Cross-cultural study of game-playing behavior in children : an interim report

Cross-cultural study of game-playing behavior in children : an

interim report

Citation for published version (APA):

Shinotsuka, H. (1975). Cross-cultural study of game-playing behavior in children : an interim report. Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1975

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75

SRI

CROSS-CULTURAL STUDY OF GAME-PLAYING BEHAVlOR IN

CHILDREN AN INTERIM REPORT

Hirorni Shinotsuka

CROSS-CLJi.TURAL STUDY CF GAHE-PLAY1NG B.EHAVIOR IN

CHILDREN:

AN INTERIM REPORT

Hiromi Shinotsuka* Technische Hogeschool te Eindhoven

1. Introduction.

This is an interim progress report on the cross-cultural study ~roject of which Prof. Toda and I have been the members of the Japaliese team. ïie had done an experiment in Japan in

1969

and ever since have been engaging in the analysis of the whole data obtained in Japan as well as in other countries. ,ie have so far obtained rnany

.

interesting results from our data analysis carried out in different levels, from very macro-scopic to very micro-scopic. Nevertheless, as yet we cannot claim that out findings fit together like the pieces of a jigsaw puzzle. So in this report I will select a few of our recent results of analysis, most of which were obtained during my stay in Eindhoven,

duction.

preceded by the general overall results as an

intro-This project was initiated by C.G.McClintock and J.M.Nuttin, Jr. Their ~otivation for the project was to compare the cultural effects upon the development of competitiveness in children. Their major finding was that Belgian boys were much less competitive than Anglo-American boys when the children in the second grade of primary schools were

---.. J:he major part of the work reported here was done under the research fellowship granted ta the author by Technische Hogeschool te Eindhoven. 'l'he .:l.uthor appreciates very much the .lr..ind hela offered b her collea.,.es

t "oU" -" J 0

a ~u, espccially by r~of.Meuwese, Dr.van Vonderen and Mr .Habbinowitsch.

·

compared, but the competitiveness of Belgian boys caught up with American boys at the sixth grade.

In addition to'these two cultures data were collected later from three more cultures, Japanese,Greek and ~iexican-American, and we found that the dyad( the pair of Ss played the game together ) differences were invariably great in any culture. ~o our interest in the data analysis has gradually shifted from the competitive motive to the general empirical model of game-playing behavior. F or example, our current major interest may oe expressed in terms of the following questions j If we are going to categorize the types of game playing

concerning individuals and dyads, what would be the most plausible clasBification? How do these types develop through the course of play? Are there some special events that may trigger a drastic change in the behavior of the players--like the outburst of competitive responses ? Ta sum up, can we predict the Ss' behavior to the later stage of the play fairly precisely by kncwing their early stage behavior? 2. Bxperimental Conditions

Payoff Hatrix: '.ehe payoff matrix in this experiment anà shown in Fig. 1 is a version of Maximizing Difference Game, abbreviate as MD, designed ~o effectively separate the two major motives, generally assumed in experimental game studies, the

motive of maximizins one's own gain and that of maximizing the difference of oue's own gain from the op~onent's. As alternative 'a' dominates

alternative tb' for each player, one cannot increase his own gain by changing his ,"'e.s~~onse from 'a' to 'b'. So i f thè shift occurs, W:re

differ-ence--- which we may simply call the competi.tive motive, and, accordingly ,we may label the respon~e 'bi as the competitive response. We mayalso call the response 'a' as the cooperative response, since the choice of

'a' makes sense, only if the two players both choose 'a', or as au invitaiional reponse intended to lead up to that cooperative state.

Ss: The experiment Ss were chosen from five cultures,

Anglo-America~, JvIexican-American'" , Belgian, Greek and Japanese. They were

boys"'· in the second, fourth and sixth grade of primary schools,as shown in Table 1.

Ap~aratus: Each player used a special response panel on which

were two buttons corresponding to the response 'a' and tb', four feed-back lamps wnich displayed the realized payoff cell when the two

players made tneir responses, and a digital counter showing the sub-total score which he had acquired up to the trial.Under the double display condition explained below, the panel had one more counter to show the op~onentls subtotal score. In addition to these counters the second' graders were also provided with glass tubes into which E ·put'the same number of poker cuips as the number aplJearing in the

counter(s) to help S grasp the meaning of tüe numer{~al ·score(s).

Display Conditions: For each culture and grade, different dyads were run under two display conditions, Single Display and Double

Display, meaning the number of counters. Under the dingle Display

---'" Those who live in California and are usually called Chicanos.

conditivn, there was only one counter for each S, displaying nis own subtotal score. Under the Double Uisplay condition anotner counter was added to suow the opponentls subtotal score. The implication of these two conditions was obvious : Ss were expected to behave more

com~etitivelyunder the Double Dislplay condition which enabled them

"directly cornpare the difference in subtotal scores, thus providing them tue outlet of tue maximizing-tne-difference motivation.

For eücn cümbination of tnree conditions, Culture X Grade X Display, at least 14 àyads were run except with Mexican-Americans. BecauGe of the difficulty in getting enough numger of Chicano subjects, They were run only under the Double Disp~ay_condition.

Experimental ~rocedure: Ss were paired to form dyads,and precautions were taken so that friends did not form a dyad. iach S Séct in front of a panel described above, which obstructed the sight of the opponent. The instruction given to the Ss was written as neutral as possible concerning their motivation, and we didn't advise them

n

to maximize the total gain cotrary to the general practice. We just

l\

told them to play the game. No monetary reward was given.

Each dyad played the game for 110 trials. '.tne last ten trials

wer~ special trials, during the period of whieh experimental assist-ants asked each B On each trial about his intentions, expectatiohs and 50 on. The analysiJof this part of data has not been under the

'.

resp~nsibilityof the Japanese team, and therefore they were omitted

from our analysis.

3.

General Results.

The dependent väriable discussed in this section is the relative frequency of competitive responses, and the eïfects of the three experimental väriables, culture, grade and display, together with the trial effect, will be graphically presented.

Fig.2. snows the effect of culture and grade. The competitive-ness increases with grade in every culture. Japanese children are the

most com~etitive, followed by Greek children,Anglo-~ericanchildren

and Belgian children, in this order. The rate of competitive responses of Cnicano children is about the same as that of Anglo-American child-ren for the second graders, but i t does not in~rease with grade as much as the children in the other cultures.

Fig.

3

shows the display X Culture X Trial effect. The solid lines are for the Single Display condition and the dotted lines are for the Double ~isplay condition. The bottom line in the American graph is for the Mexican-Americans who were run only under the Double Display condition. -Chicano children are just about the same in competi-. tivenessas the Belgian children under the sameconditioncompeti-. As trial

proceeds, Ss become more competitive in every condition. As expected, children in the Double Display conditivn are more competitive than the children in the Single Display condition.

~vidently, the most conspicuous feature of this graph is

the large difference produced by the Japanese children under the two display conditions. Japanese cllildren under the Single Display condition

show about the same ( or a bit less) competitiveness as the Greek children. Under the Douple Dis~lay condition, however from the first trial block to the second, Japanese children show a large increase in their competitiveness and their rate of competitive response

exceeds 90

%

in the last trial block. In other words, Japanese child-ren are very sensitive to the condition, whether or not they can compare their scores with those of the opponents'. This conforms witn the

common observation about Japanese culture such that Japanese are other-oriented people and use a relative criteria in evaluating their performance.

This difference in competitivenes~ between the two display

•

conditions is plotted against grade for the four cultures ( excluding Chicanos ) in Fig.4. It is natural that the difference is large with Japanese children, but its decrease with grade should not oe misinter-preted, since i t is mainly due to the fact that with higher grade Japanese children who are quite competitive even under the Single Display condition there is not much room left for the Children under

the Double Display condition to surpass them in competitiveness. The difference in the competitiveness between the two

players forming a dyad (Intra-dyad.difference) will in general indicate the response similarity between the two players. By calculating the distribution of the intradyad differences over all the Ss for each culture, which is obviously symmetric around the zero difference, we obtain the standard deviations for the first and the last 25-trial blocks as shown in table 2.

The standard deviation of Intra-dyad difference within the first 25-trial

bL-:-~k is the smallest with the Japanese Ss and the largest with the Greek

Ss. This mGans that at the beginning Japanese children tend to choose simiiar

resI'cmses to tho<> .. of the opponent.;"f _ "'-no Gree!" children show the least care

about the similarity. Note that Intra-dyad differ~ilce is strictly '~~o~u~tional

to the difference in the accumulated scores between the two players. Because of the strict parallelism between Intra-dyad difference and the score, we may

say that Japanese children do not allow large score differences, while Greek

children apparently do not mind much the score differences. The same tendency persists to the last 25-trial bloek. In comparison to the first trial block results, the standard deviations 1n the last trial block are much higher, indicating that the players tend to accept larger difference between himself and the.opponent.

Let us now seehow the players' response patterns change during the play under eaeh condition. For the purpose of making this elear, we construeted the PPD (standing for Pre-Post differenee) means for eaeh S

in the following way : Aeeording to an ~pplieation of eovariation analysis

teehnique, whieh shall not be diseussed in this paper, it turned out that we

should biseet the whole data around the 25th trial.if we a~e to do it in the

most effeetive way. So, heneeforee, we shall eall the first 25 trials the Pre stage and the remaining 75 trials the Post stage. In order to eompensate the size of the two stages, we ffiultiplied the CR (the number of eompetitive responses) of eaeh player in Pre stage by three, and subtraeted it from his CR in Post stage, to obtain this PPD. Note that the zero PPD means that thc S behaved equally c0mpetitively in bath stages. Likewise positive{negative) PPD means that the S beeame more (less) competitive in Post stage eompareà

.5

dotted lines to Double Display condition. Fig.4 shows that, in general, the_,

shift from Single to Double display condition is represented by a displacement of the distribution from left to right, meaning that the additional

information feedback in Double Display condition causes a comparatively uniform

increase in CR in Post stage in each culture, substantiated by the general

covariation analysis indicating a rather small display effect in Pre stage. The rightward displacement is most significant with Japanese children. Also the large variance of the Japanese distribution in Single Display condition

child;~",-indicates that, even though Japanese are on the average the most competitive

11.

among all the cultures studied, there still are those who show a drastic shift toward the cooperative direction in Post stage when the opponent's

score is not explicitly displayed. Fig.6 shows the relation between the

mean and the

sn

of the distributions. Black circles are for Single Display

condition, white circles for Double Display condition. Standard deviations are all similar except Japanese Single Display condition. About the means, Japanese data exhibit a dramatically large difference between the two display conditions.

L4 . Covariation Analys~~---its principle and the results

In carrying out the data analysis, we needed some analytical tool lo7ith l-]hich we can capture the overallstructure of the whole data ---- a

numerical aid or a compass that will tell us in which Jirection our analysis

·should proceed in the whole complexity of the data structure automatically

suggesting unmanagea~lymany possible alternative courses.

Covariation analysis is a data struçture analysis developed by Toda (Toda,1974) , specifically suited to analyse the kind of data we obtain

of contingency tafules.

In appearallce, the method is somewhatlike analysis of vari.ance applied to contingency tables, since it gives the information in terms of ma1n effects, interactions and residuals. Howéver, it is essentially a non-parametric, llon-statistical technique, providing no significance testing of

its own, even though one can run parallel X2 tests.

The major purpose of this method is to obtain from data the following types of information : By dtcomposing a pooled large data obtained under

e

various different conditions, this method allows us to find which conditions are more important than others, the importance in the sense of the

contribu-tion by the condicontribu-tion to the making of the given data. The method also allo~s

us to estimate which direction of further analysis appears more promising than others.

Although 1 cannot go into details of this method here, let me

just explain the major technical terms. The internal covariation is the

difference between the observed frequency alld the expected frequency calculated

from the two marginal frequencies.(See Fig. 7). The internal covariation

obtained from the pooled matrix is called the total internal covariation w t . The amount of the total internal covariation represents in its own way how much regurarity is there in the pooled matrix, demanding.an explanation in terms of the differential contributions of the conditions. This can be done

by decomposing the pooled matrix according to one or more of the conditions,

and calculating internal covariation w

1' w

z' ..

wi•• for each of the decomposeà

submatrices. Then the sum!: w. of the internal covariations of these submatrices

1

represents the regularity still left in the data af ter the conditions are

ca11 the difference between the tota1 interna1 covariation and the residual,

h 1 . .

*

t e externa covar1at10n w This external. covariation represents the amount

of the data regura1ity that is explained by the conditions used to decompose the poo1ed matrix. In other words the externa1 covariation"ascribed to each eondition (or each combination of conditions ) represents the amount of

"

contribution of the condition (ol of its combination) to the making of the

who1e data. 1be ma1n effects of and the interactions among the conditions are all defined in terms of externa1 covariations.

Various types of covariation ana1ysis have been app1ied to our data. But 1'11 show here on1y the resu1ts of the covariation ana1ysis concerning the forma11y controlled experimenta1 conditions, with which, for the ease of

interpretation, we random1y selected exact1y 14 àyads for each condition,

and omitted Mexican-American data which 1ack Single Display condition. So in tota1 336 dyads were used in this ana1ysis.

Now let me show the results • The outer circ1e of Fig.8 ShO\>1S the

tota1 interna1 covariation, the main effects of tria1( 4 b10cks of 25 trials) , display, grade and culture, their interactions. Combining them all we can exp1ain 52.6 percent of the tota1 interna1 covariation by these experimenta1 conditions alane. The remainder is the residual. The 1argest main effect is the grade, and the display main effect 1S the smallest.

When we consider the individua1 dyad as acondition variab1e and dëvide the who1e data into 336 submatrices per dyad, the dyad condition exp1ains about 77 percent out of the tota1 interna1 covariation, as shown by the inner circ1e of Fig. 8. This resu1ts ,however, indicate just the size of dyad differences when all the other conditions are neg1ected. Neverthe1ess, it certain1y points out that no satisfactory picture of the data wi11 be obtained un1ess we go much deeper in Dur ana1ysis to the level of individua1

dyads, and this consiàeraticn set our subsequent course of analysis as will

be described in the latter part of this paper. ·Besides that, the results of

covariatioll analysis is quite reasonaole, and with full of important information.

Now let us see the results ~ more closely.

Fig. 9 shows the main display effect and the conditional display

o

effects. In this figure the whole angle of 360 corresponds to the total

external covariation of 759.78 excluding the residual in the outer circle of

-tht

Fig.8. Within~innermost circular band is represented the main display effect

as the three identical sectors. Within the second circular band is represented the three conditional displayeffects ; the left sector being the display

effect given grade, the right sector that given trial Th th· d b d

• _ - . '}I e 1r an

and the bottom sector that given culture.

~----rclpresents the double conditional displayeffects, and the outermost band represents the display effect given the three other conditions. The

discre-pancies between the inner and outer sectors are due to interactio.~~.

There are numerous important messages embedded in this figure, fröm which we shall sample just a fe\-7 ; The amount of display effect is almost monopo-rized by Japanese, especially by Japanese second graders. The display effect

really emerges in the second trial bloik and keeps on. This observation

,

comforms with the graphs shown in Fig. 3,4.

Fig. 10 shows the culture effect. Tbe scale is· same as Fig. 9. From Fig. 10 we can read the followings : Tbe culture effect is greater under Double Display condition than under Single Display condition. There is little cultural differences unàer the second grade Single Display condition. The cultural difference is the greatest with the fourth graders.

Fig. 11 is for the trial effect. Trial effect is small with the

condition. The trial effects for Japanese and Anglo-American are relatively larger than those for Belgian and Greek, but the large trial effect of

Japanese is obtained mostly' under Double Display condition. Fig. 12 1S for

the grade effect. TIle scale unit is changed to the one half of the preceding

figures. The grade effect is larger for Single Display condition than for

Double Display condition, and the smallest in the first trial bloek. It is also larger for Belgian and Greek than for Japanese and American.

5. Markovian Analysis

So far we have examined effects of the formal experimental conditions culture, grade, display and trial by covariation analysis. Though the effects thus revealed are interesting and informative, there still remains a large

residual to be accounted for. In order to improve our descriptive power

of the results, what we must do is to unpool the part of the data whieh

have been pooled,i.e.,the trials (within eaeh trial bloek) and dyads.

However, a mere ~eehanieal dëeomposition of data will not serve the purpose

. What we need to do is to decompose tne data aceording to some struetural criteria eoneerning the game-playing proeesses and dyad charaeteristies. The former leads us to a Markovian analysis, and the latter to a eategoriza-tion of dyads and players. In this seeeategoriza-tion I will briefly describe what

we have done within the context of Markovian types of analyses, temporarily

disregarding differenees in experimental conditions.

Remember that the rules of the game is completely s)~etricalbetween

the two players within a dyad, and two Ss were assigned at rándom as the Row Player and the Column Player. In order to aequire some additional iIlror-mation utilizing this arbitrariness, we shall henceforce discriminate the

two players within a dyad as LC ( less competitive ) and MC ( more competitive ) according to their frequencies of competitive chcices over the whole trials.

When the number of competitive choice was identical between the two players

(though such cases are rare ) , the two players are again assigned as LC

and MC at random. And when the data are pooled, LC and MC are identified

as the row and the column players, respectively. As a result of this

proceèure, the entry of cell 2 is never less than that of cell 3. Note also

that Cell 2 is realizad when LC is cooperative and MC is competitive, and

Cell 3 is realized when the opposite is the case.

"

We then calculated the CR frequencies ~ditional to the cell number

obtained ón the immediately preceding trial, computed over 10-trial blocks.

The result is shown in Fig. 13. Note that at the beginning the CR frequency

is the lowest af ter Cell 4 and is the highest after Cell 1. The two curves

move toward the opposite directions as trial proceeds, intersect at the third

trial block and reach the respective asymptotes,i.e. about .5 for the af

ter-Cell-} curve, and .8 for the after-Cell-4 curve. Although this result is

very interesting, it should not be interpreted fo imply the corresponding direct changes took -place in conditional response probabilities. Note that

in' early trials most of the dyads may have almost equally contributed to

each conditional frequency but on the later stage of play, as each dyad has gradually made its characteristics clear, the more cooperative dyads could have monopolized the After-Cell-} frequencies and the more competitive dyads

the After-CeI1-4 frequencies.

Such is a common pitfall 1n any type of Markovian analysis,

and in order to avoid this misinterpretation it is necessary to take the

characteristics, as we sha11 do in the next section.

Now let us look at the effect of runs of the same ce11s. Fig. 14

shows the CR frequencies p10tted against the 1ength of preceding runs of the same ce11. The re1ative 10cationa1 re1ationships of the four curves are the same as in Fig. 13, though trial b10cks are pooled in this figure. Following the run of 1ength one, meaning that the preceding cell is new1y rea1ized,

the CR frequency does not much depend upon the ce11. ~~en Ce11 1 rea1izes

twice in succesion, however, the CR frequency goes do.m be10w 0.5, and it shoots up af ter two Cell-4s, going beyond 0.9 on the average af ter more than 2 Cell-4s.

Ce11 2 and ce11 3 curves both show neither trial effect (Fig.13)

nor run effect (Fig.14 ). The two curves a1so resemb1e each other as far as these p1ayer-poo1ed representations are concerned. However, a c1ear

difference emerges if one plots the resu1t of the two categories of p1ayers, LC and MC,separate1y, as shown in Fig.IS. What this figure implies are as

fo110ws : As Ce11 2 continue to occur under the sacrificing.act of LC,

MC becomes more competitive as if to exploit the LC. In contrast to this marked tendency, the immobi1ity of After-Ce11-3 curve is significant.

Concerning the after-Ce11-1 and after-Cell-4 curves,note that their

re1ative positions are reversed as the run 1ength increased, the c1ear

message of which is that the continued' rea1ization of Ce11 1 and Ce11 4 shou1d be vita1 to create cooperative and competitive dyads, respective1y. Though not reported in this paper, the idea was persued more deep1y with individua1 dyads, and we now possess a good estimate of the impacts of continued Ce11 1 and Cell 4 over the succeeding choices. In addition to these considerations,

apparent in the first trial block, despite the large trial effect shown in Fig. 13.

6. Categorization of Dyads and Players

As~we have mentioned in the preceding section, it is indispensable to categorize dyads and players to some extent in order to make really

meaningful arguments about the processes of game playing.

It is obviously desirabIe to categorize dyads and players according to some kind of "naturaI" clustering of Ss' behavior characteristics.

CR

Therefore, we have drawn quite -a few scatterograms for

Ol

frequencies of

LC and HC of all the dyads over variou

s

stages (trial-blocks)of the playc~

However, the scatterograms were always rather homegeneous, and therefore we have decided to start with a simple mechanical categorization as a start which may later be subjected to modification according to some optimization criterion.

As a matter of fact, the mechanical categorizations we employed

~

worked very weIl, and we do not expect any drastic modification as result ,\

of optimization which is now under way.

The principles of the categorization Version .1 are as follows :

The ineasure used for categorizing players is the CR frequency·in Post-stage.

A player is classified as cooperative, denoted by

Q ,

if his CR in the

Post-stage falls below 30. Likewise, a player whose measure falls within the

range of 30 to 45 (inclusive) is classified as neutral, denoted by

! '

and

one whose measure is more than 45 is classified as competitive,denoted by ~.

Once players are classified this way, the dyad categorization follows automatically according to the types of LC and MC players of each dyad,

ending up with 6 dyad categories shown in Fig.

lu.

For the sake

of mnemonic convenience we named the ~ dyad categories in the following

way :COOP for

Q

x~Q (standing for cooperators ),. COP.P for ! x ! (standing

for competitors ), .NEUT for,! x N (standing for neutrals ), ~ for

Q

x !

(cooperators put against competitors, resulting in a creation of martyrs ).

~ for 0 x N ( cooperators serving neutrals ) and EXPL for N x X

(competi-tors exploit neutrais).

Now let us see whether dyads and players show their categorical characteristics beyond their defining properties, i.e. CR frequencies. Fig.

17 shows the conditional CR probabilities given the preceding cellof LC

and MC in the first and fourth (the last) 25-trial blocks, computed separately for thc six dyad eategories. Note that, beeause of the seareity of samples for Mart dyads, their probability estimates are rather unreliable,

partieularly the one af ter Cell 3. So, even though the Mart results are

interesting, we shall in general disregards Marts in our forthcoming arguments. Now we shall first look at the Pre-stage results. There are rather large non-trivia1 differences among different eategories in the pattern of four points, indieating that, even though the eategory membership is defined on the basis of the Post-stage results, the category eharaeteristies are al ready clear in the Pre-stage. First, pay attent ion to the broken lines

connecting blaek and white triangles,corresponding to After~Cell-2 and

Ater-Cell-3 probabilities. The relative positions of these two triangles are fairly similar in every category (exeept Mart). This is the type of result

we naturally expeet as it is deserib~by the following simple rule : Note

tllat the off-diagonal cells are those whieh give one of the players and extra advantage over the other. One took advantage then will become more cooperative on the next trial, and the one who was taken advantage will beeome more

ccmpetitive ön~ the next trial. Sinee this tendeney is obviously sa natural

e

block, xven though the lengths of the broken lines become much shorter. The relative positions of the two circles on the left hand side

graphs are also similar ~n every category except caMP • So this relation

is also very general in the Pre-stage, which, however, is not at all

preserved in the last stage except for NEUT. With Caap,MART and ca~w,

the direction of toe solid arrows is practically reversed. So apparently some structural changes are taking place through the course of the play.

Now let us get back to the Pre-stage graphs and look at the

NEUT graph. The characteristics of this graph may be very simply described in such a way that each of the two players of a NEUT dyad acts with only two levels of response probability, depending only upon his own previous

response. If his previous response was competitive, his next response ~s

very likely to be cooperative, and vice versa. Remember that a NEUT dyad is formed with two N players (neutrais). The same rule also applies to

N players in other categories. So we may infer that N players are effectively represented by response alternaters ..

The same response alternation tendency is also observed with a and X players who are combined with N players, though their conditional probabilities (given onels own previous response) are of smaller variance and off centered ; those of a are more cooperative and those of X are more competitive.

Then, even though there still are more complications to

be accounted for, particularly about caap, caMP and ~UffiT, we can at least

have a fp.ir hope that a rather simple hypo thesis like the one stated above

may explain a large part of Ssls initial stage behavior.

Now the shift from the left hand side columns to the right of

for response alternating N players' two conditional probability levels is

much reduced. The response alternating tendency for X and

a

players

almost disappear. The CR probability of X players becomes quite extreme, while that of 0 players do not undergo much change. Partly corresponding to these, only those categories which involve N players do not exhibit the reversal of relative positions of the two circles.

As we have al ready seen of Fig. 14 that, when all data are poole4,

the cell-conditional CR frequencies showed a marked difference for cell-run lengths greater than one, and we had a suspicion that these differences mainly reflected the langer run lengths imposed more selectivity upon the types 01dyadS which could produce such long runs. The suspicion was duely

confirmed ; long Cell runs were produced mainly by

coap

dyads and the

majority of long Cell

4

runs were contibuted by

caMP

dyads.

So, in order to see the development of dyad category characteristics over trials in a fair, more comparable way, let me select cell-conditional

probabilities given run length and compare t~em among different dyad categories.

Fig. 18 shows these

~un

one cell-conditionals for LC and MC plotted against

25-trial blocks, aud categorical differences are apparent even with rUIl 1

,getting more and more enhanced as trial práceeds. The major characteristics

of this figure will be depicted as follows: Af ter Cell 1, the among category

differences are small ön the first trial bloek, but as trial proceeds,

COOP dyads shift toward eooperation(lower left),

caMP

dyads toward eompetition

(upper right) and MART dyads go toward lower right, ereating the characteristie MART outeomes.

CO~ dyads invariably develop increasing eompetitive tendeney

,

compared to rather small effects af ter Cells 2 and 3. EXPL dyads show a trend similar to those of CaMP , though the level of competitiveness of the former

is kept lower than the l~tter. MART dyads show the characteristic martyr

shift toward lower right _{'irrespecti~~} the preceding cello The curves

for SERV are in general similar to those of COOP ,and ~~UT shows the smallest

trial effect among the six categories.

The implications of all these results are quite complicated,

however, and hardly allow us an unequivocal, unique inteypretation. Itl-order to proceed ahead, we have to build a working model to secure our basic ground and to provide ourselves with a sort of effective information processing system that at least allows us to screen from the tremendous pile of

avail-able information those immediately processavail-able and those calling ~or further

consideration.

Before going into this topic, however, let me just briefly

mention a few other types of results obtained at this stage of our analysis. The change in competitiveness from Pre-stage to Post-stage is plotted in Fig.19 for COOP,COMP,and NEUT categories.(The CR frequencies in the Pre-stage is multiplied by three to enable a fair comparison.) What this graph indicates, among others, is that there are rnany dyads

the members of which change their response propensity rather extremely from Pre-stage to Post-stage, a change that is obviously responsible for the creation of COOP and CaMP dyads.

A detailed Markovian covariation analysis haD been carried out

with cell, trial, category and cell-runs. The overall results are very complicated and I shall not elaborate it here, except.mentioning the

following two points: The rnain effect of the preceding cell is artific.ially

positive external covariation and Cells 2 and 3 also a large negative external covariation, which mutually cancel out. When this happens, we need to deal with each cell-conditional contingency tables as separate data. Secondly,the

category factor explains about 63 percent of the total internal covariation (when cells are not separated), a value larger than the effect of all the formal experimental conditions combined described in Section 4, indicating a fair efficiency of our dyaà categorization.

At this juncture, it may be of some interest to look back at the formal experimental conditions , and see in vlhat proportion each experimental condition contributed to the creation of different categories. The result

is shown ir. Fig.20. Let us first look at the culture part. About Japanese,

CO~ dyads occupy about .68 which is the largest among the five cultures,

while their COOP is also the largest even though in a much smaller scale , leaving the rest categories to occupy only .26. The Anglo-Americans

entirely miss COOP. Belgians are ~racterized by the smallest COMP and

the largest NEUT among the five cultures, while Greeks are of the largest

MART.

Cáncerning the grade effect, it is noted that the proportion of COMP increases with grade, while the proportions of COOP and SERV decrease. The proportion of EXPL remains nearly thesame. Moving from grade to display

Q,II,CI l-x.p...

, we see that the proportion,of COMP_{v _ _1\}inc~eas~ from the Single Display

,

conditionjto the Double Display condition, !ilJ'lo LQQlaitm·

me-

sáme, and the

other categories are compressed nearly proportionally.

Now let me proceed to our categoeization of players Version 11 as a preliminary to the model buîlding. The major distinctior. of VersionII from Version I is that the players are now classified also by their

., and then goes done to ] for

as weIl as their CR frequeneies. Another minor alteration is that the elassifieation of players is no longer based on the Post-stage data alone. It is done, instead, based on various parts of the data, depending on the purpose of analysis.

Now note that the maximum possible NR is linearly goes up with ]

CR up to CR

S

2"

Max CR

]

2"

Max CR< CR,::; Max CR as shown in Fig. 21, 50 that we may only need to split

aeeording to NR the players with the middle range CR if we want to keep the number of player eategories reasonably smalle The eomparable

elassifieation sehemes for 10-trial bloek and 25-trial blöek are shown

in Fig. 2] with eategory symbols. Note that the + players in this seheme

eorrespond to Ö , =.to! and.!!. and

!:

to ~ in Version-I, if we negleet the

trial bloek upon whieh the elassifieations were made. From now on the

eategory to whieh a player belongs will be referred to~as the state of

the player on the trial bloek where the elassifieation is made.

Fig. 22 shows the relative frequeneies for the four states plotted against 10 trial bloeks.The curves demonstrate that the major state changes

oeeur among the three states --,.!!.-and~, the frequeney of the first steady

'increases andthe other two steadily deeline, while the frèqueney·of +

remains almost the same, Corresponding to the state change for individual players, we ean also plot similar results for dyad eategories represented

by the combination of the states of the two players. Fig. 23 goes along

weIl with Fig.22 as expeeted, though it also display player interacions

as weIl. Fig. 24 shows the state transition probabilities for players

the tendency to stay 1n the same state 1S the highest for all states except

L. In particular the transition- ~ - remains extremely high, suggesting

that the -- state operates as an absorbing state. On the other hand,

however, the increasing tendency of + ~ + and H ~ H imply that a sort of

response fixation takes place with + and H types of response modes as trial

proceeds. Compared to these other states, .the state L àpparently plays a

special rolè such as the middle-way station from + and H to - • This is,

of course, not much surprising, as the state L represents, by definition, the really random choice, and as such, and L-fixation behavior can hardly takes place. Now let us go a little deeper and see how these state transitions are affected by the state of the opponent. Fig.25 shows the results only

for the transition between the first and the second 25-trial blocks, as, in general, there is not much trial effect in these patterns. ASflearly seen, the transitions from the - state are very little affected by the opponent's state. The other transitions are more or less affected by the opponent's state, but the particularly noteworthy is that the - state of the opponent significantly increases the transitions to -, which, coupled with the absorbing characteristic of the state-, apparently precipitates the general rush to competition. This attraction effect of

. .

the opponent's - state, however, gradually goes down with transitions

from + and H as trial proceeds though not shown in Fig. 25, and, in

particular, reaches effectively the zero level for the transitions

from + in the period from the third to the fourth trial bloek. Instead,

however, the attraction of the opponent's L state proportionally increases,

indicatinj that the response fixation for + and H disèussed above is no

Now let us shift our attention slightly, and examine how eaeh player's initial response tendeney, represented by his state in the first

10-trial bloek and ealled'the original state, determines his later states.

The relative frequeneies öf eaeh state in every 25-trial bloek for eaeh

original state are plotted in Fig.26. Sinee the first 25-triál bloek ineluded the first ten trials used in defining the original state, there is no wonder that the same state as the original state oeeurs most frequently in the first bloek. Nevertheless, their values are far below 1.0 (except for the state - ), implying that a drastic response shifts take plaee during the period of 11 - 25 trials as suggested by Fig. 22. As trial proeeeds, most players move to the state - irrespeetive of their original states, even

though a slight tendeney remains sueh that the g~eatest population of eaeh

state is, as a rule, oeeupied by the original inhabitants.

Now we ean also define the dyad original states by eombining the player original states. The top graph in Fig. 27 shows the proportion of their remainig in the same state as the original dyad state for a few samples. The eontinuation of the state -- is the highest over all the

trials, then eomes the state ++, though its proportion nearly steadily goes

eown. The remainig dyad states shown in this graph,i.e., HH,LL and HL

show a preeipitouJdrop in the first transition. Thethree lower graphs in

I

Fig. 27 show whieh dyad states they have mainly moved t~. The superiority

of the state -- now eomes as a no surprise, but the faet is interesting that the dyads whose original state was LL showed the least resistanee.

7. Simple State-Transition Model

model ,obviously, cannot be very much sophisticated. So we decided to try out a simple state-transition model which predicts state transition

probabilities given the opponent's current state. The model involves the

following parameters: 1) The remoteness

Q

from one's current state to his

possible state in the next trial block. The value of remoteness is symmetri-cal between any pair of states, and the remoteness between the same states

is normalized as one. 2) The attractiveness K of each state. 3) Tlie effect

of the opponent's current stat~ , which is supposed to enhance the

attractiveness of the opponent's current state by being multiplied to the

corresponding!. As the states are defined under Version 11, there are

four K's, among which K+ is fixed to one for normalization. Likewise, there are six free D's, those representing the remotenesses between two different states.

For convenience, let me show the equations connecting these parameters in a form óf an example, the principles of which can be easily

extended to general cases. Suppose that one's current state is Hand the

opponent's state ~s

-the form:

Then there are four transition probabilities P of

P(+jH,-)

=

_--!:L

lra-

+t.+·H-D+H

I . ,-

, ,

P(H/H,-) =

--!lL

/LH,- +E

_H·H -DHH

'

,

(1)

~/t

P(LjH,-)

= +( DHL / H,- '- L;H, P(-/H,-)

~I<-I[H

-

+

E_.

_{H _}

HL ' , ,

where the normalization factor

LH _

_,

is expressed as

(2) + +

and

t's

are error terms such as

(3)

_LEi;H,-

_{= 0}

i

At the present stage of our model building, the purpose of the model

1S to estimate these parameters corresponding to each of the three transitions

betw~en the consecutive 25-trial blocks, whereby the left hand side transition probabilities are estimated from the corresponding relative frequencies.

The parameter estimation was done through iteration by minimizing

the disturbance (or stress) measure,

$' ,

defined as

(4)

s

₌

L

k, i,j

N' .

1)

where Nij 1S the number of players who actually made the transition from i

to

1..

The result of the parameter estimation is shown in Fig. 28.

Unfortunately, however, the author's stay in Eindhoven has ended when these first set of estimated parameter values were obtained, and their reliábility

(or unreliability due to the risks inherent in the iteration technique)

has not yet been checked. Therefore, the values shown in Fig. 28 and the

conclusions drawn from them are only tentative.

Admitting that, we can observe the following interesting facts:

The attractiveness K and the opponent's effect~ stayed fairly stabIe,

The parameter that underwent great changes we re the three remoteness measures which did not involve the state L, endorsing the hypotheses that the response fixations would be represented by increasing remoteness and that the state L played a special role of a middle way station.

Let me emphasize again the tentativeness of this conclusion and point out that the reason is not at all clear at this stage why remoteness rather than attractiveness changed with trial; whether it is really what the data dictates or an artifact due to the structure of the model.

REFERENCES

McClontock, C. G.

&

Nuttin, J. M. Jr. Development of

~

competitive game behavior ~n children across twc cultures.

J. expo soc. Psychol., 5, 203-218, 1969

Toda, M. Covariation analysis : A method of additive data

structure analysis for frequency taóles.

Column Flayer

a b

Ro\\' Player

a b

Fig. 1 The payoff matrix.

~he left-hanè number in each cell is the point for the

row player, and the right hand number for the column player.

Table 1 Sample eize Greek Culture 13elgian Japanese Ss 220

I.

I

108 I I

I

192 I i I I 168 I

I

180 ! 90

54

96

84

110 Number of dy ad anglO-.dmerican rlexican-Ameriean ~ - ~ - - - _ . ~ - _ .

I

I

-_.,~.... , 868

i

---1

Table 2

Standard deviatiolls of the distributlon of the intradyad differenees

---- --- - - 1 i'<"umber of Ss I

---=--t

192 [

!

4.25

4.15

IFirst 25

1

Last 25 , S.D. I S.D. Japanese .n.nglo-··Ameri e an

in the eompetitive responees in the first and last 25-trial bloeks.

r

-I

lIJulture

Belgiar.

4.16

4.,;4

îGS

III

1.0

IV III p 0

0.9

Pi III OJ S-t

0.8

Cl> > .r! ~-" .j...l

0.7

.r!

.

_.'

-.j...l

/ '

.-Cl> J)." .. -Pi _~

..

_'

.

-Ei

_0.6

_{" . •}

_-c:a

---0

.

..

-() / , -

--'H

_0.5

~-0 Cl> .j...l

0.4

cu Cl:<

T

•

Japanese X _Greek À Anglo-rl-merican 0 _Belgian

a

Chicano 2 4

6

Graàe

Fig. 2 Culture X Grade effect.

( Display conditions pooled )

5

4

3 2 1 Belgian .

_...

....

.

....

...

.

...-./~~/

~

5

4

3

2 American 1 5

4

3

2 Greek 1 5 4

3

2 Japanese

.

...

--.

-

.

...

...

IJ'

,

I I

/

I 1

1.0

<ll III I:l

0.9

0 Pi III Cl> S-t

0.8

Cl> > 'r!

0.7

.j...l .r! .j...l Cl> Pi

_0.6

.S 0 () 'H

0.5

0 Cl> .j...l

0.4

m p:; 20 - Trial Bloeks

{Q ~ 0 'n +> .r! 'd ~ 0 UI 0 111 (lJ _:>:. ~ cU (lJ r-i ::- P< .r! 111 +> .r! 'r! 'Cl +> (lJ 0 P< E: El +> 0 0 ~ (lJ ~ (lJ 'r; _+>

a:

(lJ (lJ 0 .0 ~ (lJ H (lJ 'H ~.~ .r! Q 0.20 0.10 O. 2 4 6 2 4 6 2 4 6 2

4

6 Graàe Anglo-American Greek Japanese

Fig.4 Display effect for Culture X Grade.

Oràinate represents àifferehecs {n the rates of competitive responses between the Double Display conàition and the Single Display conàition.

·25

Greek -~ N NVJ v..' +-+-o \.11 0 \.11 0 \.11 1 I I I ' 1 \

,

\ \ Chicano

.

... "",--,

"

.,....", Y . I

,

,

.

, " , ,".....","'''' I • PPD o -' -' 1 \.11 0 \.11 +- 1 I I "... 1 1 1 -'-'\.11 \.110 1 1 NN \.,'1 0 +-\.11 I I 1 1 1 ,

,

,

.;-_./ "'- ,. .. " \ '..,.e'--\---+--'~--1 1 V<VJ \.110 VJ \.11 I VJ +-\.11 \.11 I 1 N \.11 1 - ' \.11 I PPD ,'\

,

\

,

\

_,

\

,

,

\ PPD o , \.11 +-,

-L~~~~~:

iL\.11 \.11 \.11 4-- 1 1 1 I. - ' o

,

- ' o

,

,

N C

,

N o 1 VJ o Anglo-American

,

\.,,, o ''''~-. .,.; < .~, ... +-\.11 1 +: \.11 1 VJ \.11 1 N \.11 1 \ \

,

,

_,

\

~....,...

..

~-:x'" N VJ \.11 \.11 1 ,

,

_,

\

"'

,,

\ \

,

\

_,,

'

...

PPD

- ' \.11 1 - ' \.11

,

Belgian . __.,\-.-..- ..--- -t

PPD

" \.11 +-1 \.11 +-1 I - ' o

,

- ' o

,

1 N o

,

,

N o

,

I ,.

,

"

---,~---_.~.~,-~, ._~_.

--.\---I 1 I +- VJ +.- 0 0 - ' I

,,

~

,-

' ,1

,

- - - --'--+--4'!-=-"~""~--r-,,~m . .~--~~~-,---+~ • • 1

.15

.10 .20

.05

.00 .0) IJapanese ~

g

.25

(l) ;::s ')0 a' • L-U> H 'H ~

.15

rl +> (û .10 rl <l> ::r;

Single Display condition -- - --Double Display condition Fig. 5 The probability distributions of PPD under the two display condition

o

5

Hean

10

15

Pooled matrix all SI Rl N Decomposed matrices

,

_s' all _{. 1} r' n' 1 a' , s' 11 1 r' , n' 1 s' r' I 1 n' s" rIf = a" - 1· 1 11 -n"

WT Total internal covariation

Internal covariations . tt'

w:

External èovariation

*

W = W -T W.~

RESIDUA L

---TOTAL

I NCOV

=

1444.6~

EX C0 V ( T

x

D XG

'i.C )

=

Z

5 9.7 8

EX

C

0 V ( DYAD )(

~

)

=

~ ~2 8.65

Fig.8 The overall results of covariation analysis.

DISPLAY

EFFECT'

2'C

=

7 5 9.7 8

(E XC 0 V )

'rrial block'

,

. . '

,...

•

....

Grade 2,

4, 6

Culture Japanese ~nglo-Arnerican ~elgian Qreek

• •

•

•

CU l TURE

EF

FE CT

27t

=

759.78 (

EXCO V )

Fig.j

10

Covariation analysis Culture effect

Display 1t II

TRIAL

EFFECT

2x=759.78

(EXCOV)

I/

GRADE

EFFECT

1C

=

7 5 9. 78

(EXCOV )

Fig. 1~ Covariation analysis

1.0 0.5

-

---

_.---

....

--

-..

---..

"

_

... • ' •• - - - - ;-S- ... A -~-~ ..&~~~~~-:::-:--_-"':A>":::::':-:-'-::-_~_~_·_-:"'_~'::'A~-=-_-_- ___ 0 - - - : ..~::.._4" - · A 6 - , -..- " 4 ,,' ~-.., --~ af ter eell 1 eell 2 eell-3 eell 4 ~ CIl ç: o '.-1 .(..I '.-1

l

]

o.

0

L_-.L._--,__

~----,--~_-.L._----4---~--2 3 4 5 6 7 8 9 10 10-Trial Bloeks

Fig. 13 Conditional CR probabilities given the preeedingeell

plotted against trial bloeks (Runlength-pool)

1.0 0.5 0.0 .' ~'----~

-

--

:.- --~ 0- -_- _ -.:10- - -A-:" ....

,.-2 Runlength

_--0

4 - - - 4 af ter eell I eell 2 eell 3 eell 4

Fig. 14 Conditional CR probabilities given preeeding eell

1.0 3+

"

C eell LC _,

,

af ter

..

eell 2

"

""~ 0.5 À eell 3

•

eell 4 0.5

MC

1.0

Fig. 15 Conditional CR probability given the preeeding Cell and

the length of Run for the two players.~ Trial-pool)

.30 .25 .20 • 15 • 10 •05 -\ \ \ ~

.

o

CÖÖP NEUT COMP I I I I+:'-w +:'-OU1 - I I I I WN OU1 I I I I N -OU1 I I I I 0 -lJ1 I \ J 1 -O ' · + : ' - I -O U 1 I I I NNW 0 U 1 0 I I I W +:'-U10U1 I I I PPD

/ l

/

co~ (20~

I

/f----

---J

/ I / I /NEUT

I

EXPL (96 ) / / (72)

I

~11'>/

/ ///

r-/ -

f' /

./

./io/ ; , r , " I

.,//

i

/ .. -'

I,:

/

: ' / / j :

/ L , / ' S E R V !

L

/

_{----.~.--_.~,j---L}

caap

(

J4 )!

I

,

(

42)

I

: MART ( 8 ) i

I

75

l

CR of LC X 46 45

•

l

N 30 29

a

o

a

---a

29 30 ~ 45 46 N CR of HC

--

x

75

Fig. 16. Categorization with dyads.

The numbers are of dyads belonging to each category.

All the Mart dyads locate in the upper left quarter(the

coop

_{( 0 x 0 )} 1 . 0 0 1 - - - ~---___r first 25 trials

o

last 25 trials 1.0 ( Nx N ) 0.0 1.0 ~ "

f

'..,l'n

- - - - -

.

1.0 0.5 1.0 0 SERV (

o

x N ) last

"

, ,

,

N NEUT first

I

'--:---::-_0::---_ _1 0.0 0.5 1.0

~'Y

'~, '"4 0.0

l,

,~

~

0.0 0.5 1.0

o.

00 ~---4- ~ 0.0 N N

Rate of competitive responses (more competitive player)

Fig. 17. Conditional CR probabilities g~ven the

.0

af ter eell

preeeding eell for eaeh eategory. Estimated from

A

af ter eell 2

1.0

l

_I

I

I

l

I

I

1 L - ' . . . ---J. 0.0 0.5 1.0 MART (0 x X ) 1.0

J

x

I

0.0

L,_-+-,_

0.0 0.5

I

_I

0.5 I i I

,

o

1.0 ( N x X )

I

t

!

1.0 EXPL

i---l

0'5~

\1

I

I

V

I

r

I

i\

I

I

0.0

L--~~---J

L

__,--0.0 0.5 1.0 0.0 N al po .~ +> .~ +> al Pi -f.:l o u al po .r-! oP .~ +> al p., El o u X 1.0 COMP (X x X ) X

I

I

i---Yl

i .

~~

I

.

. I .

\

11 0.5

L

~

,.

i

h

I

I

.1

I

,

I

i

0.0

L----.

J

L __

'_..L-.-_ _

J

0.0 0.5 1.0 0.0 0.5 1.0 X

After-Cel1 til al '..-1 .~

·f

-r4 '..-1 ..0 til ..0 o I-l p., p:: ü After-Cel1 2 til al 'j

·r--..-I r4 • ..-1 ..0 cu ..0 o I-l p., p:: ü ---_._---_.-

,-/.0 CR Probabilities

SERV NEUT COOP MART EXPL COMP o

"

o

····f

/,,0

i

1 ~3 'f . ~ I After-Cell 3 After-Cell 4 /. 0 , -L

cl·

i

I

LC

i i

• S "

• S-CR Probabil~t'.L ~es

Me.

Japanese

MI \'---"'-"-~'-, -.----.~.- -- - ...•-..-.-.-- ---.- - " , _._-~

"'·'1

NEJJA! I

COOPi SERV! UT

~I

EXPL

I

COMP

I I

:..L-.-._._ _

--l_~

..__.. .._._ _'.' __

'<~

' " •• _

~

• •" , , _ _ _. ._

~.,,>c _~_

.._._, __

_ _' _ __.. .. /

\ I 1

\---r···..--··--..----·---..··-..··---··--·--···-·.._..-_.---..

~_

..

~~~~can ~UT

I

!

ExP~

I_.

~~~

__.

._

..__

, ..

Belgian

Greek

I

I

-.__._-._.-..__.

---r--·>c..

'"···,,·---~-.,·~~··,··~"'_·

-..,-.-.-..

--.--~

.."-"---._,,

~

-._-.,

····1

~

SERV

I

NEUT

!

EXP:

~o~-

...- - , - - - - -..

----.l

~J ~ERJ~;

_ __

rI:;~==

__

=-=~I-:·~=_~;_::·-:=:·===-

__

~~-_-=._=:::1

~

:----·r---..·---..

-·-T~T

....·-.."·-·-..

_··~·_····~··,--·

..._.,...,._"...

"""0"'''''' ...'>."...,'I' .'...." ....

0,

I

I

R.j

Chicano

'---'-

Pi SERV

----!..____

1 NEUT 'T EXPL COMP

l .____ ___ _...J.. ••_ _._ ~•• . _._ __ _•. .

Grade

Second

,.--._-,....,.._-_.. -- ...----..-... -- -_. -.'r'-~- '~" -- ...-. -. -.- .. -,. .. ", r '

I

I ,

IMi

\ coop

l

SERV

I

NEUT

;~j

EXPL

,_ _._.~ . ._ _...L _...._ _. ._ .• .•,. ...•. .1 __• ... ... • ..._ CaMP

---_.-_

....--.. Fourth SIXTH Display Single Double

iI

1 - - -

·--!Ml·"·----"--· .. ··-,,·

_._. ·..

···r..·"···_..··"..

~··'"···_·_···

-..

_"._~

_-_._

_"..-.- "

I1" '1;IR

I

lL§.F;By~~~2:_

.._._ __..

_--.l:.l ..

".~~~~

"'__ _._

• caMP~._.·._.._ _ ..v"'" ..._<. ....

c] .

'r _."..'"."-~.-.~" ,. _._ _ -._ _.--" I " .--..--- "'-..----.- -._-. - -., __.._ ..-- _~---~..- - ..-- _ - ._ - -..

;,l'\

NE

~;

I 1~IR UT RI E X P L ! CaMP . lfiv'

Tl

i

_ _ _ _..._ _"._ _"'...""',...~...,.."..-""'" • __" ....-..."._'...,.' .. ' -...~....,.~".~,~,...,.__~..."~··'v'~..-""'_""'~'..~,'"·,,~,',--"-,, "'...",..- ..'" •....··'··;"T\h"" "~••·,'., ...,...,'t·.,,...,·..._,,-.·,·...-..."~,.".,,,..,.,,,,_.,,,;..r.'~~-"" N'''<'

-·-1---·"·..

-r'···-··

w._ .. -. ---

-->

T-"--'-'-"-~~--"'--- .. ",." - - - -- -- - - . - - - . - . - - - .-. r I

1~11

i

I

I I R . '

L~~~L..::RV~

..

J __.__ ...

N~UT

I

Tl. __

EX~~

. . .» .

.L.

CaMP

Q)

·03

:>

...

4J C'il r-I

·2

. Q) p:::

. I -

_

....--.>---.;c.---_ _

---.I>---<---

_----0

G

+

o

1

.r

IQ-trial blocks

6 7 , .

10

Fig. 22 Relative frequencies of personal states plotted against IQ-trial blocks.

IQ-trial blocks

3

5

7

V

[J

/ i'

2t; )

(

I

/

-t-I

/

~'--"

H

- ;---_.._~

\

L

cR

\ \ {D ) l

7

o "-

J '1 '"\. 10

Fig. 21 Definition of personal states for 25-trial bloek and lO-trial bloek

.I

s

0 _-'T--t 0 ,* -A _~r L1

--}L

0---. 0-.. ---0

L

• __. _ • ~_.. J 1-71. -) 3 -""/r~

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----.--.

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~. ... .. -.-I .n Cd .n _.~ .. 0 l-i p.. ~

₃

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t--=...:...:":::: ,/._./

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P-S .

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g

'"'" Eo-<

S..2 --"

Fig. 25 Transition probabilities of personal states for the transition from the first to the seeond 25-trial bloek.

P ( S2 / SI' Si) Si : personal state in the i-the trial bloek. si : opponent's state in the trial bloek 1.

[)

I

I

'I ,.

~<:.,.

.,/,/

L

H

• .J,. • .. ->~--~.:- ~i .~ ~ ~

,

~~ o Trial bloek 4

+

.. ,I.

/

o

,

Trial bloek 3

;'/6

'/.i~

JI",i, ... !{/// --... .1.--- ..

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+

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L.

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9

I

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Trial bloek 2

L

:-(

-I-Trial 'bloek 1

?

r

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I

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L

+.,'...

H

L

1,0 Q) ':;' .~ -IJ (Ij .-I Q) p::: CIl Q) .~ t) ~ ~ .!; I-0" Q) '"'" 4-l Current state

Fig. 26 Conditional relative frequeneies of states given original states for eaeh trial bloek. 0

+

r or;j

t

-I.Ii .o.S

r

I i

or;

cin:,L [(Jrh";>::: d

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+

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--~ f- T 0

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_{---'.> H}

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A

ti

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L.L

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L.L

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I D.S

l

I

/.0 IO-trial bloeks IO-trial bloeks o HH-)

•

Hf1-è}

A H f-l ---}

x.

Hf1--}

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f

I , <~ j ,-,

x"

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1.--->3

30

I-l

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t-I i I QJ

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1-_I ;j \ ..-l cd I ~ f~ ~

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I'

,

/

/

10 .~ ;. / / 0

Dli

-..//

iJ ,////

Fig. 28 Parameter values estimated through

iteration in the simple state-transition mödel

10

~

,

i L I j

r

_I r

x

00 ,6.( o [1 - - - " " - - ...~-~...._--~~ ~----. ,_.- "._--"-".~..--l-_.. _-..

,.