On the Complementarity of Expectations:

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-

1

-

On the Complementarity of Expectations:

Coupling Parsons with Balance Theory


Kazuto Misumi


Graduate School of Social and Cultural Studies, Kyushu University

4
-
2
-
1 Ropponmatsu, Chuo
-
ku, Fukuoka 810
-
8560 Japan

kmisumi@rc.kyushu
-
u.ac.jp


Abstract

When we ar
e to introduce the interpretative dynamic role process against 'homo sociologicus,' we
are confronted with the problem how to capture the various reality systematically. By coupling
the 'complementarity of expectations' with balance theory, we formally pr
opose a cognitive
framework to analyze the double contingency role relationship. Through conceptual discussions,
we introduce two basic signed graphs: the unit graph
T

that presents the in
-
personal role
regulation and the double contingency graph
G

in whi
ch two unit graphs are connected by the two
inter
-
unit lines. While the signs of inter
-
unit lines, that inter
-
personally connect role
-

expectations, determine the complementarity in
G
, the signs of
G'
s cycles determine the balance of
G
. With regard to
G'
s balance, two theorems are derived. The first states that
T'
s imbalance is
the sufficient condition of
G'
s imbalance, and the second states that one positive cycle in
G

larger
than 3
-
cycle is the sufficient condition of the balance of
G

that is composed
of balanced
T
. Utilizing
these theorems, we systematically classify the role relationship represented by
G
, and discuss
about the substantial meaning that some paradoxical cases imply.


Key words and phrases:

Complementarity of expectations, Balance theor
y, Role



1. Introduction



Parsons and Shils
(
1951:15
)

proposed the concept of complementarity of expectations
"not in the sense that the expectations of the two actors with regard to each other's action are
identical, but in the sense that the action
of each is oriented to the expectations of the other."
In addition, "in an integrated system, this orientation to the expectations of the other is
reciprocal or complementary"
(
Parsons and Shils 1951:105
)
. In the framework of social
system, a key concept

that mediates between individual and society is role
-

expectation.
The logic is fundamentally parallel. Following Parsons and Shils
(
1951:19
-

20
)

again, "once
an organized system of interaction between ego and alter becomes stabilized, they build up
rec
iprocal expectations of each other's action and attitudes which are the nucleus of what
may be called role
-
expectations. .... The pattern of expectations of many alters .... constitutes
in a social system the institutionalized definition of ego's roles in
specified interactive
situations."


These descriptions evoke a paradox discussed by Dahrendorf
(
1968:25
)
: "At this point
where individual and society intersect stands homo sociologicus, man as the bearer of
socially predetermined roles. To a sociologi
st the individual is his social roles, but these

-

2

-

roles, for their part, are the vexatious fact of society." In other words, by constructing 'homo
sociologicus,' sociology might develop a rational understanding of society; however, it has
excluded as a res
ult individuality and freedom of the human being.


Against homo sociologicus, some researchers see the dialectical possibility of liberation
from the viewpoint of real actions
(
Tenburuck 1961, Berger and Pullberg 1965, Yamaguchi
1975, Kurioka 1980, Tana
ka 1984, Nomura 1987
)
. Others point out unrealistic and too
strong assumptions of homo sociologicus. For example, Habermas
(
1973; also see Nomura
1982: 240
-
245
)

asserts that homo sociologicus shall satisfy following three theorems:


1
)

Integration theor
em
(
high degree of the complementarity of expectations
)


2
)

Identity theorem
(
coincidence between role definition and role interpretation
)


3
)

Conformity theorem
(
realization of normative contents into actions
)

Analogously, Levinson
(
1959
)

and Morris
(
19
71
)

assert that homo sociologicus requires
unrealistic high level coincidence among three aspects; role
-
expectations, role
-
conception,
and role
-
behavior. These assertions imply that the complementarity never be rigid and
stable in the real world; rather,
it shall be inherently flexible and unstable, being founded on
interpretative processes. Although temporary stable interaction may appear through ritual
actions and role
-
playing
(
Goffman 1959,1967
)
, the complementarity is no longer a sufficient
(
even a ne
cessary
)

condition for the stability.


Our standpoint is between the extremes. As Parsonians assert, the institutionalization
shall be a basic general process and, resulting in the complementarity to some extent, it
conditions the stability of interact
ion. However, in the real world, not only the process, but
also the other interpretative processes condition the stability as anti
-
Parsonians assert.
Among those that are mostly irregular, we focus on a relatively general process, the cognitive
balance.
As we will see later, the balance theoretical perspective has primitively presented
in role theory, but the tradition has stopped before formalization.
1)

The purpose of this
paper is to develop a graphical model that takes in the viewpoint of complementar
ity and
balance at the same time, and through it, to examine conditions for the stability of role
relationship from multiple angles.


2. Conceptual Framework



Before going forward formal analysis, we introduce our conceptual framework of role. As
alrea
dy cited, some role theorists developed three concepts
(
Levinson 1959, Morris 1971,
Funatsu 1976:187
-
188, Watanabe 1981
)
:



Role
-
expectations .... Normative expectations from alter with regard to ego's action.
(
Its


referent point is functional requirements of a social system.
)



Role
-
conception .... Ego's normative expectation with regard to action of him/herself.
(
Its



referent point is functional requirements of a personality system.
)



Role
-
behavior .... Ego's action actually taken in interaction.

Following Watanabe
(
1981: 114
)
, we can consider the relationship among the three.



Role
-
consensus
(

role
-
disensus
)

....

Compatibility between role
-
expectations and role
-


-

3

-


conception in regard to normative contents.



Role
-
adaptation
(

role
-
deviation
)

.... Compatibility between normative contents of role
-


expectations and role
-
behavior.



Role
-
identification
(

role
-
distance
)

.... Compatibility between normative contents of


role
-
conception and role
-
behavior.


Morris
(
1971
)

represented the relationship as a triple graph, and Watanabe
(
1981
)
discussed the substantial meaning of each compatibility pattern. Let "
+" be
compatible and "

" be incompatible. Suppose that the sign indicates the result of
adjustment process. As three graphs that have only one negative line carry logical
problems, the following five are valid
(
Figure 1
)
.







Figure 1. Triple Graphs in Morris=Watanabe Fra
mework



This graphic presentation is of great help to classify the results of actor's internal role
regulation, although some questions arise. Suppose that an actor whose state is a) and
another whose state is
ß
) encounter. Then, certain inter
-
person
al adjustment should occur
between them; however, in this framework, we can neither describe the process, nor judge
whether the complementarity is satisfied or not between them. Moreover, the state of a)
could represent 'complementarity,' but only when th
e same role
-
expectations are shared by
alter. In order to take these interactive viewpoints into the framework, two triple graphs
shall be linked.


Kobayashi
(
1983
)
, after defining 'role
-
knowledge' as the typification knowledge based on
social expectat
ions with regard to actions, insists that it exists not only as 'stock,' but also as
'flow.' The knowledge as flow means messages of expectations that are exchanged between
actors through role
-
behavior. In the sense, role
-
behavior is an output from ego's

role
-
regulation and, at the same time, an input to alter ego's role
-
regulation. As role
-

interaction concerns plural roles that make a pair, the input is interpreted by alter as a
message with regard to the role on the alter ego's side. When ego decides

role
-
behavior to be
taken, he/she anticipates what expectation it should reciprocally indicate toward the role of
alter. In other words, ego anticipates how his/her role
-
behavior will be interpreted and, as a
result, what reaction will be taken by alter.

This is the point when Parsons says, "the
action of each is oriented to the expectations of the other."
2)


Focusing on this aspect of role
-
behavior, we refine the triple graphs as in Figure 2. We
replace 'role
-
behavior' with 'role
-
expectation toward

the other,' and designate actor i's role
-

expectation toward the other actor j as
E
ij
. We also replace original 'role
-
expectation' with
'role
-
expectation toward self,' and designate actor i's role
-
expectation toward self as
E
ii
.

-

4

-

'Role
-
conception' is the
same; however, as it is strictly 'role
-
conception of own role,' we
designate actor i's role
-
conception as
C
ii
. This refinement makes the graph to describe
consistently the cognitive relationships. More importantly, it clarifies that the role
regulation i
s not only on one role, but also on
(
at least
)

two roles that make a pair. While
the point of the former is the compatibility between
E
ii

and
C
ii

in the sense of coincidence of
normative contents, that of the latter is the anticipated reciprocity between
E
ii

and
E
ij
, as
well as between
C
ii

and
E
ij
.








Figure 2. Triple Graphs in Our Framework



With regard to a'
)
, 'conformity' is better than 'complementarity' because it describes only
in
-
personal harmonious state. The meaning of the other types ca
n be interpreted as same
as Figure 1. Suppose that there are two normative contents, 'working' and 'keeping house,'
that are not compatible with each other, but are reciprocal. For a husband
(
actor 1
)
, if both
E
11

and
C
11

are 'working' in regard to the r
ole of 'husband,' the line
E
11
-
C
11

is positive
(
compatible
)
. Moreover, if his
E
12

toward his wife
(
actor 2
)

is 'keeping house' in regard to
the role of 'wife,' both
E
11
-
E
12

and
C
11
-
E
12

are positive
(
reciprocal
)
. We can judge that this
cognitive state is
a') in Figure 2. However, if his
C
11

is 'keeping house,' not only the
compatibility between
E
11

and
C
11
, but also the reciprocity between
C
11

and
E
12
, is broken.
Then, both
E
11
-
C
11

and
C
11
-
E
12

turn negative, so that we can judge the state as d'), and so
on.


Notice that our purpose at present is not to describe and predict what role
-
behavior will
be actually taken by ego and whether or not alter ego will react against it reciprocally.
Rather, given a series of concrete interaction, we are interested i
n finding a cognitive state
that could give guidance to it, and therefore explains the observation consistently. Suppose
that we observed a husband to work till late and his wife to welcome him preparing dinner
everyday, moreover, we could know they had f
elt no difficulty in the situation. Then, how
the relationship among
E
ii
,
C
ii
, and
E
ij

shall be, that is our question. More generally, it is
our purpose to examine under what cognitive states, interaction shall be complementary and
stable, whatever the a
ctual behavioral contents.


Figure 2 will answer it, but only partially. In order to get a complete answer, as we
already suggested, two triple graphs shall be linked together. It is plausible to assume that
they are linked by two lines between
E
ii

an
d
E
ji

(
i,j=1,2 and i

j
)
, and that the
'complementarity of expectations' directly depends on the compatibility between the
expectations, that is, whether both
E
ii
-
E
ji

lines are positive or not.
3)

According to the

-

5

-

extended framework, a wife
(
actor 2
)

is as
sumed to receive
E
12

reciprocally through her
husband's
(
actor 1's
)

role
-
behavior and to check it with
E
22
. The reverse process is assumed
for her husband
(
actor 1
)
. Notice again that, at present, the complementarity is judged
from a third party's viewpo
int. In the previous example, it is neither the wife nor her
husband, but a third party
(
a researcher
)
, to judge the sign of
E
12
-
E
22
, even though the
former themselves might be in fact able to do it.
4)


Figure 3 summarizes the conceptual framework tha
t we have finally proposed, and
Figure 4 is its simplified graphical representation. This framework represents the general
cognitive relationship between two actors, but only with regard to a focused aspect of a
role
-
relation. Generally, not only that pl
ural role
-
relations
(
e.g. husband
-
wife and doctor
-

nurse
)

sometimes overlap, but also that each relation contains plural pairs of normative
contents that are essentially incompatible with each other, but reciprocal
(
e.g. 'working'
-

'keeping house,' 'workin
g'
-
'bringing up children,' and 'representing family'
-
'acting behind,' in
husband
-
wife relation
)
. In a real process of role
-
interaction, such relations and contents
might change one after another, so that a graph in Figure 4 shall be connected to another i
n
succession, making an infinite chain graph as a whole. However, if we focus only on one
pair of contents in one role
-
relation, the process circles on a graph in Figure 4, as far as each
actor never change his/her role
-
expectations and role
-
conception.
The framework, as well as
the graph, only captures a limited unchangeable aspect as such.















Figure 3. Conceptual Framework of the Cognitive Role Relationship










Figure 4. Graph of the Cognitive Role Relationship



-

6

-


We might have to
draw one more line between
E
11

and
E
22

to check the reciprocity
between them
(
i.e. institutionalization
)
. However, logically thinking, we can judge that the
institutionalization is successful only if all lines are positive in either of
E
ii
-
E
ij
-
E
jj

(
i,j=1,
2
and i

j
)
.


3. Formalization by Balance Theory



Figure 2 and Figure 4 call up the discussions of cognitive organization by Heider
(
1946
)

and its extended formalization by Cartwright and Harary
(
1956
)
, that is, balance theory.
By introducing the conce
pt of balance, we can have another analytical viewpoint that is
different from the complementarity itself in order to examine the stability
(
or instability
)

of
role relationship. The following assumptions and definitions are required for graphical
formali
zation of our framework developed in the preceding section.


Assumption 1.

The in
-
personal role regulation of actor i is represented by the triple
relationship among three aspects: role
-
expectation toward self
(
E
ii
)
, role
-
conception
(
C
ii
)
,
and role
-
expect
ation toward the other actor j
(
E
ij
)
.
(
i,j=1,2 and i

j.
)

Each relation
between them is either positive or negative.

Assumption 2.

Actor i who received
E
ji

from the other checks it with his/her own
E
ii
. The
relation between them is either positive or ne
gative.


Definition 1.

'Unit graph,'
T
, is an undirected signed triple graph that connects three nodes:
E
ii
,
C
ii
, and
E
ij
.
(
See Figure 2
)
. Actor i's unit graph is designated by
T
i
.

Definition 2.

'Double contingency graph,'
G
, is an undirected signed gr
aph in which two
unit graphs are connected by two lines between
E
ij

and
E
jj
.
(
See Figure 4
)
. We call the
lines 'inter
-
unit lines.'

Definition 3.

If both signs of the inter
-
unit lines are positive, the complementarity of
expectations is satisfied.


Addit
ionally, we confirm some graphical terminologies and an assumption with regard to the
stability/instability of
G
.


Definition 4.

A cycle is a path
(
sequence of lines
)

that returns to its node of origin without
passing the same node or line twice. The sig
n of a cycle equals the product of the signs of
the lines it contains.

Definition 5.

G

(
or its sub
-
graph,
T
)

is balanced if all cycles in
G

(
or
T
)

are positive, and is
imbalanced, otherwise.


Assumption 3.

When
G

is imbalanced, there occurs some pressure

toward a balanced state.
In that sense, imbalanced
G

is unstable, although balanced
G

is stable.



By coupling logically the complementarity with balance, we get a typification of the role
relationship. Our next task is to examine balance of G in each

type.

(

)

Perfect complementarity
: The signs of all lines in
G

are positive. It is always stable.


-

7

-

(

)

Temporary complementarity
: Both of the inter
-
unit lines are positive, but
T
i

contains at
least one negative line.


(

-
1
)

Stable


G

is 扡ba湣nd.


(

-
2
)

Unsta
ble


G

i猠業s慬a湣nd.

(

)

Half broken complementarity
: Only either of the inter
-
unit lines is negative.


(

-
1
)

Stable


G

is 扡ba湣nd.


(

-
2
)

Unstable


G

i猠業s慬a湣nd.

(

)

Broken complementarity
: Both of the inter
-
unit lines are negative.


(

-
1
)

S
table


G

is 扡ba湣nd.


(

-
2
)

Unstable


G

i猠業s慬a湣nd.


4. Theorems for the Balance of G



At first, it is noticed that imbalance of
G

is determined by imbalance of
T
, regardless of
the role relationship type. This is almost self
-
evident through
Definition 5; however, we
confirm it as a theorem.


Theorem 1.

In
G
, if both or at least either of
T
i

is imbalanced,
G

is imbalanced.



Proof


A unit graph
T

has only a 3
-
cycle
(
call it
c
)
, and as
T

is a sub
-
graph of
G
,
c

is also a
cycle of
G
. If one of
T
i

is imbalanced, then
c

must be negative through Definition 5; so that,
G

necessarily has one negative 3
-
cycle and therefore is imbalanced.



In Figure 2, imbalanced
T

is only e
)
, a dissolved case where all three lines are negative,
although Theorem 1 ho
lds even for cases in which only one line is negative. Anyway, if
either of the in
-
personal state of actors
(
represented by
T
i
)

is dissolved, the dyadic role
relationship
(
represented by
G
)

is imbalanced and unstable. This is true, not only for types
(

-
2
)

and
(

-
2
)

where the complementarity is broken, but also for type
(

-
2
)
where it is
satisfied temporarily.


When both of
T
i

are balanced, balance judgment of
G

seems not so simple. Let us
practically examine
G

that are possible in each role relationsh
ip type. It is apparent that
G

is balanced if both actors have a
)
, 'conformity'
(
i.e. for type
[

])
, because all lines
(
therefore
all cycles
)

in
G

are positive. Graphically, this is a special case of the 'positive symmetry'
where both inter
-
unit lines ar
e positive. In Figure 2, balanced
T

is exhaustively covered by
the first four, a) through d). As both inter
-
unit lines are fixed as positive under the positive
symmetry condition, possible
G

is given by 16 combinations between
T
i

as in Table 1. Here,
a
positive line is indicated by a solid line, and a negative one by a broken line. Also,
balanced
G

is marked by

, and imbalanced
G

by
×
. Table 1 shows that 8 out of 16 are
balanced. These are type
(

-
1
)
, and its special case for type
(

)

(
upper
-
left corner of
Table 1
)
.


Table 2 is for type
(

)
, broken complementarity. Type
(

)

is understood as the
opposite of

type
(

)

and
(

)
; however, graphically, it is also symmetric in a sense that




-

8

-

Table 1. Balance of
G

Composed of Balanced
T
i
: Positive Symmetry Condition












*
)

Figures are case number.


is balanced, and
×

is imbalanced. A solid line is posit
ive, and
a broken line is negative.
(
And so on for the following tables.
)

'Positive symmetry' means that
both inter
-
unit lines are positive.



Table 2. Balance of
G

Composed of Balanced
T
i
: Negative Symmetry Condition












*
)

'Negative symmetry'
means that both inter
-
unit lines are negative.


Table 3. Balance of
G

Composed of Balanced
T
i
: Asymmetry Condition












*
)

'Asymmetry' means that the signs of inter
-
unit lines are opposite to each other. The cases in
which positive inter
-
unit lin
e is lower are omitted, as the balance judgment is the same.



-

9

-

both inter
-
unit lines are negative. In fact, in Table 2, balanced cases
(
type
[

-
1
])

show the
same combination pattern as Table 1 even under this 'negative symmetry' condition.
(
Needless to say
, this does not mean that the graphs are identical in the corresponding
combination.
)


From the viewpoint of balance, the opposite of type
(

)

and
(

)
, as well as of
(

)
, is
type
(

)
, half broken complementarity. Type
(

)

is a unique case of 'asymmetry
' where
the signs of inter
-
unit lines are different from each other. It is easy to confirm that, in Table
3, balanced combinations
(
type
[

-
1
])

reveals the reversed pattern.


Thus, the 'symmetry' seems critical for balance of
G

that is composed of bala
nced unit
graphs. Balance of
G

depends on the signs of its cycles, and the signs depend on the number
of negative lines. Here, as the number is fixed as even in
T
,
G'

s balance depends on the
signs of larger cycles, which depend on the number of negative

inter
-
unit lines. Of course,
the symmetry means that the number of negative inter
-
unit lines is 0 or 2
(
even
)
, and the
asymmetry means it is 1
(
odd
)
.


The idea of symmetry/asymmetry is of help to regulate the relationship between the
types. Especiall
y, it is suggestive that the types which are substantially located oppositely
have the same pattern of balanced combinations, formally. However, as balance of
G

is not
determined only by the signs of inter
-
unit lines, this idea is not enough to discrimina
te
balanced cases in each of the combination tables. More careful examination of the tables
leads next theorem that makes balance judgment very easy.


Theorem 2.

In
G

where both of
T
i

are balanced, all cycles other than
T'
s 3
-
cycle have the
same sign.
Which means that the existence of only one positive cycle larger than or equal
to 4
-
cycle is the necessary and sufficient condition for the balance of
G
.




Proof

Generally, in
G
, there are following four cycles other than
T
.






Let us examine these graph
s under the symmetry condition where the signs of two inter
-
unit
lines,
E
12
-
E
22

and
E
21
-
E
11
, are identical. Suppose that cycle a
)

is positive, being based on the
identical sign between
C
11
-
E
11

and
C
11
-
E
12
. This implies as a rule that
E
21
-
E
22

as well as
E
12
-
E
11

must be positive, and that
E
21
-
C
22

and
C
22
-
E
22

must have the identical sign. Thus, the sign of
each line in
G

is automatically almost determined as graph e
)

below. In this graph, apparently,
cycles b
)

d
)

are all positive, that is, they have the same sign as cycle a
)
. Next, suppose that
cycle a
)

is positive, but either of
C
11
-
E
11

or
C
11
-
E
12

is negative. This implies that
E
21
-
E
22

as well
as
E
12
-
E
11

must be negative, and that either
E
21
-
C
22

or
C
22
-
E
22

m
ust be negative, which results in
graph f
)
. Again, cycles b
)

d
)

have the same positive sign as cycle a
)
.





-

10

-










Such logic comes from a fact that one of the three lines in each
T

cannot take the sign freely
because
T

is fixed as balanced, here.
Therefore, when cycle a
)

is negative, through the same
logic, all the four cycles must have the same negative sign, too. Moreover, even when we start
from another cycle other than a
)
, and even under the asymmetry condition, the same logic is
applicable as

long as
T

is fixed as balanced.




It is apparent that, through Definition 5, finding only one negative cycle is enough for
the judgment of
G'
s imbalance. Similarly, Theorem 2 guarantees that finding only one
positive cycle
(
larger than 3
-
cycle
)

is enough for the judgment of
G'
s bal
ance. These
together result that the sign of only one cycle
(
larger than 3
-
cycle
)

is critical for the
judgment of balance/imbalance of
G
.


5. Implications



The substantial implication of Theorem 1 is relatively clear. As is already mentioned,
we can
say that if either of the actors' in
-
personal states is dissolved, the dyadic role
relationship is imbalanced and unstable regardless of the complementarity. Reversely
speaking, it is not possible that dyadic role relationship is stable although either or

both of
the actors are unstable in
-
personally, even if the relationship is complementary.


On the other hand, Theorem 2 covers the cases where both actors are in
-
personally
stable. Some of them are paradoxical in the sense that
G

is complementary but

unstable, or
inversely,
G

is not complementary but stable. Theorem 2 implies that all the paradoxical
cases
(
as well as all the other cases
)

can be consistently explained based on just one cycle in
G
. As we already saw in the proof of Theorem 2, there a
re possibly four such cycles. If we
pull out one out of them through certain reasonable assumption, we could consistently and
generally explain the double contingency situations based on just two cycles in
G
, that is,
T

and the extracted one.


We have
noticed that our framework stands on a viewpoint of the third party. It is
apparent that, for a researcher who observes role
-
interaction and plans to analyze it through
our framework, the minimum cycle is the best as far as it guarantees the same balance
judgment as larger ones. Needless to say, a cycle that satisfies the condition is c
)
in the
proof of Theorem 2. We call this cycle the 'minimum contingency cycle'
(
MCC
)

and we
confirm its validity as the balance standard in the following corollary.


Corol
lary of Theorem 2.

In G, the minimum contingency cycle
(
MCC
)

is a cycle, by the sign
of which balance/imbalance of
G

(
composed of balanced
T
i
)

is determined.



-

11

-


The sign of MCC depends on two components: the signs of inter
-
unit lines,
E
ij
-
E
jj
, and
the s
igns of
E
ii
-
E
ij

(
for i,j=1,2 and i

j
)
. Coupling the two components with each other, as in
Table 4, we can compactly regulate all the combinations in Table 1 through Table 3, keeping
correspondence to the typification of role relationship.


Now, we go b
ack to the paradoxical cases previously mentioned. The first is type
(

-
2
)

where
G

is complementary but unstable. In this type, the signs of
E
ii
-
E
ij

must be


Table 4. Complementarity and Stability of
G

Composed of Balanced
T
i











different from e
ach other, therefore the MCC takes a form:










This MCC presents that, while the complementarity is kept, role
-
expectations toward self
and toward the other are not reciprocal on either side. It is also noticed that
E
11

and
E
22

are reciprocal becaus
e all lines are positive in either
E
11
-
E
12
-
E
22

or
E
22
-
E
21
-
E
11

(
This is an
institutionalized situation, in that sense.
)


Suppose again two normative contents, 'working' and 'keeping house,' that are not
compatible with each other, but reciprocal. It is
impossible in the MCC above to arrange
them with no logical contradiction. For example, under the condition of negative
E
11
-
E
12

and positive
E
22
-
E
21
, if
E
11

is 'working,'
E
12

and then
E
22

shall be 'working,' however, as
E
21

shall be also 'working,'
E
22

an
d
E
21

never be reciprocal
(
E
22
-
E
21

never be positive
)
. It is
probable that this case is actually observed with no contradiction, only if it is possible to
devise a kind of 'quasi
-
category' that satisfies the compatibility and reciprocity at the same
time,

between either of original categories. For example, 'part
-
time working' and '
(
full
-

time
)

working' can be understood as compatible with each other and reciprocal, at the same
time. Similarly 'keeping house' could be replaced by 'part
-
time housework.'



Another point of this case is that role
-
expectation toward the other never be realized
either in half or at all, even if contradictions could be avoided by utilizing a quasi
-
category.
Suppose that actor 1
(
husband
)

is 'rebellion'
(
see Figure 2
)
, therefo
re
E
11
-
E
12

is negative.

-

12

-

He rejects '
(
full
-
time
)

working'
(
E
11
)

and rebelliously expects his wife to perform it
(
E
12
)
.
Actor 2
(
wife
)

interprets the expectation as 'part
-
time working' in a compatible range of
'working.' Here,
E
12

will be realized, but only

in half, because the wife will not actually
engage in full
-
time work. On the contrary, the wife reciprocally expects her husband to
perform 'full
-
time working'
(
E
21
)
, but her expectation will not be realized at all because her
husband rejects it.


Thu
s, instability of this case implies mutual dissatisfaction. We can understand this
paradoxical case as
the unsatisfied complementarity kept by quasi
-
categories
.


The next paradoxical case is type
(

-
1
)

where
G

is not complementary but stable. In
this
type, the signs of
E
ii
-
E
ij

must be identical, and the MCC takes a form:






When both
E
ii
-
E
ij

are positive, this MCC presents that role
-
expectation toward self and
toward the other are reciprocal in each in
-
personal state, even though the complementarity

is broken. This reciprocity may come from 'conformity,' or may be just role
-
playing founded
on 'suppression.' Anyway, as
E
11

and
E
22

are not reciprocal
(
a negative line is included both
in
E
11
-
E
12
-
E
22

and
E
22
-
E
21
-
E
11
)
, the actors are conformable, but on
ly in
-
personally. A
probable situation is that both actor 1
(
husband
)

and actor 2
(
wife
)

have 'working' for
E
ii
,
and, obeying it, expect to perform 'keeping house' each other. The stability of this case is
exactly founded on the separated relationship.
Their expectations completely keep missing
each other; however, they keep their social identities, respectively. Thus, we can call this
case, which is probable in multi
-
cultural situations,
the stable separation founded on social
identities
.


When both

E
ii
-
E
ij

are negative, all relations in the MCC are broken, and
E
11

and
E
22

are
neither reciprocal
(
one and more negative lines are included in both
E
11
-
E
12
-
E
22

and
E
22
-
E
21
-
E
11
)
. If both actors are 'rebellion,' a probable situation is that actor 1
(
husban
d
)
,
rejecting 'working'
(
E
11
)
, rebelliously expects his wife to perform it; on the contrary, actor 2
(
wife
)
, rejecting 'keeping house'
(
E
22
)
, rebelliously expects her husband to perform it.
(
Though
E
11

and
E
22

are apparently reciprocal, it is not a result
of institutionalization, but
an accidental result through the role regulation.
)

As same as the previous case, the
separated relationship brings about the stability; however, in this more disordered case, the
separation is founded on personal identity rath
er than social identity. In fact, each actor
keeps his/her isolated but unshakable internal world irrespective of expectations from the
society and his/her partner. In the sense, this case is
the stable separation founded on
personal identities
.


On t
he other hand, if both actors are 'impracticability,' the situation implies the
existence of external obstacles to role performance. For example, both
E
11

and
C
11

are
'working' for actor 1
(
husband
)
, but some obstacle
(
e.g. unemployment
)

not only prevents


-

13

-

him from performing it, but also makes his
E
12

'working' unwillingly. Similarly his wife
neither be able to perform 'keeping house' following
E
22

and C
22
, and she inevitably shows
'keeping house' for
E
21
. The stability of this case clings to the interna
l consistency between
E
ii

and
C
ii

in each actor. We can call the case
the stable but inevitable separation caused by
external obstacles
.


Likewise, the other situations are also able to be explained thoroughly based on
T

and
the MCC. Having this compa
ct analytical framework, we could systematically find not only
the graphical regularity among various
G
, but also the comprehensive meaning underlying
identical cases from the viewpoint of complementarity and balance.


6. Concluding Remarks



The discus
sions of social system and homo sociologicus make sense to understand the
mechanism of stabilization of a society. On the other hand, the criticism against them also
makes sense to understand the dynamic reality which is going on even under the stability.

An important theoretical problem is that, as soon as we introduce the latter standpoint and
say that the complementarity of expectations shall be seen as inherently flexible and
unstable, we are confronted with the infinitely various world. We believe th
at the previous
analysis could suggest a possibility to grasp such world systematically and to link the two
theoretical streams that have confronted each other.


We have considered the role relationship as a cognitive framework in which actual role
inte
raction is explained from a viewpoint of the third party. In order that we stand on a
viewpoint of the actor and follow actual interaction under the framework, some specific
problems must be resolved
(
see also note 4
])
. The cognitive instability of
G

due

to its
imbalance probably reduces an actor's utility, therefore influences interaction. On the other
hand, irrational interaction will not be interrupted, but will be continued through the
revision of role relationship toward balance. It is open to furt
her discussion whether we see
the aspect in a unified utility formation process or in a dual framework of action.
Additionally, we will be confronted with the problem how to compare objective payoff with
mental comfort and distress.


With regard to the
cognitive revision, the dynamic process of
G'
s change is also our
further subject. From a formal point of view, we will be able to utilize the 'line index' of
balance to examine which
G

is easier to be transformed into which. Perhaps, it is also
required

to investigate the relationship between our model and 'expectation states theory'
(
Berger et al. 1974,1977; Fararo and Skvoretz 1986; also see note 1
])
.
5)


Our graphical formulation can easily be extended to triad and, possibly, to a general
interacti
on system that is composed of plural actors. Based on the extended
G
, as
exemplified in Figure 5, we will be able to re
-
examine Simmel's discussion, the concept of
role
-
set, and empirical findings in sociology of family as well as in ethnomethodology and
symbolic interactionism.




-

14

-











Figure 5. Examples of Extended Triad Graph


There have been so many graphical approaches to role theory
(
especially in social
network researches
)
; however, most of the discussions have been concentrated on the
'stru
ctural equivalence' and on objective role
-
relations. We hope that this paper stimulates
another graphical approach that incorporates the process of in
-
personal role regulation into
the inter
-
personal double contingency relations.


Acknowledgements

An earl
ier version of this paper was presented at European Japanese Conference on Rational Choice and
Formalization
(
Leipzig, Oct. 2001
)

and 102nd Conference of Japan Sociological Association for Social
Analysis
(
Tokuyama, Dec. 2001
)
. I appreciate all the produc
tive comments.


Notes

1
)

'Expectation states theory' developed by Berger et al.
(
1974,1977
)

should be seen as a formal extension of
this tradition. It is directly concerned with organization of status characteristics through task
performance and evaluatio
n in a task
-
oriented group; however, 'task' may be analogously replaced by
'role.' The theory also includes balance theoretical formulation. It is expected that our formulation,
that focuses on dyad and actors' internal states as well, can be linked toge
ther.

2
)

According to Kobayashi, the process consists of two aspects: 1
)

ego refers to role knowledge as an
interpretative code in expecting alter ego's reaction, and 2
)

ego refers to it as a code switch between social
role
-
expectations and role
-
conceptio
n of him/herself.
(
Also see the discussion on 'relevance' by Schutz
[
1970
]
, and the distinction between 'logic
-
in
-
use' and 'reconstructed logic' by Fukazawa
[
1990, 1994
])
.
In the conceptual framework of Watanabe
(
1981:111
)
, on which we developed our fram
ework in Figure 2,
the latter aspect was incorporated as 'role negotiation process'; however, the former was neglected. Our
refinement is capturing the aspect.

3
)

In this paper, the term 'complementarity' is limited to this inter
-
personal aspect, and is d
istinguished
from the 'anticipated reciprocity' in the individual recognition previously mentioned. The latter should
be synonymous with the anticipated 'complementarity.' However, through this conceptual distinction, we
can clearly describe those unsucc
essfully institutionalized cases as the broken complementarity, in which
E
ii
-
E
ij

is positive, but
E
ij
-
E
jj

is negative.

4
)

If we stand on the actors' viewpoint, we have to consider even judgment discrepancies between them.
Such more complex situations where

directed graphs are differently defined for the same role relationship
is within our scope of analysis, but in the future.

5
)

Also see Misumi
(
1991
)

for a graphical analysis of in
-
personal change of the role knowledge. The

-

15

-

process of cognitive revision s
hould be consistently related to the mechanism that determines
compatibility
(
or reciprocity
)

between the nodes in
G
. A Boolean role model developed by Misumi
(
2001,
2002
)

is suggestive on that point.


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