Discussion Paper No. 621

SIGNAL QUALITIES, ORDER OF DECISIONS,AND INFORMATIONAL CASCADES:
EXPERIMEMTAL EVIDENCES

Shunichiro Sasaki

October 2004

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

Signal Qualities, Order of Decisions,and Informational Cascades:
Experimental Evidences1
Shunichiro Sasaki 2
Institute of Social and Economic Research
Osaka University

1 The author acknowledges …nancial supports from Keio University and the 21st

century COE program of Osaka University.
2 Address: Institute of Social and Economic Research, Osaka University, 61 Mihogaoka, Ibaraki, Osaka 567-0047, Japan, Telephone: +81-6-6877-5111 (ex.
9179), Facsimile: +81-6-6878-2766, Email: ssasaki@iser.osaka-u.ac.jp

Abstract
This paper reports the results of informational cascades experiments where
two di¤erent decision-making systems, anti-seniority and seniority are investigated. By implementing heterogeneous signal qualities associated with
the …xed order of decisions I compare the property of each system and examine heuristics human subjects use. Major …ndings are the following: (1)complete cascades occur more frequently in seniority than in anti-seniority,(2) seniority is more e¢cient than anti-seniority, but it increases the risk forcreating negative cascades, (3) for both systems, rational complete cascades
occur less frequently than those which the Bayesian theory predicts, (4) subjects in seniority put equal weight on private signals and on predecessors’
predictions whereas those who in anti-seniority put more weight on private
signals than on predecessors’ predictions, (5) by analyzing deviations from
Bayesian posteriors, both overcon…dence and undercon…dence are identi…ed,and (6) the anchoring e¤ect is veri…ed in deviations by overcon…dence, but
is not veri…ed in deviations by undercon…dence.
Keywords: Informational cascades, experiment, signal quality, overcon…dence, undercon…dence
JEL classi…cation: C91, C92, D81.

1

Introduction

Suppose a situation where individuals have informative but noisy private signals for an underlying state and they decide in sequence appropriate actions
for that state. If individuals’ actions are publicly observable, successors caninfer their predecessors’ private signals. These publicly inferred signals areaggregated as the decision-making process goes on. When an individual’s
private signal does not correspond to the aggregated publicly inferred signals
the rational individual may ignore it and follow the actions the majority ofpredecessors have chosen. Once an identical pattern of actions is established,private signals are no longer informative, and informational cascades occur.
Bikhchandani, S., D. Hirshleifer, and I. Welch (1992) (hereafter BHW) andBannerjee, A. V. (1992) formulated such a situation and showed that informational cascades are caused by rational Bayesian update.
So far, many experiments have been conducted to test BHW’s theoryof informational cascades and its empirical applicability. In the pioneeringexperiment by Anderson, L. R. and C. A. Holt (1997), they directly tested
BHW’s model of informational cascades by implementing binary-signal andbinary-action framework. Over 56 periods where informational cascades werepossible in the symmetrically identical signal quality treatment, positive cascades occurred in 28 periods and negative cascades occurred in 13 periods1
while any cascades did not develop in 15 periods. Nöth, M. and M. Weber (1999) extended Anderson and Holt ’s experiment by introducing twodi¤erent qualities of signals and found that although informational cascades
occurred the frequency of occurrences is lower than the Bayesian theory suggests because human subjects employed a heuristic, which puts too muchweight on their private signals. This tendency of overcon…dence on private
signals is observed in hypothetical experiment (Huck, S. and J. Oechssler(2000)) and is invariant even when acquisition of signals is costly (Kraemer,C., M. Nöth, and M. Weber (2000)). Kraemer, T. and M. Nöth (2000) tried
to identify the "anchoring and adjustment" e¤ect in the Bayesian updating process. By having subjects submit their probability judgements before
making predictions, they concluded that subjects’ decisions are anchored bytheir private signals and adjusted by predecessors’ predictions. Oberhammer,
C. and A. Stiehler (2001) introduced binary-signal and continuous-action
model while Çelen, B. and S. Kariv (2004) implemented coutinuous-signal
and binary-action framework in order to …nd out subjects’ heuristics.
What we have known from these experiments is that informational cas1 In the terminology used in this paper, this includes both complete and partial cascades.

For detail, see section 4.1.

1

cades certainly occur as BHW suggests, but actual occurrences are consistently fewer than the Bayesian theory predicts. This can be attributed toheuristics human subjects use in the process of probabilistic inference. Especially, tendency to putting more weight on private signals than on predecessors’ predictions is often observed in many experiments.
In this paper I tested the occurrence of informational cascades by introducing heterogenous qualities of private signals associated with the …xed
orders in making predictions. In a series of experiments I examined two systems, anti-seniority and seniority and compared each e¢ciency, potentiality
for cascades occurrence, and ability of extracting private signals. Major …ndings are the following. Complete cascades occur more frequently in seniority
than in anti-seniority. Seniority is more e¢cient than anti-seniority, but it increases the risk for creating negative cascades. For both treatments, rational
complete cascades occur less frequently than those which the Bayesian theory
predicts. Private signals can be extracted more e¤ectively in anti-seniority
than in seniority. In the process of decision-making, subjects in seniority
put equal weight on private signals and on predecessors’ predictions whereas
subjects in anti-seniority put more weight on private signals than on predecessors’ predictions. For heuristics human subjects use, both overcon…dence
and undercon…dence are identi…ed. In addition, the anchoring e¤ect of private signals is veri…ed in deviations by overcon…dence, but is not veri…ed indeviations by undercon…dence.
The remaining paper is organized as follows. Section 2 introduces analytical framework and presents theoretical predictions, section 3 describes
experimental procedure, section 4 reports the result and discusses its implications, and section 5 concludes.

2

Analytical Framework

There are two states of the world ! 2 fA; Bg: Each state realizes equally
likely, that is Pr(A) = p = 1=2 and Pr(B) = 1 ¡ p = 1=2: Each individual
does not know which state would have realized. However, he or she receives
an informative but noisy private signal and can infer the realized state byobserving it. Let ¾ ! be the private signal which indicates that realized stateis !: Then, the assumption of noisy information leads to 1=2 < Pr(Aj¾A) < 1
and 1=2 < Pr(Bj¾ B ) < 1: Speci…cally I use heterogenous discrete signals with
six di¤erent qualities from .55 to .8 incremented by .05. That is, Pr(Aj¾ A) or
Pr(Bj¾B ) is drawn from the setf:55; :6; :65; :7; :75; :8g without replacement.
Six subjects participate in each experimental session and each of them makes
a prediction which state would have realized one by one in sequence. The
2

quality of signal is exogenously determined by the position where a subject
is assigned to. Let i 2 f1; 2; 3; 4; 5; 6g be the position, ¾i! be the signal thesubject in position i receives, and ¼i! be the prediction he or she makes.
In addition to private signals, each subject can observe all predictions
made by his or her predecessors. For example, the subject in position 2
can observe the prediction made by the subject in position 1, the subject
in position 6 (last position) can observe the …ve predictions made by the
subjects in positions from 1 to 5, and so on. Thus, the subjects assigned tolater than position 2 make predictions by referring to both private signals andtheir predecessors’ predictions whereas the subject in position 1 can observe
only his or her private signal.
In experimental sessions, two treatments anti-seniority 2 and seniority are
conducted. The di¤erence of each treatment is in the combinations of signal
qualities and orders of making predictions, which are summarized in table 1.
² Anti-seniority
In anti-seniority treatment, six subjects make predictions in ascendingorder of the signal qualities. That is, the subject who has the least precise signal makes the prediction in position 1 (Pr(Aj¾ 1A) = Pr(Bj¾1B ) =.55),
the subject who has the second least precise signal makes the predictionin position 2 (Pr(Aj¾ 2A) = Pr(Bj¾ 2B ) =.6), and in this manner, the subject who has the most precise signal makes the prediction in position 6(Pr(Aj¾ 6A) = Pr(Bj¾6B ) =.8).
² Seniority
In seniority treatment, six subjects make predictions in descending order of the signal qualities. That is, the subject who has the most precise signal makes the prediction in position 1 (Pr(Aj¾1A ) = Pr(Bj¾ 1B ) =.8),
the subjects who has the second most precise signal makes the predictionin position 2 (Pr(Aj¾2A) = Pr(Bj¾2B ) =.75), and in this manner, the subject who has the least precise signal makes the prediction in position 6(Pr(Aj¾ 6A) = Pr(Bj¾6B ) =.55).
In experimental sessions, the combinations of signal qualities and order ofdecisions are common knowledge among all subjects. Thus, rational subjects
can calculate posterior probability by using Bayes’ rule. If they do so, thesetwo di¤erent treatments would create di¤erent behavioral patterns in theaggregate as the following.
2

The term anti-seniority is used in Ottaviani, M. and P. N. Sørensen (2001).

3

In anti-seniority, imbalances between Bayesian posterior probability andprivate signal occur only at 44 out of 252 decision nodes3 . Complete informational cascades4 can occur if subjects in the …rst three consecutive positions
make the same predictions. For example, the posterior probability that state
A would have realized given that the subjects in the …rst three consecutive positions have predicted A and the subject in position 4 observes ¾ 4B is1
2 £:55£:6£:65£:3
Pr(Aj¼1A ; ¼2A; ¼ 3A; ¾:4
B ) = 1 £[(:55£:6£:65£:3)+(:45£:4£:35£:7)] = :593. Thus, in this2case the subject in position 4 should ignore his or her private signal andjoin the established pattern of predictions. Once he or she enters a cascade,subjects in later than position 4 should ignore their private signals of ¾B aswell.
In seniority, imbalances between Bayesian posterior probability and private signal occur as many as at 110 out of 252 decision nodes. More importantly, the subject in position 2 should make the same predictions as
the subject in position 1 even when the private signal of the subject in position 2 does not correspond to the prediction of the subject in position 1
1
2 £:8£:25
since Pr(Aj¼ 1A; ¾2B ) = Pr(Bj¼ 1B; ¾ 2A) = 1 £[(:8£:25)+(:2£:75)]= :571: This leads
2
to the result that subjects in a round should always make identical predictions whether they enter informational cascades or they truthfully reveal
their private signals.

3

Experimental Procedure

Subjects were recruited from undergraduate students of Keio University,
Shonan Fujisawa Campus. Six subjects participated in one session and they
played the game for 16 rounds. Each session took less than 100 minutes
including instruction and …nal questionnaire. Upon arriving the classroom,each subject drew a card for determining where to be seated. Each seat
was separated and partitioned so that anyone could see neither the othersubject’s private draw nor his or her record sheet. Before each session began, subjects were instructed about the entire structure of the experiment.
They were informed about their tasks, two sources of information they receive, combinations of signal quality and order of decisions, di¤erence of eachtreatment, payo¤ scheme, and other procedures concerning the experimentin order to make them common knowledge among all subjects5 . After instructions, subjects took a quiz about the basic structure of the experiment.
3 For the list of Bayesian posterior probability at each decision node, see Appendix C.
4 For the de…nition of complete informational cascades, see section 4.1.
5Translated version of instructions for subjects is attached in Appendix B.

4

If all the subjects submitted the correct answers, the experimental session
began from the …rst round. At the beginning of each round, the experimenter
announced which treatment to be conducted. Next, the experimenter drew
one of the two cards from the covered box. On a card a letter either ”A” or”B” was printed and the letter on the card drawn represented the state of
the world for that round. The experimenter con…rmed the letter so as not
to be seen by subjects and hid it from them until the round …nished. Then,each subject drew one of the six cards for determining the combinations of
the signal quality and the order of decision. Since each subject had beennoti…ed which treatment would be conducted at the beginning of the round,
there was no uncertainty in the combinations of signal quality and order ofdecisions for each round.
As in the procedure of Anderson and Holt’s experiment, private signals
were implemented by having subjects draw one of the twenty white and red
marbles from a covered box. Marbles were identical in size, weight, andthe feel of material. White marbles represented state A and red marbles
represented state B. Di¤erent signal qualities were created by varying the
proportion of white and red marbles in a box given the realized state, treatment, and the assigned signal quality for each subject. For example, if state A
had realized, a subject who was assigned to position 1 in anti-seniority treatment drew a marble from the box containing 11 white and 9 red marbles.
Combinations of the marbles in each position and state for each treatment
is summarized in tables 2 and 3. Note that results of the draw indicate thecorrect state more accurately than random guessing even for the least precise quality of signal because the proportion of the white and red marbles
correlates to the realized state.
A set of six boxes was used in one round and they are also identicalin size, shape, and material except codes marked on the bottom of eachboxes. They were stacked in a larger box separately by the states and were
kept in the other room. After con…rming the state for the round one of theexperimenters brought the appropriate stack from that room so that subjects
could not identify which stack was actually used.
By checking the codes on the bottom, the experimenter approached a subject one by one and presented one box which contained the exact proportionof white and red marbles for each subject. After con…rming the color of themarble drawn, each subject replaced it into the box and wrote down the state
indicated by the color in the space printed "Your Signal" on the subject’s
record sheet. Then, each subject was requested to make a prediction in thedetermined order. Subjects made a prediction by writing down one of thetwo states he or she thought which is more likely to be in the space printed"Your Prediction" on the subject’s record sheet. The experimenter checked5the prediction and wrote it down on the experimenter’s record sheet. Then,the experimenter approached the subject in the next position and showed
the sequence of predictions made by his or her predecessors 6 . In making
predictions, subjects in positions from 2 to 6 can refer to them as well as
their private signals whereas the subject in position 1 can observe only his or
her private signal. After all six subjects submitted predictions, the card theexperimenter had drawn at the beginning of the round was revealed, thensubjects and the experimenter wrote down the results of their predictions.
This process was repeated for 16 times with combinations of each treatment.
The actual order of treatment in each session is shown in table 47 .
A total of 66 subjects participated in the experiments. After all 16 rounds…nished subjects were paid payo¤s privately in cash depending on the performance in a session. For each correct prediction 200 Japanese yen (eq.
to $1.84) were paid. Average payment for AA (anti-seniority, anti-seniority)
treatments, AS (anti-seniority, seniority) treatments, and SA (seniority, antiseniority) treatment were 2176 yen, 2276 yen, and 2476 yen respectively 8 . In
addition, the minimum payment and the maximum payment were 800 yen
and 3200yen respectively.

4

Results

4.1

Aggregated Behavior

In order to compare the property of each treatment, I examine the results
based on the following four criteria.

6

In Anderson and Holt’s experiment, they announced predecessors’ predictions to all
subjects. Thus, subjects who had completed making predictions could observe predictions
made by other subjects in later positions. In my experiment, the experimenter reported
predecessors’ predictions only to the subject who was in making prediction. This is because
I intended not to have sub jects count predictions in later positions and use them as aheuristic in following rounds.
7
In session 1, only 12 rounds in anti-seniority treatment were conducted due to overtime.
In session 2, anti-seniority treatment was conducted for all 16 rounds. In sessions 3 and4, anti-seniority treatment was conducted for the …rst 12 rounds followed by seniority
treatment for the last 4 rounds. For sessions 4 to 11, each treatment is conducted by turns
for the …rst 12 rounds and the last 4 rounds.
8
In another day participants in session 1 were paid additional 544 yen which is theaverage payment of 4-round anti-seniority treatment as compensation for the early closeof the session.

6

Complete positive (negative) cascades
A complete positive (negative) cascade denotes a pattern of behaviors
such that there is at least one subject who ignores his or her private signal andall of the six subjects in a round make identical correct (incorrect) predictions.
Among complete positive (negative) cascades, a pattern of behaviors suchthat all predictions are consistent with Bayesian posterior probability is calleda rational complete positive (negative) cascade.
Partial cascades
A partial cascade denotes a pattern of behaviors such that there is atleast one subject who ignores his or her private signal but another subject
collapses the established identical predictions. As a result, predictions are
not identical in a round.
Full revelations
A full revelation denotes a pattern of behaviors such that all of the subjects make predictions consistent with their private signals. An aggregatedbehavior is counted as a full revelation whether they are consistent with
Bayesian posteriors or not, and whether all of the predictions are identicalor not.
Herds
A herd denotes identical predictions in a round. This may include complete cascades and full revelations9 . Smith, L. and P. N. Sørensen (2000) andÇelen, B. and S. Kariv (2004) emphasized that there is a signi…cant di¤erence between informational cascades and herds. Informational cascades occur when subjects ignore their private signals in making predictions, whereas
herds may occur when subjects happen to make identical predictions, notnecessarily ignoring their private signals.
I report the results as a list of observations.
Observation 1
Learning within each session was not observed.
If subjects learn a systematic way of making predictions by some reasons, patterns of predictions should be di¤erent between earlier rounds andlater rounds within a session. Tables 5 and 6 report the Mann-Whitney U
9Note that the other three criteria except herds do not overlap each other.

7test on equal proportion of predictions which is consistent with the Bayesianposterior probability between the …rst two rounds and the last two rounds
within each session for each treatment. In anti-seniority, the null hypothesesof no di¤erence cannot be rejected for all sessions. In seniority, almost all
predictions in the …rst and the last two rounds within each session are identically consistent with Bayesian posterior so that the Mann-Whitney U test
cannot be performed for those sessions. In sessions 6 and 11 where di¤erent
proportions of Bayesian consistent predictions are observed, they are not signi…cantly di¤erent between the …rst and last two rounds. Therefore, we cansay that systematic pattern of learning within each session is not observed,
so that we can pool the data for each treatment and compare the propertyof them. Results of the aggregated behavior in all of the 11 sessions for eachtreatment are summarized in table 7.
Observation 2
Each state realized equally likely within and between each treatment.
The realizations of state A are 55 out of 112 rounds in anti-seniority and31 out of 60 rounds in seniority (table 7, row 1). The two-tailed test of
population proportion cannot reject the null hypothesis that each state realized with equal frequency for each treatment (table 9). In addition, MannWhitney U test shows that the realizations of state A are not signi…cantlydi¤erent between two treatments (table 8 row1). Therefore, draws by theexperimenters are not biased within and between each treatment.
Observation 3
Subjects made more correct predictions in seniority than in anti-seniority.
Proportion of correct predictions in anti-seniority is 69.49% whereas thatin seniority is 81.94% (table 7, row 2). Mann-Whitney U test shows thisdi¤erence is statistically signi…cant (table 8 row 2). We can say that seniority
increases the subjects’ welfare in the sense that it leads to more correct
predictions.
Observation 4
Complete (positive and negative) cascades and rational complete (positive
and negative) cascades occurred more frequently in seniority than in antiseniority, whereas partial cascades occurred more frequently in anti-seniority
than in seniority.
The proportion of complete positive cascades in anti-seniority is 14.29%
whereas that in seniority is 60.00% (table 7, row 3), which is signi…cantly8di¤erent (table 8, row 3). The proportion of complete negative cascades
in anti-seniority is 1.79% whereas that in seniority is 10.00% (table 7, row4), which is also signi…cantly di¤erent (table 8 row 4). The proportion ofrational complete positive cascades in anti-seniority is 6.25% whereas thatin seniority is 60.00% (table 7, row 5), which is signi…cantly di¤erent (table
8, row 5). The proportion of rational complete negative cascades in antiseniority is .89% whereas that in seniority is 10.00% (table 7, row 6), whichis also signi…cantly di¤erent (table 8, row 6). In addition, the proportion ofpartial cascades in anti-seniority is 25.89% whereas that in seniority is 10.00%(table 7, row 7), which is signi…cantly di¤erent (table 8 row 7). Therefore
seniority enhances the occurrence of complete positive cascades but it also
increases the risk for creating complete negative cascades. Meanwhile, theestablished cascades are more likely to be collapsed in anti-seniority than inseniority.
Observation 5
For both treatments, rational complete (positive and negative) cascades
occurred less frequently than those which the Bayesian theory predicts.
For anti-seniority, rational complete cascades occur if the …rst three subjects make the same predictions as seen in section 2. By Bayes’ rule, if the…rst three consecutive subjects made same predictions, the fourth subject
cannot help making the same prediction as them even when his or her private signal does not correspond to the predecessors’ predictions. Since herds
by full revelations are not counted in rational complete positive cascades,
the probability that rational complete positive cascades occur is given by theprobability that the all six subjects draw the correct marbles subtracted fromthe probability that the …rst three subjects draw the correct marbles. Thatis, [Pr(Aj¾ 1A) £ Pr(Aj¾ 2A) £ Pr(Aj¾3A)] ¡ [Pr(Aj¾1A) £ Pr(Aj¾ 2A) £ Pr(Aj¾ 3A) £Pr(Aj¾ 4A) £ Pr(Aj¾5A) £ Pr(Aj¾ 6A)] = [Pr(Bj¾ 1B) £ Pr(Bj¾ 2B) £ Pr(Bj¾ 3B )] ¡[Pr(Bj¾ 1B ) £ Pr(Bj¾ 2B) £ Pr(Bj¾3B ) Pr(Bj¾4B ) £ Pr(Bj¾ 5B) £ Pr(Bj¾6B )] =(:55 £ :6 £ :65) ¡ (:55 £ :6 £ :65 £ :7 £ :75 £ :8) = :215 ¡ :090 = :125
On the other hand, rational complete negative cascades would occur if all of
the …rst three subjects happen to make incorrect predictions. The probability that rational complete negative cascades occur is given by the probability
that the all six subjects draw the incorrect marbles subtracted from theprobability that the …rst three subjects draw the incorrect marbles. That is,[Pr(Aj¾ 1B ) £ Pr(Aj¾ 2B) £ Pr(Aj¾ 3B)] ¡ [Pr(Aj¾1B ) £ Pr(Aj¾2B ) £ Pr(Aj¾ 3B ) £Pr(Aj¾ 4B) £ Pr(Aj¾ 5B) £ Pr(Aj¾6B )] = [Pr(Bj¾1A) £ Pr(Bj¾ 2A) £ Pr(Bj¾3A )] ¡[Pr(Bj¾ 1A)£ Pr(Bj¾2A)£ Pr(Bj¾3A ) Pr(Bj¾ 4A)£ Pr(Bj¾ 5A)£ Pr(Bj¾6A)] = (:45£:4£ :35) ¡ (:45£ :4£ :35£ :3 £ :25 £ :2) = :063 ¡ :001 = :062: For seniority, all9

the subjects later than position 2 should follow the …rst subject’s prediction,
which inevitably leads to a herd in each round. Among rational complete
cascades, whether they turn to be positive or negative depends exactly on
whether the …rst subject makes the correct prediction. As in the case ofanti-seniority, the probability that rational complete positive cascades occur
is given by the probability that the all six subjects draw the correct marbles
subtracted from the probability that the …rst subject draw the correct marble.
That is, Pr(Aj¾ 1A)¡[Pr(Aj¾1A)£ Pr(Aj¾ 2A)£ Pr(Aj¾ 3A)£ Pr(Aj¾4A)£ Pr(Aj¾5A )£Pr(Aj¾ 6A)] = Pr(Bj¾ 1B) ¡ [Pr(Bj¾1B ) £ Pr(Bj¾ 2B) £ Pr(Bj¾ 3B ) Pr(Bj¾ 4B) £Pr(Bj¾5B )£ Pr(Bj¾6B )] = (:8)¡ (:55£ :6£ :65£ :7£ :75£ :8) = :8¡:090 = :710:
On the other hand, the probability that rational complete negative cascades
occur is given by the probability that the all six subjects draw the incorrect
marbles subtracted from the probability that the …rst subject draw the incorrect marble. That is, Pr(Aj¾1B ) ¡ [Pr(Aj¾1B ) £ Pr(Aj¾2B ) £ Pr(Aj¾ 3B) £Pr(Aj¾ 4B) £ Pr(Aj¾5B ) £ Pr(Aj¾6B )] = Pr(Bj¾ 1A) ¡ [Pr(Bj¾1A) £ Pr(Bj¾2A ) £Pr(Bj¾3A ) Pr(Bj¾ 4A) £ Pr(Bj¾5A) £ Pr(Bj¾ 6A)] = (:2) ¡ (:45 £ :4 £ :35 £ :3 £:25 £ :2) = :2 ¡ :001 = :199:
However, if we look at the proportions of rational complete cascades actually observed in experiments, they are consistently lower than those which
the Bayesian theory predicts (table 7, rows 5 and 6). The one-tailed testof population proportion shows that, for both treatments, the observed proportion of rational complete (positive and negative) cascades is signi…cantly
lower than the theoretical predictions (tables 10 and 11). This …nding is consistent with previous experiments where fewer occurrences of informationalcascades than the theoretical predictions were observed such as in Anderson,
L. R. and C. A. Holt (1997), Çelen, B. and S. Kariv (2004), Huck, S. and J.
Oechssler (2000), Nöth, M. and M. Weber (2000), and Oberhammer, C. and
A. Stiehler (2001).
Observation 6
Full revelations occurred more frequently in anti-seniority than in seniority.
The proportion of full revelations in anti-seniority is 58.04% whereas thatin seniority is 20.00% (table 7, row 8), which is signi…cantly di¤erent (table
8, row 8). Thus, anti-seniority extracts private signals more e¤ectively thanseniority10 .
10Ottaviani, M. and P. N. Sørensen (2001) studied theoretical model of informationalcascades resulted from reputational concern in the framework of heterogeneous signal
qualities. They showed that anti-seniority is e¤ective in extracting private information insome circumstances, but is not always so.

10

Observation 7
Herds occurred more frequently in seniority than in anti-seniority.
Herds are caused by complete cascades or full revelations. The proportionof herds in anti-seniority is 28.57% whereas that in seniority is 83.33% (table
7 row 9), which is signi…cantly di¤erent (table 8, row 9).

4.2

Individual Behavior

We observed that aggregated behavior is di¤erent in several aspects between
two treatments. Now let us examine how each subject uses the two sourcesof information, private signal and predecessors’ predictions. In order to investigate how weight subjects place on the two sources of information, I use
Grether,M. (1989) and Hung, A. and C. Plott (2000)’s logit regression model.
1 ;:::;¼ i¡1 ;¾ i )!
!
!
Let Pr(Aj¼
be the odds of the individual who are in positioni
Pr(B j¼1! ;:::;¼i¡1! ;¾ ! )i in favor of state A over state B after observing his or her predecessors’
predictions ¼1! ; :::; ¼i¡1and his or her own private signal ¾i! . Then,
!
i
i
1
i¡1
i
Pr(Aj¼1! ; :::; ¼ i¡1Pr(A) Pr(¼1! ; :::; ¼ i¡1! ; ¾! )! ; ¾! jA)= Pr(¼! ; :::; ¼! ; ¾ ! )
=
i
i
1
i¡1
i
Pr(Bj¼ 1! ; :::; ¼i¡1Pr(B) Pr(¼1! ; :::; ¼ i¡1! ; ¾ !)
! ; ¾! jB)= Pr(¼! ; :::; ¼! ; ¾ ! )i
Pr(A) Pr(¼1! ; :::; ¼ i¡1! ; ¾! jA)=i
Pr(B) Pr(¼1! ; :::; ¼ i¡1! ; ¾! jB)
i
Pr(A) Pr(¼ 1! ; :::; ¼i¡1! jA) Pr(¾! jA)=:i
Pr(B) Pr(¼1! ; :::; ¼ i¡1! jB) Pr(¾ ! jB)

The last equation holds because of the assumption that ¼ 1! ; :::; ¼i¡1and!

¾ i! are independent. By taking logs,lni
Pr(Aj¼1! ; :::; ¼ i¡1Pr(A)Pr(¼ 1! ; :::; ¼i¡1Pr(¾ i! jA)! ; ¾! )
! jA)
=
ln
+
ln
+
ln
:
i
Pr(Bj¼ 1! ; :::; ¼i¡1Pr(B)Pr(¼1! ; :::; ¼ i¡1Pr(¾i! jB)! ; ¾ !)
! jB)
(1)
Pr(A)
Since ln Pr(B) = 0, (1) can be generalized to

Y mi = ¯ 0 + ¯ 1 ln

Pr(¼1! ; :::; ¼i¡1Pr(¾ i! jA)
! jA)
+
¯
ln
+ uim :2i jB)
Pr(¼ 1! ; :::; ¼i¡1
jB)
Pr(¾
!
!

(2)

where uim is a random disturbance.
Three possible combinations of coe¢cients are of interest as special casesof equation (2). If ¯ 0 = 0 and ¯ 1 = ¯2 , then subjects put equal weight onprivate signals and on predecessors’ predictions. If ¯ 0 = 0 and 0 < ¯2 < ¯ 1;
11then subjects put more weight on predecessors’ predictions than on private
signals. If ¯ 0 = 0 and 0 < ¯ 1 < ¯2 , then subjects put more weight onprivate signals than on predecessors’ predictions. I estimate the coe¢cientsof equation (2) by the logit regression for each treatment. The logit regression
is performed for subjects in positions from 2 to 6 since the subjects in position
1 do not observe predecessors’ predictions. The results of the logit regressions
are shown in tables 12 and 13.
Observation 8
There is no systematic bias in predicting a state.
In both treatments ¯ 0 = 0 cannot be rejected at the 95% signi…cant level.
Thus, predictions made by subjects are not biased toward one of the states.
Tables 14 and 15 show the chi-square tests of the null hypothesis thatthe estimated coe¢cients of private signals and predecessors’ predictions areequal. This hypothesis can be rejected for anti-seniority whereas cannot berejected for seniority. To summarize, we can conclude that
Observation 9
Subjects in seniority put equal weight on private signals and on predecessors’ predictions whereas subjects in anti-seniority put more weight on private
signals than on predecessors’ predictions.

4.3

Analysis of Deviation

We observed that rational complete cascades occur signi…cantly less frequently than those which the Bayesian theory predicts for both treatments.
Then, we should investigate why human subjects deviate from the predictionof the Bayesian theory. To do so, we formally de…ne three types of deviations, overcon…dence, undercon…dence, and irrational as often emphasizedin probabilistic inference problem (Edwards, W. (1968), Gri¢n, D. and A.
Tversky (1992), Tversky, A. and D. Kahneman (1971), Weinstein, N. (1980)).
Overcon…dence can be de…ned as a prediction which is not consistent with
Bayesian posterior but consistent with the subject’s private signal. Undercon…dence can be de…ned as a prediction which is consistent with neither
Bayesian posterior nor the subject’s private signal but consistent with a prediction which is made by at least one of his or her predecessors. Deviations
which is not consistent with Bayesian posterior and cannot be explained bythese two criteria are de…ned as irrational 11 .
11Therefore, an irrational prediction is consistent neither with Bayesian posterior, thesubject’s private signal, nor any predecessors’ predictions

12

The proportions of overcon…dence, undercon…dence, and irrational foreach treatment and for each position are summarized in tables 16 and 17
and depicted in …gures 1 and 2. In anti-seniority, overcon…dence cannot occur in positions 2 and 3 because obeying private signals is always consistent
with the Bayesian posteriors12 . In seniority, undercon…dence cannot occur
in position 2 because obeying the prediction of the subject in position 1 andignoring his or her private signal is always consistent with Bayesian posterior. Overcon…dence and undercon…dence certainly exist in anti-seniority
whereas only overcon…dence occurred frequently in position 2 and deviations
are rarely observed in the other positions in seniority13 .
Given the existence of overcon…dence and undercon…dence, one might
think that subjects’ predictions are "anchored" by their signal qualities14 .
That is, subjects with more precise signals tend to be overcon…dent andsubjects with less precise signals tend to be undercon…dent. To examine this,we need to investigate the relationship between qualities of private signals
and deviating behaviors for each Bayesian posterior probability. Tables 18-21
show that the percentage of deviations, for each position, categorized by …velevels of Bayesian posteriors that the state A would have realized. Note thatdeviations should occur most frequently in the level of Bayesian posterior
40-60% because subjects would not be sure which state has realized around
posteriors 50%.
If there exists the anchoring e¤ect in anti-seniority, deviation by undercon…dence would occur more in earlier positions with more broad range ofposteriors and less in later positions with less broad range of posteriors because subjects with less precise signals are undercon…dent too much about
12

For the list of Bayesian posterior probability in each decision node, see Appendix C.

13 The existence of undercon…dence would create possibility of cascades whereas the ex-istence of overcon…dence would collapse potential cascades. If there is at least one subject
who makes an overcon…dent prediction, the potential cascades will inevitably collapse.
The fact that rational complete cascades are observed less frequently than Bayesian theory predicts can be attributed to the existence of overcon…dence. This observation thatexistence of overcon…dence reduce the cascades is consistent with past several experiments,
such as Kraemer, T. and M. Nöth (2000) and Nöth, M. and M. Weber (2000).
14
Kraemer, T. and M. Nöth (2000) reported that they identi…ed the anchoring e¤ect.
In their experiment they compared posteriors evaluated by subjects and actual posteriors.
By calculating their di¤erences, they found that they are minimized when the posteriors
are close to sub jects’ private signals and concluded that this fact supports anchoring
hypothesis. However, the problem is that they did not provide subjects with any incentive
to evaluate posteriors seriously as Oberhammer, C. and A. Stiehler (2001) pointed out.
In order to investigate existence of the anchoring e¤ect, one needs to look at deviating
behavior itself resulting from over- and undercon…dence and examine the relationshipbetween deviation and signal quality for each Bayesian posteriors. The following part ofthis section tries to do it.

13the realized state compared to the actual Bayesian posteriors. On the other
hand, deviations by overcon…dence would occur more in later positions withmore broad range of posteriors and less in earlier positions with less broad
range of posteriors because subjects with more precise signals are con…dent
too much about the realized state compared to the actual Bayesian posteriors.
Observed deviations by undercon…dence in anti-seniority (table 18) certainly decrease from 9.82% in position 3 to 1.78% in position 5. However,there are two increases from 4.47% in position 2 to 9.82% in position 3 andfrom 1.78% in position 5 to 2.68% in position 6. In addition, deviations
except in the range of 40-60% occurred not in the earlier positions but inthe later positions. These two observations are not consistent with the systematic pattern of deviations discussed above so that we conclude that theanchoring hypothesis has only a partial explanation for deviations by undercon…dence in anti-seniority. For observed deviations by overcon…dence inanti-seniority (table 19), they increase from 5.36% in position 4 to 12.50%
in position 6 and deviations except 40%-60% occurred not in the earlier positions but in the later positions. In position 5, some deviations occurredin the level of posteriors 20-40 % or 60-80 %. Furthermore, in position 6,
deviations occurred even when Bayesian posteriors suggest the realization of
one state rather than the other with high probability such as in the level ofposteriors 0-20 % or 80-100 %. In fact, there were subjects who predicted"A" in spite that their posteriors that the state A would have realized wereas low as 20% and those who predicted "B" in spite that their posteriors
were as high as 85%. By looking at deviations of positions 4-6, we con…rmthat subjects with high signal qualities are more likely to deviate from theBayesian posteriors. This systematic pattern of deviations, unlike the caseof undercon…dence, supports the hypothesis of the anchoring e¤ect.
If there exists the anchoring e¤ect in seniority, by the same reasoning as
in anti-seniority, deviations by undercon…dence would occur more in later
positions with more broad range of posteriors and less in earlier positions
with less broad range of posteriors. On the other hand, deviations by overcon…dence would occur more in earlier positions with more broad range ofposteriors and less in later positions with less broad range of posteriors.
However, we observed seldom deviations in seniority as the results oflogit regression suggest. Obvious deviations occurred only in position 2 forovercon…dence when Bayesian posteriors are either 42.8% or 57.2%. This fact
moderately supports the hypothesis of the anchoring e¤ect for overcon…dence,
however few observations of deviations for undercon…dence do not support
it for undercon…dence. To summarize, we have the following observation.

14

Observation 10
The anchoring e¤ect is identi…ed in deviations by overcon…dence but not
identi…ed in deviations by undercon…dence.
Why is not the anchoring e¤ect clearly identi…ed in deviations by undercon…dence in both treatments?
A possible reason for unidenti…ability of the anchoring e¤ect in seniority
can be attributed to the intrinsic structure of posteriors in seniority. Since
posteriors that state A would have realized are extremely high or low inlater positions such as 5 or 6 in seniority, subjects with less precise signals
can predict consistent with Bayesian posteriors relatively easily by following
the established pattern of predecessors’ predictions even when their private
signals do not correspond to them. In such cases, making undercon…dent
predictions are almost irrational.
A possible reason for the unclear identi…cation of the anchoring e¤ect
in anti-seniority would be related to "the strength and weight e¤ect of thesample". Gri¢n, D. and A. Tversky (1992) studied this e¤ect in determining subject’s attitudes toward overcon…dence and undercon…dence. Througha series of hypothetical coin-spinning experiments, they found that humansubjects are more likely to be overcon…dent when the sample is small and it
shows extreme result (i.e. weight of the sample is low and strength of thesample is high), and more likely to be undercon…dent when the sample islarge and it shows moderate result (i.e. weight of the sample is high andstrength of the sample is low)15 .
If the strength and weight e¤ect of the sample exists in our informationalcascades experiment, subjects in earlier positions would be overcon…dent
whereas subjects in later positions would be undercon…dent because subjects
in earlier positions have a smaller sample (i.e. fewer predecessors’ predictions)than subjects in later positions 16 . Then, undercon…dence in earlier positions
would be o¤set by this overcon…dence and result in the unclear identi…cationof the anchoring e¤ect in anti-seniority.
15

In probabilistic inference experiments, Edwards, W. (1968) observed that subjects
were more likely to be undercon…dent whereas Tversky, A. and D. Kahneman (1971)
veri…ed that they were more likely to be overcon…dent compared to the Bayesian posterior.
However, Gri¢n, D. and A. Tversky (1992) argued that these inconsistent observations
are comprehensible as the former used a large and moderately strong sample and the latter
used a small and fairly strong sample.
16 Overcon…dence which would be caused by the strength and weight e¤ect of the sample
is observed in informational cascades experiment by Çelen, B. and S. Kariv (2004). They
found that "subjects who are early decision-makers tend to rely more heavily on their own
information in the learning process, which then becomes available to late decision-makers,
who tend more to be Bayesian."

15

However, if the above explanation holds, we have to say that this experiment confounds the anchoring e¤ect and the strength and weight e¤ectof the sample. Although the anchoring e¤ect is identi…ed in deviations by
overcon…dence in anti-seniority this should have been also canceled out by
undercon…dence resulted from the strength and weight e¤ect of the sample
to some extent. In order to evaluate each e¤ect clearly, we need to distinguish each other and investigate how these two e¤ects interact in various
situations. Because this experiment cannot do that, we need some othersettings. One possible improvement would be in a design of an experiment
where signal quality is identical in one treatment but is di¤erent amongdi¤erent treatments. By comparing patterns of deviations resulted from different treatments, one would be able to identify the anchoring e¤ect and thestrength and weights e¤ect more clearly.

5

Conclusion

By implementing heterogenous signal qualities associated with the …xed orders of making predictions in the two di¤erent decision-making systems, wehave observed the di¤erent consequences of each system and identi…ed heuristics human subjects use.
For the investigations of anti-seniority and seniority, we have observed thefollowing. Complete cascades occur more frequently in seniority than in antiseniority. Seniority is more e¢cient than anti-seniority, but it increases therisk for creating negative cascades. For both treatments, rational complete
cascades occur less frequently than those which the Bayesian theory predicts.
Private signals can be extracted more e¤ectively in anti-seniority than inseniority. In the process of decision-making, subjects in seniority put equal
weight on private signals and on predecessors’ predictions whereas subjects
in anti-seniority put more weight on private signals than on predecessors’
predictions.
By analyzing the relationship between signal qualities and deviations from
Bayesian posteriors, we have identi…ed both overcon…dence and undercon…dence. In addition, the anchoring e¤ect is veri…ed in deviations by overcon…dence, but is not veri…ed in deviations by undercon…dence. The unidenti…cation problem of the anchoring e¤ect in deviations by undercon…dence
in seniority would be due to the intrinsic structure of posteriors where subjects in later positions consistently face extreme high or low posteriors. The
unclear identi…cation problem of the anchoring e¤ect in deviations by undercon…dence in anti-seniority can be explained by the strength and weighte¤ect of the sample by which subjects with the small sample would be over16

con…dent. In order to investigate the relationship between the anchoringe¤ect and the strength and weight e¤ect of the sample more accurately, we
have to di¤erentiate each other in some other experimental settings.

17

References[1] Anderson, L. R. and C. A. Holt, "Information Cascades in the Laboratory," American Economic Review, 1997, 87 (5), 847-62.
[2] Bannerjee, A. V., "A Simple Model of Herd Behavior," Quarterly Journal of Economics, 1992, 107 (3), 797-817.
[3] Çelen, B. and S. Kariv, "Distinguishing Informational Cascades from
Herd Behavior in the Laboratory," American Economic Review,
2004, 94 (3), 484-98.
[4] Edwards, W., "Conservatism in Human Information Processing," inB. Kleinmuntz (Ed.), Formal Representation of Human Judgement,
1968, 17-52, New York, Wiley.
[5] Grether, M., "Bayes Rule as a Descriptive Model," Quarterly Journalof Economics, 1980, 95 (3), 537-57.
[6] Gri¢n, D. and A. Tversky, "The Weighing of Evidence and the Determinants of Con…dence," Cognitive Psychology, 1992, 24 (3), 411-35.
[7] Huck, S. and J. Oechssler, "Informational Cascades in the Laboratory:
Do They Occur for the Right Reasons?, " Journal of Economic Psychology, 2000, 21 (6), 661-71.
[8] Hung, A A. and C. R. Plott, "Information Cascades: Replication and
an Extension to Majority Rule and Conformity Rewarding Institutions," American Economic Review, 2001, 91 (5), 1508-20.
[9] Kraemer, T. and M. Nöth, "Anchoring and Adjustment in Information
Cascades: Experimental Evidence," 2000, mimeo.
[10] Kraemer, C., M. Nöth, and M. Weber, "Information Aggregation with
Costly Information and Random Ordering: Experimental Evidence,"
Sonder Forschungs Bereich 504 Working Paper Series 00-35, Universität Mannheim, 2000.
[11] Nöth, M. and M. Weber, "Information Aggregation with Random Ordering: Cascades and Overcon…dence," Sonder Forschungs Bereich
504 Working Paper Series 00-34, Universität Mannheim, 2000.
[12] Oberhammer, C. and A. Stiehler, "Does Cascade Behavior in Information Cascades Re‡ect Bayesian Updating? - An Experimental Study-," Humboldt-University of Berlin, 2001.
[13] Ottaviani, M. and P. N. Sørensen, "Information Aggregation in debate:
Who Should Speak First?," Journal of Public Economics, 2001, 81(3), 393-421.
18

[14] Smith, L. and P. N. Sørensen, "Pathological Outcomes of Observational
Learning," Econometrica, 2000, 68 (2), 371-98.
[15] Tversky, A. and D. Kahneman, "The Belief in the Law of Small Numbers," Psychological Bulletin, 1971, 76, 105-110.
[16] Weinstein, N. D., "Unrealistic Optimism About Future Life Events,"
Journal of Personality and Social Psycology, 1980, 39, 806-20.

19

AppendicesA

Tables and …gures
Position
1
2
3
4
5
6
Treatment Anti-seniority .55 .6 .65 .7 .75 .8
Seniority.8 .75 .7 .65 .6 .55
Table 1: Combination of signal qualities and order of decisions
in each treatment

Anti-seniority
Position (signal quality)
Realized Marbles 1 (.55) 2 (.6) 3 (.65) 4 (.7) 5 (.75) 6 (.8)
state
A
White
11
12
13
14
15
16
Red
9
8
7
6
5
4
B
White
9
8
7
6
5
4
Red
11
12
13
14
15
16
Table 2: Contents of boxes in each state and each position: Anti-seniority

Realized
state
A

Seniority
Position (signal quality)
Marbles 1 (.8) 2 (.75) 3 (7) 4 (.65) 5 (.6) 6 (.55)

White
16
15
14
13
12
11
Red
4
5
6
7
8
9
B
White
4
5
6
7
8
9
Red
16
15
14
13
12
11
Table 3: Contents of boxes in each state and each position: Seniority
Sessions
1 2 3 4 5 6 7 8 9 10
Rounds 1-12
A A A A A S A S A S
Rounds 12-16 - A S S S A S A S A
Table 4: Treatments conducted in each sessions(A: Anti-seniority, S:Seniority)

20

11
A
S

Anti-seniority
Session
1
2
3
4
5
6
z
¡:440 ¡:604:000
1:319 ¡:604 1:813
P > jzj :660
:546
1:000
:187
:546
:070
Session
7
8
9
10
11
z
¡:604 1:000 ¡1:000 :000
:604
P > jzj :546
:3173
:3173 1:000 :546
Table 5: Mann-Whitney U tests of equal proportion of predictions which isconsistent with Bayesian posterior probability between the …rst two rounds
and the last two rounds for each session: Anti-seniority
Seniority
Session6
11
z
1:000 1:000
P > jzj :3173 :3173
Table 6: Mann-Whitney U tests of equal proportion of predictions which isconsistent with Bayesian posterior probability between the …rst two rounds
and the last two rounds for each session: Seniority 17
Aggregated behavior
Anti-seniority
112 rounds,
672 decisions
Realization of state A
55 (49.11%)
Correct predictions
467 (69.49%)
Complete positive cascades
16 (14.29%)
Complete negative cascades
2 (1.79%)
Rational complete positive cascades
7 (6.25%)
Rational complete negative cascades
1 (.89%)Partial cascades
29 (25.89%)
Full revelations
65 (58.04%)
Herds
32 (28.57%)
Table 7: Aggregated behavior

Seniority
60 rounds,
360 decisions
31 (51.67%)
295 (81.94%)
36 (60.00%)
6 (10.00%)
36 (60.00%)
6 (10.00%)
6 (10.00%)
12 (20.00%)
50 (83.33%)

17
All the predictions except in sessions 6 and 11 in the …rst and the last two rounds wereidentically consistent with Bayesian posterior probability, so that Mann-Whitney test of
the equal proportions cannot be performed for those sessions.

21

Tests of equal proportion for each treatmentz
P > jzj
Realization of state A
-.319
.7497
Correct predictions-4.335 .0000
Complete positive cascades-6.204 .0000
Complete negative cascades-2.431 .0151
Rational complete positive cascades -7.736 .0000
Rational complete negative cascades -2.873 .0041
Partial cascades
2.460
.0139
Full revelations
4.767
.0000
Herds
-6.833 .0000
Table 8: Mann-Whitney U tests of equal proportion for each treatment
Realization of each stateAnti-seniority
Seniorityz=-.189, P > jzj=.8501 z=.258, P > jzj=.7963
Table 9: Tests of population proportion on equal realization of each state

Theoretical prediction and actual occurrence:
Rational complete positive cascades
Anti-seniority
Seniorityz=-2.000, P < z=.0228 z=-1.878, P < z=.0302
Table 10: Tests of population proportion on the equality between
theoretical prediction and actual occurrence:
Rational complete positive cascades

Theoretical prediction and actual occurrence:
Rational complete negative cascades
Anti-seniority
Seniorityz=-2.329, P < z=.0099 z=-1.921, P < z=.0274
Table 11: Tests of population proportion on the equality between
theoretical prediction and actual occurrence:
Rational complete negative cascades

22

Anti-seniority
Obs.: 560, Log-likelihood:-146.0963, Â2: 483.95, P > Â2 : .0000, Pseudo R2: .6235
Estimated coe¢cient Standard errorz
P > jzj¯0-.0946
.1509
-.63
.531
¯1
1.9674
.2295
8.57
.000
¯2
3.6489
.3212
11.36
.000
Table 12: Result of logit regression: Anti-seniority
Seniority
Obs.: 300, Log-likelihood:-32.1571, Â2: 349.32 P > Â2 : .0000, Pseudo R2: .8445
Estimated coe¢cient Standard errorz
P > jzj¯0-.6120
.3603
-1.70
.089
¯1
1.7908
.2923
6.13
.000
¯2
2.4707
.5487
4.50
.000
Table 13: Result of logit regression: Seniority
Test of estimated coe¢cients: Anti-seniority
H0 : ¯ 1 = ¯ 2
Â2 : 75:72
P > Â2 : :0000
Table 14: Chi-square test of the hypothesis that estimated coe¢cients ofprivate signals and predecessors’ predictions are equal: Anti-seniority
Test of estimated coe¢cients: Seniority
H0 : ¯ 1 = ¯ 2
Â2 : 3:02
P > Â2 : :0822
Table 15: Chi-square test of the hypothesis that estimated coe¢cients ofprivate signals and predecessors’ predictions are equal: Seniority

23

Deviations from Bayesian posterior by position
Position Overcon…dence Undercon…dence Irrational
1
15(13.39%)
2
5 (4.47%)
2 (1.78%)3
11 (9.82%)
0 (.00%)4
6 (5.36%)
4 (3.57%)
2 (1.78%)5
11 (9.82%)
2 (1.78%)0 (.00%)614 (12.50%)
3 (2.68%)0 (.00%)18
Total
31 (9.23% )
25 (4.46%)
19 (2.83%)
Table 16: Deviations from Bayesian posterior by position: Anti-seniority
Deviations from Bayesian posterior by position
Position Overcon…dence Undercon…dence Irrational
10(.00%)
2
9 (15.00%)
0(.00%)
30 (.00%)0(.00%)
0(.00%)
40 (.00%)1 (1.67%)0(.00%)5
1 (1.67%)0(.00%)
0(.00%)
60 (.00%)0(.00%)
0(.00%)Total
10 (3.33%)
1 (.004%)0(.00%)
Table 17: Deviations from Bayesian posterior by position: Seniority
Deviations from Bayesian posterior
by position and the level of posteriors% of deviations by position
2
3
4
5
6
Bayesian posterior
0-20.00.00 .00 .89 .00
probabilities that state
20-40 .00
.00 .89 .00 .89
A would have realized (%) 40-60 4.47 9.82 1.79 .00 .89
60-80 .00
.00 .89 .89 .00
80-100 .00.00 .00 .00 .89
Table 18: Percentage of deviations from Bayesian posterior probability foreach position and each level of posterior: Undercon…dence in anti-seniority

18

The percentage in the total deviations for tables 16 and 17 is all the observed deviations
devided by total decisions where deviations are possible.

24Deviations from Bayesian posterior
by position and the level of posterior% of deviations by position
4
5
6
Bayesian posterior
0-20.00 .00
2.68
probabilities that state
20-40 .00 2.68
.00
A would have realized (%) 40-60 5.36 3.58
2.68
60-80 .00 3.58
4.46
80-100 .00 .00
2.68
Table 19: Percentage of deviations from Bayesian posterior probability foreach position and each level of posterior: Overcon…dence in anti-seniority

Deviations from Bayesian posterior
by position and the level of posterior% of deviations by position
3
4
5
6
Bayesian posterior
0-20.00 .00 .00.00
probabilities that state
20-40 .00 .00 .00.00
A would have realized (%) 40-60 .00 .00 .00.00
60-80 .00 1.79 .00.00
80-100 .00 .00 .00.00
Table 20: Percentage of deviations from Bayesian posterior probability foreach position and each level of posterior: Undercon…dence in seniority

Deviations from Bayesian posterior
by position and the level of posterior% of deviations by position
2
3
4
5
6
Bayesian posterior
0-20.00.00 .00 .00 .00
probabilities that state
20-40
.00
.00 .00 .00 .00
A would have realized (%) 40-60 15.00 .00 .00 .00 .00
60-80
.00
.00 .00 .00 .00
80-100.00.00 .00 1.79 .00
Table 21: Percentage of deviations from Bayesian posterior probability foreach position and each level of posterior: Overcon…dence in seniority

25

.12
Overconfidence/Underconfidence
.04
.06
.08
.1
.02
2

3

4
Position
Overconfidence

5

6

Underconfidence0

Overconfidence/Underconfidence
.05
.1

.15

Figure 1: Proportions of overcon…dence and undercon…dence: Anti-seniority

2

3

4
Position
Overconfidence

5

6

Underconfidence

Figure 2: Proportions of overcon…dence and undercon…dence: Seniority

26

B

Instructions for subjects

Instruction
Welcome!

This experiment is for the study of economic decision-making. The instructions are simple, and if you follow them carefully and you make gooddecisions, you can earn a considerable amount of money which will be paid
to you in cash after the experiment.

Your task

At the beginning of each round, the experimenter will draw one of thetwo cards from covered box. On a card a letter either "A" or "B" is printed.
We call the letter printed on the card as the "state" for that round. The
experimenter will con…rm the letter so as not to be seen by you and hide it
until the round …nishes. Your task in this experiment is to predict the state
for that round.

Order of making prediction

In this experiment you will make prediction one by one in sequence. Your
position of making prediction is determined by drawing one of the six cards.
On a card you will draw a number "1", "2", "3", "4", "5", or "6" is printed.
This number is your position. For example if you draw the card on which
the number "4" is printed, you will make prediction in position 4.

Information for making prediction

In predicting the state, you will receive two kinds of information, private
signals and predecessors’ predictions. You may or may not make use of them.

Private signals

You can observe private signals by drawing one marble from a covered
box. There are twenty white or red marbles in each box. White indicatesstate "A" and red indicates state "B". The proportion of white and red varies
depending on the position, realized state, and treatment and it re‡ects thepreciseness of the signal for predicting the state. The preciseness of thesignal is printed below the number of position on the same card you willdraw. For example, suppose that under the state "A" you would have drawn
card printed "1" and ".55" in Treatment A. Then you draw one marble froma box containing 11 white and 9 red marbles. Thus, the probability that you
draw a white marble from the box is 11/20=.55. Therefore the probability
27

that your draw indicates the correct state is .55. In another example, supposethat under the state "B" you would have drawn card printed "6" and ".8"in treatment A. Then you draw one marble from a box containing 4 whiteand 16 red marbles. In this case the probability that your draw indicates thecorrect state is 16/20=.8.
2 treatments will be conducted in this experiment. In treatment A, youwill make predictions in ascending order of the signal preciseness. That is,
the subject who has the least precise signal (.55) makes the prediction inposition 1, the subject who has the second least precise signal (.6) makes
the prediction in position 2, and in this manner, the subject who has themost precise signal (.8) makes the prediction in position 6 (last position). In
treatment S, you make predictions in descending order of the signal preciseness. That is, the subject who has the most precise signal (.8) makes theprediction in position 1, the subjects who has the second most precise signal(.75) makes the prediction in position 2, and in this manner, the subject
who has the least precise signal (.55) makes the prediction in position 6 (lastposition). Which treatment is conducted is announced at the beginning ofeach round.
The following tables summarize proportions of white and red marbles foreach position, realized state, and treatment. Note that your signals indicate
the correct state more accurately than random guessing in any case becausesignal preciseness is always higher than .5.
Treatment A
Position (Signal preciseness)
State realized Marbles 1 (.55) 2 (.6) 3 (.65) 4 (.7) 5 (.75) 6 (.8)
A
White
11
12
13
14
15
16
Red
9
8
7
6
5
4
B
White
9
8
7
6
5
4
Red
11
12
13
14
15
16
Treatment S
Position (Signal preciseness)
State realized Marbles 1 (.8) 2 (.75) 3 (7) 4 (.65) 5 (.6) 6 (.55)
A
White
16
15
14
13
12
11
Red
4
5
6
7
8
9
B
White
4
5
6
7
8
9
Red
16
15
14
13
12
11

Predecessors’ predictions

In addition to private signals you will observe your predecessors’ predictions. Once a subject submits a prediction, the experimenter will record it28

on the experimenter’s record sheet and show it to other subjects in later
positions. For example, the subject in position 2 can observe the prediction
made by the subject in position 1, the subject in position 3 can observe two
predictions made by the subjects in positions 1 and 2, the subject in position 6 (last position) can observe …ve predictions made by the subjects inpositions from 1 to 5, and so on.
Thus, the subjects assigned to later than position 2 make predictions
by referring to both private signals and all of their predecessors’ predictions
whereas the subject in position 1 can observe only his or her private signal.

Procedure of the experiments

We call a sequence of making predictions by six subjects a round. In
today’s experiment you will make predictions for sixteen rounds. In eachround, experiment is conducted in the following procedure.
1. The experimenter announces which treatment to be conducted for thatround.
2. The experimenter draw a card which determines the state for thatround. After con…rming the letter the experimenter hide it from you.
3. You draw one card on which the position of making prediction and thesignal preciseness are written.
4. The experimenter approaches you one by one from position 1 to position
6 and present one box which contains the exact proportion of white andred marbles for that position, realized state, and treatment.
5. You draw one marble from the box.
6. After con…rming the color of the marble, you replace it into the box
and write down the state indicated by the color of the marble in thespace printed "Your Signal" on the subject’s record sheet. At the sametime the experimenter also record it on the experimenter’s record sheet.
7. 3-6 is repeated until all of you observe the private signals.
8. You make your prediction when the experimenter approaches you andtells you to do so. Before making a prediction, the experimenter shows asequence of your predecessors’ predictions to you. You write down themin the space printed "Your Predecessors’ Predictions". A prediction ismade by writing down one of the two states you think which is more
29

likely to be in the space printed "Your Prediction" on the subject’s
record sheet. Do not write your prediction until you are told to
do so.
9. Once a prediction is made, the experimenter checks it and write it downon the experimenter’s record sheet. Then, the experimenter approaches
the subject in the next position and show the sequence of his or her
predecessors’ predictions.
10. 8-9 is repeated until all of you make predictions.
11. The card the experimenter have drawn is revealed. The experimenter
record the correct and incorrect predictions for each subject on theexperimenter’s record sheet.

Payment

Your payo¤ for participating this experiment depends on the number ofcorrect predictions. You will be paid 200 yen for a correct prediction and0 yen for an incorrect prediction. Sum of the results for all 16 predictions
will be paid to you privately after the experiment. The theoretical possible
maximum payment is 3200 yen and the minimum is 0 yen.

Your information

Your private information concerning this experiment is not known toanyone. Your name, address, student ID number, predictions, payo¤s are
con…dential.

NoticeThis experiment is conducted based on scienti…c method. Please notefollowings.
² Do not talk with others.
² Do not see others’ draw and record sheet.
² Do not write your signals and predictions until you are told to do so.
If you do not follow these instructions, you will be dismissed. In this case,
you cannot receive payo¤ you earned until then.
If you have any questions, raise your hand.

30

C

Bayesian posterior probability that state A
would have realized at each decision node19

C.1

Anti-seniority

Pr(Aj¾1A )Pr(Aj¾1B )Pr(Aj¼1A ; ¾2A)
Pr(Aj¼1A ; ¾2B )Pr(Aj¼1B ; ¾ 2A)
Pr(Aj¼1B ; ¾ 2B)
Pr(Aj¼1A ; ¼2A; ¾ 3A)
Pr(Aj¼1A ; ¼2A; ¾ 3B)
Pr(Aj¼1A ; ¼2B ; ¾3A)
Pr(Aj¼1A ; ¼2B ; ¾3B )Pr(Aj¼1B ; ¼ 2A; ¾3A)
Pr(Aj¼1B ; ¼ 2A; ¾3B )Pr(Aj¼1B ; ¼ 2B; ¾ 3A)
Pr(Aj¼1B ; ¼ 2B; ¾ 3B)
Pr(Aj¼1A ; ¼2A; ¼ 3A; ¾4A)
Pr(Aj¼1A ; ¼2A; ¼ 3A; ¾4B )Pr(Aj¼1A ; ¼2A; ¼ 3B; ¾ 4A)
Pr(Aj¼1A ; ¼2A; ¼ 3B; ¾ 4B)
Pr(Aj¼1A ; ¼2B ; ¼3A; ¾ 4A)
Pr(Aj¼1A ; ¼2B ; ¼3A; ¾ 4B)
Pr(Aj¼1B ; ¼ 2A; ¼3A; ¾ 4A)
Pr(Aj¼1B ; ¼ 2A; ¼3A; ¾ 4B)
Pr(Aj¼1A ; ¼2B ; ¼3B ; ¾4A)
Pr(Aj¼1A ; ¼2B ; ¼3B ; ¾4B )Pr(Aj¼1B ; ¼ 2B; ¼ 3A; ¾4A)
Pr(Aj¼1B ; ¼ 2B; ¼ 3A; ¾4B )Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¾4A)
Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¾4B )Pr(Aj¼1B ; ¼ 2B; ¼ 3B; ¾ 4A)
Pr(Aj¼1B ; ¼ 2B; ¼ 3B; ¾ 4B ).55
.45.647059.44898.55102.352941.772973.496774.602105.30496.69505.397895.503226.227027.888199.593361.697283.297297.779292.393398.841727.494134.505866.158273.702703.302717.606602.220708.406639.111801

Pr(Aj¼ 1A; ¼2A; ¼ 3A; ¼4A; ¾ 5A)
Pr(Aj¼ 1A; ¼2A; ¼ 3A; ¼4A; ¾ 5B)
Pr(Aj¼ 1A; ¼2A; ¼ 3A; ¼4B ; ¾5A)
Pr(Aj¼ 1A; ¼2A; ¼ 3A; ¼4B ; ¾5B )Pr(Aj¼ 1A; ¼2A; ¼ 3B; ¼ 4A; ¾5A)
Pr(Aj¼ 1A; ¼2A; ¼ 3B; ¼ 4A; ¾5B )Pr(Aj¼ 1A; ¼2B ; ¼3A; ¼ 4A; ¾5A)
Pr(Aj¼ 1A; ¼2B ; ¼3A; ¼ 4A; ¾5B )Pr(Aj¼ 1B; ¼ 2A; ¼3A; ¼ 4A; ¾5A)
Pr(Aj¼ 1B; ¼ 2A; ¼3A; ¼ 4A; ¾5B )Pr(Aj¼ 1A; ¼2A; ¼ 3B; ¼ 4B; ¾ 5A)
Pr(Aj¼ 1A; ¼2A; ¼ 3B; ¼ 4B; ¾ 5B )Pr(Aj¼ 1A; ¼2B ; ¼3B ; ¼4A; ¾ 5A)
Pr(Aj¼ 1A; ¼2B ; ¼3B ; ¼4A; ¾ 5B )Pr(Aj¼ 1B; ¼ 2B; ¼ 3A; ¼4A; ¾ 5A)
Pr(Aj¼ 1B; ¼ 2B; ¼ 3A; ¼4A; ¾ 5B )Pr(Aj¼ 1A; ¼2B ; ¼3A; ¼ 4B; ¾ 5A)
Pr(Aj¼ 1A; ¼2B ; ¼3A; ¼ 4B; ¾ 5B )Pr(Aj¼ 1B; ¼ 2A; ¼3B ; ¼4A; ¾ 5A)
Pr(Aj¼ 1B; ¼ 2A; ¼3B ; ¼4A; ¾ 5B )Pr(Aj¼ 1B; ¼ 2A; ¼3A; ¼ 4B; ¾ 5A)
Pr(Aj¼ 1B; ¼ 2A; ¼3A; ¼ 4B; ¾ 5B )Pr(Aj¼ 1A; ¼2B ; ¼3B ; ¼4B ; ¾5A )Pr(Aj¼ 1A; ¼2B ; ¼3B ; ¼4B ; ¾5B )Pr(Aj¼ 1B; ¼ 2A; ¼3B ; ¼4B ; ¾5A )Pr(Aj¼ 1B; ¼ 2A; ¼3B ; ¼4B ; ¾5B )Pr(Aj¼ 1B; ¼ 2B; ¼ 3A; ¼4B ; ¾5A )Pr(Aj¼ 1B; ¼ 2B; ¼ 3A; ¼4B ; ¾5B )Pr(Aj¼ 1B; ¼ 2B; ¼ 3B ; ¼4A; ¾5A )Pr(Aj¼ 1B; ¼ 2B; ¼ 3B ; ¼4A; ¾5B )19.959732.725888.814042.327231.873582.434327.913738.540643.941019.639344.559322.123596.754374.254425.876404.440678.660508.17775.82225.339492.745575.245626.360656.058981.459357.080262.565673.126418.672769.185958

Bayesian posteriors are typed in italics if imbalance between Bayesian posterior andprivate signal occurs.

31

Pr(Aj¼ 1B ; ¼2B ; ¼3B ; ¼4B ; ¾5A )Pr(Aj¼ 1B ; ¼2B ; ¼3B ; ¼4B ; ¾5B )Pr(Aj¼ 1A; ¼2A ; ¼3A; ¼4A ; ¼5A; ¾ 6A)
Pr(Aj¼ 1A; ¼2A ; ¼3A; ¼4A ; ¼5A; ¾ 6B)
Pr(Aj¼ 1A; ¼2A ; ¼3A; ¼4A ; ¼5B ; ¾6A)
Pr(Aj¼ 1A; ¼2A ; ¼3A; ¼4A ; ¼5B ; ¾6B )Pr(Aj¼ 1A; ¼2A ; ¼3A; ¼4B ; ¼ 5A; ¾6A)
Pr(Aj¼ 1A; ¼2A ; ¼3A; ¼4B ; ¼ 5A; ¾6B )Pr(Aj¼ 1A; ¼2A ; ¼3B ; ¼4A; ¼ 5A; ¾6A)
Pr(Aj¼ 1A; ¼2A ; ¼3B ; ¼4A; ¼ 5A; ¾6B )Pr(Aj¼ 1A; ¼2B ; ¼ 3A; ¼4A; ¼ 5A; ¾6A)
Pr(Aj¼ 1A; ¼2B ; ¼ 3A; ¼4A; ¼ 5A; ¾6B )Pr(Aj¼ 1B ; ¼2A; ¼ 3A; ¼4A; ¼ 5A; ¾6A)
Pr(Aj¼ 1B ; ¼2A; ¼ 3A; ¼4A; ¼ 5A; ¾6B )Pr(Aj¼ 1A; ¼2A ; ¼3A; ¼4B ; ¼ 5B; ¾ 6A)
Pr(Aj¼ 1A; ¼2A ; ¼3A; ¼4B ; ¼ 5B; ¾ 6B)
Pr(Aj¼ 1A; ¼2A ; ¼3B ; ¼4B ; ¼5A; ¾ 6A)
Pr(Aj¼ 1A; ¼2A ; ¼3B ; ¼4B ; ¼5A; ¾ 6B)
Pr(Aj¼ 1A; ¼2B ; ¼ 3B; ¼ 4A; ¼5A; ¾ 6A)
Pr(Aj¼ 1A; ¼2B ; ¼ 3B; ¼ 4A; ¼5A; ¾ 6B)
Pr(Aj¼ 1B ; ¼2B ; ¼3A; ¼ 4A; ¼5A; ¾ 6A)
Pr(Aj¼ 1B ; ¼2B ; ¼3A; ¼ 4A; ¼5A; ¾ 6B)
Pr(Aj¼ 1A; ¼2A ; ¼3B ; ¼4A; ¼ 5B; ¾ 6A)
Pr(Aj¼ 1A; ¼2A ; ¼3B ; ¼4A; ¼ 5B; ¾ 6B)
Pr(Aj¼ 1A; ¼2B ; ¼ 3A; ¼4B ; ¼5A; ¾ 6A)
Pr(Aj¼ 1A; ¼2B ; ¼ 3A; ¼4B ; ¼5A; ¾ 6B)
Pr(Aj¼ 1A; ¼2B ; ¼ 3A; ¼4A; ¼ 5B; ¾ 6A)
Pr(Aj¼ 1A; ¼2B ; ¼ 3A; ¼4A; ¼ 5B; ¾ 6B)
Pr(Aj¼ 1B ; ¼2A; ¼ 3A; ¼4A; ¼ 5B; ¾ 6A)
Pr(Aj¼ 1B ; ¼2A; ¼ 3A; ¼4A; ¼ 5B; ¾ 6B).274112.040268.989619.856287.913738.398329.945976.522533.965085.633373.976943.725888.984572.799544.660508.108415.835443.240876.924727.434327.965944.639344.754374.16104.886135.327231.824802.227345.876404.307087

Pr(Aj¼ 1B ; ¼2B ; ¼3B ; ¼4A; ¼ 5B; ¾ 6A)
Pr(Aj¼ 1B ; ¼2B ; ¼3B ; ¼4A; ¼ 5B; ¾ 6B )Pr(Aj¼ 1B ; ¼2B ; ¼3B ; ¼4B ; ¼5A; ¾ 6A)
Pr(Aj¼ 1B ; ¼2B ; ¼3B ; ¼4B ; ¼5A; ¾ 6B )Pr(Aj¼ 1B ; ¼2B ; ¼3B ; ¼4B ; ¼5B ; ¾ 6A)
Pr(Aj¼ 1B ; ¼2B ; ¼3B ; ¼4B ; ¼5B ; ¾ 6B).477467.054024.601671.086262.143713.01038132

Pr(Aj¼1B ; ¼ 2A; ¼3A; ¼ 4B; ¼ 5A; ¾6A)
Pr(Aj¼1B ; ¼ 2A; ¼3A; ¼ 4B; ¼ 5A; ¾6B )Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¼4A; ¼ 5A; ¾6A)
Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¼4A; ¼ 5A; ¾6B )Pr(Aj¼1A ; ¼2A; ¼ 3B; ¼ 4B; ¼ 5B ; ¾6A)
Pr(Aj¼1A ; ¼2A; ¼ 3B; ¼ 4B; ¼ 5B ; ¾6B )Pr(Aj¼1A ; ¼2B ; ¼3A; ¼ 4B; ¼ 5B ; ¾6A)
Pr(Aj¼1A ; ¼2B ; ¼3A; ¼ 4B; ¼ 5B ; ¾6B )Pr(Aj¼1A ; ¼2B ; ¼3B ; ¼4A; ¼ 5B ; ¾6A)
Pr(Aj¼1A ; ¼2B ; ¼3B ; ¼4A; ¼ 5B ; ¾6B )Pr(Aj¼1A ; ¼2B ; ¼3B ; ¼4B ; ¼5A ; ¾6A)
Pr(Aj¼1A ; ¼2B ; ¼3B ; ¼4B ; ¼5A ; ¾6B )Pr(Aj¼1B ; ¼ 2A; ¼3A; ¼ 4B; ¼ 5B ; ¾6A)
Pr(Aj¼1B ; ¼ 2A; ¼3A; ¼ 4B; ¼ 5B ; ¾6B )Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¼4A; ¼ 5B ; ¾6A)
Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¼4A; ¼ 5B ; ¾6B )Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¼4B ; ¼5A ; ¾6A)
Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¼4B ; ¼5A ; ¾6B )Pr(Aj¼1B ; ¼ 2B; ¼ 3A; ¼4A; ¼ 5B ; ¾6A)
Pr(Aj¼1B ; ¼ 2B; ¼ 3A; ¼4A; ¼ 5B ; ¾6B )Pr(Aj¼1B ; ¼ 2B; ¼ 3A; ¼4B ; ¼5A ; ¾6A)
Pr(Aj¼1B ; ¼ 2B; ¼ 3A; ¼4B ; ¼5A ; ¾6B )Pr(Aj¼1B ; ¼ 2B; ¼ 3B; ¼ 4A; ¼5A ; ¾6A)
Pr(Aj¼1B ; ¼ 2B; ¼ 3B; ¼ 4A; ¼5A ; ¾6B )Pr(Aj¼1A ; ¼2B ; ¼3B ; ¼4B ; ¼5B ; ¾ 6A)
Pr(Aj¼1A ; ¼2B ; ¼3B ; ¼4B ; ¼5B ; ¾ 6B)
Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¼4B ; ¼5B ; ¾ 6A)
Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¼4B ; ¼5B ; ¾ 6B)
Pr(Aj¼1B ; ¼ 2B; ¼ 3A; ¼4B ; ¼5B ; ¾ 6A)
Pr(Aj¼1B ; ¼ 2B; ¼ 3A; ¼4B ; ¼5B ; ¾ 6B).921394.422835.948727.536279.360656.034056.463721.051273.577165.078606.692913.123596.565673.075273.672769.113865.772655.175198.759124.164557.83896.245626.891585.339492.200456.015428.274112.023057.366627.034915

C.2

Seniority

Pr(Aj¾1A )Pr(Aj¾1B )Pr(Aj¼1A ; ¾2A)
Pr(Aj¼1A ; ¾2B )Pr(Aj¼1B ; ¾ 2A)
Pr(Aj¼1B ; ¾ 2B)
Pr(Aj¼1A ; ¼2A; ¾ 3A)
Pr(Aj¼1A ; ¼2A; ¾ 3B)
Pr(Aj¼1A ; ¼2B ; ¾3A)
Pr(Aj¼1A ; ¼2B ; ¾3B )Pr(Aj¼1B ; ¼ 2A; ¾3A)
Pr(Aj¼1B ; ¼ 2A; ¾3B )Pr(Aj¼1B ; ¼ 2B; ¾ 3A)
Pr(Aj¼1B ; ¼ 2B; ¾ 3B)
Pr(Aj¼1A ; ¼2A; ¼ 3A; ¾4A)
Pr(Aj¼1A ; ¼2A; ¼ 3A; ¾4B )Pr(Aj¼1A ; ¼2A; ¼ 3B; ¾ 4A)
Pr(Aj¼1A ; ¼2A; ¼ 3B; ¾ 4B)
Pr(Aj¼1A ; ¼2B ; ¼3A; ¾ 4A)
Pr(Aj¼1A ; ¼2B ; ¼3A; ¾ 4B)
Pr(Aj¼1B ; ¼ 2A; ¼3A; ¾ 4A)
Pr(Aj¼1B ; ¼ 2A; ¼3A; ¾ 4B)
Pr(Aj¼1A ; ¼2B ; ¼3B ; ¾4A)
Pr(Aj¼1A ; ¼2B ; ¼3B ; ¾4B )Pr(Aj¼1B ; ¼ 2B; ¼ 3A; ¾4A)
Pr(Aj¼1B ; ¼ 2B; ¼ 3A; ¾4B )Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¾4A)
Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¾4B )Pr(Aj¼1B ; ¼ 2B; ¼ 3B; ¾ 4A)
Pr(Aj¼1B ; ¼ 2B; ¼ 3B; ¾ 4B ).8.2.923077.571729.428571.076923.965517.837209.756757.363636.636364.243243.162791.034483.981132.937799.905222.734694.852459.626198.764706.485149.514851.235294.265306.094778.373802.147541.062201.018868

Pr(Aj¼ 1A; ¼2A; ¼ 3A; ¼4A; ¾ 5A)
Pr(Aj¼ 1A; ¼2A; ¼ 3A; ¼4A; ¾ 5B)
Pr(Aj¼ 1A; ¼2A; ¼ 3A; ¼4B ; ¾5A)
Pr(Aj¼ 1A; ¼2A; ¼ 3A; ¼4B ; ¾5B )Pr(Aj¼ 1A; ¼2A; ¼ 3B; ¼ 4A; ¾5A)
Pr(Aj¼ 1A; ¼2A; ¼ 3B; ¼ 4A; ¾5B )Pr(Aj¼ 1A; ¼2B ; ¼3A; ¼ 4A; ¾5A)
Pr(Aj¼ 1A; ¼2B ; ¼3A; ¼ 4A; ¾5B )Pr(Aj¼ 1B; ¼ 2A; ¼3A; ¼ 4A; ¾5A)
Pr(Aj¼ 1B; ¼ 2A; ¼3A; ¼ 4A; ¾5B )Pr(Aj¼ 1A; ¼2A; ¼ 3B; ¼ 4B; ¾ 5A)
Pr(Aj¼ 1A; ¼2A; ¼ 3B; ¼ 4B; ¾ 5B )Pr(Aj¼ 1A; ¼2B ; ¼3B ; ¼4A; ¾ 5A)
Pr(Aj¼ 1A; ¼2B ; ¼3B ; ¼4A; ¾ 5B )Pr(Aj¼ 1B; ¼ 2B; ¼ 3A; ¼4A; ¾ 5A)
Pr(Aj¼ 1B; ¼ 2B; ¼ 3A; ¼4A; ¾ 5B )Pr(Aj¼ 1A; ¼2B ; ¼3A; ¼ 4B; ¾ 5A)
Pr(Aj¼ 1A; ¼2B ; ¼3A; ¼ 4B; ¾ 5B )Pr(Aj¼ 1B; ¼ 2A; ¼3B ; ¼4A; ¾ 5A)
Pr(Aj¼ 1B; ¼ 2A; ¼3B ; ¼4A; ¾ 5B )Pr(Aj¼ 1B; ¼ 2A; ¼3A; ¼ 4B; ¾ 5A)
Pr(Aj¼ 1B; ¼ 2A; ¼3A; ¼ 4B; ¾ 5B )Pr(Aj¼ 1A; ¼2B ; ¼3B ; ¼4B ; ¾5A )Pr(Aj¼ 1A; ¼2B ; ¼3B ; ¼4B ; ¾5B )Pr(Aj¼ 1B; ¼ 2A; ¼3B ; ¼4B ; ¾5A )Pr(Aj¼ 1B; ¼ 2A; ¼3B ; ¼4B ; ¾5B )Pr(Aj¼ 1B; ¼ 2B; ¼ 3A; ¼4B ; ¾5A )Pr(Aj¼ 1B; ¼ 2B; ¼ 3A; ¼4B ; ¾5B )Pr(Aj¼ 1B; ¼ 2B; ¼ 3B ; ¼4A; ¾5A )Pr(Aj¼ 1B; ¼ 2B; ¼ 3B ; ¼4A; ¾5B )33.987342.971963.957655.909513.934754.864266.896552.793893.829787.684211.80597.648649.614173.414343.351351.19403.715328.527591.472409.284672.585657.385827.315789.170213.206107.103448.135734.065246.090487.042345

Pr(Aj¼ 1B ; ¼2B ; ¼3B ; ¼4B ; ¾5A )Pr(Aj¼ 1B ; ¼2B ; ¼3B ; ¼4B ; ¾5B )Pr(Aj¼ 1A; ¼2A ; ¼3A; ¼4A ; ¼5A; ¾ 6A)
Pr(Aj¼ 1A; ¼2A ; ¼3A; ¼4A ; ¼5A; ¾ 6B)
Pr(Aj¼ 1A; ¼2A ; ¼3A; ¼4A ; ¼5B ; ¾6A)
Pr(Aj¼ 1A; ¼2A ; ¼3A; ¼4A ; ¼5B ; ¾6B )Pr(Aj¼ 1A; ¼2A ; ¼3A; ¼4B ; ¼ 5A; ¾6A)
Pr(Aj¼ 1A; ¼2A ; ¼3A; ¼4B ; ¼ 5A; ¾6B )Pr(Aj¼ 1A; ¼2A ; ¼3B ; ¼4A; ¼ 5A; ¾6A)
Pr(Aj¼ 1A; ¼2A ; ¼3B ; ¼4A; ¼ 5A; ¾6B )Pr(Aj¼ 1A; ¼2B ; ¼ 3A; ¼4A; ¼ 5A; ¾6A)
Pr(Aj¼ 1A; ¼2B ; ¼ 3A; ¼4A; ¼ 5A; ¾6B )Pr(Aj¼ 1B ; ¼2A; ¼ 3A; ¼4A; ¼ 5A; ¾6A)
Pr(Aj¼ 1B ; ¼2A; ¼ 3A; ¼4A; ¼ 5A; ¾6B )Pr(Aj¼ 1A; ¼2A ; ¼3A; ¼4B ; ¼ 5B; ¾ 6A)
Pr(Aj¼ 1A; ¼2A ; ¼3A; ¼4B ; ¼ 5B; ¾ 6B)
Pr(Aj¼ 1A; ¼2A ; ¼3B ; ¼4B ; ¼5A; ¾ 6A)
Pr(Aj¼ 1A; ¼2A ; ¼3B ; ¼4B ; ¼5A; ¾ 6B)
Pr(Aj¼ 1A; ¼2B ; ¼ 3B; ¼ 4A; ¼5A; ¾ 6A)
Pr(Aj¼ 1A; ¼2B ; ¼ 3B; ¼ 4A; ¼5A; ¾ 6B)
Pr(Aj¼ 1B ; ¼2B ; ¼3A; ¼ 4A; ¼5A; ¾ 6A)
Pr(Aj¼ 1B ; ¼2B ; ¼3A; ¼ 4A; ¼5A; ¾ 6B)
Pr(Aj¼ 1A; ¼2A ; ¼3B ; ¼4A; ¼ 5B; ¾ 6A)
Pr(Aj¼ 1A; ¼2A ; ¼3B ; ¼4A; ¼ 5B; ¾ 6B)
Pr(Aj¼ 1A; ¼2B ; ¼ 3A; ¼4B ; ¼5A; ¾ 6A)
Pr(Aj¼ 1A; ¼2B ; ¼ 3A; ¼4B ; ¼5A; ¾ 6B)
Pr(Aj¼ 1A; ¼2B ; ¼ 3A; ¼4A; ¼ 5B; ¾ 6A)
Pr(Aj¼ 1A; ¼2B ; ¼ 3A; ¼4A; ¼ 5B; ¾ 6B)
Pr(Aj¼ 1B ; ¼2A; ¼ 3A; ¼4A; ¼ 5B; ¾ 6A)
Pr(Aj¼ 1B ; ¼2A; ¼ 3A; ¼4A; ¼ 5B; ¾ 6B).028037.012658.989619.984572.976943.965944.965085.948727.945976.921394.913738.876404.856287.799544.924727.891585.835443.772655.660508.565673.398329.307087.886135.83896.754374.672769.824802.759124.725888.639344

Pr(Aj¼ 1B ; ¼2B ; ¼3B ; ¼4A; ¼ 5B; ¾ 6A)
Pr(Aj¼ 1B ; ¼2B ; ¼3B ; ¼4A; ¼ 5B; ¾ 6B )Pr(Aj¼ 1B ; ¼2B ; ¼3B ; ¼4B ; ¼5A; ¾ 6A)
Pr(Aj¼ 1B ; ¼2B ; ¼3B ; ¼4B ; ¼5A; ¾ 6B )Pr(Aj¼ 1B ; ¼2B ; ¼3B ; ¼4B ; ¼5B ; ¾ 6A)
Pr(Aj¼ 1B ; ¼2B ; ¼3B ; ¼4B ; ¼5B ; ¾ 6B).051273.034915.034056.023057.015428.01038134

Pr(Aj¼1B ; ¼ 2A; ¼3A; ¼ 4B; ¼ 5A; ¾6A)
Pr(Aj¼1B ; ¼ 2A; ¼3A; ¼ 4B; ¼ 5A; ¾6B )Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¼4A; ¼ 5A; ¾6A)
Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¼4A; ¼ 5A; ¾6B )Pr(Aj¼1A ; ¼2A; ¼ 3B; ¼ 4B; ¼ 5B ; ¾6A)
Pr(Aj¼1A ; ¼2A; ¼ 3B; ¼ 4B; ¼ 5B ; ¾6B )Pr(Aj¼1A ; ¼2B ; ¼3A; ¼ 4B; ¼ 5B ; ¾6A)
Pr(Aj¼1A ; ¼2B ; ¼3A; ¼ 4B; ¼ 5B ; ¾6B )Pr(Aj¼1A ; ¼2B ; ¼3B ; ¼4A; ¼ 5B ; ¾6A)
Pr(Aj¼1A ; ¼2B ; ¼3B ; ¼4A; ¼ 5B ; ¾6B )Pr(Aj¼1A ; ¼2B ; ¼3B ; ¼4B ; ¼5A ; ¾6A)
Pr(Aj¼1A ; ¼2B ; ¼3B ; ¼4B ; ¼5A ; ¾6B )Pr(Aj¼1B ; ¼ 2A; ¼3A; ¼ 4B; ¼ 5B ; ¾6A)
Pr(Aj¼1B ; ¼ 2A; ¼3A; ¼ 4B; ¼ 5B ; ¾6B )Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¼4A; ¼ 5B ; ¾6A)
Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¼4A; ¼ 5B ; ¾6B )Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¼4B ; ¼5A ; ¾6A)
Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¼4B ; ¼5A ; ¾6B )Pr(Aj¼1B ; ¼ 2B; ¼ 3A; ¼4A; ¼ 5B ; ¾6A)
Pr(Aj¼1B ; ¼ 2B; ¼ 3A; ¼4A; ¼ 5B ; ¾6B )Pr(Aj¼1B ; ¼ 2B; ¼ 3A; ¼4B ; ¼5A ; ¾6A)
Pr(Aj¼1B ; ¼ 2B; ¼ 3A; ¼4B ; ¼5A ; ¾6B )Pr(Aj¼1B ; ¼ 2B; ¼ 3B; ¼ 4A; ¼5A ; ¾6A)
Pr(Aj¼1B ; ¼ 2B; ¼ 3B; ¼ 4A; ¼5A ; ¾6B )Pr(Aj¼1A ; ¼2B ; ¼3B ; ¼4B ; ¼5B ; ¾ 6A)
Pr(Aj¼1A ; ¼2B ; ¼3B ; ¼4B ; ¼5B ; ¾ 6B)
Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¼4B ; ¼5B ; ¾ 6A)
Pr(Aj¼1B ; ¼ 2A; ¼3B ; ¼4B ; ¼5B ; ¾ 6B)
Pr(Aj¼1B ; ¼ 2B; ¼ 3A; ¼4B ; ¼5B ; ¾ 6A)
Pr(Aj¼1B ; ¼ 2B; ¼ 3A; ¼4B ; ¼5B ; ¾ 6B).633373.536279.522533.422835.692913.601671.577165.477467.463721.366627.360656.274112.434327.339492.327231.245626.240876.175198.227345.164557.16104.113865.108415.075273.200456.143713.123596.086262.078606.054024