THE FORMALIZATION 0F BALANCE: ' _
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ABSTRACT

THE FORMALIZATION OF BALANCE THEORY AS AN

EXPLANATION OF ONE-STEP DEPENDENCE

BY
Philip S. Hart

Through the technique of formalization, or the
process of making explicit the logical structure of a set
of assertions, the author is attempting to account for the
'lack of fit' between theory and empirical results in the
Camilleri-Berger model of rational decision-making. The
theory predicted that a trial-by-trial independence pro-
cess would be generated.. However, the data revealed a
one-step dependent trials process which as an underlying
dynamic is not accounted for in the original theoretical
formulation.

The concept of balance as postulated by Fritz
Heider and later formalized by Dorwin Cartwright and Alvin
Zander is the methodological tool employed as a possible
means of accounting for the one-step dependence process.

A central conceptual and methodological derivative of

balance theory which is utilized is that of the degrees

Philip S. Hart

of balance in a system of relations. This concept is ex-
panded to provide a system of ordering predictions. Pre-
dictions are made within and between systems, and such an
ordering provides further explanation for the dependence

process.

It is pointed out that the ad-hoc nature of the
original theory did not approach with sufficient clarity
the dynamism of the verifying device, i.e., the experimen-
tal design. This inability aids in accounting for the
discrepancy between the theory and results. The author
proposes that a dual level of analysis is necessary if the
original theory is actually valid and/or malleable. These
levels are; (a) the conceptualization and measurement of
intra-individual balance, and (b) the conceptualization
and measurement of intra-systemic balance. It is pr0posed
that with a thorough grounding of these two levels of
balance the generation and understanding of an independent
trials process would result.

The formalization of balance theory as outlined
here thus serves both an explanatory and heuristic function.
It is explanatory insofar as an accounting is provided for
a dependence process-in-action. The formalization is
heuristic insofar as its function as a generative technique
for further theory and model-building. Such a model has
been explored by the author in another paper, and the model
under consideration is a modification of Bernard P. Cohen's

model of conforming behavior, but is deemed a 'balance model'

THE FORMALIZATION OF BALANCE THEORY AS AN

EXPLANATION OF ONE-STEP DEPENDENCE

BY

Philip S. Hart

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF ARTS

Department of Sociology

1969

TABLE OF CONTENTS

Chapter Page
I HISTORICAL I O O I O O O O O O O O O O O O 1
Introduction . . . . . . . . . . . . . . l

The Original Experimental Design . . . . 4

The Problem Under Investigation . . . . 8

II A MOVE TOWARD EXPLANATION . . . . . . . . 11
Use of the Diagrams . . . . . . . . . . 25

III BALANCE/IMBALANCE: SYSTEMS AT WORK . . . 45
IV CONCLUSION . . . . . . . . . . . . . . . . 50
NOTES . . . . . . . . . . . . . . . . . . . . . . 54

REFERENCES 0 O O I O O O O I O O O O O O I O O O O 5 7

ii

CHAPTER I
HISTORICAL

Introduction

 

Social psychological research literature has been
the scene of a number of research designs incorporating
the concepts of subjective value1 and utility, either as
central tOpics of investigation or as components of a
larger problem. Through utilizing the concept of utility
and an attempt to measure2 it, there has been the attend-
ant concern with subjective probabilities. For once one
is able to measure the utility attached to performance
(or non-performance) of a particular action-choice, then
the subjective probability of making such a choice can
also hOpefully be measured. This concern with the meas-
urement of utility and subjective probabilities is a vital
one in statistical decision-theory. And Sidney Siegel in
"Level of aspiration and decision making"3 states that a
"behavioral model of decision-making should contain not
only subjective probability but also utility whose main
concepts are level of aspiration and reinforcement of
effects."

One such decision-making model to which my con-

cern will be directed here is incorporated into a research

study carried out by Santo F. Camilleri and Joseph Berger,
"Decision-making and social influence: A model and an
experimental test."4 A key point in the model and the
experimental design constructed to test it, is the Actor's
subjective structure. For the authors make the point that
if this subjective structure is known then Actor's choices
should be predictable. Camilleri and Berger attach utili-
ties to three sources of potential feedback to Actor and
these utilities are fundamental in the decision situation.
These utilities are used in the construction of models of
choice behavior in the attempt to predict the probability
of particular responses in experimental conditions.

There occurred a breakdown between the theory and
the data in this research study. The original formulation
predicted that over a sequence of trials in the experi-
mental situation that an independence5 process would ensue.
However the data mirrored more of a dependence process in
action. Thus my concern here will be with develoPing
viable means to view and interpret the data gathered in
relation to 'testing' the Camilleri-Berger decision-making
model.

A Markov6 process seems to be in effect in which
P 1 depends upon Pn 92ly_and not on the earlier values

n+

of the function p (where P = df the probability of a

 

n+1
particular choice on trial n+1). This Markov prOperty

of the model defines the phenomenon such that "the subject's

performance in trial n+1, although dependent on his level
of behavior in the preceding trial (as measured by Pn),
is independent of the past record of performance leading
to trial n."7

Thomas L. Conner in his M.A. thesis (Stanford
University, October, 1963) "First order dependence and
Self-other expectation theory" also made note of the de-
pendence process spoken of and he tried to demonstrate
that "the very structure of the experimental situation
is such that one could never have an independent trials
process."8 And in the attempt to explain why dependence
rather than independence prevails, Conner utilized par-
ticular of the notions of Fritz Heider's9 balance theory
as formalized by Dorwin Cartwright and Frank Harary.10
This attempt to account for theoretical and empirical
divergence Conner attacked by revising the original
theory. In his revision, Connor considered the two ex-
perimental states "where the individual holds high ex-
pectations for self and low expectations for other [+-],
and where the individual holds low expectations for self
and high expectations for other [-+]." (Page 19)

This revision attempts to lay the dependent re-
lations as found in the lap of (a) states of imbalance
leading to tension which foster alterations in perfor-

mance in the attempt to restore balance in the microsystem,

i.e., if a particular choice leads to imbalance, then the

probability of that choice on the next trial will be less-
ened. By viewing the situation in such a light, dependence

is necessarily an assumption because the choice situation

 

is a function of the preceding trial. That is, an explana-
tion based upon the notion of balance makes dependence a
gizgn as well as an a postereori finding. The second
point which Conner felt fostered dependence was (b) the
structure of the situation. And he goes on to state in
relation to these two points that "the crucial question
focuses upon accounting for the dependence in terms of
imbalance produced by the structure of the choice situa-
tion itself and not by fixed conditions of the situation.
Only if one could in some sense remove the tension would
it be possible for independence to exist." (Page 19) I
would concur somewhat with Conner on these two reasons
why independence exists. And in concurring with Conner
on imbalance-produced tension and the structure of the
situation as producing dependence, I will attempt to View
these in a somewhat different light and also to build

further on these ideas.

The Original Experimental Design

 

The original experimental procedure was intro-

duced by Berger and J. Laurie Snell in "A stochastic

ll

theory of Self-other expectations" in which the purpose

was to "explicate a stochastic theory describing the

relation between specific and defined 'self' and 'other'
expectations and the pattern of disagreement resolutions
in a two-man group."12
This description of the design used by Camilleri
and Berger is taken from "Decision-making and Social In-
fluence: A model and an Experimental test“ with certain
modifications and deletions. In the Berger experimental
situation, a pair of subjects is asked to make binary
choices in two steps. The subjects are shown a series
of slides presenting two alternatives. They are given
the instruction to make a provisional, independent de-
cision between the alternatives (the initial choice);
then, after being told what the other has decided, each
is to take this information into account as he sees fit
in making a final decision on the trial. A subject's
final choices are not communicated to his partner. The
subjects are led to believe that the final decisions are
being evaluated as 'correct' or 'incorrect' and that
such evaluations constitute their performance score in
the task situation. These evaluations, however, are not
communicated by the experimenter to the subjects during
the series of trials.
The exchange of information about initial de-
cisions involves a deception, for the communicated initial
choices are not necessarily the actual choices made but

follow a schedule determined by the experimenter according

to whether or not he desires to have disagreement created
between the two subjects. This involves experimental
manipulation by which the machination used is controlled
by the experimenter such that feedback can be arranged
to mirror continuous (or whatever schedule is desired)
disagreement over a series of twenty slides on the ini-
tial choices. Whetheréasubject makes a final choice
which is inconsistent with his provisional choice (an
'other' response), or one that is consistent with his
initial choice (a 'self' response) on any trial of the
process is regarded as an indication of whether he ac—
cepts influence or not.

The task is an ambiguous one in that the slides
used are in the form of a larger rectangle composed of
smaller black and white rectangles each of which color
comprises fifty per cent of the total area. In the
artificial "Spatial Judgment Ability Test" the subjects
are told that there is a right and a wrong answer, i.e.,
in some rectangles there is more black than white, and
they are to act on this information in making a choice
(choosing either black or white). The subjects make
their choices under various structural conditions. They
are first induced in the manipulation phase to believe
that each of them has a certain amount (not necessarily
the same) of the ability (Spatial Judgment) needed to

make the choices presented. Each subject is also

cognizant of the amount of ability possessed by his
partner. These beliefs constitute the 'expectation
state' of a subject. And since each subject is induced
to believe that he has either superior or below average
ability, four expectation states are distinguished:
high ability for self and low ability for other is de-
noted by [+-]; high self and high other by [++]; low
self and low other by [--]; and low self and high other
by [-+].

The task conditions are so defined that the sub—
jects are working as a team. They are told that perform-
ance scores, based on their final decisions, are being
calculated for the team as a unit. Further, subjects
may differ in their control over the team score. There
are situations of (i) no control (ii) full control, and
(iii) equal control. The situation of interest here is
that of equal control where the final choices of the two
subjects are weighted equally in determining the team's'
performance. In this case, the subjects are 'equals' in
their decision-making rights. The data to be viewed in
this paper will be that of the equal-control situations.

The specific experiments to which the Camilleri-
Berger decision-making model was applied were carried out
on university undergraduates who were paid volunteers.
The pair of subjects in an experiment were of the same

sex and usually previously unaquainted with each other.

In all cases the subjects were randomly assigned to one
of the four expectation states and to one of the three
control conditions (full, equal or no control).

The equal control experiments were carried out
both at Michigan State University and at Stanford Uni-
versity. In this paper I will only be looking at the
equal-control data gathered in the Stanford experiments.
There was a difference in the number of trials run at
each locale. At Michigan State there were a series of
twenty,continuous disagreeing trials, while at Stanford
subjects were asked to make initial and final choices on
twenty-five trials. On twenty—three of these trials,
each subject found himself in disagreement with his
partner on his provisional choice. This difference be—
tween the number of trials run should not effect the data
itself (i.e., give rise to differences between the Stan-
ford and MSU data), but rather should only indicate some
differences in the suspicion rates between Stanford and

MSU subjects.

The Problem Under Investigation

 

My concern will be with the attempt to lend a
more functional interpretation to the existence of the
dependence process. In doing so, I will be building
upon conceptions introduced by Heider, Cartwright and

Harary, and by Conner. That is there will be the effort

to explicate more fully why trial-by-trial (or one-step)
dependence does exist in the experimental data and to
posit some sort of functional relationship between vari-
ables affecting such a process. And with such a function
as a base of inquiry, I will attempt to postulate a
"quasi-program" outlining a procedure for ordering pre-
dictions on a trial-by-trial basis. In constructing

such a function, I will be incorporating into the im-
balance-produced tension interpretation a key tenet con—
tained in the formalization of balance theory. Reference
here is directed to the idea of the degree of balance
(denoted as b(G) within a system modeled after Heider's
p-o-x paradigm.

Such a conception will lend to the interpretation
more of a quantitative character than that afforded by
Conner. This will not be quantification just for the
sake of quantification, but a concerted effort to con-
struct a measure that will enable me to order predictions.
As suggested, such a function will be one measurable at
the ordinal level and the philOSOphical mode of analysis
will be that of explication. This method of analysis will
involve making more precise the idea of tension (such
precision entering in under the guise of a function con-
structed between tension and degrees of balance) in the
universe of discourse. In Conner's usage, tension is an

informal context. It is (i) incomplete and (ii)

10

inconsistent, thus there is an inability to apply the
term. This necessitates transference to a more formal
context in which an explication is defined which is more
precise. The concern here will be with some sort of
quantitative confirmation, or a measure of some degree
of support for the one-step dependence as accounted for

by the mode of analysis I will attempt.

CHAPTER II
A MOVE TOWARD EXPLANATION

It has been noted by Conner that in the high-low
and low-high expectation patterns, given continuous dis-
agreements, the individual never changes his expectations
for self or other. Under the assumption of independence,
the process as outlined by Camilleri and Berger thus be-
comes a Bernoulli independent trials process,13 where
P(S-—)S) = p(O-—9S); [P(S)-probability of a Self response;
P(O)-probability of an Other response] in which P(S) re-
sponse is a constant probability p, independent of previous
trials. That is, "the probabilities of success (self re-
sponse) and failure (other response) remain the same

throughout the trials."14

All of this seems theoretically
sound, the only drawback is the data. It does not indi-
cate independence in action, rather one-step dependence

is observed.

Independence takes the form,

n+1
S‘ O
I
S a : l-a
n ---4——---
O a : l—a

where P(S) = p(§/3) = P(S/O) = a.

ll

12

This is a stochastic matrix which mirrors the probabili-
ties associated with making a particular choice on trial
n+1, gizgn_a particular choice on trial n, e.g., P(S/S)
symbolizes the probability of making a self-response on
trial n+1 giygn a self-response on trial n. A stochastic
matrix thus mirrors conditional probabilities. At this
point I will utilize the formalization of balance theory
in the attempt to explain the dependent relations found.
This will first involve a look at the balance diagrams
used by Conner (as derived from Cartwright and Harary)
and then a discussion of the diagrams which I will use.
A step-by-step mirroring of each stage of the decision
process will be presented with an interpretation of the
components of the system. Once the difference is noted,
I will go into an exposition as to why my balance dia-
grams (and attendant interpretation) are more suitable
for analysis.

Conner's balance diagrams contain, at most, five
elements or entities, these are;

p = self (the reference individual in the system),

p' = self as he views himself as an object,

o = other,

A,B = choice alternatives A and B, respectively of
the series of mutually exclusive binary

choices.

13

As in Heider's theory of balance, two relationships may
obtain between elements, (a) sentiments and (b) unit re-
lationships. In the diagrams used by Conner, a directed
line will represent a sentiment relationship (in this
case a specific evaluation of a choice alternative or

of a person) and a directed brace will represent a unit
relationship (in this case, selection or rejection of

a choice alternative). Thus,

p ——>p' p evaluates himself

p -——>o p evaluates others

p —-§A p evaluates A

p’/‘5A p selects or rejects A

0 ——9A p perceives that o evaluates A

o”"A p perceives that 0 selected or
rejects A

p“’\’A p sees himself as having selected or
rejected A

p'-—9A p sees himself as having evaluated A.

A particular diagram will be referred to as a
choice structure and balance in such a structure will be
defined as Cartwright and Harary define it in their for-
malization, i.e., all cycles must be counted and all
cycles must be positive. It will further be assumed
that on any step of the decision sequence, only those

relationships can occur which maintain balance in the

14

total structure for that point in time. Once p has made
his initial choice he exchanges information with o and
discovered that o's preliminary choice differs from his

own . . . this will be called by Conner as act-of-

 

disagreement.
+
A A
+ +
P P
B B
p positively evaluates p selectes the alternative
alternative A and nega- he positively evaluates
tively evaluates alter— and rejects and one he
native B negatively evaluates

Figure l.--Balance diagram of p selecting the alternative
which he positively evaluates.

15

 

p receives the information that 0
has selected alternative B and re-
jected A . 3 . he sees himself as
having selected A and rejected B

W V
||

p associates the disagreement on
choices with different evaluations
of alternatives by himself and 0

Figure 2.--Balance diagram of p inferring o's evaluation
of alternatives.

Conner at this point notes that it is sufficient
to state that p will always assign unit evaluations of
persons or alternatives in a manner which maintains

balance in his choice structure.

 

 

p positively evaluates and selects
the same alternative he selected
initially (and hence negatively
evaluates and rejects the other

 

alternative)
+ A '
+ If} -
Pkup+-¢O
-1}.

p, as a consequence of his evalua-
tions of the alternatives, posi-
tively evaluates 0, which balances
the total structure

Figure 3.--Balance diagram of p assigning unit evaluations
of persons after evaluating the alternatives.

l7

 

 

p first assigns unit evaluations of

persons by deciding he is wrong and
o is right

 

 

As a consequence of his unit evalua-
tions of persons, p positively
evaluates and selects o's initial
choice (he negatively evaluates and
rejects his own initial choice)

Figure 4.--Balance diagram of p evaluating the alterna-

tives after designing unit evaluations of
persons.

18

In such a setup, the consequences of p's unit
evaluations of persons, on any stage, is that the possi-

bility then exists which did not exist previously for

 

him to 'assign' states of C* (which denotes a change,
the same as an Other response). Moreover, when p does
'assign' states of C the positively evaluated state of

C will always be assigned to the person who was given
the positive unit evaluation and the negatively evaluated
to the person who is negatively evaluated. Stability is
not problematical if p believes himself to be more com-
petent than 0 and makes an S-response, or believes him—
self to be less competent and makes a C-response. This
situation arises because p's unit evaluations of persons
in the above-mentioned cases are consistent with his
assignment of states of C. If, however, p believes him-
self more competent and makes a C—response, or believes
himself less competent and makes an S-response, his unit

evaluations of persons are inconsistent with his assign-

 

ment of states of C. P thus less frequently makes re-
sponses which lead to inconsistent evaluations. And
stability is primarily a function of a change in the way

p interprets the unit evaluations of persons.

 

*Assigning states of C occurs after each unit
evaluation. A state of C is the same as an O-response.
These states are assigned consistent with the maintenance
of balance in the total structure.

19

With this brief synopsis of the formalization
used by Conner concluded, I will next turn my attention
to the balance diagrams as I used them which point up
certain modifications and differences in interpretation.
As in the original formalization, this system is composed
of five elements, these being;

p' = p's perception of himself (analogous to an

external self—evaluation),
p = the reference actor in the system, self,
0 = other,
A,B = choice alternatives of the series of mutually

exclusive binary choices.

This formalization will be composed of sentiments,
unit relationships and a third component which can be
termed 'meta-sentiments' in that this is a relationship
positing p's view of himself as evaluating a particular
alternative. And this relationship shall be designated
by a broken or dashed line. A difference in interpreta-
tion between the two systems can be noted among one of
the components, i.e., my interpretation of p'. Thus we
have

p“——a'p p's evaluation of himself, e.g., p's

perception of himself is one of a
particular level

p-———)o p evaluates other

20

p———§A p makes a particular initial choice
(either selecting or rejecting A)
o-——€>A. p perceives o as making a particular

initial choice (a choice always the

 

opposite of p's choice)

p”~73A p selects or rejects A on the final
choice

o’“*‘3A p perceives (implicitly) that o selects
or rejects A on the final choice,
though the experimental design does
not call for final choice feedback

p'--)'A,B p's views of himself evaluating A(B)

Differences thus do arise between interpretations
of the two systems. In the [-+] and [+-] conditions, ex-
pections remain constant throughout the series of trials.
However in the [++] and [--] conditions does expectation
change with continuous disagreement? And does this change
in expectation states follow any consistent pattern as in
the [+-] and [-+] conditions? I would posit that any
changes in expectations would vary across individuals and
it would be difficult to isolate aggregate patterns. Even
upon considering subjects who are team-oriented, it would
be difficult to assess a consistent aggregate pattern of
a resultant change in expectation state over the series

of trials. What this is saying in relation to the balance

21

diagram presented is that once manipulation is induced

then the relationships between p' and p, and p and o are

constant.

Thus we have,

 

 

 

Figure 5.--(l) Based upon the manipulation induced in
Phase I, p perceives himself (p'—-"'—) p) as
having high ability and perceives o (p--=9Io)
as having low ability (this expectation re-
mains constant over the series of trials) and
(2) p evaluates A and B, and selects A on his
initial choice, (3) p views himself as posi—

tively evaluating A and negatively evaluating
B.

 

 

 

Figure 6.--(l) p receives information that 0 has posi-
tively evaluated and selected B (rejected A)
on his initial choice. (2) An act of dis-
agreement 18 created.

 

 

 

Figure 7.—-(l) Given this act of disagreement outlined in
Figure 6, a decision-making situation is thus
created. (2) p reacts by selecting alterna-
tive A (rejecting B) on the final choice. (3)
The system remains stable.

Figure 8.--(l) p implicitly perceived o as selecting B
(rejecting A) on his final choice. (2) p
thus perceives o as maintaining systemic
stability also.

 

A

/77

+//’ :r
4 Alp

p'<

   

Thus Figure 8 is the complete system once the
final choices are made by both p and o. In this system
p does not actually perceive o's final choice. O's final
choice is not communicated to p. However p's implicit
perception of 0's final choice revolves around the con-

cept of systemic stability within o's system. P perceives

23

0'3 quest for stability as being no different from his
(p's) quest. This whole matter becomes more complex as
we enter the arena of intra-individual and intra-systemic
balance. Perhaps this is not a Erng_model of reality,
but it allows for analysis and resultant measurement

which is a valid approximation of reality. At best these

 

balance diagrams can only serve as approximations. And
all attempts should be made to have such a measure ap-
proach with as great a degree of accuracy as possible,
the actual situation. These diagrams serve to reflect
the general phenomenon of concern, and attempts to de—
fine the system more precisely will only take us into
the realm of individual psychology.

My system is thus (i) more condensed in that
the choice structures are contained within the same
sub-cycle. In the original system the initial/final

choice structure is such,

initial

I o 0
choice P final ch01ce

w<——*u——->:>

whereas in my system,

initial choice P final choice

 

\
B

Also (ii) my system contains the final choice of 0,
which is a convenient device for more elaborate systemic
interpretation, (iii) I have a different interpretation
of p', and (iv) there is a constant relation between p',
p and 0 which is a function of the manipulation induced
in Phase I. Thus there are constant valences between
particular components in the system, and one is not lost
in the process of assigning variable states which may be
more subjective and arbitrary than is demanded.

The latter diagrams constructed are thus more
malleable in that there are fewer subjective variable
relationships to consider, and they serve as a closer
approximation to the actual state of nature. This en-
ables one to isolate those relationships that do vary
and to account for this variance which comes about with
the help of a concept which mirrors the magnitude of
balance in the total system. Here I am referring to the
degrees of balance, b(G), which will be utilized in
conjunction with the informal context (as it is now used)

of tension to make ordering predictions.

25

Use of the Diagrams

 

With the four experimental conditions, [++], [+—],
[-+] and [--], diagrams have been constructed and b(G)

has been computed. And in such a system as this one,

there are seventy sub-cycles which go into computing

b(G), where

b(G) = —(—5-+g(g)
b(G)~- degrees of balance,

where 0 < b(G) s l,
+c(G) - the number of positive cycles,

C(G) - total number of cycles.

An example of such a computation will be illustrated

here in relation to the [+-] O-response situation,

 

 

the number of cycles are counted and we find,

Cycle Number of

 

+ ..

2-cycles 2 2
3-cycles 6 6
4-cycles 20 10
5-cycles lg ‘12
Total 40 30

thus b(G) = 40/70 = .57.

These diagrams and the computation of b(G) will
be explicitly used for ordering the cells in the sto-
chastic matrix. And inherent in such a predictive
scheme will be the notion of 'tension' as it will be—
come more refined briefly, i.e., it will take on a more
formal context as the mode of analysis requires.

Next I will turn to an analysis of the individual
systems beginning with,

(A) [+-]

 

 

27

the situation in which p makes an S response. In this
situation, b(G) = 1.00 (where l.00f= perfect balance),

in that the system is locally balanced at p, i.e., every
semi-cycle containing p is balanced. Thus, in the [+-]
situation where p makes an S response, the system is
balanced, i.e., there is very little tension. The fact
that the system is perfectly balancedidoes not mean that
p will not be prone to make an 0 response, although he
will not do this too frequently. In doing this, he is
attempting to keep the entire system in balance (under
the assumption that he cannot be correct all of the time),
whereas a string of S responses preserves intra-individual
balance.i This situation of balance is thus rewarding to

p, an 0 response is punishing.

Where p makes an 0 response, we have,

 

 

 

in which b(G) = .57 and the system is imbalanced. This
imbalance produces tension and p can alleviate this ten-
sion by making an 8 response on the next trial. Thus we

have,

28

1.00

S response: b(G)

0 response: b(G) = .57

P(S—-) S) , b(G) constant

P(S—fi O) , b(G) decrease

P(O-—§ S), b(G) increase

P(O—) O) , b(G) constant
therefore a directional hypothesis based upon b(G) would
be that,

P(O—5 S) > P(S——)’S) > P(S—) O) > P(0——>0) .

In words this would mean that the probability of
an S response on trial n+1 given an 0 response on trial
n is greater than the probability of an S response on
trial n+1 given an S response on trial n. These are
conditional probabilities and they are a function of
the balance in the system. Of course the subjects do
not know the magnitude of balance in a system, they are
only cognizant of the tension. And this tension arises
due to the imbalance, i.e., the more balance, the less
tension and the resultant pressure to change on the next
trial is lessened.

Based upon these assumptions of balance then, a
Markov process seems to be in effect where,

thus the transition matrix in this instance would

become,

where
P(S/S)

P(S/O) = b
P(O/S) = l-a, and
P(O/O) = 1-b

II
D.)

which is a stochastic matrix depicting one-step dependence.

Based upon the concept of degrees of balance, in
the [+-] situation the magnitude of the balance in the
system was determined. The magnitude was implicitly re-
lated to tension, i.e., the greater the degree of balance
(the closer the system approached perfect balance) the
less tension is there in the system and thus there is
less pressure to alter one's choice on the next trial.
One-step dependence is thus handled in relation to the
degree of balance, and this magnitude serves as a key in
attempting to understand and explain the divergence from
independence in the original experimental data.

This interpretation of imbalance-produced ten-
sion is not in direct accord with Heider's theoretical
propositions. For he states in one of his propositions
that: (Tension states) "If a change is not possible,
the state of imbalance will produce tension." The as-

sumption in our system is that on any one trial the

30

system may be imbalanced and this produces tension. This
tension is carried over to the next trial, and the subject
has the opportunity before him to restore balance (reduce
the tension) depending upon the choice he makes (whether

an S response or an 0 response). So in essence the indi-

 

vidual controls the system.

 

(B) Much that was said regarding the previous
situation can be repeated here, however we are more in-
terested in the differences that result in this [-+]

situation.

 

 

In this system, b(G) = .49. The system is im-
balanced and this makes for tension on this trial. In
other words, p by making an S response creates tension
in the system and the intensity of this tension will have
a bearing on his performance on the next trial.

Thus we have,

 

 

31

the situation where p makes 0 response. Here b(G) = .57
and the system is a little more in balance than with an

S response. But the system is still imbalanced. There
is a little less tension and p will be prone to make
variable responses. The near equality of b(G) in both
situations dictates that this situation will approach

an independence process more so than the other three
conditions. And in that b(G) in the two above systems
are very similar, it would appear that the amount of
tension in each would be similar. The subjects would

not be capable of discerning such a negligible difference,
thus it seems as if there should be an equal probability
of S or 0 responses. However, from the observer's view-
point (one which knows the exact magnitude of b(G) there
would be a difference, slight though it may be. And this
difference is some increment, A, which is a function of

b(G), e.g.,

P(S/O) - P(S/S) = A

where A is defined by b(G) in the S response system and
in the 0 response system.

According to the "quasi-program" laid out in
relation to ordering predictions, in the [-+] case
P(S———9S) > P(O———9S) and P(S———$O) > P(O———90), but the

mathematics of matrices makes this an impossibility.

32

Thus it is to other sources that prediction in this case
must be made, and I will go into this briefly.

The degree of balance, b(G), can thus be viewed
as an indicator of the tension in the system, i.e., the
greater is b(G) then the less tension is there in the
system. Seemingly an inverse linear relationship can
be posited between these two variables and I will con-
cern myself with whether or not tension is in actuality

a function of b(G), e.g.,

’F

tension

\

.25 .50 .75 1.037
b(G)

The attribution of values (beyond merely qualitative

 

ones) to the dependent variable of tension presents a
problem. For beyond being cognizant of the magnitude

of b(G) in any one system, the quantitative aspect of
tension cannot be discerned. Thus the function as it
exists is a "loose" one in the sense that the independent
variable is quantitative, while the dependent one is
qualitative, e.g., a high amount of tension, etc. The
function thus becomes defined upon observing the choice

structure, i.e., the P(S) response on trial n+1 given an

33

0 response or S response on trial n. That is, the vari-
able of tension can enter the quantitative realm a
postereori based upon assumptions made previous to veri-
fication. And in that such a function seemingly exists,
for each value of b(G) there is a unique "value" attri-
buted to the tension variable.

From here I will turn to,

. Wk

P '<———->P———+ 0

\\ _
-\
\

B.

(C)[++]

 

This is the system in which both p and 0 have high
ability and p makes an S response. In this system,
calculations indicate that b(G) = .71. Tension is thus
in the system, but its force is not as great as in the
[-+] situation and its force is less than in the [+-]

situation. Thus it would be expected that P(S) follows

this ordering
[+-] > [++] > [-+] ,

i.e., more S response in [+-], fewer in [++] and fewer

still in [-+J-

34

Upon looking at the system in which p makes an

0 response, we have

 

 

and calculations reveal that b(G) .49. This system is
thus in greater imbalance than is the previous one and
this would lead one to expect that there would be more
pressure to change on the following trial.

Thus we have,

.71

S response: b(G)

.49

0 response: b(G)

and a directional hypothesis based upon b(G) would be

that,

P(O—-—)-S) > p(s——)5) > P(s——90) > P(O—-—)O) .

As stated previously, the subjects do not know the mag-
nitude of b(G). They are only aware of the fact that
tension is in the system and based upon the intensity
of the tension, pressure ensues to alter one's choice
on the next trial. Based upon b(G), the P(O) response

can also be ordered between systems, such that

35

[+-] = [-+] > [++]

and these orderings would lead us to believe that predic-

tions can be made between systems as to the probability
of particular choices.

The final system to be considered is

(D) [--]

 

 

where p and 0 have low ability and in this situation,

p makes an 8 response. As before, b(C) is computed and
it is found to be equal to .66. By including this sys-
tem in the between-system ordering procedure based upon
b(G), we have (ordering according to the degree of

balance),
[+-] > [++] > [--1 > [-+]

which predicts that the probability of S responses fol-
lows the above ordering. In other words, there will be
a greater number of S response in the [+-] situation

than in the [++] situation, symbolically,

(S) response [+-] > SR[++] > SR{--] > SR[-+] .

36

Now it is necessary, as before, to turn to the

situation where p makes an 0 response.

 

here b(G) = .53. Thus there is more tension in this
system in that it is less balanced than is the S re-
sponse case. we would therefore expect that p would be
under more pressure to change his response on the fol-
lowing trial given an 0 response on trial n than he
would if an S response were on trial n. Symbolically,

we have,

S response: b(G) .66

.53.

0 response: b(G)

And it would be hypothesized that,

P(O-——9S) > P(S———)S) > P(S-——90) > P(O———90) .

Here I will also view the between-system ordering

predictions based upon b(G). And they indicate that

OR = OR > OR > OR
[+-] [+-] [---l [++]

It can thus be seen that my.postulation hinges

upon tension as being a function of b(G) in that b(G)

37

indicates the degrees of balance in a system. And a
state of imbalance produces tension which can and often
is relieved on the following trial by the selection of

a particular alternative. As noted before, the indivi-
dual controls the system insofar as he has it within his
power to ease the tension within the system, i.e., by
making a choice which puts the system into a more balanced
state. However, there is an asymptotic level of balance
capable of being attained within each system as defined
by b(G). And the individual is powerless to go beyond
this limit, other than withdrawing from the situation
entirely.

Thus particular propositions can be stated in
relation to b(G) and tension as they provide an inverse
functional relationship in the explanation of choice-
structures in the balance diagrams.

Proposition I: An actor will tend to move to a state
that is more balanced.

 

Proposition II: An actor will be more likely to move
to a state that is more balanced, than
from one state to another that has the
same magnitude of balance.

 

Proposition III: An actor will not tend to move to an
imbalanced state from a balanced one.

 

The exceptions to this latter proposition which
do arise will be discussed in more detail later in rela-
tion to the topic of intra-systemic and intra-individual

balance.

38

Proposition IV: An actor will be more likely to move
between equal states of balance than
from a balanced state to an imbalanced
one.

 

This proposition states that an individual will
be more likely to make a choice on the following trial
which preserves the balance in the system, rather than
making a choice on trial n+1 which will lessen the de—
gree of balance in the system.

Given the stochastic matrix which indicates con-

ditional probabilities,

U)
0

then mathematically if b 2 a, then l-a 2 l-b. That is
either both of the top entries (a and l-a) or both of
the bottom entries (b or l-b) cannot exceed 2222.0f the
other entries, e.g., a situation where b > a and l-b > l-a
would be mathematically impossible in accordance with the
mathematics of this stochastic matrix.

By viewing the matrices for each condition we
can get a representation of what the function defined
between b(G) and tension indicates in mathematical terms.

Thus we have,

39

[+-]

where b > l-a, and l-a > l-b, and b should be the largest

entry in the matrix.

[-+]

n+1
S O
S a : l-a
n ----f----
O b : l-b

where b > a, and l-a > l-b. This ordering prediction
violates the prOpositions as stated. But the internal
inconsistency of the matrix is a function of the near
equality of b(G) in both choice situations and an iné
terpretation of this structure as being a reflection of
intra—systemic balance at its height. For as the between
system predictions pointed out, the P(S) response in the
[-+] condition would be less than the P(S) response in
the other three situations. This is a result of the
manipulation induced which imposes more of an intre-
systemic balance proclivity to Actor in the [-+] condi-

tion, and this means that entry l-a will be the largest

40

one in the matrix. And if we have a situation where

l—a > 1-b, then b must be greater than a. This situation
is a hybrid in the sense that it does not follow from the
propositions, but rather is a function of a team-
orientation on the part of p (Actor), between system

ordering predictions and mathematical induction.

[++]

n+1
s o
S a : l-a
n ——--r—---
o b :l-b
I

where b > a, and l-a > l-b and b should be the largest
entry.

[--1

where b > a, and l-a > l-b.

This attempt to utilize the formalization of
balance theory to account for one-step dependence thus
gives rise to a number of assumptions to be stated in
more precise terms. And at this point I will state these

assumptions in a more formal manner.

41

Assumption I: The dependent trials process is a
function of the tension inherent in
the structure of the situation.

 

Assumption II: Given imbalance in the system, there
is the opportunity to restore balance
on the following trial, i.e., a one-
step dependence trials process.

 

Assumption III: Tension is a function (an inverse
linear one) of the degree of balance,
and given the notion of transitivity
we have that the dependent trials
process is a function of the degree
of balance.

 

 

From these more all-inclusive assumptions we now move
to assumptions involving specific response situations.
Thus,

Assumption IV: In the [+-] situation, P(S/O) > P(S/S)
and P(O/O) > P(O/S)

 

which is contrary to the original assumption of inde-

pendence where,

P(S/O) = P(S) and P(S/S) = P(S)

because if independence is assumed, our formulation says

that P(S) > P(S), which is contradictory.

Assumption V: In the [-+] situation, P(S/O) > P(S/S)
and P(O/O) > P(O/S).

 

Assumption VI: In the [++] situation, P(S/O) > P(S/S)
and P(O/O) > P(S/S).

 

Assumption VII: In the [--] situation, P(S/O) > P(S/S)
and P(O/O) > P(O/S).

 

42

Thus in all of the conditions we have the situation hypo-

thesized that P(S/O) > P(S/S) and P(O/S) > P(O/O).

Assumption VIII: Balance is a reward and imbalance is a
cost.

 

These assumptions based primarily upon systemic
balance provide a possible explanation for the one-step
dependence. They also indicate the relative magnitudes
which a particular dyad of responses will assumed rela-
tive to another distinct dyad, e.g., P(S/O) > P(S/S) in

the [++] situation. These predictions are thus situa-

 

tionally specific. By looking at the equal control data

 

from the Stanford studies we have,

[+-]

n+1
S O
S .76 i .24
n ----r----
O .84 E .16

where b > a > l-a > l-b, which holds with predictions
(Assumption IV).
Turning now to,

[-+]

43

where l-a > b + l-b > a, which holds with predictions
(Assumption V). The salience of preserving systemic
balance (i.e., p is cognizant of 0's higher ability and
an 0 response is "rewarding" to p) would outweigh the
desire to maintain ones' integrity (an S response viewed
as a cost). As noted, there is not perfect balance in
the system where p makes an 0 response, for b(G) = .57
and this divergence from perfection in the system could
effect p's response to tension.

[++]

n+1
S O
S .63 i .37
n ---—T----
O .76 : .24

where b > a > l-a > l-b, which holds with predictions
(Assumption VI).

One-step dependence is thus seen to be in
existence in the above three conditions, but the for-
malization of balance theory as a predictive device is
not stringent throughout in that a 're-interpretation'
or hybrid situation had to be posited in order to account
for the matrix ordering in the [-+] situation.

Before delving into a consideration of a possible

explanation, I will turn finally to the

44

n+1
S O
S .63 E .37
n ----T----
O .71 : .29

where b > a > l-a > l-b. This situation is also consis-
tent with the predictions (Assumption VII).

Thus the ordering predictions are met upon look-
ing at the Stanford equal control data. Now let us take
a retrospective view of the original theory and the
language of the theory. What we come across is Homans'
conception of cost and reward, which he uses rather am-
biguously despite attempts otherwise through a quasi-
equation, i.e., profit = reward - cost. By incorporating
these "concepts" into my explanation, it is seen by
Assumption VIII that the notion of balance is a cost.
But at what level or to what meaning of balance is
reference being made? Granted, tension leads to imbalance
within the system, but there are two systems at work.
Perhaps there needs to be a stricter definition of balance

in relation to p and to the system in its entirety.

CHAPTER III

BALANCE/IMBALANCE: SYSTEMS AT WORK

Within the realm of social psychology the concern
is with the interrelated efforts of an individual upon an
aggregate (with all the attendant structurings), i.e.,
how a person x effects and is effected by a group Y
(attributing syntality to the group). Experimental social
psychology makes the focus even sharper and this is what
Erving Goffman refers to as 'focused gatherings' and what
Hans Zetterberg refers to as artificial situations. In
these situations, variables to be studied are isolated
with utmost care and there is the attempt to eliminate
extraneous variables of no particular interest. What
seems to have arisen in the Camilleri-Berger laboratory
experiment which the theory did not take account of was
the two levels of systems within which cost/reward are
defined. In looking at the notion of balance, I feel
that it is essential to consider (a) intra-systemic
balance, and (b) intra-individual balance. Is there a
threshold beyond which an individual sacrifices his in-
tegrity in order to work for group harmony (in terms

used here, harmony = balance). For it is seen that in

45

46

all systems except the [-+] that b(G) in the S response

systems is greater.

 

 

TABLE 1
+- S response: b(G) = 1.00) 43
0 response: b(G) = .57 ‘
++ S response: b(G) = .71) 22
0 response: b(G) = .49 '
-- S reSponse: b(G) = .66) 13
O response: b(G) = .53 °
-+ S response: b(G) = .49) _ 08
0 response: b(G) = .57 ‘

 

What this says is that the degree of balance is greater
with an S response in all systems, except the [-+] one
and even here the difference is not that large. Thus
individual integrity seems to be a key concern. Seeming-
ly there is more balance in a system given an S response,
indicating a great reluctance to make an 0 response (in
that there is more tension). And even in a situation
where it would be expected that individual x would tend
to make more 0 response([-+]) the difference in tension
in the system is not great. Even perceptibly so, the
[-+] 0 response situation has less tension and in making
this choice the individual has moved to considerations

of intra-systemic balance, i.e., his interest is in pre—

serving balance in the system at the expense of 'balance

47

within himself.‘ The other systems indicate Actor pre-
serving intra-individual balance, and in these cases we
would expect a greater number of S responses. At this
point I am not referring to conditional probabilities,
but rather S responses over a sequence of trials.

This is important in the sense that the [-+] case
is the only one 'contrary' to the propositions. Thus
this seems to be the system where intra-systemic balance
is more a concern because p perceives himself as having
low ability and o as having high ability. Therefore he
succumbs to the idea of his 'wrongness,‘ and thus works
for the group, i.e., he is more team-oriented than what
is seemingly the case in the other situations, despite
experimental instructions.

As was stated earlier, it was assumed that bal-
ance is rewarding to actor and imbalance is punishing
(i.e., tension-producing). The nature of the data in-
dicates that perhaps this is not true, or else the multi—
tude of ramifications indigenous to costs and rewards are
not fully understood. This seems to imply that the notion
of a cost in relation to the system needs to be defined
more specifically, as does the notion of reward. For in
some instances it seems as if what we view as a cost
(e.g., an 0 response in the [+-] situation) may in
actuality be viewed as a reward by the actor (he believes

he cannot be correct all of the time, so he makes an O

48

response in order to preserve intra-systemic balance).
Thus it is that the notions of intra-systemic and intra-
individual balance are operative on each trial and we
can possibly infer from overt acts the particular type
of systemic balance that the individual was attempting
to attain or preserve. Attaching probabilities to being
in one or the other 'balance-predispositions' would be
a function of b(G) in relation to each alternative (the
system as defined by the choice of a particular mutually
exclusive alternative), the preceding decision and a
cost/reward matrix defined by some subjective means.
This model of behavior would involve assumptions simi-
lar to Cohen's* conformity model in that one would be
able to derive probabilities by observing which state
an actor was in. This paradigm would be more inclusive
in that it would incorporate the cost/reward notions as
varying within and between experimental situations.
Here the main concern has been with balance at these two
levels and asymptotic points beyond which particular
states will not be sacrificed.

One-step dependence is thus a reality in the data
and here I attempted to account for this dependence using

the formalization of balance theory and the use of a key

 

*Bernard P. Cohen, Conflict and Conformity: A
Probability Model and Its Applications. Cambridge, Mass.

 

 

49

idea from this theory, that of the degrees of balance
b(G). The dependence was explained, but in so doing we
were swept into another realm of concern which transcended
my original intentions, i.e., the idea of intra-individual
and intra-systemic balance. But this realm must be con-
sidered, for it may serve as a key to unlock the mystery

behind the entire theory.

CHAPTER IV
CONCLUSION

The work done here was an attempt to account for
the dependence process generated by the Camilleri-Berger
experimental design. Perhaps a change in this design
would lead to the generation of data more in accordance
with the original model and theory. However, the con-
cern in this paper was with explaining why a dependence
process did result. Crucial to this explanation is the
formalization of balance theory and one of its key com-
ponents, that of the degree of balance within a system.

Thomas L. Conner in his M.A. thesis raised a
crucial issue when he stated that the structure of the
(experimental) situation itself precludes an independence
process. For inherent to this structure is the concept
of tension. Tension was posited in my work as a function
of the degree of balance. And this inverse linear func-
tion served to order predictions both within and between
systems.

It was found that through the utilization of such
a function, the ordering predictions did hold. The only

discrepant situation was that of the low-high condition.

50

51

The ordering predictions as made in this condition moved
beyond the functional interpretation into the realm of
intra-systemic and intra-individual balance. This involved
a transcendance which stamps the dependence process as a
reality while at the same time it calls for a further
analysis of the meaning of balance. Such an analysis
must delve more deeply into the idea of a cost-reward sys-
tem as a function of intra-systemic and intra-individual
balance. Such a function would have to consider every
conceivable individual as well as systemic 'pre-disposi-
tion.‘ For it will only be through a thorough grounding
of the intra-individual and intra-systemic levels that a
design can be constructed which will indeed be an inde-
pendence process. The inability to make thisnecessary
distinction between the two levels of balance further
confused the original theoretical verification.

Further implications of such an effort could
possibly involve research studies in the area of 'socio-
existential' concerns. What I am alluding to here is
the existential idea on the part of an individual of his
own unique worth. This embodies an 'I am my worth' con-
cern which moves beyond the 'lboking-glass self' and
'we-ness' ideas spoken of by George H. Mead and C. H.
Cooley. This 'I-ness' is revealed through the between-
system ordering predictions. It was seen that in every

condition except the low-high there were more S-responses.

52

There was more of an intra-individual emphasis. This
was despite experimental instructions which set up a
team situation. What this is saying is that every con-

dition except the low-high approached the high-low con-

 

dition in which the self was more worth preserving than
was the group. Individual integrity was seemingly the
principal concern, and it would not be sacrificed at the
expense of intra-systemic balance. In a sense then, the
actors 'existentialized' themselves to the point where a
series of continuous disagreements could not convince
them of their 'wrongness.‘

Such an individual response cannot allow for an
independence process and this is where the within system
ordering predictions come in. The heuristic value of
this intra-individual emphasis is abundant. Contemporary
America is the scene of traditional group-stripping in
which there is at least the overt pronouncement of 'doing
my own thing.' This implies a discarding of the concep-
tion of a larger level of conformity and the construction
and preservation of intra-individual balance is becoming
a more potent force. Heuristic implications exist not
only in the social psychological17 realms of conformity,
identity and decision-making studies but also in the area
of mental health. Such an interest enters in the realm
of mental health insofar as personality decomposition is

concerned. This involves studies of breakdowns say in

53

combat situations and everyday tension-producing situa-
tions. Generally it is found that loyalty to a group18
is the last defensive layer surrendered prior to a total
breakdown. Such traditional intra—systemic loyalty is
overtly declining, yet is the incident of mental break-
down increasing?

Questions such as these can perhaps be explored
and answered. I see these ideas of intra-individual and
intra-systemic balance as having far-reaching heuristic
value in contemporary society. What is needed is a re-
finement of these ideas to the point that they are
malleable concepts. The inability to account for and
thus define and refine these ideas rendered the Camilleri-

Berger model and theory invalid in relation to the experi-

mental design.

1.

NOTES

A mention of a few of such studies would include,
Harold Berman, "Probability, Statistical Decision
Theory and Accounting," Accounting Review, 1962. v.
37, pp. 400-405; N. T. Feather, "Success Probability
and Choice Behavior," Journal of Experimental Psy-
chology, 1959. v. 58, pp. 257-266; William S. Vickrey,
"Utility, Strategy and Social Decision Rules," Quar-
terly Journal of Economics, 1960. v. 74, pp. 507-735;
R. T. Feather, "Subjective Probability and Decision
under uncertainty," Psychological Review, 1959. v.
66, pp. 150-164; Ernest W. Adams and Robert F. Fagot,
"A model of Riskless Choice," Behavioral Science,
1959. v. 4, pp. 1-10; Ward Edwards, "Utility, Subjec-
tive Probability, Journal of Conflict Resolution,
1962. v. 6, pp. 42-51; Sidney Siegel, "Level of As-
piration and Decision making," Psychological Review,
1957, v. 64, pp. 253-262. Subjective probability
arises when a decision-maker is ignorant of the
statistical frequencies of events in a particular
situation, yet he still tends to behave as though

he has assigned numerical probabilities to the events.

Adams, Ernest W. and R. F. Fagot in "A model of
Riskless Choice," Behavioral Science, 1959, v. 4,

pp. l-lO. Their model provides a method of measur-
ing 'subjective value' or 'utility' which under some
conditions leads to an ordered metric scale and under
still more restrictive conditions to an interval
scale of utility (a cardinal utility function).

Siegel, Sidney, "Level of Aspiration and Decision
Making," Psych. Review, 1957, v. 64, pp. 253-262.

Santo F. Camilleri and Joseph Berger, "Decision-
making and Social Influence: A Model and an Ex—
perimental Test," Sociometry, 1967, v. 30(1), pp.
365—578.

Two trials (of the same or different experiments)
are independent if the outcome of one does not ef-
fect the outcome of the other. Symbolically, then,
two events in the same probability space are inde-
pendent if and only if P(AB) = P(A)P(B). An

54

10.

ll.

12.

l3.

14.

15.

55

independence process is thus one that yields a
series of independent trials.

A finite Markov chain is a stochastic process which
moves through a finite number of states, and for

which the probability of entering a particular state
depends only on the last state occupied. From John
G. Kemeny and J. Laurie Snell, Finite Markov Chains.

 

Samuel Goldberg. Introduction to Difference Equa-
tions. p. 104.

 

Thomas L. Conner, "First Order Dependence and Self-
other Expectation Theory." Masters thesis, Stanford
University, Oct. 1963.

Fritz Heider, "Attitudes and Cognitive Organization,"
Journal of Psychology, v. 21, pp. 107-293.

Dorwin Cartwright and Frank Harary, "Structural
Balance: A Generalization of Heider's Theory,"
Psych. Review, v. 63 (Sept. 1956), pp. 277-293.

Joseph Berger and J. Laurie Snell, "A Stochastic
Theory of Self-other Expectations," Technical Re-
port, under the auspices of the National Science
Foundation Grant c-13314, Dept. of Sociology, Stan-
ford, Calif., Feb. 1961.

Joseph Berger, et a1., Types of Formalization,
p. 116.

An experiment which has only two possible outcomes
is called a Bernoulli trial. In an n independent
Bernoulli trials process, the word 'repeated' is
meant to indicate that the probabilities of success
and failure remain the same throughout the trials.
The natural generalization of the notion of inde-
pendent Bernoulli trials is the notion of Markov
dependent Bernoulli trials. For further exposition,
see Emanuel Parzen, Modern Probability Theory and
Its Application, Chapter 3.

 

Ibid., Parzen, p. 100. (The parentheses are my own).

A matrix is defined as stochastic if the sum of the
entries in any row is equal to one. A matrix is
defined as doubly stochastic if in addition the sum
of the entries in any column is equal to one.

l6.

17.

18.

56

Given two events, A and B, by the conditional prob-
ability of the event B, given the event A, denoted
by P(B/A), we mean intuitively the probability that
B will occur, under the assumption that A has oc-
curred. In other words, P(B/A) represents our re-
evaluation of the probability of B in the light of
the information that A has occurred. For further
exposition see Parzen, Chapter 2.

See Jan Hojda, "Ambivalence and Social Relations,"
Sociological Focus, winter 1968, vol. 2(2). pp. 21-
26; Erving Goffman, “Alienation from Interaction,"
Human Relations, 10(1957), pp. 47-59; Robert K.
Merton and Elinor Barber, "Sociological Ambivalence,"
in Sociological Theory, Values, and Sociocultural
Chan e. New York: The Free Press of Glencoe, 1963,
pp. 31-120.

For example, see; Maj. R. Sobel, "Anxiety-depressive
Reactions After Prolonged Combat experience--the old
sergeant syndrome." Bull. U.S. Army Fed. Dept.,
Combat Psychiat. Suppl., Nov. 1949, pp. 137-146.
Sobel found in studying the eventual breakdowns of
the army personnel who had been most resistant to
personality decomposition that such individuals
seemed to have been protected by five 'defensive
layers.’ These were surrendered progressively in
the face of too severe stress and threat. Distant
ideals like 'democracy' and 'the four freedoms' went
first. Loyalty to the group was the last to be
given up. Pride in self was the second-to-last
thing given up.

REFERENCES

 

10.

11.

REFERENCES

Berger, Joseph, et a1. Types of Formalization.
Houghton-Mifflin Company, Boston, 1962.

 

Camilleri, Santo F. and Berger, Joseph. "Decision-
making and social influence: A model and an
experimental test." Sociometry, 1967, v. 30(1).

 

Coleman, James S. Introduction to Mathematical
Sociology. The Free Press of Glencoe, London,

 

Conner, Thomas L. "First order dependence and Self-
other expectation theory," M.A. Thesis, Stanford
University, Oct. 1968.

Fentinger, Leon. A Theor of Coggitive Dissonance.
Stanford UniverSity Press, 1 7}

Goldberg, Samuel. An Introduction to Difference
Equations. Science Editions, Inc., New York,
1966.

 

 

Harary, Frank, et al. Structural Models. John Wiley
and Sons, Inc., New York, 1965.

 

Homans, George C. Social Behavior: Its Elementapy
Forms. New York, Harcourt, Brace and WOrld,
Inc., 1961.

 

Kemeny, John G. and Snell, J. Laurie. Finitg_Markov
Chains. D. Van Nostrand Company, Inc., Princeton,
New Jersey, 1960.

 

Lewin, Kurt. A Dynamic Thegry of Personality. New
York, McGraw-Hill Co., Inc., 1935.

Mosteller, Frederick, et a1. Probability With Statis-
tical Applications. Addison-Wesley Publishing
Co., Inc., Rea ing, Mass., 1961.

 

57

58

12. Parzen, Emanuel. Medern Probability Thepry and Its
Applications. John WiIey and Sons, Inc., 1960.

13. Wasserman, Paul and Silander, Fred S. Decision-Makin :
An Annotated Bibliography, Supplement, I958-I963.
Graduate School of Business and Public Administra-
tion. Cornell Univ., Ithaca, New York, 1964.

 

 

 

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