with a 1% significance level by using two-sample t-tests.

20

i = 1, 2, 3 are 0, 0.5, and 0.5, respectively, and the overall average sincere bid rate is
2/6 = 1/3 = 33.33%. We primarily use the overall average sincere bid rate for thestatistical analysis.
Figure 7 plots the valuations and the corresponding bids for each treatment. Panels(a) and (b) show the data for Units 1 and 2, respectively. Note that sincere bids lie along
the 45◦ green line. The eight graphs in the figure reveal a concentration of valuations andbids on the 45◦ line, representing sincere bids. The data points located above and below
this line indicate overbids and underbids, respectively. Panel (a) shows that some bids
for Unit 1 reached a maximum of 2,000. Furthermore, the frequencies of these maximum
bids are higher in HHnohint and HRnohint than in HHhint and HRhint. Conversely, Panel(b) shows that some bids for Unit 2 are at a minimum value of 0, and these minimum
bids are also more common in HHnohint and HRnohint than in HHhint and HRhint.
Figure 8 shows the frequencies of sincere bids, overbids, and underbids for eachtreatment.10 Panels (a)–(c) display results of the first and second units. HHnohint is thebenchmark environment in which no hints about SP are provided, and the subjects bid
against human rivals. Panel (a) shows that in this treatment, the total overbidding rate
is approximately 53%, and the total sincere and underbidding rates are lower, 25% and21%, respectively. This observation is consistent with findings in experiments on themulti-unit Vickrey auction by Manelli et al. (2006), Engelmann and Grimm (2009), andKagel and Levin (2009). Panel (b) illustrates a more pronounced overbidding tendency
for the first unit, whereas Panel (c) indicates that this tendency is weaker for the second
unit. HRnohint, which excludes the human interaction present in HHnohint, exhibitsa tendency similar to HHnohint. This implies that even without the impact of humaninteraction, such as strategic interaction and social preference, many subjects overbid,
and few bid sincerely. As summarized in Subsection 3.2, most subjects have a goodunderstanding of Vickrey auction rules. Thus, most subjects in HHnohint and HRnohint
overbid, although they have a good understanding of Vickrey auction rules.
10

We asked subjects how they bid in the auction through a post-experimental survey. We confirmedthat there is a positive correlation between their reported bidding behavior in the survey and theiractual bidding behavior in the experiments. This suggests that subjects honestly reported theirbidding strategies in the survey. For detailed results from the post-experimental survey, please refer
to Online Appendix A.

21

Figure 7: Scatter plots between bids and valuation in Periods 1–10 and 11–20 by treatment and units(a) Unit 1(b) Unit 2

22

Figure 8: Bid category by treatments and units(a) Both units(b) Unit 1(c) Unit 2

Notes: The error bars correspond to the 95% confidence interval.

23

In summary, in the absence of hints about SP, subjects tend to overbid in the Vickrey
auction, even when most of them have a good comprehension of Vickrey auction rules,
and factors such as strategic interaction and social preference are removed. Thus, overbidding in Vickrey auctions cannot be solely attributed to insufficient comprehension of
auction rules or human interaction.
However, the bidding tendencies of HHhint and HRhint differ. This implies thatthe hints about SP influenced the subjects’ bidding behavior. In Subsection 3.4, we
statistically examine this effect.

3.4

Effect of hint on strategy-proofness

Table 2 summarizes the sincere bidding rates for (i) all subjects, (ii) subjects with perfectscores in Quiz 2, and (iii) subjects with imperfect scores in Quiz 2 in each treatment.
First, to assess whether SP hints have a significant impact on the sincere bid rate, we
conducted statistical tests, specifically two-sample t-tests. The summarized findings are
as follows:
Result 1 (Effect of hints about SP )(i) Hints about SP significantly increase sincere bids in Vickrey auctions when considering data from both units. The mean increase is 23.528% from HHnohint to HHhint (95%
confidence interval, 20.518% to 26.537%).
(ii) The effectiveness of hints persists in HHnohint and HHhint, even when focusing solely
on subjects with perfect scores or those with imperfect scores.
Support. To establish (i), Hypothesis 1 is rejected at p < 0.001. To establish (ii), we
analyze data separately for subjects with perfect and imperfect scores in Quiz 2. For(ii), Hypothesis 1 is rejected at p < 0.001 for subjects with perfect and imperfect scores.

Result 2 (Effect of hints about SP with robot rivals)(i) In environments with no human interaction (HRnohint and HRhint), hints about SP
are effective when considering data from both units. The mean increase is 28.463% from
HRnohint to HRhint (95% confidence interval, 25.344% to 31.582%).

24

(ii) Hints are effective when focusing only on the data of subjects with perfect scores.
However, the hints become ineffective and even decrease sincere bids when focusing only
on data of subjects with imperfect scores.
Support. We first establish (i). Thus, Hypothesis 2 is rejected at p < 0.001. Regarding(ii), Hypothesis 2 is rejected at p < 0.001 regarding subjects with perfect scores. Nevertheless, it is rejected at p < 0.05 in a direction that is different from what we expected
regarding subjects with imperfect scores.
The results indicate that subjects provided with SP hints via Quiz 2 w/ hints havesignificantly higher rates of sincere bidding. This suggests that understanding SP is an
important factor in sincere bidding. We postulate that the subjects achieving perfectscores in Quiz 2 w/ hints possess a better understanding of SP, resulting in higher rates ofsincere bids compared to those with imperfect scores. This postulation was statisticallytested. In HHhint, subjects with perfect scores in Quiz 2 w/ hints submit sincere bids
more frequently than those with imperfect scores across all data points for both units.
The mean increase is 10.392% (95% confidence interval, 5.420% to 15.363%). Similarly,
in HRhint, even in the absence of human interaction, perfect scores have a significant
impact on the rate of sincere bids. Using all data from both units, the mean increase is58.544% (95% confidence interval, 54.078% to 63.011%). These results further support
our hypothesis that understanding SP is an important factor in sincere bidding.

3.5

Effect of human interaction

Based on the data presented in Table 2, we analyze the effects of human interaction.
We summarize our findings on human interaction as follows:
Result 3 (Effect of Human interaction)(i) Removing human interaction increases the sincere bid rate in Vickrey auctions whenno hints about SP are provided. This is based on the dataset for both units. The
mean increase from HHnohint to HRnohint is 7.839% (95% confidence interval, 4.892%
to 10.786%).
(ii) When hints about SP are provided, removing human interaction leads to an increase25

Table 2: Sincere bidding rates by treatment, Quiz 2 scores, and unit
(4)
HRhint(3) − (1)(4) − (2)0.333 <*** 0.617(0.011)(0.011)1720
1920
0.339 <*** 0.751(0.012)(0.011)
1600
1480
∨∗∗
∨∗∗∗
0.250 >** 0.166(0.040)(0.018)120440

0.078∗∗∗

0.128∗∗∗

0.089∗∗∗

0.230∗∗∗

Imperfect score(S.E.)
obs.

0.254 <*** 0.489(0.010)(0.012)
1920
1800
0.249 <*** 0.522(0.011)(0.014)
1640
1240
∧
∨∗∗∗
0.282 <*** 0.418(0.027)(0.021)280
560−0.032−0.252∗∗∗(b)Unit 1
All(S.E.)
obs.

0.253 <*** 0.461(0.014)(0.017)960
900

0.301 <*** 0.606(0.016)(0.016)860
960

0.048∗∗

0.145∗∗∗

0.248 <*** 0.508(0.015)(0.020)
820
620
∧
∨∗∗∗
0.286
< 0.357(0.038)(0.029)140280

0.295 <*** 0.736(0.016)(0.016)
800
740
∧
∨∗∗∗
0.383 >*** 0.168(0.063)(0.025)60
220

0.047∗∗

0.228∗∗∗

0.098−0.189∗∗∗0.255 <*** 0.518(0.014)(0.017)960
900
0.251 <*** 0.535(0.015)(0.020)820
620∧∨
0.279 <*** 0.479(0.038)(0.030)140280

0.364 <*** 0.628(0.016)(0.016)860
960
0.383 <*** 0.766(0.017)(0.016)
800
740
∗∗∗
∨
∨∗∗∗
0.117
< 0.164(0.042)(0.025)60
2200.109∗∗∗

0.110∗∗∗

0.131∗∗∗

0.231∗∗∗−0.162∗∗−0.315∗∗∗Treatments(a)Both units
All(S.E.)
obs.
Perfect score(S.E.)
obs.

Perfect score(S.E.)
obs.
Imperfect score
(S.E)
obs.
(c)Unit 2
All(S.E.)
obs.
Perfect score
(S.E)
obs.
Imperfect score
(S.E)
obs.

(1)
HHnohint(2)
HHhint(3)
HRnohint

Notes: a) In HHnohint, subjects compete against human rivals and take Quiz 2 w/o hints. In HHhint,
subjects compete against human rivals and take Quiz 2 w/ hints. In HRnohint, subjects compete
against robot rivals and take Quiz 2 w/o hints. In HRhint, subjects compete against robot rivals
and take Quiz 2 w/ hints. b) We compare the sincere bidding rates between treatments using two
sample t-tests. c) * ** and *** denote significance at the 10%, 5%, and 1% levels, respectively. d)
Numbers in parentheses are standard errors. e) The unit of observations is the decision-making ineach period. The total number of observations is the number of subjects × number of periods ×
number of units in each treatment.

26

in the sincere bid rate in Vickrey auctions, again based on the dataset for both units.
The mean increase from HHhint to HRhint is 12.774% (95% confidence interval, 9.604%
to 15.945%).
(iii) Eliminating human interaction results in an increased rate of sincere bids in Vickrey
auctions. However, this increase is only observed in the data from subjects who achieved
perfect scores in either Quiz 2 w/o hints or Quiz 2 w/ hints. No such increase is observed
when analyzing the data from subjects who scored imperfectly in either quiz.
Support. We first establish (i) and (ii): Hypotheses 3a and 3b are rejected at p <0.001. Regarding (iii), Hypotheses 3a and 3b are rejected at p < 0.001 for the data ofsubjects with perfect scores. For the subjects with imperfect quiz scores, Hypothesis 3a
is supported. Hypothesis 3b is rejected at p < 0.001 in a direction that is different fromour expectations.
The results in this subsection imply that human interaction is also a non-negligible
factor in sincere bidding. In particular, human interactions and understanding SP affectthe rate of sincere bidding. Thus, our findings suggest that various factors, including
human interactions and differing levels of understanding SP, cause insincere bidding in
Vickrey auctions.

3.6

Learning Effect

Table 3 summarizes the sincere bidding rates for (i) the first round, (ii) the first ten
rounds, and (iii) the last ten rounds of each treatment. We statistically test whether
SP hints increase the sincere bid rate during the first period by conducting two sample
t-tests.
Result 4 (First period )(i) Hints about SP increase the sincere bids in the Vickrey auctions in the first period. The mean increase is 25.000% from HHnohint to HHhint (95% confidence interval,
11.419% to 38.581%).
(ii) In environments with no human interaction (i.e., in HRnohint and HRhint), hints
about SP are effective in the first period. The mean increase is 31.347% from HRnohint
to HRhint (95% confidence interval, 17.417% to 45.277%).
27

(iii) Hints are effective when focusing only on the data of the subjects with perfect scores.
However, they are ineffective when focusing only on subjects with imperfect scores.
The results indicate that subjects provided with SP hints via Quiz 2 w/ hints havesignificantly higher rates of sincere bidding in the first period. This suggests that anunderstanding of SP is an important factor in sincere bidding, even if the bidders do
not have much experience with auctions.
We use two-sample t-tests to statistically test whether learning leads to increased
sincere bid rates by comparing the rates between the first and last 10 periods.
Result 5 (Learning effect)(i) There is no significant difference in sincere bidding rates between the first and last
10 periods in all treatments (p = 0.463 for HHnohint; p = 0.962 for HHhint; p = 0.357
for HRnohint; p = 0.241 for HRhint).
(ii) Focusing only on the data of the subjects with perfect scores, there is no significant
difference in sincere bidding rates between the first and last 10 periods in HHnohint(p = 0.530), HHhint (p = 0.776) and HRnohint (p = 0.342). There is a 10% significant
difference in HRhint.
(iii) Focusing only on the data of the subjects with imperfect scores, there is no significant
difference in sincere bidding rates between the first and last 10 periods in all treatments(p = 0.692 for HHnohint; p = 0.608 for HHhint; p = 1.000 for HRnohint; p = 0.701 forHRhint).
The results indicate that the subjects do not increase their sincere bidding overtime. This suggests that while an understanding of SP is an important factor in sincere
bidding, learning does not effectively increase sincere bidding rates.

3.7

Regression analysis of sincere bidding

We conducted a logistic regression analysis to examine the treatment effect on sincere
bidding. The analysis uses a sincere bidding dummy variable as the dependent variable,
which takes the value of 1 if the subject bids sincerely in a given period and 0 otherwise.
The independent variables include an SP hint dummy (SP hint), which is set to 1 for28

Table 3: Sincere bidding rates of both units by treatment, Quiz 2 scores and periods
Treatments(1)
HHnohint(2)
HHhint(3)
HRnohint(4)
HRhint(3) − (1)(4) − (2)

0.041

0.104

0.044

0.184∗∗−0.048−0.2010.075∗∗∗

0.115∗∗∗

0.085∗∗∗

0.207∗∗∗−0.021−0.234∗∗∗0.082∗∗∗

0.140∗∗∗

0.094∗∗∗

0.253∗∗∗−0.043−0.269∗∗∗(a) Period 1
All(S.E.)
obs.
Perfect score(S.E.)
obs.

0.250 <*** 0.500(0.044)(0.053)96
90
0.244 <*** 0.532(0.048)(0.064)82
62∧∨
Imperfect score 0.286< 0.429(S.E.)(0.125)(0.095)obs.
14
28

0.291 <*** 0.604(0.049)(0.050)86
96
0.288 <*** 0.716(0.051)(0.053)
80
74
∨
∨∗∗∗
0.333
> 0.227(0.211)(0.091)
622(b) Former 10 periods
All(S.E.)
obs.
Perfect score(S.E.)
obs.

0.247 <*** 0.489(0.014)(0.017)960
9000.243 <*** 0.526(0.015)(0.020)
820
620
∧
∨∗∗∗
Imperfect score 0.271 <*** 0.407(S.E.)(0.038)(0.029)
obs.
140
280

0.322 <*** 0.604(0.016)(0.016)860
960
0.328 <*** 0.732(0.017)(0.016)
800
740
∨
∨∗∗∗
0.250
> 0.173(0.056)(0.026)60220(c) Latter 10 periods
All(S.E.)
obs.
Perfect score(S.E.)
obs.

0.261 <*** 0.490(0.014)(0.017)960
900
0.256 <*** 0.518(0.015)(0.020)
820
620
∧
∨∗∗
Imperfect score 0.293< 0.429(S.E)(0.039)(0.030)
obs.
140
280

0.343 <*** 0.630(0.016)(0.016)860
960
0.350 <*** 0.770(0.017)(0.015)
800
740
∨
∨∗∗∗
0.250
> 0.159(0.056)(0.025)60
220

Notes: a) In HHnohint, subjects compete against human rivals and take Quiz 2 w/o hints. In
HHhint, subjects compete against human rivals and take Quiz 2 w/ hints. In HRnohint, subjects
compete against robot rivals and take Quiz 2 w/o hints. In HRhint, subjects compete against robot
rivals and take Quiz 2 w/ hints. b) We compare the sincere bidding rates between treatments using
two-sample t-tests. c) * ** and *** denote significance at the 10%, 5%, and 1% levels, respectively.
d) Numbers in parentheses are standard errors. e) The unit of observations is the decision-making
in each period. The total number of observations is the number of subjects × number of periods ×
number of units in each treatment.

29treatments where Quiz 2 w/ hints is administered to the subjects, and a robot dummy,which is set to 1 for treatments involving robot rivals. We also included a control
variable, a perfect score dummy, which takes the value of 1 if subjects achieved a perfectscore on Quiz 2, and Period, which ranges from 1 to 20. We clustered the robust
standard errors at the individual level. The results of Unit 1, Unit 2, and both units are
summarized in Table 4.
The logistic regression results demonstrate that the SP hint dummy has a statisticallysignificant positive impact on sincere bidding—administering Quiz 2 w/ hints enhances
sincere bidding. The perfect score dummy also has a significant positive impact onsincere bidding. These results confirm the conclusion drawn in Subsection 3.4 that
understanding SP is an important factor in sincere bidding. However, the robot dummy
is not statistically significant, indicating that the human interaction factor does not
significantly affect sincere bidding. This contradicts the conclusion in Subsection 3.5.
In our logistic regression analysis, the human interaction factor is not as robust as theunderstanding of SP. Additionally, the period variable is not statistically significant,indicating that learning does not significantly affect sincere bidding. This is consistent
with the results presented in Subsection 3.6.
Next, we performed a regression analysis to examine the impact of the subjects’
performance in Quiz 2, with and without hints, on sincere bidding. We conducted an
OLS linear regression, considering average sincere bids for Units 1 and 2, as well as bothunits collectively, with robust standard errors. The dependent variable is the averagerate of sincere bids, which ranges from 0 to 1. The independent variables included the
Quiz 2 scores, which range from 0 to 10, and a dummy variable for treatments withrobot rivals, referred to as the robot dummy. The results are summarized in Table 5,organized by Quiz 2 type and units.
The results indicate that higher scores in Quiz 2 w/ hints are associated with more
frequent sincere bidding in Units 1 and 2, as well as when both units are combined.
However, the scores in Quiz 2 w/o hints do not seem to influence the rate of sincere
bids. Based on these findings, we suggest that our design of Quiz 2 w/ hints effectively
enhances sincere bidding and serves as a reliable method for measuring the level of30

Table 4: Regression analysis of treatment dummy on sincere bidding
Both
(1)

(2)

Unit 1(3)(4)

Unit 2

(5)

(6)

(7)

(8)

(9)

SP hint

1.034*** 1.230*** 1.230*** 0.926** 1.110*** 1.110*** 1.142*** 1.351*** 1.351***(0.347)(0.375)(0.375) (0.371) (0.399) (0.399) (0.356) (0.389)(0.389)
Robot
0.380
0.320
0.321
0.240
0.181
0.181
0.513
0.454
0.454
(0.374)(0.381)(0.382) (0.408) (0.419) (0.419) (0.390) (0.395)(0.395)
SP hint × Robot 0.140
0.138
0.138
0.347
0.351
0.351
−0.060 −0.067 −0.067(0.513)(0.517)(0.517) (0.552) (0.557) (0.558) (0.531) (0.536)(0.536)
Perfect score1.040*** 1.040***
1.012** 1.012**
1.071*** 1.071***(0.370)(0.370)(0.402) (0.402)(0.378)(0.378)Period
0.006
0.008
0.005
(0.004)(0.006)(0.005)Constant−1.077***−1.998***−2.063***−1.082***−1.978***−2.060***−1.071***−2.021***−2.069***(0.262) (0.457)(0.457) (0.270) (0.479) (0.482) (0.270) (0.472)(0.471)
Obs.
Cluster
Wald Chi2
Prob > Chi2

7,360
184
20.71
0.000

7,360
184
26.58
0.000

7,360
184
29.86
0.000

3,680
184
18.85
0.000

3,680
184
22.74
0.000

3,680
184
24.68
0.000

3,680
184
20.15
0.000

3,680
184
25.82
0.000

3,680
184
28.50
0.000

Notes: a) We conducted logit regression on the sincere bidding dummy, considering the SP hintdummy and robot dummy, conditional on the Quiz 2 perfect score dummy, with robust standard
errors clustered at the individual level. b) Sincere bidding dummy equals 1 if subjects submitted asincere bid in a period and 0 otherwise. SP hint dummy equals 1 if subjects take Quiz 2 w/ hints
and 0 if they take Quiz 2 w/o hints. Robot dummy equals 1 if subjects compete against robot rivals
and 0 if they compete against human rivals. Perfect score equals 1 if subjects get a perfect scorein Quiz 2 and 0 otherwise. Period ranges from 1 to 20. c) * ** and *** denote significance at the
10%, 5%, and 1% levels, respectively. d) Numbers in parentheses are standard errors. e) The total
number of observations is the number of subjects × number of periods × number of units in eachtreatment.

Table 5: Regression analysis of Quiz 2 scores on average sincere bids
w/ hints (HHhint & HRhint )Both
Unit 1
Unit 2

Quiz 2 score
Robot
ConstantObs.
R-squared

w/o hints (HHnohint & HRnohint )Both
Unit 1
Unit 2

(1)

(2)

(3)

(4)

(5)

(6)

0.078***(0.016)0.132(0.082)−0.230(0.160)0.081***(0.016)0.150*(0.087)−0.281*(0.154)0.076***(0.017)0.114(0.084)−0.178(0.171)

0.005(0.016)0.078(0.078)0.205(0.157)−0.006(0.024)0.049(0.084)0.309(0.228)0.016(0.017)0.107(0.084)0.101(0.172)

93
0.140

93
0.137

93
0.122

91
0.012

91
0.004

91
0.023

Notes: a) We conducted OLS linear regression on the average sincere bids for units 1 and 2, as
well as both units collectively, regressed on Quiz 2 scores with robust standard errors. b) Average
sincere bids indicate the mean of sincere bidding over 20 periods for each subject, ranging from 0
to 1. Quiz 2 scores range from 0 to 10. Quiz 2 w/ hints scores are used in Model (1-3), and Quiz 2
w/o hints are used in Model (4-6). Robot dummy is 1 if subjects compete against robot rivals and
0 if subjects compete against human rivals. c) * ** and *** denote significance at the 10%, 5%,
and 1% levels, respectively. d) Numbers in parentheses are standard errors. e) The total number ofobservations is the number of subjects in HHhint & HRhint or in HHnohint & HRnohint.

31

understanding of SP in Vickrey auctions.
The OLS results indicate that higher scores in Quiz 2 w/ hints are associated withmore frequent sincere bidding in Units 1 and 2 as well as when both units are combined.
Nevertheless, the scores in Quiz 2 w/o hints do not appear to influence the rate ofsincere bids. As confirmed in Subsection 3.2, there is no statistical difference in theearnings levels and comprehension levels of the auction rules between Quiz 2 w/ hints
and Quiz 2 w/o hints. Thus, these results confirm the conclusion that understanding SP
is an important factor in sincere bidding. However, the robot dummy is not statisticallysignificant, except for in Unit 1 of HHhint & HRhint (10%). Similar to the logistic
regression analysis, our OLS analysis suggests that the human interaction factor is not
as robust as the understanding of SP. Notably, our OLS analysis further indicates thatour design of Quiz 2 w/ hints effectively enhances sincere bidding and serves as a reliable
method for measuring the level of understanding of SP in Vickrey auctions.

3.8

Ratios of sincere and insincere bidders

We categorize subjects into the following groups based on their sincere bidding frequencies x specifically: “x = 0%”; “0% < x < 20%”; “20% ≤ x < 40%”; “40% ≤ x < 60%”;“60% ≤ x < 80%”; “80% ≤ x < 100%”; “x = 100%.” Each subject submits 40 bids
(i.e., 20 periods × 2 units per period). For instance, a subject who bids sincerely 20
times is categorized as “40% ≤ x < 60%.” Similarly, a subject who bids sincerely 25
times is categorized as “60% ≤ x < 80%.” Figure 9 shows the ratios of the subjects
who bid sincerely at various frequencies. Figure 9a shows the results for the overall
dataset, whereas Figure 9b presents the outcomes for the data of subjects with perfectand imperfect scores in Quiz 2.
Figure 9 illustrates two distribution concentrations. One is the group of subjects
whose sincere bidding frequencies are greater than or equal to 80%, and the other isthe group of subjects whose sincere bidding frequencies are less than 20%. We callbidders in the first group “sincere bidders” and those in the second group “insincere
bidders.”11 Subsequently, we employ a two-sample t-test to compare the ratios of sincere
11

We calculate the correlation between bidder types and Quiz 2 w/ hints scores using the Spearman
correlation method. We find a significant positive correlation between “sincere bidders” and Quiz32

Figure 9: The ratios of sincere and insincere bidders(a) Overall(b) Imperfect score vs Perfect score33

and insincere bidders across the treatments.

3.8.1

Sincere bidders

First, we analyze the impact of SP hints on the ratios of sincere bidders. Overall, ourfindings reveal that the ratios of sincere bidders increases from 17% in HHnohint to 36%
in HHhint at the 5% significance level. The ratio increases from 21% in HRnohint to
54% in HRhint at the 1% significance level. For subjects with perfect scores, the ratio
increases from 17% in HHnohint to 39% in HHhint at the 5% significance level and from
23% in HRnohint to 68% in HRhint at the 1% significance level. Conversely, for subjects
with imperfect scores in Quiz 2, the ratios of sincere bidders remains similar between
HHnohint (14%) and HHhint (29%) with no significant difference (p = 0.494). Similarly,
there is no significant difference between HRnohint (0%) and HRhint (9%) with respect
to the ratios of sincere bidders (p = 0.621).
Second, we analyze the impact of human interaction on the ratios of sincere bidders.
For subjects given Quiz 2 w/o hints, we observe no statistically significant difference inthe ratios of sincere bidders between HHnohint and HRnohint in the overall dataset (17%in HHnohint vs 21% in HRnohint, p = 0.607), perfect score dataset (17% in HHnohint vs
23% in HRnohint, p = 0.546), and imperfect score dataset (14% in HHnohint vs 0% in
HRnohint, p = 0.545). Nevertheless, when subjects are given Quiz 2 w/ hints, we find astatistically significant increase in the ratios of sincere bidders, from 36% in HHhint to
54% in HRhint, at the 10% significance level. This increase is also pronounced among
subjects with perfect scores in Quiz 2, with the ratio increasing significantly from 39%
in HHhint to 68% in HRhint at the 5% significance level. Contrarily, no such increase isobserved among subjects with imperfect scores in Quiz 2, with the ratio being 29% in
HHhint and 9% in HRhint (p = 0.244).
In summary, the SP hints exert a positive influence, particularly on subjects withperfect scores, compelling them to bid sincerely frequently. A substantial ratio of thesesubjects comprises sincere bidders when competing against robot rivals. Conversely, theeffects of human interaction are mixed. We observe no significant impact on the ratios
2 w/ hints scores (ρ = 0.305, p < 0.01) and a negative correlation between “insincere bidders” andQuiz 2 w/ hints scores (ρ = −0.361, p < 0.01).

34

of sincere bidders when subjects are given Quiz 2 w/o hints but significant impacts whensubjects receive Quiz 2 w/ hints.

3.8.2

Insincere bidders

First, we analyze the impact of SP hints on the ratios of insincere bidders. Overall, ourfindings reveal that the ratios of insincere bidders decrease from 65% in HHnohint to
38% in HHhint at the 1% significance level. The ratios decrease from 56% in HRnohint to
33% in HRhint at the 5% significance level. For subjects with perfect scores, the ratios
decrease from 66% in HHnohint to 32% in HHhint at the 1% significance level and from
58% in HRnohint to 19% in HRhint at the 1% significance level. Conversely, for subjects
with imperfect scores in Quiz 2, the ratios of insincere bidders remain similar between
HHnohint (57%) and HHhint (50%) with no significant difference (p = 0.772) and also
between HRnohint (33%) and HRhint (82%) with no significant difference (p = 0.115).
Second, we analyze the impact of human interaction on the ratios of insincere bidders.
For subjects given Quiz 2 w/o hints, we observe no statistically significant difference inthe ratios of insincere bidders between HHnohint and HRnohint in the overall dataset(65% in HHnohint vs 56% in HRnohint, p = 0.399), perfect score dataset (66% in HHnohint vs 58% in HRnohint, p = 0.446), and imperfect score dataset (57% in HHnohint
vs 33% in HRnohint, p = 0.545). When subjects are given Quiz 2 w/ hints, we find nostatistically significant difference in the ratios of insincere bidders between HHhint and
HRhint in the overall dataset (38% in HHhint vs 33% in HRhint, p = 0.659), perfect score
dataset (32% in HHhint vs 19% in HRhint, p = 0.211), and imperfect score dataset (50%in HHhint vs 82% in HRhint, p = 0.108).
In summary, the SP hint has a positive impact on reducing the ratio of insincere
bidders. However, the factor of human interaction does not influence the ratios ofinsincere bidders, regardless of whether subjects are given Quiz 2 w/ hints or Quiz 2
w/o hints.

35

3.8.3

Converting insincere bidders to sincere bidders

The above results imply that SP hint converts a substantial ratio of insincere bidders
into sincere bidders. This conversion explains the increase of the sincere bidding rates
caused by SP hint, which is reported in Subsection 3.4. Conversely, the above results
imply that human interaction does not make such clear conversion. This differs fromthe results in Subsection 3.5 that human interaction has non-negligible effects on sincere
bidding rates but is consistent with results in Subsection 3.7 that the impact of humaninteraction is not as robust as SP hints.

4

Efficiency, bidders’ payoffs, and seller’s revenue

4.1

Efficiency measure

We follow Kagel and Levin (2009)’s efficiency measure. For the details of the efficiency
measures, please refer to Masuda et al. (2022). In each period, if bidders i and j are,respectively, the winners with the first and second highest bids, then the efficiency isgiven byr= (v 1 + v 1 )⧸(v[1] + v[2]) if i ̸= j
i

j

,


 (vi1 + vi2 )⧸(v[1] + v[2]) if i = j
where v[1] and v[2] denote the two highest valuations among {v11 , v12 , v21 , v22 , v31 , v32 }.12
We apply the above efficiency formula to calculate the efficiencies of HHnohint and
HHhint, where the experiments involve only human bidders, which we refer to as observed efficiency. In the auctions of HRnohint and HRhint, two of the three bidders arerobot bidders, whose bids are automatically generated from a specified probability distribution. It does not make sense to compute efficiency by using the bid data of robot
bidders. Thus, to calculate the efficiency of HRnohint and HRhint only for the bid dataof human subjects, we conduct simulations as explained in Subsection 4.2. We refer tothe simulation results as simulated efficiency.
We divide the dataset into two parts: one containing subjects with perfect scores in
12

It might be the case that i = j when one bidder’s two bids are the first and second highest. Ties
are broken with equal probability.

36

Quiz 2 and the other containing subjects with imperfect scores in Quiz 2. Subsequently,
we calculate the simulated efficiencies of each part of the data. Thus, we analyze theefficiency when only subjects with perfect scores are matched with each other, as wellas when only subjects with imperfect scores are matched.

4.2

Simulation method

In each treatment, let G be the largest integer, such that 3 ∗ G is less than or equal to
the number of subjects in the treatment group. For example, if the number of subjects
in the treatment is 43, as in HRnohint, then G = 14 and 3 ∗ G = 42. For each treatment
and each period t of the experiment, we calculate the simulated efficiency, simulated
bidders’ payoffs, and simulated seller’s revenue using the following steps:
Step 1: We randomly draw 3 ∗ G subjects from the human subjects in the treatmentgroup and match them into G groups, each of which comprises three subjects. The
groups are named 1, · · · , G.
Step 2: In each group g ∈ {1, · · · , G}, we have three subjects with valuations of the two
1 , v 2 , v 1 , v 2 , v 1 , v 2 ). Let v [1] and v [2]units observed in period t, that is, (v1gtgtgt
1gt 2gt 2gt 3gt 3gt
1 , v 2 , v 1 , v 2 , v 1 , v 2 }.
be the first- and second-highest valuations among {v1gt
1gt 2gt 2gt 3gt 3gt

Step 3: In each group g ∈ {1, · · · , G}, we also have the submitted bids of the threesubjects observed in period t. In other terms, (b11gt , b21gt , b12gt , b22gt , b13gt , b23gt ). Let bidders
i and j be the winners, with the first and second highest bids among{b11gt , b21gt , b12gt , b22gt , b13gt , b23gt }. Ties are broken with equal probability.
Step 4: In each group g ∈ {1, · · · , G}, we calculate efficiency rgt in period t as follows:

rgt =



 (v 1 + v 1 )⧸(vgt [1] + vgt [2]) if i ̸= j
igt
jgt

,


1 + v 2 )⧸(v [1] + v [2]) if i = j (vigt
gt
gt
igt

We also calculate the bidders’ payoffs and seller’s revenue from period t.
Step 5: We compute the efficiency averaged across all groups and periods as follows:
1 P20 PG
r = G×20
t=1
g=1 rgt . We also compute the mean of the bidders’ payoffs and seller’s
revenue for all groups and periods.
Step 6: We repeat Steps 1–5 1,000 times—we perform bootstrap sampling 1,000 times.

37Subsequently, we obtain the efficiency, the bidders’ payoffs, and the seller’s revenue foreach bootstrap sample B = 1, 2, 3, ..., 1, 000, where B is the bootstrap sampling index.
Step 7: We calculate simulated efficiency, which is the the mean efficiency of all bootstrap
1 P1,000
sampling by 1,000
B=1 rB , and its bootstrapped standard error by taking the meanof standard errors of each bootstrap sample. We then similarly calculate the simulated
bidders’ payoffs, the simulated seller’s revenue, and their bootstrapped standard errors.

4.3

Efficiency: observed and simulated

Table 6 summarizes the results for the observed and simulated efficiencies. Overall, theobserved efficiency is higher for HHhint (0.992), where subjects are given Quiz 2 w/
hints compared to HHnohint (0.984). The simulated efficiencies of HHnohint (0.982) and
HHhint (0.991) exhibit similar trends. The simulated efficiencies in HRnohint (0.983)
and HRhint (0.982) are closer, although the difference is statistically significant at the
1% level.
For the dataset of subjects with perfect scores in Quiz 2, the simulated efficiencies
are higher when Quiz 2 w/ hints is taken than when Quiz 2 w/o hints is taken—higher in
HHhint (0.991) than in HHnohint (0.984) and higher in HRhint (0.993) than in HRnohint(0.983). This implies that SP hints consistently increase the efficiency of subjects withperfect scores in Quiz 2.
Conversely, for subjects with imperfect scores in Quiz 2, while the simulated efficiency
in HHhint (0.988) is higher than that in HHnohint (0.970), it is lower in HRhint (0.951)
than in HRnohint (0.990). In particular, simulated efficiency in HRhint is notably low,which accounts for the overall low simulated efficiency of all subjects in HRhint.

4.4

Bidder’s payoff and seller’s revenue: observed and simulated

Table 7 summarizes the results for observed bidders’ payoffs and sellers’ revenues andsimulated bidders’ payoffs and sellers’ revenue. Overall, the observed bidders’ payoffs
are higher in HHhint (152.044) than in HHnohint (143.490), although the difference is
not statistically significant.
For the simulation results, the impact of SP hints on the simulated bidders’ payoffs

38

Table 6: Efficiency by treatment and Quiz 2 scores: observed and simulated
Treatments
Observed efficiency
All
obs.
Simulated efficiency
All
obs.
Perfect Score
obs.
Imperfect Score
obs.

(1)
HHnohint(2)
HHhint(4)
HRhint(3) − (1)(4) − (2)0.983 >∗∗∗ 0.982(0.002)(0.002)280,000
320,000
0.983 <∗∗∗ 0.993(0.002)(0.001)
260,000
240,000
0.990 >∗∗∗ 0.951(0.001)(0.003)20,00060,0000.001∗∗∗−0.008∗∗∗−0.002∗∗∗0.002∗∗∗

0.019∗∗∗−0.037∗∗∗(3)
HRnohint

0.984 <∗∗ 0.992(0.003)(0.002)320300

0.982 <∗∗∗ 0.991(0.002)(0.001)
320,000
300,000
0.984 <∗∗∗ 0.991(0.002)(0.001)
260,000
200,000
0.970 <∗∗∗ 0.988(0.002)(0.001)40,00080,000Notes: a) We compare the observed and simulated efficiencies between treatments using two-sample
t-tests. b) * ** and *** denote significance at the 10%, 5%, and 1% levels, respectively. c) Numbers
in parentheses are standard errors for efficiency and bootstrapped standard errors for simulated
efficiency. d) For efficiency, the total number of observations is the number of groups × 20 periods
in each treatment. For simulated efficiency, the number of observations is the number of simulatedgroups in each treatment × 20 periods × 1,000 bootstraps.

are mixed. When their rivals are humans, subjects who take Quiz 2 w/ hints receive lower
payoffs in HHhint (145.385) than those who take Quiz 2 w/o hints in HHnohint (147.785).
However, this trend is reversed when the rivals are robots in HRnohint (139.638) and
HRhint (148.560).
When we divide the subjects based on their Quiz 2 scores, different patterns arefound. Subjects with perfect scores in Quiz 2 w/ hints earn higher payoffs (143.745 in
HHhint; 162.900 in HRhint ) than those with perfect scores in Quiz 2 w/o hints (142.690
in HHnohint; 139.536 in HRnohint ), irrespective of whether they face human or robot
rivals. Conversely, subjects with imperfect scores experience reversed outcomes andreceive low payoffs (100.679) in HRhint.
Overall, the observed seller’s revenue is higher in HHhint (1117.333) than in HHnohint(1098.531), although the difference is not statistically significant. For the simulation
results, the impacts of SP hints on the simulated seller’s revenue are mixed, and thepatterns are reversed for bidders’ payoffs. In other words, SP hints increase simulated
seller’s revenue when bidders face human rivals (1085.880 in HHnohint vs 1129.619 in

39

Table 7: Bidder’s payoffs and seller’s revenues by treatment and Quiz 2 Score: observed
and simulated
Treatments(1)
HHnohint(2)
HHhintObserved bidder’s payoffAll
143.490(6.032)obs.
320< 152.044(5.467)
300

Simulated bidder’s payoff
All147.785 >∗∗∗ 145.385(3.392)(3.074)obs.
320,000
300,000
Perfect score
142.690 <∗∗∗ 143.745(3.353)(2.970)obs.
260,000
200,000
Imperfect score
169.884 >∗∗∗ 147.427(3.551)(3.342)
obs.
40,00080,000(4)
HRhint(3) − (1)(4) − (2)139.638 <∗∗∗ 148.560(3.326)(3.302)280,000
320,000
139.536 <∗∗∗ 162.900(3.360)(3.021)
260,000
240,000
132.833 >∗∗∗ 100.679(3.006)(3.888)20,00060,000−8.148∗∗∗3.175∗∗∗−3.154∗∗∗19.155∗∗∗−37.051∗∗∗−46.748∗∗∗41.813∗∗∗−37.234∗∗∗

32.217∗∗∗−74.756∗∗∗

57.531∗∗∗70.325∗∗∗(3)
HRnohint

Observed seller’s revenueAll
1098.531 < 1117.333(21.298)
20.620obs.
320
300
Simulated seller’s revenueAll1085.880 <∗∗∗ 1129.619(12.029)(11.253)
obs.
320,000
300,000
Perfect score1099.216 <∗∗∗ 1144.788(12.042)(10.982)
obs.
260,000
200,000
Imperfect score1021.469 <∗∗∗ 1098.884(12.320)(11.764)
obs.
40,00080,0001127.693 >∗∗∗ 1092.384(12.239)(11.281)280,000
320,000
1131.434 >∗∗∗ 1070.032(12.277)(10.977)
260,000
240,000
1079.000 <∗∗∗ 1168.208(13.826)(11.959)
20,00060,000Notes: a) We compare the bidder’s payoff, simulated bidder’s payoff, seller’s revenue, and simulated
seller’s revenue between treatments using two-sample t-tests. b) * ** and *** denote significanceat the 10%, 5%, and 1% levels, respectively. c) Numbers in parentheses are standard errors for
efficiency and bootstrapped standard errors for simulated efficiency. d) For bidder’s payoff andseller’s revenue, the total number of observations is the number of groups × 20 periods in eachtreatment. For simulated bidder’s payoff and simulated seller’s revenue, the number of observations
is the number of simulated groups in each treatment × 20 periods × 1,000 bootstraps.

40

HHhint ) but decrease when bidders face robot rivals (1127.693 in HRnohint vs 1092.384 in
HRhint ). When focusing on subjects with perfect scores, we observe the same pattern
as the overall simulated seller’s revenue. Conversely, when focusing on subjects withimperfect scores, SP hints lead to increased simulated seller’s revenue, regardless ofwhether the rivals are human or robot bidders.

5

Conclusion

In our experiment, we focused on two key factors that influence sincere bidding inhomogeneous multi-unit Vickrey auctions: the subjects’ understanding of SP and humaninteraction, which includes social preferences (spite and altruism), responses to strategic
uncertainty, and tacit collusion. To analyze the effect of understanding SP, we compared
the bidding behavior of subjects who took a quiz with SP hints and those who took aquiz without SP hints. To analyze the effect of human interaction, we compared thebidding behavior of subjects competing against robot rivals and those competing against
human rivals. Our study demonstrates that a better understanding of SP is a crucial
and strong driver of sincere bids. Conversely, although human interaction had a nonnegligible impact, it did not have as much influence as the understanding of SP. This led
us to conclude that the main factor causing insincere bidding was not human interaction
but a lack of understanding of the concept of SP.
We raise several questions for future research. First, is the difficulty in understanding
SP confined to Vickrey auctions, or is it a broader issue applicable to other types ofauctions? As discussed in Section 1, several researchers have offered insights based on
their experiments in which they aid subjects in recognizing SP. Recently, Gonczarowski
et al. (2023) proposed menu descriptions of mechanisms to enable subjects to understandSP more easily. Second, what are the most effective ways to educate subjects about SP
to encourage more sincere bidding?
In summary, our study contributes new insights to existing research on Vickrey
auctions by focusing on the factors that lead to insincere bidding. Our findings highlightthe importance of understanding SP as a key determinant of sincere bidding. This lays
the groundwork for future research to delve deeper into the causes of insincere bidding
41

and paves the way for targeted educational and design solutions that encourage moresincere bidding practices in auctions.

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44

Online Appendix to “Toward an Understanding of Insincere
Bidding in a Vickrey Auction Experiment”
Shigehiro Serizawa∗, Natsumi Shimada†, Tiffany Tsz Kwan Tse‡
February 16, 2024

Contents
A How did subjects bid in the auction, and why?

2

B Advice effect4

C Cognitive ability, risk aversion, and loss aversion

8

D Experimental instruction and quizzes

11

E Post-experiment survey

33∗

Corresponding author: Faculty of Economics, Osaka University of Economics
Organization for IT Initiatives, Shinshu University‡
Institute of Social and Economic Research, Osaka University
†

1

A

How did subjects bid in the auction, and why?

Based on Questions 5 and 6 of our post-experimental survey, we examine subjects’
bidding policies in the auction and the motivations for their policies. Question 5 is about
the subjects’ bidding policies. It asks, “How did you bid in the auction?” subjects chose
one of four answers:(a) Bid the amount of valuation(b) Bid higher than valuation(c) Bid lower than valuation(d) Other
Table A1: Survey of bidding policies (Question 5)
Treatments

HHnohint

HHhintHRnohint

HRhintProportion of selection of choice (a)Average sincere bid rates among subjects who
select choice (a)

27%
0.642

40%
0.864

35%
0.81

60%
0.947

Proportion of selection of choice (b)Average overbid rates among subjects who select choice (b)

46%
0.814

51%
0.682

44%
0.754

27%
0.808

Proportion of selection of choice (c)Average underbid rates among subjects who
select choice (c)

4%
0.4

0%

5%
0.938

6%
0.758

Proportion of selection of choice (d)

23%

9%

16%

6%

obs.

48

45

43

48

Notes: The number of observations is that of subjects in each treatment.

Table A1 summarizes the proportion of subjects selecting each answer in survey
question 5 and their average sincere bid rates, overbid rates, and underbid rates. We report the correlation between “Choice (a) Bid the amount of valuation” and the average
sincere bid rates using pairwise correlation analyses, specifically the Spearman correlation method. We find a significant correlation between “Choice (a)” and the average
sincere bid rates obtained in the experiments (ρ = 0.793, p < 0.01). Additionally, using
the Spearman correlation method, we find significant correlations between “Choice (b)”and the average overbid rates (ρ = 0.735, p < 0.01), as well as between “Choice (c)”and the average underbid rates (ρ = 0.302, p < 0.01). Therefore, we conclude that mostsubjects are likely to report their bidding policies honestly.
Question 6 concerns the motivations of subjects’ bidding policies. It asks subjects
why they bid as they did in Question 5, and subjects choose one of four answers:2

(a) To maximize your earnings in the auction(b) To maximize your winning probability rather than to maximize your earnings in
the auction(c) To make the earnings of other bidders smaller(d) To make the earnings of other bidders larger(e) Other
Choice (a) corresponds to the motivation assumed by standard auction theory.
Choice (b) corresponds to the joy of winning (Cooper & Fang, 2008). Choices (c)and (d), respectively, correspond to social preferences of spite and altruism.
Table A2 summarizes the proportion of subjects selecting each choice in Question
6. The proportion of subjects selecting (a) was approximately half for each treatment.
Moreover, the proportion of subjects who chose option (b) was approximately onethird for each treatment. These violations of the fundamental assumptions of auction
theory may have caused insincere bidding. However, only a few subjects reported social
preferences for spite and altruism. Thus, such social preferences may have had a limited
impact on inducing insincere bidding.
Table A2: Survey of motivations of bidding policies (Question 6)
Treatments

HHnohint

HHhintHRnohint

HRhintProportion of selection of choice(a)Proportion of selection of choice(b)
Proportion of selection of choice(c)
Proportion of selection of choice(d)Proportion of selection of choice
(e)

48%

64%

58%

65%

33%

31%

28%

25%

2%

2%

5%

0%

0%

0%

0%

2%

17%

2%

9%

8%

obs.

48

45

43

48

We found that 21% of the subjects in HHnohint, 38% in HHhint, 28% in HRnohint,
and 52% in HRhint chose (a) for Questions 5 and 6. These subjects understood that asincere bidding strategy could maximize their earnings. The introduction of SP hints(Quiz 2 w/ hints about SP) led to an increase in the proportion of such subjects. Thisaligns with the results presented in Section 3. We also found that 29% of subjects in
HHnohint, 29% in HHhint, 26% in HRnohint, and 19% in HRhint chose (b) for Questions
3

5 and 6. The objective of these subjects was to win the auction instead of maximizing
payoffs, and indeed, they did so. Thus, these data reveal that the joy of winning (Cooper& Fang, 2008) is also an important factor in insincere bidding. SP hints (Quiz 2 w/
hints about SP) did not decrease the proportion of such subjects.
Notably, 13% of the subjects in HHnohint, 20% in HHhint, 16% in HRnohint, and 6%
in HRhint chose (b) for Question 5 and (a) for Question 6. These subjects assumed thatoverbidding could maximize their earnings—they failed to understand SP. The averagescores for Quiz 2 of these subjects were 10 in HHnohint, 8.2 in HHhint, 10 in HRnohint,
and 8 in HRhint. The proportions of these subjects who got perfect scores in Quiz 2
are 100% in HHnohint, 56% in HHhint, 100% in HRnohint, and 33% in HRhint. These
data suggest that subjects who misunderstood SP and believed that overbidding could
maximize their earnings tended to score low in Quiz 2 w/ hints. Additionally, only afew subjects achieved a perfect score.

B

Advice effect

Masuda et al. (2022) demonstrated that providing advice to subjects can increase
the rate of sincere bidding from 20% to 47%. In their study, they assessed subjects’comprehension of Vickrey auction rules through a quiz similar to Quiz 1 in our study;
however, their quiz did not measure or aim to enhance subjects’ understanding of SP.
To investigate the effects of advice in our experimental environment, we designed two
additional treatments. These two treatments follow the design of HHhint and HRhint,
both of which include administering Quiz 2 w/ hints to subjects and providing themwith the same advice used in the study by Masuda et al. (2022). In other words, we
perform the following additional treatments:• Treatment HHhintA: only human subjects competed, and each subject was given
Quiz 2 with SP hints and advice• Treatment HRhintA: each subject competed with two robot rivals and was given
Quiz 2 with SP hints and advice
Figure B1 illustrates the flow of the experiments in HHhintA and HRhintA. The
procedure for these experiments closely resembles that of HHhint and HRhint, with one
exception. After the subjects completed Quiz 2 with hints and listened to an audio
explanation of the answer key, we distributed a paper containing the written advice.
The text of the advice is as follows:“The following advice is about the auction in which you are participating.
Please consider carefully whether this advice is true or false. It is completely
up to you whether you follow the advice or not.”4You can maximize your earnings by bidding your valuations as they are,regardless of what others bid.
Subsequently, the subjects listened to an audio recording of this advice.
Figure B1: Flow of the experiment

Table B1: Summary of treatments
Treatments

HHhintA

HRhintA

Type of Quiz 2
Nature of rivals
No. of sessions
Duration (min)No. of rounds
No. of subjects
Avg. payment (JPY)
Avg. score of Quiz 1
Avg. score of Quiz 2 w/ hints

w/ hints
human
2
150
20
42
5083
9.310
8.786

w/ hints
robot
2
150
20
40
5089
9.725
9.650

Table B1 summarizes the basic data for HHhintA and HRhintA. Table B2 lists thesincere bidding rates for HHhint, HRhint, HHhintA, and HHhintA. Using data from bothunits displayed in Panel (a), we examine whether the advice leads to higher rates ofsincere bidding among subjects with a better understanding of SP. Specifically, we use
two-sample t-tests to compare the rates of sincere bidding between HHhint and HHhintA
when rivals are human and between HRhint and HRhintA when rivals are robots.
Overall, our findings indicate that providing advice effectively increases the rate ofsincere bidding, although the effect in our experiment is less than that in Masuda et al.
5(2022). The mean increase from HHhint to HHhintA is 8.675% (95% confidence interval,
5.367% to 11.982%), and that from HRhint to HRhintA is 13.094% (95% confidence interval, 10.019% to 16.169%). As mentioned in Appendix A, approximately one-third of
subjects reported that they overbid to increase their winning probabilities rather thantheir earnings. Therefore, there is a limit to increasing sincere bidding through understanding SP. We hypothesize that this limit made the effect of advice in our experimentless effective than that in Masuda et al. (2022) because the sincere bidding rates in
HHhint and HRhint were already high and had less potential for further improvement.
This advice has been demonstrated to be highly effective when we focus on data
pertaining to subjects who do not achieve perfect scores. For subjects with imperfectscores in Quiz 2 w/ hints, the mean increases from HHhint to HHhintA are 17.679%(95% confidence interval, 11.903% to 23.454%), and those from HRhint to HRhintA are
34.481% (95% confidence interval, 28.073% to 40.888%). For subjects with perfect scoresin Quiz 2 w/ hints, the mean increase from HHhint to HHhintA is 4.519% (95% confidence
interval, 0.495% to 8.543%). The mean increase from HRhint to HRhintA is 4.713%(95% confidence interval, 1.610% to 7.816%). This outcome aligns with our hypothesis,suggesting that subjects with imperfect scores had lower initial sincere bidding rates
and more potential for improvement.
Our results suggest that the advice effectively promotes sincere bidding. This positive effect is observed among subjects with higher and lower levels of understanding of
SP. Moreover, the advice appears to be particularly impactful for those with imperfectscores—a lower understanding of SP.

6

Table B2: Sincere bidding rates by treatment, Quiz 2 scores, and unit
(1)
HHhint(2)
HRhint(3)
HHhintA(4)
HRhintA(3) − (1)(4) − (2)(4) − (3)(a)Both units
All(S.E.)
obs.
Perfect score(S.E.)
obs.
Imperfect score(S.E.)
obs.

0.489(0.012)1800
0.522(0.014)1240
0.418(0.021)560

0.617(0.011)
1920
0.751(0.011)1480
0.166(0.018)440

0.576(0.012)1680
0.567(0.015)1120
0.595(0.021)560

0.748(0.011)1600
0.798(0.011)1320
0.511(0.030)280

0.087∗∗∗

0.131∗∗∗

0.172∗∗∗

0.045∗∗

0.047∗∗∗

0.232∗∗∗

0.177∗∗∗

0.345∗∗∗−0.084∗∗(b)Unit 1
All(S.E.)
obs.
Perfect score(S.E.)
obs.
Imperfect score
(S.E)
obs.

0.461
(0.017)900
0.508(0.020)620
0.357(0.029)280

0.606(0.016)960
0.736(0.016)740
0.168(0.025)220

0.562(0.017)840
0.555(0.021)560
0.575(0.030)280

0.759(0.015)800
0.808(0.015)660
0.529(0.042)1400.101∗∗∗

0.153∗∗∗

0.197∗∗∗

0.047

0.071∗∗∗

0.252∗∗∗

0.218∗∗∗

0.360∗∗∗−0.046(c)Unit 2
All(S.E.)
obs.
Perfect score
(S.E)
obs.
Imperfect score
(S.E)
obs.

0.518
(0.017)900
0.535(0.020)620
0.479(0.030)280

0.628(0.016)960
0.766(0.016)740
0.164(0.025)220

0.590(0.017)840
0.579(0.020)560
0.614(0.029)280

0.738(0.016)800
0.789(0.016)660
0.493(0.042)140

0.073∗∗∗

0.109∗∗∗

0.147∗∗∗

0.043

0.023

0.211∗∗∗

0.136∗∗∗

0.329∗∗∗−0.121∗∗Treatments

Notes: a) In HHhint, subjects compete against human rivals and take Quiz 2 w/ hints. In HRhint,
subjects compete against robot rivals and take Quiz 2 w/ hints. In HHhintA, subjects compete
against human rivals, take Quiz 2 w/ hints, and receive advice. In HRhintA, subjects compete
against robot rivals, take Quiz 2 w/ hints, and receive advice. b) We compare the sincere bidding
rates between treatments by using two-sample t-tests. c) * ** and *** denote significance at the
10%, 5%, and 1% levels, respectively. d) Numbers in parentheses are standard errors. e) The unit ofobservations is the decision-making in each period. The total number of observations is the numberof subjects × number of periods × number of units in each treatment.

7

C

Cognitive ability, risk aversion, and loss aversion

Understanding SP may be related to subjects’ cognitive abilities. We examine thecorrelation between subjects’ cognitive abilities and their quiz scores. Approximately
half of the subjects participated in an experiment conducted by Hanaki et al. (2022).
From their dataset, we obtained data on the subjects’ cognitive abilities, risk aversion,and loss aversion.
For cognitive ability, we used the subjects’ International Cognitive Ability Resource(ICAR) scores, as proposed by Condon and Revelle (2014)1 . Hanaki et al. (2022)
used a three-dimensional (3D) rotation measure (four questions) and a matrix reasoning
measure (four questions) from those included in ICAR-16 (Condon and Revelle 2014,
Table 4). Subjects were given 40 seconds to answer each of the 3D rotation questions
and 30 seconds to answer each matrix reasoning question. The maximum possible score
was 8, and the subjects correctly answered an average of 2–3 questions.
For risk aversion, we collected the subjects’ risk attitudes from Hanaki et al. (2022)’s
experiments, in which they used the designs of Noussair et al. (2014) and Masuda andLee (2019). The elicitation task comprised five questions related to risk aversion. For
each question, subjects were asked to choose between two lotteries. The measure of risk
aversion is the number of safe options that a subject chooses from five questions. The
maximum score was 5, and the subjects selected an average of three safe options.
For loss aversion, Hanaki et al. (2022) used the experimental task proposed byKöbberling and Wakker (2005) to measure the degree of loss aversion. They asked
subjects to choose between a sure zero payment and a lottery in which they would get
600 JPY with a 50% chance or lose X JPY with a 50% chance, where X = 120, 240,
360, 480, 600, or 720. They assumed that loss-averse individuals tend to choose a surezero-payment option. Loss aversion was measured as the number of safe options in thesix questions. The maximum score was 6, and the subjects selected an average of threesafe options.
We report the correlations between ICAR and quiz scores using pairwise correlation
analyses, specifically the Spearman correlation method. We found no correlation between ICAR scores and Quiz 1 scores (ρ = 0.085, p = 0.399) or between ICAR scores
and Quiz 2 scores w/o hints (ρ = −0.024, p = 0.864). However, we found a significant
correlation between ICAR and Quiz 2 w/ hints scores (ρ = 0.373, p < 0.01). We conclude that achieving a perfect score in Quiz 2 w/ hints requires cognitive ability. Thus,
understanding SP also appears to require cognitive abilities.
1

See the ICAR team website, https://icar-project.com, for further details.

8

Table C1: Demographic data
Treatments

HHnohint

HHhintHRnohint

HRhint(i) Cognitive ability (ICAR scores)
obs.
(ii) Risk aversion (Noussair et al., 2014)
obs.
(iii) Loss aversion (Köbberling & Wakker, 2005)
obs.

2.129
31
3.323
31
3.226
31

2.318
22
3.182
22
2.727
22

3.143
21
3.217
23
2.381
21

2.815
27
3.586
29
3.852
27

Total number of subjects

48

45

43

48

Table C2: Regression analysis of treatment dummy on sincere bidding by controlling
preferences

SP hint
Robot
SP hint × Robot
Cognitive ability
Risk aversion
Loss aversion
Constant

Observations
Wald Chi2
Prob >Chi2

Both
(1)

Unit 1(2)

Unit 2(3)

1.279***(0.488)0.159(0.524)-0.045(0.741)0.147(0.125)0.030(0.140)-0.130(0.118)-1.177*(0.699)1.036**(0.528)-0.284(0.590)0.622(0.820)0.167(0.133)0.0452(0.146)-0.136(0.126)-1.280*(0.727)1.528***(0.495)0.534(0.551)-0.647(0.767)0.130(0.124)0.0143(0.145)-0.125(0.122)-1.076(0.698)

3,920
13.81
0.032

1,960
12.61
0.050

1,960
14.34
0.026

Notes: a) We conducted logit regression of the sincere bidding dummy on the SP hint dummy androbot dummy conditional on cognitive ability, risk aversion, and loss aversion with robust standard
error clustered at the individual level. b) Sincere bidding dummy equals 1 if subjects submitted asincere bid in a period and 0 otherwise. SP hint dummy equals 1 if subjects take Quiz 2 w/ hints
and 0 if they take Quiz 2 w/o hints. Robot dummy equals 1 if subjects compete against robot rivals
and 0 if they compete against human rivals. Cognitive ability ranges from 0 to 8, risk aversion
ranges from 0 to 5, and loss aversion ranges from 0 to 6. c) * ** and *** denote significance at the
10%, 5%, and 1% levels, respectively. d) Numbers in parentheses are standard errors. e) The total
number of observations is the number of subjects × number of periods × number of units in eachtreatment.

Some may argue that factors other than understanding SP, such as subjects’ prefer9

ences, could affect bidding behavior. To address this, we run the regression presented
in Table 4 of the main paper, which additionally controls for subjects’ cognitive ability
levels, risk aversion, and loss aversion. Table C2 presents these results. These results
confirm our original findings—the SP hint dummy has a statistically significant positive
impact on sincere bidding. This suggests that providing SP hints in Quiz 2 w/hints
promotes sincere bidding, even after controlling for cognitive ability, risk aversion levels,and loss aversion levels.

10

D

Experimental instruction and quizzes

This section contains experimental materials. The materials are originally in Japaneseand translated to English. From next page, subsection titles and page numbers are now
in the header of each page.
D.1 HHnohint and HHhint (competing against human rivals)
D.2 HRnohint and HRhint (competing against robot rivals)
D.3 Quiz 1
D.4 Quiz 2 w/ hint
D.5 Quiz 2 w/o hint

11

D.1 HHnohint and HHhint (competing against human rivals)

12

Instructions
Notice:· Please follow the instructions of the experimenter, but do not operate PC otherwise.
· Please do not communicate or exchange memos with other subjects.
· Please do not watch the behaviors and screens of other subjects.
· Please turn off your mobile phone and put it into your bag.
· Please raise your hand when you have a question.
· You can take a memo using the pen on your desk. You can also use the calculator on yourdesk.
Overview of experiment• Today’s Experiment is a sequence of auctions. In an auction, subjects bid for items thatthey wish, and subjects with the highest bids win the items. Subjects’ bids also determine
the winners’ payment. Hereafter, a subject bidding for items is called a “bidder.”• In this experiment, auctions are conducted for 20 rounds, and three bidders participate
in each auction. You play the role of one of the three bidders, and the other two bidders
are other subjects. You will be randomly matched with the other two subjects in eachround of auction. You will not know who you are matched with either during or after theexperiment.
• Your reward after the experiment is based on the results of 20 rounds of auction in which
you participate.

Procedure of experiment(1) Two units of an identical object are auctioned off in every period. Three bidders, including
yourself, participate in each round of the auction.
(2) If a bidder wins an item, they will draw satisfaction from the item. Such satisfaction
is assumed to be measured by money, and the satisfaction as measured by money is called“valuation.” The satisfaction from the first unit is called the valuation of the first unit, andthat from the second unit is called the valuation of the second unit.
(3) In this experiment, your valuations in each round of auction are chosen by PC. In eachround, PC randomly draws two values from the interval 10 JPY to 1,000 JPY in increments of
10 JPY. The higher value is your valuation of the first unit, and the lower one is your valuationof the second unit. This assumes that your satisfaction with the second unit is less than thatobtained from the first.
The valuations of the other subjects are determined similarly in each round. Thus, different
subjects have different valuations. However, in each round, the valuation of each bidder fromthe second unit is lower than that of the first unit. The other bidders will not know yourvaluations, and you will not know others’ valuations.
1

D.1 HHnohint and HHhint (competing against human rivals)

13(4) In this experiment, you are required to submit your “bid for the first unit” and “bid for thesecond unit” to the experimenter. Please note the following three points:· Your “bid for the first unit” must exceed your “bid for the second unit.”· Your bids must be at least 0 JPY.
· Your bids must be in increments of 10 JPY.
(5) In this experiment, each of the three bidders submit two bids. Thus, there is a total of
six bids. The two highest bids are winning bids, and a bidder who submits the winning bid(s)wins the corresponding object(s). In the scenario of a tie among the highest bids, the program
selects the winning bids with equal probabilities. This procedure determines the number ofunits won by each bidder.
(6) How to determine the payments of winning bidders is explained later. The earnings of awinning bidder is the sum of the valuations of the units they win minus their payment. The
earnings of a non-winning bidder is zero.

Calculation of payments and earnings
We explain how to calculate a bidder’s payments and earnings using numerical examples.
There are three bidders: A, B, and C. Here, we focus on bidder A. In the following examples,amounts displayed are in JPY.
(1) In the scenario where a bidder wins one unit: A bidder who wins one unit pays thehighest bid among the other bidders’ losing bids.
Example 1. Suppose A’s valuations for the first and second units are 680 and 480, respectively. The table below shows the bids from the three bidders.
Bidder

Bid for the first unit

Bid for the second unit

A

500

450

B

600

300

C

400

250

The winning bids are “A’s 500” and “B’s 600.” The losing bids are “A’s 450,” “B’s 300,”“C’s 400,” and “C’s 250.” Thus, for bidder A, the other bidders’ losing bids are “B’s 300,”“C’s 400,” and “C’s 250.” Bidder A pays the highest bid among these losing bids—400. This
payment differs from A’s bid for the first unit. Bidder A earns the following amount:
680 (valuation for the first unit) − 400 (payment) = 280

2

D.1 HHnohint and HHhint (competing against human rivals)

14

Example 2. Suppose A’s values for the first and second units are 680 and 480, respectively.
The table below displays the bids from the three bidders.
Bidder

Bid for the first unit

Bid for the second unit

A

800

350

B

750

300

C

700

250

The winning bids are “A’s 800” and “B’s 750.” Thus, bidder A wins one unit. The losing
bids are “A’s 350,” “B’s 300,” “C’s 700,” and “C’s 250.” Thus, for bidder A, the other bidders’losing bids are “B’s 300,” “C’s 700,” and “C’s 250.” Bidder A pays the highest bid among theselosing bids—700. Note that this payment differs from A’s bid for the first unit. Subsequently,
bidder A earns the following amount:
680 (valuation for the first unit) − 700 (payment) = −20(2) In the scenario where a bidder wins two units: A bidder who wins two units pays
the sum of the highest and second-highest bids from among the losing bids.
Example 3. Suppose A’s values for the first and second units are 680 and 480, respectively.
The table below displays the bids from the three bidders.
Bidder

Bid for the first unit

Bid for the second unit

A

600

550

B

500

300

C

400

250

The winning bids are “A’s 600” and “A’s 550.” The losing bids are “B’s 500,” “B’s 300,”“C’s 400,” and “C’s 250.” Bidder A pays the sum of the highest and second-highest bids among
the losing bids—400 + 500 = 900. This payment differs from that in A’s bids. Subsequently,
bidder A earns the following amount:
680 (valuation for the first unit) + 480 (valuation for the second unit) − 900 (payment) = 260

3

D.1 HHnohint and HHhint (competing against human rivals)

15

Example 4. Suppose A’s values for the first and second units are 680 and 480, respectively.
The bids of the three bidders are as follows:
Bidder

Bid for the first unit

Bid for the second unit

A

900

850

B

800

600

C

700

550

The winning bids are A’s 900 and A’s 850. The losing bids are “B’s 800,” “B’s 600,” “C’s
700,” and “C’s 550.” Thus, bidder A pays 1,500, the sum of the highest bid of 800 and thesecond highest bid of 700 from the losing bids. A’s payment differs from A’s bids. A’s earnings
are calculated as follows:
680 (valuation for the first unit)+480 (valuation for the second unit) −1500 (payment) = −340

The earnings calculation method is summarized as follows:• In the scenario where you win one unit, and the third highest bid is yours:Your earnings = your valuation for the firs tunit − fourth highest bid• In the scenario where you win one unit, and the third highest bid is not yours:Your earnings = your valuation for the first unit − third highest bid
Note that when the payment is higher than the value of the first unit, your earnings arenegative.
• In the scenario where you win two units:Your earnings = (your valuation for first unit + your valuation for second unit)− (third highest bid + fourth highest bid)
Note that when your payment is higher than the sum of your values of the first and second
units, your earnings will be negative.
• If you do not win anything, your earnings are 0.

Rewards.
• We pay the sum of your earnings over all 20-round auctions as a reward.
• Before the auctions, you will answer Quizzes 1 and 2. Each quiz comprises ten questions. In
addition to the earnings from auctions, you will be paid according to your scores in Quizzes 1
and 2.
• The total reward is the sum of your earnings from auctions and rewards based on quiz scores.
4

D.2 HRnohint and HRhint (competing against robot rivals)

16

Instructions
Notice:· Please follow the instructions of the experimenter, but do not operate PC otherwise.
· Please do not communicate or exchange memos with other subjects.
· Please do not watch the behaviors and screens of other subjects.
· Please turn off your mobile phone and put it into your bag.
· Please raise your hand when you have a question.
· You can take a memo using the pen on your desk. You can also use the calculator on yourdesk.
Overview of experiment• Today’s Experiment is a sequence of auctions. In an auction, subjects bid for items thatthey wish, and subjects with the highest bids win the items. Subjects’ bids also determine
the winners’ payment. Hereafter, a subject bidding for items is called a “bidder.”• In this experiment, auctions are conducted for 20 rounds, and three bidders participate
in each auction. You play the role of one of the three bidders, and the other two bidders
are played by PC.
• Your reward after the experiment is based on the results of 20 rounds of auction in which
you participate.
Procedure of experiment(1) Two units of an identical object will be auctioned off in every period. Three bidders,
including yourself, participate in each round of the auction. However, bidders other than you
are robots.
(2) If a bidder wins an item, they will draw satisfaction from the item. Such satisfaction isassumed to be measured by money, and satisfaction as measured by money is called “valuation.”
The satisfaction from the first unit is called the valuation from the first unit, and that from thesecond unit is called the valuation of the second unit.
(3) In this experiment, your valuations in each round of auction is chosen by PC. In each round,PC randomly draws two values from the interval from 10 JPY to 1,000 JPY in increments of 10
JPY. The higher value is your valuation from the first unit, and the lower one is your valuationfrom the second unit. This assumes that your satisfaction with the second unit is less thanthat with the first.
(4) In this experiment, you will submit your “bids for the first ” and “second units” to theexperimenter. Please note the following three points:· Your “bid for the first unit” must exceed your “bid for the second unit.”· Your bids must be at least 0 JPY.
· Your bids must be in increments of 10 JPY.
1

D.2 HRnohint and HRhint (competing against robot rivals)

17(5) As a bidder, a robot will randomly draw two values from the interval 10 JPY to 1,000 JPY
with increments of 10 JPY. The higher value is the bid for the first unit of the robot as abidder, and the lower one is the bid for the second unit. The bids of the other bidder (also a
robot) are determined similarly.
(6) In this experiment, you are required to submit two bids, and the other two bidders (played
by robots) also submit two bids. Thus, there will be a total of six bids. The two highest
bids are winning bids, and a bidder who submits the winning bid(s) wins the corresponding
object(s). In case of a tie among the highest bids, the program selects the winning bids withequal probabilities. This procedure determines the number of units won by each bidder.
(7) How to determine the payments of winning bidders is explained later. The earnings of awinning bidder is the sum of the valuations of the units that they win minus their payment.
The earnings of a non-winning bidder is zero.

Calculation of payments and earnings
We explain how to calculate a bidder’s payments and earnings using numerical examples.
There are three bidders: A, B, and C. Here, we focus on bidder A. In the following examples,amounts displayed are in JPY.
(1) In the scenario where a bidder wins one unit: A bidder who wins one unit pays thehighest bid among the other bidders’ losing bids.
Example 1. Suppose A’s valuations for the first and second units are 680 and 480, respectively. The table below shows the bids from the three bidders.
Bidder

Bid for the first unit

Bid for the second unit

A

500

450

B

600

300

C

400

250

The winning bids are “A’s 500” and “B’s 600.” The losing bids are “A’s 450,” “B’s 300,”“C’s 400,” and “C’s 250.” Thus, for bidder A, the other bidders’ losing bids are “B’s 300,”“C’s 400,” and “C’s 250.” Bidder A pays the highest bid among these losing bids—400. This
payment differs from A’s bid for the first unit. Bidder A earns the following amount:
680 (valuation for the first unit) − 400 (payment) = 2802

D.2 HRnohint and HRhint (competing against robot rivals)

18

Example 2. Suppose A’s values for the first and second units are 680 and 480, respectively.
The table below displays the bids from the three bidders.
Bidder

Bid for the first unit

Bid for the second unit

A

800

350

B

750

300

C

700

250

The winning bids are “A’s 800” and “B’s 750.” Thus, bidder A wins one unit. The losing
bids are “A’s 350,” “B’s 300,” “C’s 700,” and “C’s 250.” Thus, for bidder A, the other bidders’losing bids are “B’s 300,” “C’s 700,” and “C’s 250.” Bidder A pays the highest bid among theselosing bids—700. Note that this payment differs from A’s bid for the first unit. Subsequently,
bidder A earns the following amount:
680 (valuation for the first unit) − 700 (payment) = −20(2) In the scenario where a bidder wins two units: A bidder who wins two units pays
the sum of the highest and second-highest bids from among the other bidders’ losing bids.
Example 3. Suppose A’s values for the first and second units are 680 and 480, respectively.
The table below displays the bids from the three bidders.
Bidder

Bid for the first unit

Bid for the second unit

A

600

550

B

500

300

C

400

250

The winning bids are “A’s 600” and “A’s 550.” The losing bids are “B’s 500,” “B’s 300,”“C’s 400,” and “C’s 250.” Bidder A pays the sum of the highest and second-highest bids among
the losing bids—400 + 500 = 900. This payment differs from A’s bids. Subsequently, bidder A
earns the following amount:
680 (valuation for the first unit) + 480 (valuation for the second unit) − 900 (payment) = 2603

D.2 HRnohint and HRhint (competing against robot rivals)

19

Example 4. Suppose A’s values for the first and second units are 680 and 480, respectively.
The bids of the three bidders are as follows:
Bidder

Bid for the first unit

Bid for the second unit

A

900

850

B

800

600

C

700

550

The winning bids are A’s 900 and A’s 850. The losing bids are “B’s 800,” “B’s 600,” “C’s
700,” and “C’s 550.” Thus, bidder A pays 1,500, the sum of the highest bid of 800 and thesecond highest bid of 700 from the losing bids. A’s payment differs from A’s bids. A’s earnings
are calculated as follows:
680 (valuation for the first unit)+480 (valuation for the second unit) −1500 (payment) = −340

The earnings calculation method is summarized as follows:• In the scenario where you win one unit, and the the third highest bid is yours:Your earnings = your valuation for the firs tunit − fourth highest bid• In the scenario where you win one unit, and the third-highest bid is not yours:Your earnings = your valuation for the first unit − third highest bid
Note that when your payment is higher than your value for the first unit, your earnings will
be negative.
• In the scenario where you win two units:Your earnings = (your valuation for first unit + your valuation for second unit)− (third highest bid + fourth highest bid)
Note that when your payment is higher than the sum of your values for the first and second
units, your earnings will be negative.
• If you do not win anything, your earnings are 0.

Rewards• We pay the sum of your earnings over all 20-round auctions as a reward.
• Before the auctions, you will answer Quizzes 1 and 2. Each quiz comprises ten questions. In
addition to the earnings from auctions, you will be paid according to your scores from Quizzes
1 and 2.
• The total reward is the sum of your earnings from auctions and rewards based on quiz scores.
4

D.3 Quiz1

20

Quiz 1Seat number
Please answer all the following questions. You will earn 100 JPY for each correct answer.
Please refer to the instructions if necessary.
Assume that the bids from the three bidders and bidder A’s evaluation are as given in the tablebelow. Please answer Questions 1–4 based on the table.
First unit

Second unit

Bidder A’s bid

800

500

Bidder B’s bid

1000

700

Bidder C’s bid

600

500

Bidder A’s evaluation

900

700

[Question 1] (Two) Winning bids

Answer:[Question 2] Bidder A’s payment

Answer:[Question 3] Bidder A’s earning (Their evaluation on the items they win – Payment)
Answer:[Question 4] Bidder C’s payment

Answer:Assume that the bids of the three bidders and bidder B’s evaluation are as given in the tablebelow. Please answer Questions 5–7 based on the table.
First unit

Second unit

Bidder A’s bid

650

500

Bidder B’s bid

1000

700

Bidder C’s bid

600

500

Bidder B’s evaluation

950

800

[Question 5] (Two) Winning bids

Answer:[Question 6] Bidder B’s payment

Answer:[Question 7] Bidder B’s earning (Their evaluation on the items they win – Payment)
Answer:.

Assume that the bids of the three bidders and bidder C’s evaluation are as given in the tablebelow. Please answer Questions 8–10 based on the table.
First unit

Second unit

Bidder A’s bid

800

500

Bidder B’s bid

1000

700

Bidder C’s bid

900

500

Bidder B’s evaluation

500

400

[Question 8] (Two) Winning bids

Answer:[Question 9] Bidder B’s payment

Answer:[Question 10] Bidder B’s earning (Their evaluation on the items they win – Payment)
Answer:1

D.4 Quiz 2 w/ hint

21

Seat number

Quiz 2

Please answer all the following questions. You will earn 100 JPY for each correct
answer. Please refer to the instructions if necessary.
Assume that you are bidder B, and your valuations of the first and second units are
700 and 400, respectively. Maintain these assumptions for Questions 1–10 below. For
each question, select the correct answers from the choices provided in the table below,
where the the first and second figures in each choice are the bids of the first and second
units, respectively. If several choices are correct, select all of them.
Choice a: (900, 820)

Choice b: (780, 500)

Choice c: (700, 400)

Choice d: (400, 390)

Choice e: (310, 200)

Choice f: (660, 200)

The calculation sheets for these questions are attached on pages 4–6. Use them ifnecessary. (However, the calculation sheets will not be marked.)
In Questions 1 and 2, assume that you expect the other two bidders (A and C) tobid as shown in the table below. The calculation sheet for the two questions is shownon page 4.
Bidder

Bid for 1st unit

Bid for 2nd unit

A

380

300

C

300

250

[Question 1] Select the choices that make your earnings (the sum of valuations of theunits you win - your payments) zero. If no such choice exists, answer “none.”
Answer:[Question 2] Select the choices that maximize your earnings (the sum of valuations of
the units you win – your payments). (Note that if the earnings from all choices arenonpositive, the maximized earnings may be zero.)

1

Answer:D.4 Quiz 2 w/ hint

22

Seat number

In Questions 3–6, assume that you expect the other two bidders (A and C) to bid
as shown in the table below. The calculation sheet for the three questions is shown on
page 5.
Bidder

Bid for 1st unit

Bid for 2nd unit

A

650

300

C

750

450

[Question 3] Select the choices that make your earnings (the sum of valuations of theunits you win – your payments) zero. If no such choice exists, answer “none.”
Answer:[Question 4] Select the choices that make your earnings (the sum of valuations of theunits you win – your payments) negative. If no such choice exists, answer “none.”
Answer:[Question 5] Select the choices that maximize your earnings (the sum of valuations of
the units you win – your payments). (Note that if the earnings from all choices arenonpositive, the maximized earnings may be zero.)Answer:[Question 6] Select the choices that are the correct answers of both Questions 2 and5—the choices that maximize your earnings for both of the two expectations about thebids of the two other bidders. If no such choice exists, answer “none.”
Answer:2

D.4 Quiz 2 w/ hint

23

Seat number

In Questions 7–9, assume that you expect the other two bidders (A and C) to bid
as shown in the table below. The calculation sheet for the three questions is shown on
page 6.
Bidder

Bid for 1st unit

Bid for 2nd unit

A

800

660

C

750

500

[Question 7] Select the choices that make your earnings (the sum of valuations of theunits you win - your payments) zero. If no such choice exists, answer “none.”
Answer:[Question 8] Select the choices that make your earnings (the sum of valuations of theunits you win - your payments) negative. If no such choice exists, answer “none.”
Answer:[Question 9] Select the choices that maximize your earnings (the sum of valuations of
the units you win - your payments). (Note that if the earnings from all choices arenonpositive, the maximized earnings may be zero.)Answer:[Question 10] Select the choices that are the common answers of all of Questions 2, 5,
and 9—the choices that maximize your earnings for all of the three expectations about
the bids of the two other bidders. If no such choice exists, answer “none.”
Answer:3

D.4 Quiz 2 w/ hint
Seat number

Calculation sheets for Questions 1 and 2
Bid for 1st unit

Bid for 2nd unit

Bidder A

380

300

Bidder C

300

250

Choice a

900820

Choice b

780

500

Choice c

700

400

Choice d400

390

Choice e

310

200

Choice f

660

200Your evaluation

700

400

Computation for Choice a: (900, 820)
Units you win

Sum of valuations of units you win

Payment

Earning

Payment

Earning

Payment

Earning

Payment

Earning

Payment

Earning

Payment

Earning

Computation for Choice b: (780, 500)
Units you win

Sum of valuations of units you win

Computation for Choice c: (700, 400)
Units you win

Sum of valuations of units you win

Computation for Choice d: (400, 390)
Units you win

Sum of valuations of units you win

Computation for Choice e: (310, 200)
Units you win

Sum of valuations of units you win

Computation for Choice f: (660, 200)
Units you win

Sum of valuations of units you win

424

D.4 Quiz 2 w/ hint
Seat number

Calculation sheets for Questions 3–6
Bid for 1st unit

Bid for 2nd unit

Bidder A

650

300

Bidder C

750

450

Choice a

900820

Choice b

780

500

Choice c

700

400

Choice d400

390

Choice e

310

200

Choice f

660

200Your evaluation

700

400

Computation for Choice a: (900, 820)
Units you win

Sum of valuations of units you win

Payment

Earning

Payment

Earning

Payment

Earning

Payment

Earning

Payment

Earning

Payment

Earning

Computation for Choice b: (780, 500)
Units you win

Sum of valuations of units you win

Computation for Choice c: (700, 400)
Units you win

Sum of valuations of units you win

Computation for Choice d: (400, 390)
Units you win

Sum of valuations of units you win

Computation for Choice e: (310, 200)
Units you win

Sum of valuations of units you win

Computation for Choice f: (660, 200)
Units you win

Sum of valuations of units you win

525

D.4 Quiz 2 w/ hint
Seat number

Calculation sheets for Questions 7–9
Bid for 1st unit

Bid for 2nd unit

Bidder A

800

660

Bidder C

750

500

Choice a

900820

Choice b

780

500

Choice c

700

400

Choice d400

390

Choice e

310

200

Choice f

660

200Your evaluation

700

400

Computation for Choice a: (900, 820)
Units you win

Sum of valuations of units you win

Payment

Earning

Payment

Earning

Payment

Earning

Payment

Earning

Payment

Earning

Payment

Earning

Computation for Choice b: (780, 500)
Units you win

Sum of valuations of units you win

Computation for Choice c: (700, 400)
Units you win

Sum of valuations of units you win

Computation for Choice d: (400, 390)
Units you win

Sum of valuations of units you win

Computation for Choice e: (310, 200)
Units you win

Sum of valuations of units you win

Computation for Choice f: (660, 200)
Units you win

Sum of valuations of units you win

626

D.5 Quiz 2 w/o hint

27

Seat number

Quiz 2

Please answer all the following questions. You will earn 100 JPY for each correct
answer. Please refer to the instructions if necessary.
Assume that you are Bidder B in Questions 1–10. For each question, select thecorrect answers from the choices given in the table below, where the first and second
figures in each choice are the bids of the first and second units, respectively. If several
choices are correct, select all of them.
Choice a: (900, 820)

Choice b: (780, 500)

Choice c: (700, 400)

Choice d: (400, 390)

Choice e: (310, 200)

Choice f: (660, 200)

The calculation sheets for these questions are attached on pages 4–6. Use them ifnecessary. (However, the calculation sheets are not marked.)
In Questions 1–3, assume that your valuations of the first and second units are 500
and 300, respectively, and you expect the other two bidders (A and C) to bid as shownin the table below. The calculation sheet for the three questions is shown on page 4.
Bidder

Bid for 1st unit

Bid for 2nd unit

A

380

300

C

300

250

[Question 1] Select the choices that make your earnings (the sum of valuations of theunits you win - your payments) zero. If no such choice exists, answer “none.”
Answer:[Question 2] Select the choices that make your earnings (the sum of valuations of theunits you win - your payments) negative. If no such choice exists, answer “none.”
Answer:[Question 3] Select the choices that make your earnings (the sum of valuations of theunits you win - your payments) positive. If no such choice exists, answer “none.”
Answer:1

D.5 Quiz 2 w/o hint

28

Seat number

In Questions 4–6, assume that your valuations of the first and second units are 710
and 290, respectively, and you expect the other two bidders (A and C) to bid as shownin the table below. The calculation sheet for the three questions is shown on page 5.
Bidder

Bid for 1st unit

Bid for 2nd unit

A

650

300

C

750

450

[Question 4] Select the choices that make your earnings (the sum of valuations of theunits you win - your payments) zero. If no such choice exists, answer “none.”
Answer:[Question 5] Select the choices that make your earnings (the sum of valuations of theunits you win - your payments) negative. If no such choice exists, answer “none.”
Answer:[Question 6] Select the choices that make your earnings (the sum of valuations of theunits you win - your payments) positive. If no such choice exists, answer “none.”
Answer:2

D.5 Quiz 2 w/o hint

29

Seat number

In Questions 7–9, assume that your valuations of the first and second units are 730
and 370, respectively, and you expect the other two bidders (A and C) to bid as shownin the table below. The calculation sheet for the three questions is shown on page 6.
Bidder

Bid for 1st unit

Bid for 2nd unit

A

800

660

C

750

500

[Question 7] Select the choices that make your earnings (the sum of valuations of theunits you win - your payments) zero. If no such choice exists, answer “none.”
Answer:[Question 8] Select the choices that make your earnings (the sum of valuations of theunits you win - your payments) negative. If no such choice exists, answer “none.”
Answer:[Question 9] Select the choices that make your earnings (the sum of valuations of theunits you win - your payments) positive. If no such choice exists, answer “none.”
Answer:[Question 10] Select the choices that are the common answers of all of Questions 1, 4,
and 7—the choices that make your earnings zero for all of the three expectations about
the bids of the two other bidders. If no such choice exists, answer “none.”
Answer:3

D.5 Quiz 2 w/o hint
Seat number

Calculation sheets for Questions 1 and 2
Bid for 1st unit

Bid for 2nd unit

Bidder A

380

300

Bidder C

300

250

Choice a

900820

Choice b

780

500

Choice c

700

400

Choice d400

390

Choice e

310

200

Choice f

660

200Your evaluation

500

300

Computation for Choice a: (900, 820)
Units you win

Sum of valuations of units you win

Payment

Earning

Payment

Earning

Payment

Earning

Payment

Earning

Payment

Earning

Payment

Earning

Computation for Choice b: (780, 500)
Units you win

Sum of valuations of units you win

Computation for Choice c: (700, 400)
Units you win

Sum of valuations of units you win

Computation for Choice d: (400, 390)
Units you win

Sum of valuations of units you win

Computation for Choice e: (310, 200)
Units you win

Sum of valuations of units you win

Computation for Choice f: (660, 200)
Units you win

Sum of valuations of units you win

430

D.5 Quiz 2 w/o hint
Seat number

Calculation sheets for Questions 4–6
Bid for 1st unit

Bid for 2nd unit

Bidder A

650

300

Bidder C

750

450

Choice a

900820

Choice b

780

500

Choice c

700

400

Choice d400

390

Choice e

310

200

Choice f

660

200Your evaluation

710

290

Computation for Choice a: (900, 820)
Units you win

Sum of valuations of units you win

Payment

Earning

Payment

Earning

Payment

Earning

Payment

Earning

Payment

Earning

Payment

Earning

Computation for Choice b: (780, 500)
Units you win

Sum of valuations of units you win

Computation for Choice c: (700, 400)
Units you win

Sum of valuations of units you win

Computation for Choice d: (400, 390)
Units you win

Sum of valuations of units you win

Computation for Choice e: (310, 200)
Units you win

Sum of valuations of units you win

Computation for Choice f: (660, 200)
Units you win

Sum of valuations of units you win

531

D.5 Quiz 2 w/o hint
Seat number

Calculation sheets for Questions 7–9
Bid for 1st unit

Bid for 2nd unit

Bidder A

800

660

Bidder C

750

500

Choice a

900820

Choice b

780

500

Choice c

700

400

Choice d400

390

Choice e

310

200

Choice f

660

200Your evaluation

730

370

Computation for Choice a: (900, 820)
Units you win

Sum of valuations of units you win

Payment

Earning

Payment

Earning

Payment

Earning

Payment

Earning

Payment

Earning

Payment

Earning

Computation for Choice b: (780, 500)
Units you win

Sum of valuations of units you win

Computation for Choice c: (700, 400)
Units you win

Sum of valuations of units you win

Computation for Choice d: (400, 390)
Units you win

Sum of valuations of units you win

Computation for Choice e: (310, 200)
Units you win

Sum of valuations of units you win

Computation for Choice f: (660, 200)
Units you win

Sum of valuations of units you win

6

32

E

Post-experiment survey

1. Please indicate your seat number for today’s experiment.
2. Please select your gender.

□ Female□ Male□ Do not answer

3. Please select your major.
4. Did you understand the rules of today’s auction before it started?
□ Very well
□ Well□ Not so much
□ Little□ Other:
5. How did you bid in the auction?
□ Bid the amount of valuation□ Bid higher than valuation□ Bid lower than valuation□ Other:
6. Select the reason for bidding as you did in 5) from the choices below.
□ To maximize your earnings in the auction.
□ To maximize your winning probability rather than to maximize your earnings in the auction.
□ To make the earnings of other bidders smaller.
□ To make the earnings of other bidders larger.
□ Other:33

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Hanaki, N., K. Inukai, T. Masuda, and Y. Shimodaira (2022): “Comparing behavior between a large sample of smart students and Japanese adults,” The Japanese
Economic Review, 1–39.
Köbberling, V., and P. P. Wakker (2005): “An index of loss aversion,” Journal of
Economic Theory, 122, 119–131.
Masuda, T., and E. Lee (2019): “Higher order risk attitudes and prevention under
different timings of loss,” Experimental Economics, 22, 197–215.
Masuda, T., R. Mikami, T. Sakai, S. Serizawa, and T. Wakayama (2022): “Thenet effect of advice on strategy-proof mechanisms: an experiment for the Vickrey
auction,” Experimental Economics, 25, 902–941.
Noussair, C. N., S. T. Trautmann, and G. Van de Kuilen (2014): “Higher orderrisk attitudes, demographics, and financial decisions,” Review of Economic Studies,81, 325–355.

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