A Mouse-Tracking Bandit Experiment on
Meaningful Learning in Weighted Voting1
N. Watanabe∗∗ Keio Business School

July 13, 2022 at University of Padova, Italy (online).
17th European Meeting on Game Theory (SING 17)

1

revised on Aug 25, 2023
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Suppose that you were a subject.

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choice stage (30 sec.); no choice within 30 seconds → zero point.
Each number in red can be viewed by a mouse-click.

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Figure: Choice stage.
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feedback stage 1 (30 sec.); OK button → feedback stage 2. Each
number in red can be viewed by a mouse-click.

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Figure: Feedback stage 1.
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feedback stage 2 (30 sec.); OK button → next round. Each
number in red can be viewed by a mouse-click.

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Figure: Feedback stage 2.
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I Answer 60 binary choice problems (60 rounds).

When you finish 60 rounds, you are asked to wait
until the others finish.
I 1 point = 1 JPY (about 115 JPY/ USD in 2017),

added to the payment for participation (1000 JPY)
mouse-tracking to capture what information subjects view on theirmonitorI When measuring gaze with eye trackers, subjects often

unconsciously move their eye-sight to places other than themeasurement targets. (a kind of noise)

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1. Introduction: Research Question
I Question: What information did subjects view, when they

could (not) “meaningfully learn” the latent feature ofweighted voting in a “2-armed bandit experiment”?
I

Felsenthal and Machover (1998): It would be difficult (forpeople) to see the underlying relationship between the actual
voting powers and the nominal voting weights.

I When people generalize what they have learned in a situation

to a similar but different one, this higher order concept oflearning is called meaningful learning (Rick and Weber, 2010).
I 2-armed bandit experiment: Subjects choose one of two

weighted voting games repeatedly. Payoffs are determinedstochastically according to a payoff-generating function that is
hidden from subjects.

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1. Introduction: Previous Research
I Guerci et al. (2017, TD) could not observe meaningfullearning by subjects who received immediate feedback
information on their current payoffs.
Withholding feedback information induced meaningfullearning (introspective thinking). Did the subjects get
confused by the feedback information?
I Ogawa et al. (2021, mimeo.) reconfirmed a similar result atfour different experimental sites.
I

I ref. Guerci et al. (2014, SCW): subjects played 4-person

weighted voting games
not focused on their learning but compared frequencies of their“mistakes” observed in bargaining protocols in voting.
(single approval or multiple approval...)I Esposito et al. (2012, mimeo.): subjects chose a weighted
voting game and then they played the game they chose.
→ subjects’ learning was affected by others’ voting behavior.
I

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1. Introduction: Main Result
We measured how often or long each subject views informationnecessary for his or her choice without duplication of viewing time.
The feedback Information about cumulative payoffs promoted
meaningful learning, whereas the information on current payoffs
did not, or even hindered it.
I The latter part of this result may explain the reason whymeaningful learning was not observed by Guerci et al. (2017),
because in their experiment only the current payoffs wereshown to the subjects.
search behavior: win-stay-lose-shift (WSLS), random choice of runs
→ outcomes: current payoff and failure in meaningful learning

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2. Experimental Design
I Subjects choose one of two weighted voting games.

[q; v1 , v2 , v3 , v4 ]; q is the quota, vi is the voting weight.
Each subject acts as Member 1, who has v1 .
I clearly informed that the other three members are all fictitious.

I

I

I one binary choice problem (Problem) for the first 40 rounds,

and a similar but different one in the following 20 rounds;
I

This information was not provided to subjects.

I four sequences: A → B, B → A, C → D, D →C
I payoff-generating function: based on Deagan-Packel index

e.g., Problem A, Choice 1 → 40 points with Prob=2/3,
0 point with Prob=1/3
Table: Binary choice problems and expected payoffs for Member 1
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)
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choice stage (30 sec.); no choice within 30 seconds → zero point.
Each number in red can be viewed by a mouse-click.

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Figure: Choice stage.
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feedback stage 1 (30 sec.); OK button → feedback stage 2. Each
number in red can be viewed by a mouse-click.

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01234",5!"!#"$%&'()*+"$,-()*+"$ .

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Figure: Feedback stage 1.
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feedback stage 2 (30 sec.); OK button → next round. Each
number in red can be viewed by a mouse-click.

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Figure: Feedback stage 2.
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I Subjects answer 60 binary choice problems.

→ a message that asks them to wait until the others finish(→ Questionnaire and Raven test)
I 1 point = 1 JPY, added to the payment for participation(1000 JPY)

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3-1. Analysis: Session Details

I December 17 in 2014 to January 17 in 2017
I at the University of Tsukuba
I undergraduate students who did not know voting indices
I 40 subjects for each sequence (160 subjects in total)
I the average amount paid as a reward was 2507 JPY

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(a) A → B(b) B → A(c) C → D(d) D → C

Figure: RED: fraction of subjects who chose the correct answer among
those who made choices before the time limit. BLUE: fraction of those
who failed to make their choice before the time limit.

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I FRi : the relative frequency of rounds in which subject i chosekthe correct answer within the k-th block of 5 consecutive
rounds.
I The change in the relative frequencies that subject i chose thecorrect answer between the l-th block and the m-th block isdefined as∆FRil,m = FRil − FRim ,where l > m.
I ∆FRl,m (FRl , FRm ): a vector the i-th component of which is∆FRil,m (FRil , FRim ).
I

When the elements of FRl are, on average, significantly larger
than those of FRm , we write this as ∆FRl,m > 0.

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3-2. Results: Aggregate Data

Definitions: For each binary choice problem, we consider that
subjects learned the “correct” answers if ∆FR8,1 > 0 was
statistically confirmed and that those who learned in the binary
choice problem meaningfully learned the “underlying structure” ofweighted voting games if ∆FR9,1 > 0 in the same binary choiceproblem was statistically confirmed.
Result 1: For all binary choice problems, the subjects learned thecorrect answer. For all binary choice problems except Problem B,
the subjects meaningfully learned the underlying structure ofweighted voting.

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∆FR2,1∆FR8,1∆FR9,1

Problem A
0.04150.01430.0147

Problem B
0.0086<0.0001
0.2498

Problem C
0.0396<0.0001<0.0001

Problem D
0.6943<0.0001
0.0230

Table: P-values for the two-sided signed-rank test (normalized). The null
hypotheses are ∆FR2,1 = 0, ∆FR8,1 = 0, and ∆FR9,1 = 0, respectively

I An auxiliary experiment was conducted at Kansai University:

Cumulative payoffs were shown to the subjects’ monitors aswell as their current payoffs, and similar results to Results 1
and 2 were observed. → The use of mouse trackers would notaffect subjects’ choice in the aggregate level.

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3-3. Results: Individual Data
I two groups of subjects G 91 (s) and G 91 (s) for binary choice01
problem s = A, B, C , D: ∆FRi9,1 = FRi9 − FRi1 > 0 or not
I canonical discriminant analysis: the first 40 rounds to identify

important mouse-tracked variables that had a large effect on
the classification of the two groups.
I

Box’s M test on variance-covariance matrices → Mahalanobis
distance was applied to identify the discriminant function

variable
A count:
A time:decision time:
A decision time:
main2 ok:
main3 ok:no info:judgment:
judgment2 :numbers of views of A; A = vr , wr , xr , pr (r = 1, 2, 3, 4), yyy, zzz
cumulative time for viewing A
time spent for the final decision in the choice stage
relative length of time spent for viewing A up to the final decisiontime spent in feedback stage 1time spent in feedback stage 2
dummy variable that takes a value of 1 if the individual subject views
v1 , . . . , v4 or w1 , . . . , w4 even once; otherwise, 0
dummy variable that takes a value of 1 if ∆FRi8,1 > 0; otherwise, 0
dummy variable that takes a value of 1 if ∆FRi9,1 > 0; otherwise, 0

Table: Major variables.

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Definition: For each sequence of binary choice problems, weconsidered that subject i learned the “correct” answers if∆FRi8,1 > 0 and that those who learned in a binary choice problem
meaningfully learned the “underlying structure” of weighted voting
games if ∆FRi9,1 > 0 in another binary choice problem in thesequence.
I the numbers of subjects who succeeded in learning(meaningful learning) were 20, 30, 32, and 27 (14, 13, 19,17), respectively, under the following definitions.
I For every binary choice problem s = A, B, C , D, we could
reject the null hypothesis of no difference between G081 (s) andG181 (s) (between G091 (s) and G191 (s)) for independent

variables, because Wilks’ lambda < 0.001.

21 / 27A

zzz count
0.678

p1 time
0.627

v4decision time
-0.486yyy count-0.484p3 time-0.378B

decision time
-0.827x1 decision time
-0.552x1 count
0.521

zzz count
0.506

yyy count
0.241

C

x1 count-1.359x3 count
1.040

yyy time-0.549w4 time
0.543

v3 time
0.473

D

p4 time0.749

zzz count
0.479

p1 time0.315

no info-0.298yyy count-0.142

Table: Standardized coefficients in the canonical discriminant functions: thefour largest absolute values observed in the first 40 rounds in the discriminantfunction that separated well G091 (s) and G191 (s) for binary choice problem
s = A, B, C , D.

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Result 2:
The information on subjects’ cumulative payoffs promoted
meaningful learning, whereas the information on their own current
payoffs did not, or even hindered it.
I The latter part of this result may explain the reason whymeaningful learning was not observed by Guerci et al. (2017),
because in their experiment only the current payoffs wereshown to the subjects.
I

Watanabe (2022): reconfirmation of meaningful learning
without any feedback information (correct answers werereversed in the first 40 rounds and the following 20 rounds).

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Subjects’ Search Behavior
In what follows, we examine the null hypotheses in 20 rounds.
For each individual subject i, let ai denote the number of rounds inwhich yyy count − zzz count ≥ 0 and denote bi the number ofrounds in which yyy time − zzz time ≥ 0.
I We say that in 20 consecutive rounds, subject i paid more

attention to the cumulative payoffs than to the current
payoffs, if ai ≤ 10 and bi ≤ 10 in those rounds.
I If either ai > 10 or bi > 10, we say that subject i paid more

attention to the current payoffs.

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Define current payoff/rundruns as the number of subjects who
paid more attention to the current payoffs among those who chosethe runs randomly and meaningful/randruns as the number ofsubjects who succeeded in meaningful learning among those whochose the runs randomly.
In the next table, the values in the parentheses are the numbers ofsubjects who paid more attention to the cumulative payoffs among
subjects who chose the runs of options randomly and the numbersof those who failed in meaningful learning among those who chosethe runs randomly.
I Under the null hypothesis in the (Wald-Wolfowitz) runs test,

the number of runs of options chosen by a subject is a randomvariable.
I

Computer scientists consider the optimal length of runs ofoptions in which decision-makers should continue to choose
the same options for accumulating the feedback information.
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Result 3:
It was plausible that more subjects paid attention to cumulative
payoffs but failed in meaningful learning when they chose the runsof options randomly.

rounds
A→B
B→A
C→D
D→C

current payoff/randruns
1-20
21-40
41-60
11 (17) 10 (13) 6 (18)
11 (19)7 (9)8 (14)
12 (8)
6 (14)
7 (15)
18 (12) 10 (18) 10 (15)meaningful/randruns
1-20
21-40
41-60
9 (19) 8 (15) 7 (17)13 (17) 14 (14) 8 (14)10 (10) 9 (11) 12 (10)
11 (19) 9 (19) 6 (19)Table: Frequencies of observations: current payoff, meaningful learning, andrandom choice of runs. The rejection of the null hypothesis at the 5%
significance level in the one-sided binomial test is indicated in boldfaced value.

Rounds 21-40 are more important than rounds 1-20 due to thedefinition of learning.
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4. Final Remarks
Note: We could not observe that individual subjects exactly took
Win-stay-lose-shift strategy (Nowak and Sigmund, 1993).
Summary:
I Information on subjects’ cumulative payoffs promoted

subjects’ deep understanding (meaningful learning) theunderlying structure of weighted voting.
I Even if subjects paid more attention to their cumulative

payoff, they would fail in meaningful learning when they chosethe runs of options randomly.
I future researchlength of memory → case-based decision theory(Gilboa and Schmeidler (1995, 2001)
I response time in the process of learning(ref. Rubinstein, 2013)I

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