A School Choice Experiment:
Cognitive Ability and Information
N. Watanabe1 , T. Kawamura2 , and K. Ogawa3
1 Keio U., 2 Tezukayama U., 3 Kansai U.

June 17, 2021 at Seoul National Uviv. (on-line)
SAET annual meeting

1 / 48

1. Introduction: Today’s Research Question▶ In a school choice (college admissions) experiment(Chen-Kesten 2019, GEB),▶

How does cognitive ability of subjects and information
provided to them aﬀect the allocative eﬃciency andsubjects’ behavior? · · · stability and equilibria: omitted today▶ environments: 4 schools (6 schools) with capacity of 1 and 4

stuednts (6 students); one student has one district school.
▶ Boston mechanism (Bos)vs. deferred acceptance mechanism (DA)
▶ 4 (= 2 × 2) treatments of information × 2 groups of subjects

in cognitive ability scores▶

complete vs. incomplete info on students’ preferences,
clear vs. unclear info on schools’ priority orders
CK2019’s treatment = complete and unclear info
▶ high vs. low scores in Raven’s APM test
2 / 48

1. Introduction: Main Results I (4 schools - 4 students)▶ Which information is important for the allocative eﬃciency?
▶

Complete or Incomplete Info on the other students’ preferences
over schools does not significantly aﬀect the eﬃciency underboth Bos and DA.
▶ Clear or Unclear Info on schools’ priority orders over students
does aﬀect the eﬃciency under DA.
▶ Subjects’ behavior is sensitive to both types of info. except

high score holders under Bos.
▶
▶

About 50-70% of choices were dominant strategies.
About 90% of choices were representing true preferences orchoosing district schools as the top choice in every session.

▶ The result in Chen-Kesten (2019) can be supplemented:

Bos ≺ DA in eﬃciency under CK2019’s treatment of
information in each group of cognitive ability scores▶
▶

In CK2019, Bos ∼ DA in eﬃciency.
We could replicate Bos ∼ DA by mixing the observed
behaviors of 2 high score holders and 2 low score holders.
3 / 48

1. Introduction: Main Results II (6 schools - 6 students)
still in progress... (delayed due to the COVID-19), but we havealready had the following result under CK2019’ treatment of
information.
▶ The result in Chen-Kesten (2019) can be supplemented:

Bos ∼ DA in eﬃciency by matching 4 high score holders and2 low score holders and in the environment of 6 high (low)score holders.
▶
▶

In CK2019, Bos ≻ DA in eﬃciency.
We could replicate Bos ≻ DA by matching 2 high (3 high)score holders and 4 low (3 low) score holders.

▶ ⇒ Matching subjects with diﬀerent groups of cognitive ability

score generates diﬀerent results on allocative eﬃciency.
▶

We should control subjects’ cognitive ability inexperiments.

▶ In the environment of 6 low score holders, truth-telling wasobserved less than that observed in the other environment.
4 / 48

2. Experimental Design: Students’ Preferences▶ Students’ preferences over schools are represented by payoﬀs.
▶ (4 schools - 4 students) Each student obtains 11 points when

he or she is matched with his or her district school (secondly
preferred school).
▶

Assume that there is a utility function that represents theirpreference orderings.
Table: Students’ preferences.

student \ school
1
2
3
4

A
11
5
7
5

B
7
11
16
16

C
5
7
11
7

D
16
16
5
11

In a complete information treatment, students’ preferences areshown the instruction, whereas the others’ preferences are notshown there in an incomplete information treatment.
5 / 48

2. Experimental Design: Schools’ Priority Orders▶ In Chen-Kesten (2019), priority orders over students are

determined by one of 4-digit (6-digit) queues.
1. Each student is given the first priority by his or her district
school.
2. (4 schools - 4 students) Given a 4-digit number, say 2341,
school B gives the second priority to student 4 whichcorresponds to 1 in 2341 (smallest among 2, 4, and 1).
Similarly, school B gives the third priority to student 1 whichcorresponds to 2 in 2341 (smaller between 2 and 4). 2413 is
the priority order of students for school B▶ In summary, school B gives priorities to students according to(

1 2 3 42 3 4 1),

where the numbers in the first row represent student IDs.
▶ Subjects have to compute those priority orders by themselves.
6 / 48In CK2019’s treatment of information, priority orders are notdisplayed on subjects’ monitor; unclear information. The above
Figure shows a subject’s monitor in the treatment with clear
information.
7 / 48go back to Introduction: Another Literature
Pais-Pintér (2008, GEB): Information provision in the case of 5
“teachers” and 3 schools (with priority orders over teachers)▶ zero info setting = Incomplete and Uncertain info
▶ low info setting = Incomplete and Certain info
▶ partial info setting = Complete and Uncertain info
▶ full info setting = Complete and Certain info

A surprising result: percentile of choosing the dominant strategy isordered in zero > low > “partial = full” under both Bos and DA.
▶ We could not observe this remarkably clear relationship.
▶ It was, however, partly observed that withholding informationinduced more subjects to choose dominant strategies (e.g.,
low score holders under Bos).

8 / 48

2. Experimental Design: Subject Pool
▶ Raven test: measuring subjects’ ability of pattern recognition.
▶

In each question of the test, eight patterns are drawn, and thesubject selects a pattern that matches those visual patterns
from the options.
▶ Colored Progressive Matrices (CPM), Standard Progressive
Matrices (SPM), and Advanced Progressive Matrices (APM),
in ascending order of diﬃculty.
▶ The APM version of the test is composed of 48 questions in
total. We used 16 questions excerpted from the test.
▶ The Raven scores of the subjects had been taken prior to thedates for our sessions, who were recruited from all over thecampus at Kansai University in Japan.
▶ Our subjects were recruited from that subject pool.
▶ Subjects were categorized into two groups, high score group(12-16 points) and low score group (4-11 points).
9 / 48

2. Experimental Design: Subjects’ Raven Scores
The case of 4 schools and 4 students
Mean score11.135
13.12
9.151

Standard Deviation
2.418
0.982
1.686

Min
4
12
4

Max
16
16
11

20

Frequency
40

60

Obs.
384
192
192

0

All
High
Low

1

2

3

4

5

6

7

8
9 10
Raven score

low score group

11

12

13

14

15

16

high score group10 / 48

2. Experimental Design: Raven Score of Non-students
Num. of Obs.
934

Mean
7.935

Std. Dev.
3.283

Min.
0

Max.
16

0

.05

density.1.15

2016.4-2018.3

0

5

10

15

Raven’s score

11 / 48

2. Experimental Design: Treatments and Categories▶ There are 4 experimental treatments of information: complete

and clear info (CC), complete and unclear info (CU),incomplete and clear info (IC), and incomplete and unclear
info (IU).
▶ In each treatment of information, we have 4 environments (2× 2 factorial structure).
high
low

Boston12x2 subjects
12x2 subjects

DA
12x2 subjects
12x2 subjects▶ 384 (=12 subjects x 2 sessions x 4 treatments x 4

environments) participants (4-4 case).
▶ 180 (=18 subjects x 2 mechanisms x 5 combinations)participants (6-6 case, only CK2019’s treatment)▶ Each session, which lasted approximately 100 minutes. Everysubject had not ever participated in school choice experiments.
12 / 48

go back to Introduction: Subjects’ Groups

Baskeck-Mantovani (2018, GEB) first argued the cognitive
ability in a school choice (not college admissions) experiment,using the Raven SPM test,▶ but their criterion for the classification into 2 groups is not

fixed; diﬀerence from our subject control.
▶ our experimental site: Exp. Econ. Lab. at Kansai University

has taken the score of a cognitive ability test from almost all
subjects since April in 2016; no eﬀect on random recruiting

13 / 48

go back to Introduction: Bos vs. DA
Featherstone-Niederle (2016, GEB): 4 schools with capacity of
1 and 5 students, Incomplete and Clear info, uniform distribution
over preferences.
▶ Bos ≻ DA in eﬃciency.
▶

Schools has the same preference for two types of students, Topor Average. (Clear info.)▶ Truth-telling is a unique weakly undominated strategy underBos. Subjects could learn that truth-telling is better.
▶ CK2019

4 schools - 4 students; Bos ∼ DA in eﬃciency. District schools
are the 2nd preferred ones.
▶ 6 schools - 6 students; Bos ≻ DA in eﬃciency. District schools
are the 2nd-5th preferred ones. Truth-telling is not necessarily
a dominant strategy.
▶ Thus, the reasoning behind Bos ≻ DA should be diﬀerent fromthat derived by Featherstone-Niederle. (See the following
slides) ⇒ subjects’ cognitive ability▶

14 / 48

3. Part 1

The case of 4 schools and 4 stundents
We could replicate CK2019’s result (Bos ∼ DA) by mixing theobserved behaviors of 2H2L (2 high score holders and 2 low scoreholders).

15 / 48

3. Part 1: Stable Matching and Nash Eqm▶ There exists a unique stable matching

(

1

µ =

1 2 3
A B C

4
D

)
,

which is the Nash equilibrium outcome under Bos and DA(district schools as the top choices under Bos and truth-tellingunder DA).
▶ Under DA, there is the other Nash equilibrium outcome,which is not stable but eﬃcient,()
1 2 3 42µ =,A D C B
which is obtained by P = (P1 , P2 , P3 , P4 ), where
P1 = (A, ∗, ∗, ∗), P2 = (D, B, ∗, ∗), P3 = (C , ∗, ∗, ∗), andP4 = (B, D, ∗, ∗).
⇒ Intuitively, CK2019 would obtain Bos ≺ DA, but actually they
had Bos ∼ DA. subjects’ cognitive ability or unclear information?
16 / 48

▶ Questions from the subjects were answered privately.
▶ The experiment was programmed in z-Tree (Fischbacher,2007).
▶ Each session has 20 rounds in total. Priority orders arechanged in every 5 rounds. There is no practice round.
▶ 14 review questions; the first four are the computation of thematching in Bos or DA (Q1). The answers to those questions
are shown in the monitor.
▶ Subjects are randomly matched every round in such a waythat there are 4 types of students in each group.
▶ Payment: 1000 JPY for show-up +# of correct answers inreview questions x 20 JPY + 1 point earned in each round x 5
JPY. Subjects were paid privately in cash.
▶ average payment = about 2365 JPY.

17 / 48

3. Review Questions

clear, comphighlowunclear, comphighlowclear, incomp
highlowunclear, incomp
high
low

Bos
(Q1)

DA
(Q1)

Bos
(Q2-Q11)DA(Q2-Q11)

22
21

13
14

0.91670.75420.87080.7792

23
22

17
14

0.95500.89500.92500.8750

21
23

12
13

0.93750.73750.92920.7750

22
22

14
8

0.87920.90420.82080.820818 / 48

3. Part 1: Eﬃciency
normalized eﬃciency rate =max group rank (14) − actual group rank
max group rank (14) − min group rank (6)

Bos
mean
comp, unclear(CK2019)highlow
p-value (BM)0.75310.71670.0016(std. dev.)(0.0573)(0.1231)DA
mean

0.78750.75730.0013(std. dev.)

p-value(BM)(0.1106)(0.0693)<0.0001<0.0001▶ Observation A: For both groups of cognitive ability, theeﬃciency rates under the DA were higher than those underthe Bos in CK2019’ treatment (Complete and Uncertain info).
Subjects’ cognitive ability matters.
19 / 48

3. Part 1: Allocation

A matching is observed frequently under Bos()
1 2 3 4µ3 =,A C B D
where the sum of ranks is 8 and the eﬃciency rate is 0.750.
The eﬃciency rate of
(
4

µ =

1 2 3
D B C

4
A

)

is 0.625. It is 0.750 for µ1 and 1.000 for µ2 , respectively.

20 / 48

The sample size = 120 for each category; observed frequencies of
equilibria were few.
Complete, Unclear (CK2019)(b) High, DA

89

17
4
ABCD
DBCA

4

0 10 20 30 40 50 60 70 80 90 100 110

0 10 20 30 40 50 60 70 80 90 100 110(a) High, Bos
89

21
2

ACBD
ADCB

ABCD
DBCA

num dist = 12 in ABCD

num TT = 38 in ABCD

ACBD
ADCB

num P in ADCB = 10(d) Low, DA

70

17
9

4
ACBD
ADCB

num dist = 5 in ABCD

0 10 20 30 40 50 60 70 80 90 100 110

0 10 20 30 40 50 60 70 80 90 100 110(c) Low, Bos

ABCD
DBCA

1

98

3
ABCD
DBCA

num TT = 18 in ABCD

8

6
ACBD
ADCB

num P in ADCB = 3
21 / 48

The sample size = 120 for each category; observed frequencies of
equilibria were few.
Complete, Clear(b) High, DA

87

17
7
ABCD
DBCA

4

0 10 20 30 40 50 60 70 80 90 100 110

0 10 20 30 40 50 60 70 80 90 100 110(a) High, Bos
106

8

ACBD
ADCB

num dist = 8 in ABCD

num TT = 31 in ABCD

91

15
3
ABCD
DBCA

ACBD
ADCB

num P in ADCB = 0(d) Low, DA

1
ACBD
ADCB

num dist = 16 in ABCD

0 10 20 30 40 50 60 70 80 90 100 110

0 10 20 30 40 50 60 70 80 90 100 110(c) Low, Bos

3

2

ABCD
DBCA

99

15
0
ABCD
DBCA

num TT = 6 in ABCD

0
ACBD
ADCB

num P in ADCB = 6
22 / 48

3. Part 1: Sensitivity to Information Provision

Table: P-values for the Shapiro-Wilk test for normality of samples

asin(sqrt(BosHdis))asin(sqrt(BosLdis))asin(sqrt(DAHtru))asin(sqrt(DALtru))

CC
0.00330.42290.1096
0.2968

Shapiro-Wilk
CU
IC
0.1972 0.2230
0.9143 0.2619
0.1583 0.1753
0.1693 0.3035

LeveneIU
0.44040.00130.65960.04790.66250.81320.60450.9532▶ In the Shapiro-Wilk tests, the null hypothesis is that the

samples are normally distributed.
▶ In the Levene tests, the null hypothesis is that the variances ofsamples are equal among four treatments.

23 / 48

Table: Results of a two-way ANOVA.

factor U
factor I
U·I

BosHdisp-value
0.12170.13980.7944

BosLdisp-value
0.8401<0.0001<0.0001

DAHtrup-value
0.05010.01210.0039DALtrup-value
0.0029<0.0001<0.0001

Observation B:▶ District schools as the top choice made by high score holders

under Bos was not sensitive to information provision.
▶ low score holders under DA is most sensitive to information

provision.

24 / 48

3. Part 2

The case of 6 schools and 6 students▶ As for the allocative eﬃciency and subjects’ behavior, we

could replicate CK2019’s result (Bos ≻ DA, distribution ofTT) in 2H4L (2 high score holders and 4 low score holders).
▶ Bos ∼ DA in 6H, 4H2L, and 6L.

25 / 48

Table: Students’ preferences.

student \ school
1
2
3
4
5
6

A
9
16
9
16
16
16

B
16
11
16
7
13
13

C
11
5
7
9
11
11

D
13
13
11
13
7
5

E
7
9
5
5
9
7

F
5
7
13
11
5
9

District schools are 2nd to 5th preferred by students in thisenvironment.

26 / 48

3. Raven score in the case of 6 schools and 6 students
Standard Deviation
2.345
1.028
1.813

Min
4
12
4

Max
16
16
11

10

Frequency
20

30

40

Mean score11.17813.000
9.356

0

All
High
Low

Obs.
180
90
90

1

2

3

4

5

6

7

8
9 10
Raven score

low score group

11

12

13

14

15

16

high score group27 / 48

3. Results 2. 6 schools and 6 students
1
Average Efficiency.1 .2 .3 .4 .5 .6 .7 .8 .9
00

Average Efficiency.1 .2 .3 .4 .5 .6 .7 .8 .9

1Complete, Unclear (CK2019’s treatment)(a) normalized eﬃciency, 4H2L(b) normalized eﬃciency, 3H3L

1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930
PeriodBOS

p = 0.4403 (BM)1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930
PeriodDA

BOS

p = 0.0092 (KS)p = 0.0063 (BM)p = 0.0011 (KS)The proportion of Truth Telling.1 .2 .3 .4 .5 .6 .7 .8 .91(d) truth-telling, 3H3L

00

the proportion of Truth Telling.1 .2 .3 .4 .5 .6 .7 .8 .91(c) truth-tellling, 4H2LDA

1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930
PeriodBOS

p < 0.0001 (BM)1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930
PeriodDA

p < 0.0001 (KS)BOS

p < 0.0001 (BM)DA

p < 0.0001 (KS)28 / 48

1
Average Efficiency.1 .2 .3 .4 .5 .6 .7 .8 .9
00

Average Efficiency.1 .2 .3 .4 .5 .6 .7 .8 .9

1Complete, Unclear (CK2019’s treatment)(a) normalized eﬃciency, 2H4L(b) normalized eﬃciency, 6H

1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930
PeriodBOS

p < 0.0001 (BM)1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930
periodDAp = 0.0033 (KS)BOSDA ∼ Bos

Average Efficiency.1 .2 .3 .4 .5 .6 .7 .8 .91(d) normalized eﬃciency, 6L00

Proportion of Truth Telling.1 .2 .3 .4 .5 .6 .7 .8 .91(c) truth-tellling, 2H4L

DA

relatively low in period 11-20 under Bos

1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930
periodBOS

p < 0.0001 (BM)1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930
periodDA

p < 0.0001 (KS)BOSDA ∼ BosDA

relatively low in period 11-20 under Bos

29 / 48

1
Average Efficiency.1 .2 .3 .4 .5 .6 .7 .8 .9
00

Average Efficiency.1 .2 .3 .4 .5 .6 .7 .8 .9

1Complete, Unclear (CK2019’s treatment)(a) normalized eﬃciency, 6H(b) normalized eﬃciency, 6L

1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930
periodBOS

p = 0.3373 (BM)1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930
periodDA

BOS

p = 0.0149 (KS)p = 0.5735 (BM)proportion of Truth Telling.1 .2 .3 .4 .5 .6 .7 .8 .91(d) truth-telling, 6L

00

proportion of Truth Telling.1 .2 .3 .4 .5 .6 .7 .8 .91(c) truth-tellling, 6HDA

p = 0.1163 (KS)1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930
PeriodBOS

p < 0.0001 (BM)1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930
PeriodDA

p < 0.0001 (KS)BOS

p < 0.0001 (BM)DA

p < 0.0001 (KS)30 / 48

▶ We counted the # of observed frequency of TT with > 0.5(> 0.2) under DA (Bos) in every 5 rounds.
▶

There were no significant diﬀerences in the distribution of
those observations between CK2019 and 2H4L. (diﬀerences
observed in 3H3L)▶ it was more plausible that we could replicate the same result

as the one shown in CK2019 in the case of 2H4L.
Table: P-values for the KS test: (TT under DA and Bos)

round
1−5
6 − 1011 − 1516 − 20
21 − 25
26 − 30
p-value (KS)DA
CK2019
4
4
1
1
2
1
−−−

> 0.5
2H4L
3
3
0
5
3
1
0.8928Bos
CK2019
0
0
0
1
3
0
−−−

> 0.2
2H4L
1
0
0
1
0
0
0.44131
31 / 48

4. Final Remarks

Subjects’ cognitive ability scores and informationaﬀect allocative eﬃciency and subjects’ behavior.
▶ Subjects’ cognitive ability and information on school’s priority

orders over students matter in comparison of allocativeeﬃciency between Bos and DA.
▶ As for sensitivity of behavior, their cognitive ability andinformation on their preferences over schools are important.

32 / 48

4. Final Remarks: More on Subjects’ Behavior under DA▶ In the case of 4 students and 4 schools, many subjects chosetruth-telling strategy or district schools as the top choice; e.g.,
89.6 % of observed choices made by high score holders in the
CU treatment (CK2019)▶ skipping down strategy (Kawagoe et al. 2018, GEB):
stating preferences in order of schools that value them more
highly▶

In this experiment with district schools, skipping down strategy= choosing district schools as the top choice.

▶ 3-school and 5-student, Complete and Clear info under plain

DA (not DA with majority quota or with minority reserve):
average 62.5-69.44% of preference statements wereskipping-down strategy.

33 / 48

relative frequencies of skipping-down strategy observed in
CK2019’s treatment (Complete and Unclear info.)
Complete, Unclear (CK2019)

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

(b) 6-schools, DA

0

0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

(a) 4-schools, DA

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20
PeriodHighhigh = 0.1729, low = 0.2125

Low

p = 0.1199 (BM)1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930
PeriodHighhigh = 0.0926, low = 0.1574

Low

p = 0.0012 (BM)34 / 48

relative frequencies of skipping-down strategy observed in 4 schools
and 4 students.1 .2 .3 .4 .5 .6 .7 .8 .91(b) skipping-down, Complete, Unclear (CK2021)

00Skipping Down strategy.1 .2 .3 .4 .5 .6 .7 .8 .91(a) skipping-down, Complete, Clear

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20
PeriodHighhigh = 0.2458, low = 0.2896

1

2

3

4

5

6

7

8

9

Low

p = 0.1258 (BM)high = 0.1729, low = 0.2125

Low

p = 0.1199 (BM)(d) skipping-down, Incomplete, Unclear
Skipping Down strategy.1 .2 .3 .4 .5 .6 .7 .8 .900Skipping Down strategy.1 .2 .3 .4 .5 .6 .7 .8 .91(c) skipping-down, Incomplete, Clear1

10 11 12 13 14 15 16 17 18 19 20
Period

High

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20
PeriodHighhigh = 0.1688, low = 0.2250

Low

p = 0.0282 (BM)

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20
PeriodHighhigh = 0.2458, low = 0.4250

Low

p < 0.0001 (BM)35 / 48

In the case of 4 schools and 4 students▶ It is intuitively understandable that the relative frequencies ofdropping-down strategy were higher for both high and lowscore holders in treatment with incomplete and unclear
information than in the other treatment.
▶ The apparent diminishing tendency was observed in every

treatment of information, regardless of groups of subjects incognitive ability scores. ... learning?

36 / 48

4. Final Remarks: More on Subjects’ Behavior under Bos▶ sincere player and sophisticated player in Pathak andSönmez (2008, AER): players who represent their true
preferences and those who play the Nash equilibrium strategy▶

In the case of 4 schools and 4 students, choices made bysophisticated or sincere players are, e.g., 95.8 % in the highscore group in the CU treatment (CK2019)

Sorry! We are still struggling to find all the Nash equilibria in theenvironment of 6 schools and 6 students under Bos.
... They are too many...

37 / 48

4. Final Remarks: Software
Search with “Naoki Watanabe, Keio”. At the bottom of mywebsite there are some softwares:
Excel for Two-Sided Matching ver.3.2
http://labs.kbs.keio.ac.jp/naoki50lab/sample3_v2.xlsm

The User’s Manual
http://labs.kbs.keio.ac.jp/naoki50lab/manualEn2Sh.pdf

A Brief Note on a Personnel Assignment in a Japanese Electrical
Company · · · The full version is sold as a case material of KeioBusiness School; Case 91-18-3199
http://labs.kbs.keio.ac.jp/naoki50lab/personnel2EnSh2.pdf

Softwares for TTC and VCG are also there.

38 / 48

Appendix A. Experimental Results: An Overview
1
Average Efficiency.1 .2 .3 .4 .5 .6 .7 .8 .9
00

Average Eficiency.1 .2 .3 .4 .5 .6 .7 .8 .9

1Complete, Unclear (CK2019’s treatment)(a) normalized eﬃciency, Bos(b) normalized eﬃciency, DA

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20
periodHIghp = 0.0016 (BM)

1

2

3

4

5

6

7

8

9

Low

p = 0.0135 (KS)Low

p = <0.0001 (KS)p = 0.0013 (BM)(d) truth-telling, DA
Fraction.1 .2 .3 .4 .5 .6 .7 .8 .900Fraction.1 .2 .3 .4 .5 .6 .7 .8 .91(c) district school, Bos1

10 11 12 13 14 15 16 17 18 19 20
period

High

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20
periodHigh

p = <0.0001 (BM)

1

2

3

4

5

6

7

8

9

Low

p = 0.0047 (KS)10 11 12 13 14 15 16 17 18 19 20
periodHighp = 0.2720 (BM)Low

p = 0.7937 (KS)39 / 48

Appendix B. Truth-telling under DA (CK2019)Bos(uncond)DA(uncond)Bos(Q1 correct)

DA(Q1 correct)

a = 460
b = 440
153/a = 0.33
166/b = 0.38
0.1641c = 340d = 280254/c = 0.75
208/d = 0.74
0.9264a = 440b = 440155/a = 0.35
107/b = 0.24
0.0005c = 280
d = 160
190/c = 0.68
86/d = 0.54
0.0041

comp, unclear(Chen-Kesten2016)high (H)low (L)
p-value (Fisher)s = 480s = 480

156/s = 0.33
166/s = 0.35
0.5384347/s = 0.72
319/s = 0.67
0.0293incomp, unclears = 480s = 480high (H)low (L)
p-value (Fisher)

178/s = 0.37
136/s = 0.28
0.0048295/s = 0.62
211/s = 0.44
<0.0001

H0 : no diﬀerence in relative frequencies observed between H and L.
P-values for two-sided (one-sided) Fisher exact test (DA (uncond)).
40 / 48

Appendix B. District School as Top Choice under Bos(CK2019)Bos(uncond)DA(uncond)Bos(Q1 correct)

DA(Q1 correct)

a = 460
b = 440
288/a = 0.63
255/b = 0.58
0.1728c = 340d = 280
56/c = 0.17
54/d = 0.19
0.3984a = 440b = 440
269/a = 0.61
315/b = 0.72
0.0013c = 280
d = 160
55/c = 0.20
63/d = 0.39
<0.0001

comp, unclear(Chen-Kesten2016)highlow
p-value (Fisher)s = 480s = 480304/s = 0.63
249/s = 0.52
0.000483/s = 0.17
102/s = 0.21
0.0703incomp, unclears = 480s = 480highlowp-value (Fisher)

286/s = 0.60326/s = 0.68
0.0088118/s = 0.25
204/s = 0.43
<0.0001

H0 : no diﬀerence in relative frequencies observed between H and L.
P-values for two-sided (one-sided) Fisher exact test (DA (uncond)).
41 / 48

Appendix B. Wrap-up▶ We can remove the significant diﬀerences in subjects’ behavior

between two groups of cognitive ability observed in CK2019’s
treatment.
▶ Confine our analysis to the data taken from the subjects withcorrect answer to Q1.

42 / 48

Appendix C: DA algorithm
1. Each student has his or her preference over schools to which
he or she is to be assigned, and each school has its priority
orders for students. The preferences and priority orders are
represented by ranking without ties.
2. Quota: the maximal number of students that can be acceptedby each school, i.e., capacity.
An assignment of students to schools is called a matching anddenoted by µ, such as()
a b c d e fµ=,
X Z Z X Y Z
where, e.g., students a and d are assigned to school X .

43 / 48

▶ A matching is deemed unstable if it sends a student to aschool when there is another school that is preferred by thestudent, and either has room for him or her or could makeroom by rejecting someone else it prefers less.
▶ For each two-sided matching problem, in general, there aresome stable matchings.
▶ Stability implies (Pareto) eﬃciency for both sides.

44 / 48

Example
Table: Students’ preferences over schools

a:
b:
c:

X
Z
Z

Y
Y
Y

Z
X
X

d:
e:
f:

Y
Y
Z

Z
Z
X

X
X
Y

Table: Schools’ priority orders for students

(2)
(1)
(3)

X:
Y:
Z:

b
a
f

c
d
e

a
c
c

d
e
d

f
f
b

e
b
a

45 / 48

A stable matching
∗

µ =

(

a
X

b
X

c
Z

d
Y

e
Z

f
Z

)

is computed by the student-proposing DA algorithm as follows.
Table: Computation process of µ∗ by the student-proposing DA algorithm

(2)
(1)
(3)

X
Y
Z

round 1ad, eb, c, fround 2adb , c, f eround 3ad, bc, f , eround 4a, bdc, f , e46 / 48

The student-proposing DA algorithm guarantees the following
properties.
▶ Strategy-proofness for students: For each student,

whatever other students report their preferences over schools,
it is never assigned to the more preferable school for him or
her by misrepresenting their true preference.
The next property is, unfortunately, not guaranteed at the stablematching computed by the student-proposing DA algorithm.
▶ (Pareto) eﬃciency among students: In order to reassign astudent to a higher ranked school, there is at least one
student who must be transferred to a lower ranked school forhim or her.

47 / 48

▶ Deferred acceptance (replaceability of applications) induces

students to tell their true preferences and guarantees
eﬃciency for both sides but sometimes fail in attaining
eﬃciency among students.
▶ In fact, CK2019 reported that there was no significant

diﬀerence in eﬃciency between Bos and DA in the case of
4 schools and 4 students and that Bos is better in terms ofeﬃciency (measured by an numerical index) than DA in thecase of 6 schools and 6 students.

48 / 48