ソシオネットワーク戦略ディスカッションペーパーシリーズ
第 62 号

ISSN 1884-9946

2018 年 6 月

RISS Discussion Paper Series
No.62

June, 2018

Supplementary Results for“Meaningful Learningin Weighted Voting Games: An Experiment”

Naoki Watanabe

文部科学大臣認定 共同利用・共同研究拠点

関西大学ソシオネットワーク戦略研究機構
Research Institute for Socionetwork Strategies,
Kansai University
Joint Usage / Research Center, MEXT, Japan
Suita, Osaka, 564-8680, Japan
URL: https://www.kansai-u.ac.jp/riss/index.html
e-mail: riss@ml.kandai.jp
tel. 06-6368-1228
fax. 06-6330-3304

Supplementary Results for “Meaningful Learning in Weighted
Voting Games: An Experiment”∗
Naoki Watanabe†
November 16, 2021

Abstract
This paper clarifies subjects’ search behavior of correct options behind the experimental results shown by Guerci et al. (2017). In the experiment, subjects were asked tochoose one of two weighted voting games repeatedly and their payoﬀs are stochasticallydetermined for each of their choice according to a payoﬀ-generating function that was
hidden from subjects. The main results are as follows. (1) In the additional sessions
conducted for the treatment without any payoﬀ-related feedback information, it wasreconfirmed that subjects learned to choose the correct option that generates higher
expected payoﬀs for them and generalized what they had thought introspectively in abinary choice problem to a similar but diﬀerent one. (2) Feedback information about
payoﬀs given immediately after subjects’ choice often confused their inference on therelationship between nominal voting weights and actual payoﬀs so that they took thewin-stay-lose-shift strategy in some sessions. (3) Immediate payoﬀ-related feedback information sometimes induced subjects to randomly choose the runs of options.
Keywords: meaningful learning, bandit experiment, weighted voting, search behavior,
win-stay-lose-shift strategy
JEL Classification: C91, D72, D83∗

This paper is a revised version of Kansai University RISS Discussion Paper Series No. 62 issued in
2018. The author wishes to thank Eric Guerci and Nobuyuki Hanaki, for giving him their permission toreuse the data, and Daniel Friedman, Tetsuya Kawamura, Midori Hirokawa, Yoichi Hizen, Tatsuya Kameda,
Naoko Nishimura, and Kazuhito Ogawa for their comments on that previous work which motivated him
to provide these supplementary results. This research could not have been completed without the research
assistance of Yoichi Izunaga, Toru Suzuki, Hiroshi Tanaka, and Yosuke Watanabe. Financial support from
Foundation for the Fusion of Science and Technology (FOST), MEXT Grants-in-Aid 24330078 and 25380222,
JSPS-ANR bilateral research grant “BECOA” (ANR-11-FRJA-0002), and Joint Usage/Research of ISER at
Osaka University are gratefully acknowledged.
†
Graduate School of Business Administration, Keio University, 4-1-1 Hiyoshi Kohoku, Yokohama, Kanagawa 223-8526, Japan; E-mail: naoki50@keio.jp; Phone: +81-45-564-2039

1

1

Introduction

It would be truly diﬃcult for people to deal with weighted voting; Felsenthal and Machover(1998, pp.164-165) noted that it might be diﬃcult even for the policy makers and oﬃcials
who designed and re-designed the system to see the latent relationship between the actual
voting powers and the nominal voting weights, and Gelman et al. (2004) stated that thestandard theoretical indices of voting power do not predict the actual voting outcomes.
Weighted voting is, however, a popular collective decision-making system. In order for us
to use this system better, it is important to examine whether people can learn from theirexperiences the underlying structure of weighted voting.
When people generalize what they have learned in a situation to a similar but diﬀerent
one, that higher order concept of learning is called meaningful learning (Rick and Weber(2010)).1 In a two-armed bandit experiment, Guerci et al. (2017) could not observe meaningful learning of subjects when immediate payoﬀ-related feedback information was given tothem, whereas they observed it only in sessions for the treatment without any payoﬀ-related
feedback information. Feedback information is essentially incorporated also in the standard
theory of reinforcement learning (e.g., Erev and Roth, 1998), belief-based learning (e.g.,
Cheung and Friedman, 1997), and experience weighted attraction learning (Camerer andHo, 1999). What hindered the human subjects from meaningful learning when immediate
feedback information was given to them? For taking a new step to the future research, thispaper clarifies some features on subjects’ search behavior behind the experimental results
shown by Guerci et al. (2017), reexamining the data in their sessions and additional sessions.
In strategic situations, feedback information provided to each subject contains the outcomes generated by unplanned or exploratory behavior of other subjects, and thus individual
inferences might be confused mutually among subjects. Guerci et al. (2017) thus drasticallysimplified the experimental design to remove subjects’ learning through their strategic interaction. The experimental design is as follows. In each session subjects choose one of two
weighted voting games (options) repeatedly and obtain their payoﬀs which are stochastically generated for each choice they make according to a voting theory. The binary choiceproblems are diﬀerent between the first and second parts of the session, but the payoﬀ generating function in binary choice problems remains the same. Subjects thus have a chance
to learn something underlying the situation they face in the first part and to apply what
they learned in the first part to their decision in the second part.
1

Meaningful learning is also called “transfer of learning” (Cooper and Kagel, 2003, 2008) or “epiphany”(Dufwenberg et al., 2010).

2

There are three treatments; no feedback, partial feedback, and full feedback. For the
no-feedback treatment, subjects are not informed of any payoﬀs they receive as the resultof their choice until the session ends. For the partial-feedback treatment, each subject
is informed of his or her own payoﬀ. For the full-feedback treatment, after each choicesubjects are informed of the payoﬀs including others. Intuitively, for each binary choiceproblem, it is said that meaningful learning is observed in the problem when there is asignificant increase in number of subjects who chose the correct option between subjects
who experienced similar but diﬀerent problem and subjects who did not.
Guerci et al. (2017) observed that (i) subjects learned to choose the correct optionthat generates higher expected payoﬀs for them even without any payoﬀ-related feedback
information and that (ii) in each sequence that starts with an easy binary choice problem andthen gives a diﬃcult one there was statistically significant evidence of meaningful learning
only in sessions for the treatment with no payoﬀ-related feedback information. In their
sessions, however, meaningful learning with no feedback information might be a consequenceof inertia on subjects’ choice, because the correct options were the same choice (Choice 2)
in the first and second parts of the sequences of binary choice problems. We should thus
confirm whether observation (ii) was not the consequence of the inertia on subjects’ choiceby changing the correct options between the two parts.
Hypothesis 1. Meaningful learning without any payoﬀ-related feedback information is not
the consequence of the inertia on subjects’ choice, and it is observed in each sequence thatstarts with an easy binary choice problem and then gives a diﬃcult one.
Another major question is, as stated above, is to investigate what hindered subjects fromgaining a deeper insight on the underlying structure of weighted voting when immediate
feedback information was given to them. In the post-experimental questionnaire, somesubjects reported that they changed their choices when they received zero points, while
they did not when they received positive points. It would be plausible that subjects took
this “win-stay-lose-shift” strategy (Nowak and Sigmund, 1993).
Hypothesis 2. Subjects took the win-stay-lose-shift strategy, when feedback information isprovided immediately after their choice.
Together with Hypothesis 2, we should confirm whether they searched for the correct
options with some certain rule, when they were given immediate feedback information.
Hypothesis 3. Immediate payoﬀ-related feedback information induced subjects to randomly
choose the runs of options.
3This paper shows the following results, reexamining the data used in Guerci et al. (2017)and additional sessions conducted by the author. (1) It was reconfirmed that without any
feedback information, subjects meaningfully learned to choose the correct option in one oftwo sequences in which Guerci et al. (2017) observed subjects’ meaningful learning, eventhough the correct option was changed from Choice 1 in the first part to Choice 2 in thesecond part in the sequence. The answers to the post-experimental questionnaire show
their proper reasoning of their choice. Accordingly, meaningful learning was not observedby chance. (2) For treatments with immediate feedback information, it was confirmed that
subjects took the win-stay-lose-shift strategy in some sessions, but (3) in other sessions they
did not search for the correct options with any certain rules.
In particular, we found the following search behavior of the subjects in a binary choiceproblem where meaningful learning was observed in the treatment with no payoﬀ-related
feedback information. Subjects chose their options in almost the same thinking time, regardless of the treatments in which they participated. In the partial-feedback treatment,
subjects did not search for the correct options with any certain rules. In the full-feedback
treatment, experienced subject took the win-stay-lose-shift strategy. There was no significant diﬀerence in their thinking time among all treatments. Accordingly, we do not haveto care about the eﬀect of thinking time on their behavior in the comparison.
In summary, Hypothesis 1 was aﬃrmatively confirmed and Hypotheses 2 and 3 werepartly confirmed. Result (1) was not obtained by chance. In the post experimental questionnaire, subjects noted the proper reasoning for their choice. According to Results (2)and (3), subjects’ inference on the relationship between nominal voting weights and actual payoﬀs might be mistaken or confused when payoﬀ-related feedback information wasgiven immediately after their choice. Withholding such misleading or confusing information
allows subjects to deliberate the underlying structure of weighted voting.
The rest of this paper proceeds as follows. Section 2 describes the experimental design.
Sections 3 provides observations and Result (1) on subjects’ learning and meaningful learning. Section 4 presents Results (2) and (3), where we deal with the length of time subjects
spent for thinking, win-stay-lose-shift strategy, and their random choice of runs in eachsequence of the options. Section 5 notes two brief remarks for further research, based on
our results and observations. One is the verification of implications derived by theoreticalmodels, and the other is the relationship between subjects’ cognitive ability and their meaningful learning. Appendix A provides the instructions of the experiment, and Appendix B
summarizes the comments of subjects who participated in the sessions for Guerci et al.
(2017) for reference.
4

2

Experimental Design

This section describes the experimental design. Subsection 2.1 explains the session outline,
Subsection 2.2 shows what subjects see on their monitors in the session, and Subsection 2.3
refers to the experimental procedure. The instructions given to subjects are attached inAppendix A, although the instructions for bandit experiments are extremely simple.

2.1

Session Outline

There are three treatments, which are explained later at the end of this subsection. Eachsession in those treatments consists of 60 periods. At each period, the subject is asked tochoose one of two committees of four members who are supposed to divide 120 points among
them. Let N = {1, 2, 3, 4} be the set of the members (players). A committee (a weighted
voting game) is represented by [q; v1 , v2 , v3 , v4 ], where q is the quota (the minimum numberof votes required for an allocation to be adopted) and vi is the voting weight (the number ofvotes) allocated to player i ∈ N . Both committees have the same numbers of total votes andquota as well as the same number of votes for player 1. Every subject acts as player 1 andfaces a binary choice problem for the first 40 periods, while in the following 20 periods he or
she faces another but diﬀerent one; for example, in the first 40 periods subjects face a choice
between [14; 5, 3, 7, 7] and [14; 5, 4, 6, 7], while in the following 20 periods they are faced with
[6; 1, 2, 3, 4] and [6; 1, 1, 4, 4]. Subjects are not asked to play the weighted voting games thatthey choose.2 In the instruction, subjects are clearly informed that the other members of
the committees are all fictitious. This experiment is thus regarded as a two-armed bandit
experiment with contextual information on weighted voting.
The payoﬀ each subject obtains from his or her choice is externally determined according
to a power index called DPI (Deegan and Packel, 1978), which is defined as follows. Givena weighted voting game, a non-empty subset S of N is called a coalition, and a coalition is∑called a winning coalition if i∈S vi ≥ q; otherwise, it is called a losing coalition. A minimum winning coalition (MWC) is a winning coalition such that deviation by any memberof the coalition turns its status from winning to losing. In the experiment, for each period,
one MWC is drawn with equal probability from all the possible MWCs for the committeethat the subject has chosen. If the subject is a member of the drawn MWC, then he or she
receives an equal share of the total payoﬀ with the other members.
2This setting was made to avoid the complexities mentioned at the beginning of the Introduction; subjects
are simultaneously learning to play a weighted voting game from interfering with the other subjects learning
about the underlying relationship between their nominal voting weights and their expected payoﬀs.

5

The binary choice problems we use are listed in Table 1. We denote each MWC by thevotes apportioned to its members; the MWCs in a committee [14; 5, 3, 7, 7] are written as(5, 3, 7), (5, 3, 7), and (7, 7). When [14; 5, 3, 7, 7] is chosen by a subject (player 1), player 1
has a 2/3 chance of being on the MWCs and will receive 1/3 of the total payoﬀ (120 points)
for being on each of the MWCs, while there is a 1/3 probability that player 1 will not beon the MWC and will receive nothing. Subjects are not informed of this underlying payoﬀ
generating function; they are simply told that payoﬀs were determined based on a theory ofdecision-making in committees. Note, however, that we use binary choice problems in which
the better committees for the subjects are the same regardless of whether we employ DPIor other power indices such as BzI (Banzhaf, 1965) and SSI (Shapley and Shubik, 1954).3
ProblemA
B
C
D

Choice 1[14; 5, 3, 7, 7][6; 1, 2, 3, 4][14; 3, 5, 6, 8][9; 1, 3, 5, 6](Expected payoﬀ)(120 × 2/3 × 1/3)
(120 × 1/3 × 1/3)
(120 × 2/3 ×1/3)(120 × 1/3 × 1/3)

Choice 2[14; 5, 4, 6, 7][6; 1, 1, 4, 4][14; 3, 6, 6, 7][9; 1, 2, 6, 6](Expected payoﬀ)(120 × 3/4 × 1/3)
(120 × 2/3 × 1/3)
(120 × 3/4 × 1/3)
(120 × 2/3 × 1/3)

Table 1: Four binary choice problems used in the experiment. In each committee, subjects are all assigned
to player 1, and the number of votes given to player 1 is shown in bold. Choice 2 generates a higher expected
payoﬀ for the subjects in all binary choice problems. In Problem A and Problem C, one option has two“large” voters who can form an MWC on their own, whereas the other option does not. In Problem B and
Problem D, there is no such clear diﬀerence between the two options.

Note that in Problems A and C, one option has two large voters who can form an MWC
on their own, whereas the other option does not. In Problems B and D, there is no suchclear diﬀerence between the two options as there are two large voters who can form an
MWC by themselves in both options. We hereafter refer to Problems A and C as “easy”
binary choice problems and Problems B and D as “diﬃcult” binary choice problem.
Subjects are faced with one of the following sequences of binary choice problems: A →B, B →, C → D , or D → C (the order of problems is indicated by the arrows), where
the first problem is used in the first 40 periods, and the second problem is used in thesubsequent 20 periods. In Guerci et al. (2017), as shown in Table 1, the committee thatgenerates a higher expected payoﬀ for subjects (correct option) is Choice 2 for all problems.
In the additional sessions, however, the correct option is Choice 1 in the first 40 periods
and it is Choice 2 in the subsequent 20 periods. Subjects are not informed of what binary
choice problems being given before those problems are shown on their monitors.
3

Guerci et al. (2017) adopted DPI as their payoﬀ generating function because, in all the experiments
reported in Montero et al. (2008), Aleskerov et al. (2009), Esposito et al. (2012), Guerci et al. (2014), and
Watanabe (2014), the most frequently observed winning coalitions were MWCs.

6

As noted at the beginning of this subsection, there are three treatments: (1) no feedback,(2) partial feedback, and (3) full feedback. For the no-feedback treatment, subjects are notinformed of anything as the result of their committee choice until the session ends. For
the partial-feedback treatment, each subject is informed of his or her own payoﬀ in thecommittee he or she chooses. For the full-feedback treatment, after each choice subjects are
informed of the payoﬀs of all four players in the committee they chose. There is a 30-second
time limit for the choice stage and a 10-second limit for the feedback stage, regardless of
the amount of feedback information. If a subject does not choose a committee within the
30 seconds of the choice stage, then he or she obtains zero points for that period. In thiscase, regardless of the treatment, in the feedback stage the subject receives a notice that heor she obtained nothing as a result of their failure to make a choice within the time limit.
If a subject makes an early choice, then a waiting screen is shown until all the subjects
in the session have made their decisions. If all the subjects make their choice before theend of the 30-second time limit, then they all enter the feedback stage. For the no-feedback
treatment, during the 10-second feedback stage subjects are shown a screen conveying themessage “Please wait until the experiment continues.” For the full-feedback and partialfeedback treatments, the relevant payoﬀ information is displayed during these 10 seconds.
Cooper and Kagel (2003, 2008) found that discussion among subjects promotes meaningful learning in signaling games. In this experiment, any communication with the others
is prohibited during each session. Subjects are also prohibited from note-taking, because it
is considered as communication with themselves.

2.2

Subject’s Monitor

The instructions for bandit experiments are intentionally simplified, because they examine
whether subjects learn something from their experiences. For readers’ convenience, we
here illustrate the examples of what subjects see on their monitors in this experiment. In
each session for the full-feedback treatment, subjects are provided a problem and otherinformation on their monitors as follows.
Please choose one out of the following two committees (Choice 1 or Choice 2). Eachcommittee decides a distribution of 120 points among four members. You are Member 1.
In both committees, 22 votes are apportioned to those members and you have 5 votes.
Any proposals of point distributions need 14 votes in favor to be adopted.

Choice 1[14; 5, 3, 7, 7],

Choice 27[14; 5, 4, 6, 7]

When subjects choose Choice 2 and MWC (5, 6, 7) appears, they see, for instance, thefollowing results on their monitor, regardless of any treatments.
You chose the following committee.

Choice 2: [14; 5, 4, 6, 7].
Next, in the full-feedback treatment, subjects see
The committee decided to distribute 120 points this time as follows.
You obtained 40 points this time.

(40, 0, 40, 40)
on their monitors. In the partial-feedback treatment, the payoﬀ distribution is not shown,
but rather the following note is shown on their monitors:
You have obtained 40 points this time.

In the no-feedback treatment, the payoﬀ distribution is not shown and simply
Please wait for a while.

is shown on the subjects’ monitors.

2.3

Experimental Procedure

The sessions for Guerci et al. (2017) were conducted at the Institute of Social and Economic
Research (ISER) at Osaka University in June 2014 and at the University of Tsukuba in
November 2014. The subjects were undergraduate students recruited from all over thecampus, but third- or fourth-year economics majors were excluded. Each session in Osaka(June) involved 20 subjects and each session in Tsukuba (Nov) involved 10 subjects, andthus 360 subjects participated in those sessions.
The sessions in Tsukuba (Nov) were conducted for a robustness check to the results obtained in the sessions in Osaka (June). The same results were obtained at two experimentalsites, and thus the main results were reported in Guerci et al. (2017) with the pooled data.
For the no-feedback treatment, the additional sessions were conducted also at the ISER in
September 2014. Each session in Osaka (Sept) involved 10 subjects, and thus we had 40
subjects in total there. No one has ever participated in this experiment and every subject
participated once in this experiment.
8

The experiment was computerized with z-Tree (Fischbacher, 2007). Each session lasted
around 60 minutes including the time for administering the instructions and the postexperiment questionnaire. At the beginning of each session, subjects were provided with awritten instruction upon arrival, and then the experimenter read it aloud. No communication among subjects was allowed. Subjects were allowed to ask questions regarding the
instruction and they were given the answers which other subjects could hear. Thereafter,
any information available to the subjects was provided through their computer screens.
At the end of each session, subjects were asked to write the reasoning behind theirown choice in free format. For the payment scheme, we followed other bandit experiments(Meyer and Shi, 1995; Hu et al., 2013). In the instruction, each subject was informed thatin addition to the show-up fee of JPY 1000, he or she would receive payment according to
the total points he or she obtained over all 60 periods at a rate of 1 point = JPY 1. The
average earning of our subjects was JPY 2500 (about 18 USD in 2014.)

3

Results

In Subsections 3.1 and 3.2, we reconfirm learning and meaningful learning of subjects who
participated in sessions for the no-feedback treatment.
No-fb(a) A → B(b) B → A(c) C → D(d) D → C

Figure 1: Time series of the percentage of subjects who chose the correct option among those who chosewithin the time limit in the additional sessions. 10 out of 40 subjects failed to make a choice within thetime limit at least once in the 60 periods.

9Figure 1 presents the time series of the percentage of subjects in additional sessions
who chose the correct option among those who chose within the time limit in each of thefour sequences; the dotted vertical line in each panel separates Period 40 and Period 41 toindicate that the problems were diﬀerent in the periods before and after the line. In the
additional sessions, as noted in Subsection 2.1, the correct option was Choice 1 in the first
40 periods, which was changed to Choice 2 in the subsequent 20 periods.
Recall that in Subsection 2.1, Problems A and C are named easy binary choice problems
and Problems B and D are names diﬃcult ones. In fact, at Period 1, as is seen in Figure 1,
subjects found the correct option easier in Problems A and C than in Problems B and D,
which was reported also in Guerci et al. (2017) for all the three treatments. For each binary
choice problem, we could not reject the null hypothesis that the percentage of subjects whochose the correct option among those who chose within the time limit at Period 1 was equal
between the additional sessions and the sessions conducted for Guerci et al. (2017); the pvalues for the two-sided χ2 test are 0.526, 0.559, 0.853, and 0.673 in Problems A, B, C, andD, respectively. Thus, it is not inappropriate to compare the additional data with thoseused in Guerci et al. (2017) in the following subsections. The features of the additionalsessions are summarized in Table 2. There was no large diﬀerence in the average amountsof payments among Osaka (June), Osaka (Sept), and Tsukuba (Nov).
In what follows, no-feedback, partial feedback, and full-feedback treatments are abbreviated as No-fb, Part-fb, and Full-fb, respectively. We sometimes indicate the number ofsubjects who participated in sessions: No-fb (10) for Osaka (Sept), No-fb(30)), Part-fb(30),and Full-fb (30) for Osaka (June) and Tsukuba (Nov).

session
no.
1
2
3
4

sequence ofbinary choice
A →B
B →A
C →D
D →C

Table 2: Features of the additional sessions.
show-uppoint-to- # ofsessionfee (JPY) JPY ratio subj.
date
1000
1.0
10
Sept.24, 2014
1000
1.0
10
Sept.24, 2014
1000
1.0
10
Sept.25, 2014
1000
1.0
10
Sept.25, 201410

avg. payment
to subject
2552
2442
2628
2472

3.1

Learning the Correct Option in Additional Sessions

In Subsections 3.1 and 3.2, we analyze the data in additional sessions. Let FRik denote therelative frequency of periods in which subject i chose the correct option within the k-th
block of 5 consecutive periods, that is, from 5(k − 1) + 1 to 5k. FRi2 is, e.g., the numberof times subject i chose the correct option from Period 6 to Period 10, divided by 5. Thechange in the relative frequencies that subject i chose the correct option between the l-th
block and the m-th block is defined as∆FRil,m = FRil − FRim .
Let ∆FRl,m denote a vector whose i-th component is ∆FRil,m . For each of 4 sequences ofbinary choice problems, we had 10 subjects in Osaka (Sept) in which No-fb was conducted,
and thus we have 10 observations of ∆FRl,m .
We expect experience in introspective thinking to improve learning for choosing thecorrect option even without any payoﬀ-related feedback information. For each binary choiceproblem, the p-value for the one-tailed signed-rank (SR) test for each feedback treatment
is thus reported in Table 3, where the null hypothesis is ∆FR2,1 ≤ 0 (∆FR8,1 ≤ 0) and thealternative hypothesis is ∆FR2,1 > 0 (∆FR8,1 > 0).

∆FR2,1∆FR8,1

Problem A
0.29480.0339Problem B
0.17630.0463

Problem C
0.19900.0125

Problem D
0.33920.0142

Table 3: P-values of one-tailed SR test in the sessions for the no-feedback treatment.
As shown in Table 3, for all problems, values in ∆FR2,1 were not significantly greaterthan zero but values in ∆FR8,1 were significantly greater than zero. Thus, we see that,
overall, the distributions of ∆FR8,1 lie towards the right of those for ∆FR2,1 .
Observation 1. It was reconfirmed for all binary choice problems that subjects learned
to choose the option with higher expected payoﬀs even without any feedback information
regarding their payoﬀs.
Guerci et al. (2017) reported that in Problems B and D, subjects learned to choose theoption with higher expected payoﬀs even without any payoﬀ-related feedback information.
In the next subsection, Problems B and D are thus the candidates of the binary choiceproblems in which subjects meaningfully learned the underlying feature of weighted voting.
11

3.2

Meaningful Learning (Learning Transfer) in Additional Sessions

We say that subjects are experienced if they have completed their choice in the first 40
periods. For each binary choice problem, we say that meaningful learning was observed if
the following criteria (a) and (b) were both satisfied; (a) There was a significant increase in
number of subjects who chose the correct option between Period 1 and Period 41. (b) There
was a significant increase from FR1 to FR9 . Note that in criteria (a) and (b), subjects whochose at Period 1 or for FR1 and those at Period 41 or for FR9 are faced with diﬀerent
sequences of binary choice problems.

Problem A
Inexperienced
Experienced
p-value (χ2 )Problem B
Inexperienced
Experienced
p-value (χ2 )Problem C
Inexperienced
Experienced
p-value (χ2 )Problem D
Inexperienced
Experienced
p-value (χ2 )

No-fb
(10)

No-fb
(30)

Part-fb(30)

Full-fb
(30)

7
7
1.000

20
15
0.190

25
16
0.012

22
17
0.176

4
5
0.653

9
21
0.002

13
8
0.176

9
10
0.781

4
9
0.019

19
23
0.260

22
22
1.000

19
19
1.000

2
7
0.025

8
21
<0.001

5
16
0.003

9
15
0.114

Table 4: Numbers of subjects who chose the correct option at Period 1 and Period 41 for the binary choiceproblems used in (No-fb (10) and No-fb(30), Part-fb(30)), and Full-fb (30). For each treatment, the numberof subjects who participated in sessions for the treatment is noted in parentheses. The p-values for χ2 tests
are reported for comparison between inexperienced and experienced subjects.

First, we consider whether criterion (a) is satisfied. The left-most column No-fb (10) in
Table 4 presents the numbers of subjects who chose the correct option at Period 1 and Period
41 in each binary choice problem in the additional sessions for the no-feedback treatment.
Table 4 also lists those numbers reported in Guerci et al. (2017) in the columns No-fb (30),Part-fb (30), and Full-fb (30) for reference.
12

The p-values for the χ2 test are also reported in Table 4, where the null hypothesisis that the percentages of inexperienced subjects and experienced subjects who chose thecorrect options are the same when they first encountered the same problem. Significantly
more experienced subjects chose the correct options than inexperienced subjects in No-fb(30) for Problem B (p = 0.002), in No-fb (10)) for Problem C (p = 0.019), and in No-fb (10),No-fb (30), Part-fb (30) for Problem D (p = 0.025, p < 0.001, and p = 0.003, respectively).
Thus, those five cases satisfied criterion (a).

Problem A
p-value (perm, MW)p-value (KS)Problem B
p-value (perm, MW)p-value (KS)Problem C
p-value (perm, MW)p-value (KS)Problem D
p-value (perm, MW)p-value (KS)

No-fb
(10)

No-fb
(30)

Part-fb(30)

Full-fb
(30)

0.445

0.867
0.913

0.854
0.889

0.547
0.283

0.376

0.049
0.030

0.336
0.299

0.538
0.297

0.107

0.052
0.009

0.453
0.536

0.994
0.297

0.001

<0.001<0.001

0.264
0.103

0.791
0.990

Table 5: P-values for testing FR1 = FR9 . Guerci et al. (2017) reported the p-values for the Mann-Whitney
U-test (MW) and the Kolmogorov-Smirnov test (KS) for No-fb (30), part-fb (30), and Full-fb (30). For
No-fb (10), the p-values are not calculated with the MW test and the KS test but with the permutation test
(perm) due to the small sample size. In No-fb (10), criteria (a) and (b) are satisfied only for Problem D.

Next, we consider whether criterion (b) is satisfied by comparing the relative frequency inwhich inexperienced subject i chose the correct option within the first block of 5 consecutive
periods (Periods 1 to 5) with the relative frequency in which experienced subject j chosethe correct option within the first block of 5 consecutive periods when they faced the sameproblem within the ninth block of 5 consecutive periods (Periods 41 to 45). Table 5 lists
p-values for the Mann-Whitney U-test (MW) test as well as the Kolmogorov-Smirnov test(KS) of the distributions of FR1 (inexperienced subjects) and FR9 (experienced subjects)
for each feedback treatment, where the null hypothesis is FR1 = FR9 .

13

In Part-fb (30) and Full-fb (30), the p-values for the MW and KS tests in Table 5
suggest that there be no significant diﬀerence between FR1 and FR9 for all problems. Thus,
meaningful learning was not observed for the partial-feedback and full-feedback treatments.
In sessions for No-fb (30), however, there were significant diﬀerences between FR1 andFR9 in Problem B (p = 0.049 for MW and 0.030 for KS, respectively) and in Problem D(p < 0.001 for MW and < 0.001 for KS, respectively). For Problems B and D, criteria (a)and (b) were thus both satisfied. Therefore, Guerci et al. (2017) reported that meaningfullearning was observed only for the no-feedback treatment in Problems B and D.
In No-fb (10), there was a significant diﬀerence between FR1 and FR9 in Problem D(p = 0.001 for perm), whereas there was no such a diﬀerence in Problem B (p = 0.376
for perm). For Problem C, criterion (a) was truly satisfied, as was confirmed above but
criterion (b) was not. Thus, only for Problem D, criteria (a) and (b) are both satisfied.
Thus, we have the following observation. This observation was not made by chance.
For Problem D in No-fb (10), the percentage of subjects who chose the correct option was
70-90% in B9 in sequence C → D (Figure 1). In the post experimental questionnaire, atleast 6 subjects who participated in sequence C → D noted the proper reasoning for theirchoice (Appendix B).
Observation 2. Statistically significant evidence of meaningful learning was reconfirmed
in Problem D in the additional sessions for the no-feedback treatment.
Guerci et al. (2017) reported that meaningful learning was observed in Problems B andD, but in additional sessions it was observed solely in Problem D. Recall that at Period 1 in
the additional sessions, the percentage of subjects who chose the correct option in Problems
D was lowest (Figure 1). Under criteria (a) and (b), we might be able to detect meaningfullearning more easily in Problem D than in Problem B. These findings are summarized in
the following statement.
Result 1. Hypothesis 1 was aﬃrmatively confirmed in a sequence that starts with an easy
problem and then gives a diﬃcult one.
According to Observation 2, we have the following result. We conclude that meaningfullearning without any payoﬀ-related feedback information is not the consequence of theinertia on subjects’ choice, because the correct option was Choice 1 in the first 40 periods
and it was changed to Choice 2 in the subsequent 20 periods.

14

4Subjects’ Search Behavior

Meaningful learning was observed only in No-fb, according to Observation 2 and Guerci
et al. (2017). Then, what hindered subjects from meaningfully learning the underlying
structure of weighted voting, when immediate feedback information was given to them?
For each binary choice problem, this section compares the behavior of subjects who weresearching for the correct options among treatments. Prior to the comparison, we refer tothe length of thinking time subjects spent for their decision, because we do not have to careabout the eﬀect of thinking time on their behavior in the comparison if they spent the samelength of time for choosing options among sessions for all treatments.

4.1

Length of Thinking TimeWe recorded the remaining time in seconds until the time limit (30 seconds); for eachsubject, his or her thinking time is computed by subtracting the remaining time from thetime limit. Table 6 presents the average lengths of remaining time measured in seconds
for subjects to make their choice until the time limit in the 1st block and the 9th block of
5 consecutive periods in sessions for each treatment. The p-values for the Kruskal-Wallis
test are provided in the rightmost column, where the null hypothesis is that the lengths ofremaining time for subjects to choose in the same 5 consecutive periods when they werefirst faced with a binary choice problem are on average the same among all treatments. In
Problems C and D, there was no significant diﬀerence in average length of remaining time
experienced subjects had among all treatments.
Table 7 presents p-values for the one-sided Steel test for multiple comparison of average
lengths of remaining time subjects had in 1st block and 9th block of 5 consecutive periods.
The alternative hypothesis is that as compared to No-fb (10) (control group), they had
a longer remaining time on average in No-fb (30), Part-fb (30), and Full (30) (treatmentgroups), respectively. For every pair of the control group and the treatment group, no
significant diﬀerence was observed in average length of remaining time experienced subjects
had in Problems C and D. Thus, we note this point as a remark.
Remark: We found that there was no significant diﬀerence in length of thinking time theexperienced subjects who were faced with Problem D spent among treatments.
As stated in Observation 2, meaningful learning was reconfirmed in Problem D in No-fb.
In the same length of thinking time, what occurred in subjects’ behavior when they were
given feedback information? We note a comment on this point in subsection 4.3.
15

No-fb
(10)

No-fb
(30)

Part-fb(30)

Full-fb(30)

Problem A
Inexperienced
Experienced
p-value (BM)14.26018.2000.086

15.38015.5330.812

20.547
20.1020.816

17.38022.287<0.001

0.003<0.001

Problem B
Inexperienced
Experienced
p-value (BM)12.06017.7000.002

13.98017.1200.039

16.75322.567<0.001

18.11323.373<0.001

0.004<0.001

Problem C
Inexperienced
Experienced
p-value (BM)14.60020.7600.049

12.24719.149<0.00117.90721.3530.004

16.21321.167<0.001<0.0010.240

Problem D
Inexperienced
Experienced
p-value (BM)14.24019.8000.016

13.40018.913<0.00115.84720.800<0.00117.087
20.667
0.022

0.046
0.195

p-value (KW)

Table 6: Average lengths of remaining time in seconds for subjects to choose in the 1st block (for inexperienced subjects) and the 9th block (for experienced subjects) of 5 consecutive periods. The p-values forthe two-sided Brunner-Munzel test (BM) are provided for the comparison of average lengths of remaining
time between inexperienced subjects and experienced subjects who were faced with the same binary choiceproblem. The p-values for the Kruskal-Wallis test (KW) are also provided for testing whether there is nodiﬀerence in average lengths of remaining time subjects had in the same 5 consecutive periods among all
treatments.

4.2

Switching Options

The statistical test results shown in Subsection 3.2 imply that meaningful learning was notobserved in sessions for both the partial-feedback and the full-feedback treatments. Then,
what hindered subjects from meaningfully learning in sessions for those treatments withfeedback information? As noted in Guerci et al. (2017), in the post-experimental questionnaire, some subjects in sessions for the partial-feedback and full-feedback treatments
reported that they changed their choices when they received zero points, while they did
not when they received a non-zero amount of points. If subjects had not yet been convinced of what they had learned, then it would be plausible for them to take this type of“win-stay-lose-shift (WSLS)” strategy (Nowak and Sigmund, 1993).
16

No-fb(30)

Part-fb(30)

Full-fb(30)

Problem A
Inexperienced
Experienced

No-fb (10)No-fb (10)

0.402
0.968

0.008
0.238

0.066
0.034

Problem B
Inexperienced
Experienced

No-fb (10)No-fb (10)

0.480
0.604

0.030
0.004

0.008
0.004

Problem C
Inexperienced
Experienced

No-fb (10)No-fb (10)

0.946
0.916

0.204
0.620

0.427
0.315

Problem D
Inexperienced
Experienced

No-fb (10)No-fb (10)

0.890
0.661

0.383
0.078

0.223
0.287

Table 7: P-values for the one-sided Steel test for multiple comparison of average lengths of remaining time
for subjects to choose that were observed in 1st block and 9th block of 5 consecutive periods, where thealternative hypothesis (H1 ) is that as compared to the control group No-fb (10), they had a longer remaining
time on average in the treatment group No-fb (30), Part-fb (30), and Full-fb (30), respectively.

In this experiment, subjects received 0 points or 40 points, as shown in Table 12 in
Subsection 4.3), in all binary choice problems. Tables 8 - 11 list the frequencies of observing 0
points and 40 points (freq), the frequencies of switching choices immediately after observing
0 points and 40 points (switch), the ratios of switch to freq (ratio), for the partial-feedback
treatment and the full-feedback treatment in the 1st (B1), 8th (B8), 9th (B9), and 12th(B12) blocks of 5 consecutive periods, respectively.
The p-value for the two-sided Fisher exact test is reported for each binary choice problem, where the null hypothesis is that switching choices immediately after observing 0 points
and switching choices immediately after observing 40 points were equally likely to be observed. In the following analysis, we confine attention to Problem D in sequence C → D,
because meaningful learning was observed in No-fb (30) as well as in No-fb (30) for Problem
B but it is observed only in No-fb (10) for Problem D (Observation 2).

17

Table 8: Frequency of switching choices in B1: two-sided Fisher test
Part-fb (30)Full-fb (30)0 point 40 points
0 point
Problem A freq
43
77
Problem A freq
44
switch23
18
switch
25
ratio
0.535
0.234
ratio
0.568
p-value
0.001
p-value
B freq
53
66
B freq65
switch30
15switch
36
ratio
0.566
0.227
ratio
0.554
p-value<0.001p-value
C freq27
90
C freq39
switch1230
switch
20
ratio
0.444
0.333
ratio
0.513
p-valuep-value
0.361
D freq57
62
D freq62
switch37
21
switch
37
ratio
0.649
0.339
ratio
0.597
p-value
0.001
p-value

40 points
76
19
0.250
0.001
55
13
0.236
0.001
81
28
0.346
0.111
57
13
0.228
<0.001

Table 9: Frequency of switching choices in B8: two-sided Fisher test
Part-fb (30)Full-fb (30)0 point 40 points
0 point
Problem A freq
45
105
Problem A freq
49
switch15
23
switch
15
ratio
0.333
0.219
ratio
0.306
p-value
0.155
p-value
B freq
59
91
B freq72
switch20
10switch
21
ratio
0.339
0.120
ratio
0.292
p-valuep-value
0.001
C freq43
107
C freq39
switch1031switch
12
ratio
0.233
0.290
ratio
0.282
p-value
0.547
p-value
D freq52
97
D freq62
switch15
13
switch
37
ratio
0.289
0.134
ratio
0.452
p-value
0.028
p-value

40 points
101
22
0.218
0.313
78
9
0.115
0.008
111
18
0.162
0.063
57
13
0.210
<0.00118

Table 10: Frequency of switching choices in B9: two-sided Fisher test
Part-fb (30)Full-fb (30)0 point 40 points
0 point
Problem A freq
28
92
Problem A freq
40
switch15
13
switch
12
ratio
0.536
0.130
ratio
0.300
p-value<0.001p-value
B freq58
62
B freq55
switch23
7
switch
28
ratio
0.397
0.113
ratio
0.509
p-value<0.001p-value
C freq
36
83
C freq35
switch16
16
switch
20
ratio
0.444
0.193
ratio
0.571
p-valuep-value
0.007
D freq
54
66
D freq52
switch20
14switch
31
ratio
0.370
0.212
ratio
0.566
p-value
0.068
p-value

40 points
80
12
0.150
0.088
65
14
0.215
0.001
85
18
0.212
<0.00168
12
0.177<0.001

Table 11: Frequency of switching choices in B12: two-sided Fisher test
Part-fb (30)Full-fb (30)0 point 40 points
0 point 40 points
Problem A freq
60
85
Problem A freq
63
87
switch23
11
switch
19
9
ratio
0.383
0.129
ratio
0.302
0.104
p-value<0.001p-value
0.003
B freq36
114
B freq48
102
switch415switch
10
12
ratio
0.111
0.132
ratio
0.208
0.118
p-value
0.215
p-value>0.999C freq56
93
C freq54
95
switch24
17switch
18
11
ratio
0.429
0.183
ratio
0.333
0.116
p-value
0.002
p-value
0.002
D freq41
108
D freq45
105
switch925
switch
22
25
ratio
0.220
0.232
ratio
0.489
0.238
p-value>0.999p-value
0.004

19

In sequence C → D in Part-fb (30), it is not plausible that in B1 and B8 inexperienced
subjects took the WSLS strategy in Problem C (p = 0.361 and p = 0.290) and also it is
not plausible that subjects who experienced Problem C from B1 to B8 and encountered
Problem D in B9 took the WSLS strategy (p = 0.068). Moreover, the null hypothesis was
not rejected also in B12 (p > 0.999).
Observation 3. In the partial-feedback treatment, subjects did not take the win-stay-loseshift strategy in Problem C and subjects who experienced Problem C also did not take thatstrategy in Problem D.
It is not plausible that inexperienced subjects learned something in Problem C (p =0.768 for the one-sided SR test for the null hypothesis ∆FR8,1 ≤ 0) according to Figure 3(c) in Guerci et al. (2017). As noted above, however, it is also not plausible that subjects
who experienced Problem C took the WSLS strategy in Problem D. What search behavior
did they take? The answer is shown in subsection 4.3.
In sequence C → D in Full-fb (30), it is not plausible that in B1 subjects took the WSLS
strategy in Problem C (p = 0.111) and yet they did in B8 (p = 0.063), as in the case ofPart-fb (30). It is, however, inferred that subjects who experienced Problem C from B1 to
B8 and encountered Problem D took the WSLS strategy (p < 0.001) in B9 and that they
still took the WSLS strategy (p = 0.004) in B12. We have the following observation.
Observation 4. In the full-feedback treatment, inexperienced did not take the win-staylose-shift strategy in Problem C but subjects who experienced Problem C took the win-staylose-shift strategy in Problem D.
Let us look at another sequence. It is inferred that in B9 subjects who experienced
Problem A from B1 to B8 and encountered Problem B took the WSLS strategy (p < 0.001),
where “lose-shift” was chosen at about 40% (=0.397) after 0 points realized while “win-stay”
was chosen at about 89% (=1-0.113) after 40 points realized, although the null hypothesis
was rejected in B12. It is inferred that subjects who experienced Problem A from B1 to B8
and encountered Problem B took the WSLS strategy (p = 0.001) in B9, although the null
hypothesis was rejected in B12. All test results are summarized in Table 13 in subsection 4.3
together with the test results of random choice of runs in each sequence of the options.
We summarize these test results including the ones stated in Observations 3 and 4 in thefollowing result; there were some sequences in which subjects took the win-stay-lose-shift
strategy, when feedback information is provided immediately after their choice.
Result 2. Hypothesis 2 was partly verified.
20

4.3

Random Choice of Runs

In reference to Observation 3, we could not conclude that in sequence C → D, the WSLS
strategy hindered subjects from meaningful learning in sessions for the partial-feedback
treatment. In this subsection, we consider whether subjects randomly chose the runs ofoptions in their search for the correct ones.
Problem A

Choice 1(71 , 72 )(5, 3, 71 )(5, 3, 72 )[14; 5, 3, 7, 7](0, 0, 60, 60)(40, 40, 40, 0)(40, 40, 0, 40)

Problem B

Choice 1(2, 4)(3, 4)(1, 2, 3)
Choice 1(6, 8)(3, 5, 6)(3, 5, 8)[6; 1, 2, 3, 4](0, 60, 0, 60)(0, 0, 60, 60)(40 40, 40, 0)[14; 3, 5, 6, 8](0, 0, 60, 60)(40, 40, 40, 0)(40, 40, 0, 40)

Choice 1(3, 6)(5, 6)(1, 3, 5)[9; 1, 3, 5, 6](0, 60, 0, 60)(0, 0, 60, 60)(40, 40, 40, 0)

Problem C

Problem D

Choice 2(5, 4, 6)(5, 4, 7)(5, 6, 7)(4, 6, 7)
Choice 2(41 , 42 )
(11 , 12 , 41 )
(11 , 12 , 42 )Choice 2(3, 61 , 62 )(3, 2, 61 )(3, 2, 62 )
(61 , 62 , 7)
Choice 2(61 , 62 )(1, 2, 61 )(1, 2, 62 )[14; 5, 4, 6, 7](40, 40, 40, 0)(40, 40, 0, 40)(40, 0, 40, 40)(0, 40, 40, 40)[6; 1, 1, 4, 4](0, 0, 60, 60)(40, 40, 40, 0)(40, 40, 0, 40)[14; 3, 6, 6, 7](40, 40, 40, 0)(40, 40, 0, 40)(40, 0, 40, 40)(0, 40, 40, 40)[9; 1, 2, 6, 6](0,0, 60, 60)(40, 40, 40, 0)(40, 40, 0, 40)

Table 12: MWCs and payoﬀ vectors. The MWCs in [14; 5, 3, 7, 7] are, e.g., written here as (5, 3, 71 ),(5, 3, 72 ), and (71 , 72 ) by the votes apportioned to the members, not with references to the specific players.

Under the null hypothesis, the number of runs of the options each subject chooses isa random variable. When the payoﬀs associated with options (shown in Table 12) arestochastically determined, however, the null hypothesis is not rejected as often in the runs
test, even if the sequence of options is generated by the WSLS strategy.4 We thus prioritize
the test result for the WLSL strategy over the result in the runs test, when it is inferredthat subjects took the WSLS strategy in both B1 and B8 or in both B9 and B12.
4In Table 12, denote by 1-1, 1-2, and 1-3 payoﬀ vectors (0, 60, 0, 60), (0, 0, 60, 60), and (40, 40, 40, 0),
respectively when Choice 1 is chosen in Problem D and by 2-1, 2-2, and 2-3 (0, 0, 60, 60), (40, 40, 40, 0),and (40, 40, 0, 40), respectively when Choice 2 is chosen. Assume that when Choice k (= 1, 2) is successively
chosen, payoﬀ vectors realize in the order of k-1→ k-2 → k-3 → k-1 → · · · , and assume in addition that when
the alternative option is once chosen after observing payoﬀ vector k-i (i=1, 2, 3) and then Choice k is chosen
again, the order resumes from the payoﬀ vector next to k-i. Given that Choice 1 is chosen at Period 1, the
WSLS strategy generates the following sequence of 20 choices: 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 1, 2, . . ..
The p-value for the runs test applied to the sequence of those choices is 0.8391, and thus the null hypothesis
was not rejected, although they were generated systematically with the WSLS strategy.

21

Table 13 presents the numbers of subjects each of whom is counted if the null hypothesis
was rejected in the runs test for each binary choice problem in sessions for each treatment.
The sample size of 5 consecutive periods would be too small to conduct the runs test. The
runs test was thus applied to a sequence of the options each identical subject chose in abinary choice problem in Periods 21-40 and in another one in Periods 41-60, respectively.
In the table, “W” is put when the WSLS strategy was observed, while “w” was used when
it was observed only in the first 5 consecutive periods in each binary choice problem.

Problem A
Inexperienced
Experienced
Problem B
Inexperienced
Experienced
Problem C
Inexperienced
Experienced
Problem D
Inexperienced
Experienced

No-fb
(10)

No-fb
(30)

Part-fb(30)

Full-fb
(30)

5
3

22
15

12
9

W

11
15

4
5

22
13

13
8

W
w and rand

118Ww and rand

5
4

14
18

9
10

rand
W

12
13

W

7
5

21
15

12
7

W
rand

7
7

W
W

Table 13: Numbers of subjects each of whom is counted if the null hypothesis (H0 ) was rejected, where H0
is that the number of runs in a sequence of the options each subject chose in Periods 21-40 (in Periods 41-60)
is a random variable. “W” indicates that we can infer that the WSLS strategy was taken in both B1 and B8
for the inexperienced subjects and in both B9 and B12 for the experienced subjects, respectively, and “w”
was used when the WSLS strategy was observed only B9. Another mark “rand” indicates that subjects did
not search the correct options with any certain rules, according to the test result for the random choice ofruns. Letters in bold indicates the test results in sequence C → D.
As shown in Subsection 3.1, for all four binary choice problems we examined, it wasreconfirmed that subjects learned to choose the correct options in sessions for No-fb (10)(Observation 1). Table 13 shows that in Problem B, 40% of inexperienced subjects searched
for the correct options with a certain rule in sessions for No-fb (10). Thus, we presume in
the following analysis that subjects did not search for the correct options with any certain
rules, if at most 30% of subjects are counted for each of whom the null hypothesis was
rejected in the runs test and if they did not take the WSLS strategy. In Table 13, “rand”
is put when subjects did not search for the correct options with any certain rules.
22

We found in subsection 4.1 that there was no significant diﬀerence in length of thinking
time the experienced subjects spent in Problem D among treatments (Remark). In almost
the same length of thinking time, how did those subjects search for the correct options, when
they were given feedback information? Observation 3 states that in Part-fb (30) subjects
who experienced Problem C did not take the WSLS strategy in Problem D. Table 13 shows,
however, that there were only 7 experienced subjects in Problem D are counted for eachof whom the null hypothesis was rejected in the runs test. Observation 4 states that in
Full-fb (30) subjects who experienced Problem C did take the win-stay-lose-shift strategy
in Problem D. We have the following observation, according the 30% criterion noted above
and the priority of the test result for the WLSL strategy.
Observation 5. In the partial-feedback treatment, subjects did not search for the correct
options with any certain rules in Problem D. In the full-feedback treatment, experienced
subject took the win-stay-lose-shift strategy in Problem D.
In Problem D, experienced subjects chose their options in almost the same thinking time,
regardless of the treatments in which they participated. Observation 5 thus more neutrally
implies the reason why subjects could not meaningfully learn the underlying relationshipbetween payoﬀ distributions and vote apportionments, when payoﬀ-related feedback information was provided immediately after their choice.
Finally, let us consider the cases of Problem B, because meaningful learning was observed
also in No-fb (30). What occurred in Part-fb (30) and Full-fb (30)? In each of those feedback
treatments, as shown in Table 13, experienced subjects did take the WSLS strategy in B9,and there were only 8 experienced subjects who were counted for each of whom the null
hypothesis was rejected in the runs test. Thus, we have the following observation.
Observation 6. In both partial-feedback treatment and the full-feedback treatment, experienced subjects took the win-stay-lose-shift strategy in some consecutive periods after they
were first faced with Problem B but finally they did not search for the correct options witha certain rule.
Table 13 indicates that when subjects received immediate feedback information, there
are some cases where we could not reject the null hypothesis that that they did not search
for the correct options with a certain rule. We summarize those findings including Observations 5 and 6 as the following result. Immediate payoﬀ-related feedback information
sometimes induced subjects to randomly choose the runs of options.
Result 3. Hypothesis 3 was partly verified.
23

5

Final Remarks

In this section, we note brief remarks on the improvement of our experimental design andpossible questions for future research, based on our results and observations. One is theverification of implications derived by theoretical models, and the other is the relationshipbetween subjects’ cognitive ability and their meaningful learning.

Potential Theoretical Models
Results 2 and 3 might suggest that subjects’ inference on the relationship between nominal
voting weights and actual payoﬀs be mistaken or confused when feedback information wasgiven immediately after their choice. This idea is still unverified, but if it is true, then we cansay that in the sessions for no-feedback treatment, subjects might deeply infer the underlying
structure of weighted voting without such misleading or confusing information by being kept
away from the feedback information. Even in sessions for full-feedback treatments, however,there were some subjects who could succeed in meaningful learning.
What feedback information then induced those subjects to meaningfully learn the underlying structure of weighted voting meaningfully? In this experiment, subjects were notgiven the information about cumulative payoﬀs but provided with instantaneous payoﬀsgiven immediately after their decision, and they were prohibited from taking any notes
during the sessions. By this lack of suﬃcient memory on the outcomes that were realized by their previous choice, immediate payoﬀ-related feedback information might confuse
subjects’ inference on the underlying structure of weighted voting. From the viewpointof the eﬀect of subjects’ memory on their learning, case-based decision theory (CBDT)
might also provide theoretical reasoning for some patterns of the search behavior observedin the partial-feedback and full-feedback treatments. (Gilboa and Schmeidler (2001) is acomprehensive review of the CBDT written by the founders of the theory.)
The CBDT describes how people make an analogy of the circumstances surrounding
them by the past experiences when they are ignorant of the structure. If we assumed
that subjects were memoryless, then the CBDT would provide a possible explanation of
the WSLS strategy of subjects who gave up learning. On the other hand, if subjects in
our experiment could take notes on realized payoﬀ distributions, then they might infer
the correct options. At the beginning of the experimental instruction, however, subjects
were informed that they were prohibited from taking notes during the session. Thus, theinformation on the cumulative payoﬀs they earned might assist subjects in inferring the
expected payoﬀs of the options they had chosen. We cannot verify what memory subjects
24have during the sessions, but we can record what information subjects see. Therefore,a “mouse-tracking” experiment is an alternative way for capturing the eﬀect of subjects’memory on their leaning behavior.5
There is another mathematical model to test the theoretical implications behind ourobservations. Grant et al. (2017), for example, proposed a model of learning in which
complete lack of information regarding the underlying data generating process is expressed
as a (maximal) family of priors. In the replication studies of human learning, Arifovic et al.
(2006) showed that standard models fail to replicate human behavior in a repeated gameof battle of the sexes, and Erev et al. (2010) reported that those models do not perform
well in predicting how people behave in market entry games. It would thus be valuable todetect the implications derived from these theoretical models.

Subjects’ Cognitive Ability
Albeit on a trial basis, subjects’ cognitive ability scores were measured by the Raven’s APM
test in the sessions for Guerci et al. (2017) and the additional sessions conducted by theauthor. The Raven’s test is a well-known test that measures subject’s ability for visual
pattern recognition, and there are three versions; Colored Progressive Matrices (CPM),
Standard Progressive Matrices (SPM), and Advanced Progressive Matrices (APM), in ascending order of diﬃculty. The Raven’s APM test is composed of 48 questions in total, but
in those sessions 16 questions were selected so that that the subjects could answer thosequestions within 10 minutes after the bandit experiment.6 The average score was about
12.3 for subjects who participated in the sessions for Guerci et al. (2017) and it was 12.1
for those who participated in the additional sessions conducted by the author.
Ogawa et al. (2020) conducted the same experiment as the one designed by Guerci et al.
(2017) at four experimental sites the subjects’ characteristics of which were diﬀerent, andthey reported that meaningful learning was observed only at an experimental site where
subjects had, on average, higher scores (11.6) of the Raven’s APM test than those had atother sites. Accordingly, it would be inferred that meaningful learning could be observedat the experimental sites where the subjects’ cognitive ability scores were relatively high.
This feature should be reconfirmed by designing a more comprehensive experiment.
5

In the mouse-tracking experiment, votes and payoﬀs of the committee members were “hidden” fromsubjects in windows on their monitors; using a computer mouse, he or she needed to bring the cursor to thewindows and click on them to view the hidden information. Almost all other aspects are the same as thosefor the full-feedback treatment in our experiment.
6
The question numbers are 1, 4, 7, and 10 from Set I and questions 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31,
and 34 from Set II.

25

Declarations
Conflict of Interest
The author declares that there is no conflict of interest. Eric Guerci and Nobuyuki Hanaki
gave their permission to reuse the data to the author. Their permission emails can be shown
to the editors of this journal.

Availability of Data and Material
All raw and processed data as well as zTree codes are available upon requests.

Code Availability
All data were processed with Excel Statistics. The software information is available at thefollowing website, although the contents are written in Japanese.
https://bellcurve.jp/ex/

Ethical Approval
All procedures performed in studies involving human participants were in accordance withthe ethical standards of the institutional and/or national research committees and with the1964 Helsinki declaration and its later amendments or comparable ethical standards. This
article does not contain any studies with animals performed by any of the authors.

26

Appendix A: Instructions
The instructions for bandit experiments are in general less informative to subjects. We
attach the instructions here for showing to the readers that they were actually simple.

Instructions
Welcome! Thank you for participating in this experiment today. You will be paid
1000 JPY for your participant and an additional reward that ranges from 0 to 3200 JPY
depending on your choice and performance in the experiment. At first,• Please follow the instructions of the experimenter.
• Please do not take notes during this session.
• Please remain quiet and especially do not talk with other participants.
• Please do not look at what other participants are doing.
• During the experiment, please maintain an upright posture without leaning on thebackrest.
• Do absolutely nothing other than the operation that you are instructed to do.
• Please turn oﬀ your mobile phone and definitely refrain from using it.
• If you have any questions or require assistance, please silently raise your hand.
You will be asked to repeatedly make a simple choice between two options. Imagine thatyou need to represent your interests within a voting committee. This committee decides
how to divide 120 points among its members. The committee has three other members, andeach member has a predetermined number of votes, which may be diﬀerent from one to theother. The committee will make a decision only when a proposal receives the predetermined
required number of votes. You will be told what is the required number of votes. If morethan one proposal is put before the committee, the members cannot vote for multiple
proposals by dividing their allocated number of votes. A member can vote for only one
proposal, and all of his/her votes must be cast for that proposal.
You are asked to choose which of the two possible committees you prefer to join. You will
be informed of the number of votes allocated to each of the four members of the committee(including you), and the number of votes required for a proposal to be approved. The
number of votes you have will always be indicated with the label YOU.
27

full-feedback treatment
There are a total of 60 periods. At each period, you have 30 seconds to make your choice
between the two committees. If you do not make a choice within the 30 seconds at one
period, you will receive zero points for that period. When a choice is made, the chosencommittee will automatically allocate 120 points among the four members. The outcomesmay vary from one period to another, but are based on a theory of decision-making incommittees. Once the allocation is made, you will immediately be shown the resulting
allocation. At the end of the experiment, you will be paid according to your total earnings
during the 60 periods, at an exchange rate of 1 point = 1 JPY.

partial-feedback treatment
There are a total of 60 periods. At each period, you have 30 seconds to make your choice
between the two committees. If you do not make any choice within the 30 seconds at one
period, you will receive zero points for the period. When a choice is made, the chosencommittee will automatically allocate 120 points among the four members. The outcomesmay vary from one period to another, but they are based on a theory of decision-making incommittees. Once the allocation is made, you will be shown the number of points allocatedto you. You will not see the allocations to the other members of the committee. At the endof the experiment, you will be paid according to your total points at an exchange rate of 1
point = 1 JPY.

no-feedback treatment
There are a total of 60 periods. At each period, you have 30 seconds to make your choice
between the two committees. If you do not make any choice within the 30 seconds at one
period, you will receive zero points for the period. When a choice is made, the chosencommittee will automatically allocate 120 points between the four members. The outcomesmay vary from one period to another, but they are based on a theory of decision-making
in committees. You will not see the resulting allocation after each period. However, atthe end of the experiment, you will be told the total points you have obtained during the
60 periods, and you will be paid according to the points earned over the 60 periods at an
exchange rate of 1 point = 1 JPY.
If you have any questions, please raise your hand.

28

Appendix B: Subjects’ Comments
In additional sessions, 10 subjects participated in sequence C → D. As shown in Figure 1, atleast 6 subjects chose the correct option in Periods 41-45. Below are 6 answers to the postexperimental questionnaire of those subjects who succeeded in meaningful learning. The
answers show that they had the proper reasoning for their choice, although the questionnaire
was unfortunately not structured and the answers were written in a free format. Recall thatin the additional sessions, however, the correct option is Choice 1 in the first 40 periods andit is Choice 2 in the subsequent 20 periods.
Question: Which option did you mainly choose? Why did you choose that option?
Please explain the reason behind your choice.
• I realized that I could not obtain any reward without the approval of three voters
including myself. Thus, in the second half, I chose the option in which it was lesslikely to be approved by two large voters only. (2 subjects)• I chose the options that have more cases where three voters could win by themselves.
(3 subjects)• In Periods 1-40, there was a case where two large voters could collect 14 votes bythemselves in Choice 2, but there was no such a case in Choice 1. Thus, I chose
Choice 1. But, I sometimes chose Choice 2, because I was not sure about how 120
points would be distributed. In Periods 41-60, there were two cases where two voters
exept me could collect 9 votes by themselves in Choice 1, but there was one case where
two voter could collect 9 votes by themselves. Thus, I chose Choice 2 many times.
But, I sometimes chose Choice 1, because I did not know how 120 points would be
distributed. (1 subject)29

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