A Formal Theory of Interaction in Social Groups
Author(s): Herbert A. Simon
Source: American Sociological Review, Vol. 17, No. 2 (Apr., 1952), pp. 202-211
Published by: American Sociological Association
Stable URL: http://www.jstor.org/stable/2087661
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202
AMERICAN SOCIOLOGICAL REVIEW
characteristics which are considered by our
respondents as Jewish. Also, this same
image of the Jew may be detached from
some of the actual Jews living in the community. The same phenomenon, in a less
extensive manner, has occurred with respect
to the Negro image.
Because of such confused identity it is
impossible to predict from a knowledge of
verbal expressions by Maple County re-
spondents how the same respondent will behave toward specific members of a minority
group. The particularity of situation, role,
and specificity of image all lead to this
inability to predict in Maple County. For
adequate prediction it would be necessary
to know the specific images which the
majority peoples have of the minority persons or groups and in what specific types of
situations behavior is likely to occur.
A FORMAL THEORY OF INTERACTION IN SOCIAL GROUPS
HERBERT A. SIMON *
Carnegie Institute of Technology
o a person addicted to applied mathematics, any statement in a non-mathematical work that contains words like
"increase," "greater than," "tends to," constitutes a challenge. For such terms betray
the linguistic disguise and reveal that underneath the words lie mathematical objectsquantities, orderings, sets-and hence the
possibility of a restatement of the proposition in mathematical language. But what
purpose, other than an aesthetic one, does
such a restatement serve? In this paper I
shall attempt to show, by means of a concrete
example, how mathematization of a body of
theory can help in the clarification of concepts, in the examinationof the independence
or non-independence of postulates, and in
the derivation of new propositions that suggest additional ways of subjecting the theory
to empirical testing.
The example we shall use is a set of propositions that constitutes a part of the theoretical system employed by Professor George
C. Homans, in The Human Group,' to explain some of the phenomena that have been
observed of group behavior. This particular
example was selected for a number of reasons: first, although non-mathematical, it
shows great sophistication in the handling of
systems of interdependentvariables; second,
Professor Homans takes care with the operational definition of his concepts, and these
concepts appear to be largely of a kind that
can be measured in terms of cardinal and
ordinal numbers; third, Professor Homans'
model systematizes a substantial number of
the important empirical relationships that
have been observed in the behavior of human
groups. Whether his theory, in whole or part,
turns out to be correct or incorrect (and
this is a question we shall not raise in the
present paper), it will certainly receive careful attention in subsequent research on the
human group.
* I am indebted, for stimulation, assistance, and
suggestions in the formulation of this theory, to my
colleagues in a research project on administrative
centralization and decentralization sponsored at
Carnegie Institute of Technology by the Controllership Foundation, and particularly to Professor
Harold Guetzkow, who has worked closely with
me at every stage of the theory formulation. Valuable help has also been received from Professor
George C. Homans of Harvard University, and
from seminars at Columbia University and the University of Chicago, and a session at the 1951 annual
meetings of the American Sociological Society, where
various portions of the paper were read and
discussed.
The system will be described in my own
language. After I have defined the variables
and set forth the postulates, I will discuss
what I believe to be the relationship between
the system and the language that Homans
employs in his book.
The Variables.We consider a social group
(a group of persons) whose behavior can
be characterizedby four variables, all functions of time:
T
THE SYSTEM:
CONCEPTS AND POSTULATES
1 New York: Harpers, 1950.
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A FORMAL THEORY OF INTERACTION IN SOCIAL GROUPS
I(t)-the intensity of interactionamongthe
members;
F(t)-the level of friendliness among the
members;
A(t)-the amount of activity carriedon by
memberswithin the group;
E(t)-the amountof activity imposedon the
group by the external environment
(the "external system")
This particular set of variables includes
most of those employed by Homans in the
first part of his book (he adds others in his
later chapters), and the underlined terms
are the ones he uses. In this paper we will
assume that operationaldefinitions (Homans'
or others) have been assigned to the variables, such that the behavior of a group at
any moment in time can be measured in
terms of the four real numbers I, F, A, and
E. For our purposes, we need to make only
two points clear about these operational
definitions.
First, since the units in which such variables can be measured are somewhat arbitrary, we shall try to make use only of the
ordinal properties of the measuring scalesthe relations of greater or less-and, perhaps,
of certain "natural" zero points.
Second, since the variables refer to the
behavior of a plurality of human beings, they
clearly represent averages or aggregates.
For the interaction variable, I, let Iij represent the number of interactions per day (or
the time, per day, spent in interaction), of
the ith member of the group with the jth
member. Then we could define I as the
average rate of interaction per memberi.e., as 1/n times the sum of Iij over the
whole group, where n is the number of
members. Similarly, we could define F as
the average friendliness between pairs of
members; and A might be defined as the
average amount of time spent per member
per day in activity within the group.2
Finally, E might be defined as the average
2 The concept of "activity within the group"
might require rather sophisticatedtreatment. For
example, time spent by a worker in daydreaming
about his family or outside social relations might,
ideally, be excluded from his activity within the
group. For some purposes,we might wish to regard
as "activity within the group" uniformitiesof behavior among group members-that is, the degree
to which activity lies within the group might be
measuredby similarity of behavior. On this point,
see Homans, op. cit., pp. 119-121.
203
amount of time that would be spent per
member per day in activity within the group
if group members were motivated only by
external pressures.3
The Postulates. We postulate three sets of
dynamic relations among the variables, treating I(t), F(t) and A(t) as endogenous
(dependent) variables whose values are determined within the system: while E(t) is
an exogenous (independent) variable.
(1) The intensity of interaction depends
upon, and increases with, the level of friendliness and the amount of activity carried on
within the group. Stated otherwise, we postulate that interaction is produced, on the one
hand, by friendliness, on the other, by the
requirements of the activity pattern; and
that these two causes of communication are
additive in their effect. We will postulate,
further, that the level of interaction adjusts
itself rapidly-almost instantaneously-to
the two variables on which it depends.
(2) The level of group friendliness will
increase if the actual level of interaction is
higher than that "appropriate"to the existing level of friendliness. That is, if a group
of persons with little friendliness are induced
to interact a great deal, the friendliness will
grow; while, if a group with a great deal of
friendliness interact seldom, the friendliness
will weaken. We will postulate that the adjustment of friendliness to the level of interaction requires time to be consummated.
(3) The amount of activity carried on by
the group will tend to increase if the actual
level of friendliness is higher than that "appropriate"to the existing amount of activity,
and if the amount of activity imposed externally on the group is higher than the
existing amount of activity. We will postulate that the adjustment of the activity level
to the "imposed" activity level and to the
actual level of friendliness both require time
for their consummation.
These three relations can be represented
3 This formulationreveals that the direct measurement of E might pose greater problems than
the direct measurementof the other variables. In
most cases, we would attempt to measure E indirectly in terms of the magnitude of the force
producingE-in somewhatthe same manneras the
force of the magnetic field is sometimes measured
by the strength of the current producingit. The
problem is by no means insoluble, but we do not
wish to deal with it in detail here.
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204
AMERICAN SOCIOLOGICAL REVIEW
The next section of this paper will be
by the following equations, where dx repre- devoted to an
analysis of the system represents the derivatives of x with respect to sented by equations (1)-(3). It should be
emphasized again that this system is only a
time.
partial
representation of the complete sys(1.1) I(t)=aF(t)+a2A(t)
tem of hypotheses proposed by Homans,
[I (t)-PF (t)
(1.2) dF(t)b
and, of course, an even sketchier representation of reality. Furthermore,the assumption
dAt)
d( )-cl[F(t)
(1.3)
of linear relations in the equations is a
-A(t)]-c2[E(t)-A(t)]
All constants in these equations are as- serious oversimplification, which will be
remedied in a later section of the paper.
sumed to be positive.
the system incorporatesseveral
Nevertheless,
If we look at equation (1.2), we see that
of
the
important
relationships that might be
interof
amount
the
B8Fmay be regardedas
hypothesized
as
holding among the four
action "appropriate" to the level, F, of
variables
and
which
Homans found did, in
friendliness. For if I=,8F, then F will have
hold
in
the
fact,
situations
he investigated.
no tendency either to increase or decrease.
is,
The reciprocal of the coefficient /3, that
THE SYSTEM: DERIVATIONS FROM THE
1/,3, might be called the "congeniality coeffiPOSTULATES
cient" since it measures the amount of
A number of well-known techniques may
friendliness that will be generated per unit be applied to derive consequences from the
of interaction.
system of postulates that could be tested by
Similarly, from equation (1.1) we see that comparisonwith empirical data.
a1F may be regarded as the amount of
(1) The equations might be solved exinteraction generated by the level, F, of plicitly to give the time path the system
friendliness in the absence of any group would follow from any particular initial
activity. That is, if A=0, then I-a1F.
position. This presents no mathematical difFurther, the coefficient a2 measures the ficulties, since systems of linear differential
amount of interaction generated per unit of equations with constant coefficients can be
group activity in the absence of friendliness. solved completely and explicitly. On the
Hence, a, and a2 might be called "coefficients other hand, the solutions would be useful
of interdependence."
for prediction only if the constants of the
Finally, from equation (1.3) we see that equations were known or could be estimated.
the reciprocal of the coefficient y measures For this reason, the explicit solutions would
the amount of activity that is generated per seem to be of interest at a later stage in the
unit of friendliness,in the absence of external development of measurement instruments
pressure. We may call 1f}y a coefficient of and testing of the theory, and we will not
"spontaneity." The remaining coefficients, dwell on them here.
b, cl and c2, determine how rapidly the
(2) The equilibrium positions, if any, of
system will adjust itself if it starts out from the system might be obtained, and their
a position of disequilibrium.
properties examined. This would permit us
Relation to Homans' System. These equa- to make certain predictions about the betions, and their verbal interpretations, ap- havior of the system when it was in or near
pear to represent with reasonable accuracy equilibrium.
the larger part of the generalizations about
(3) The conditions for stability of the
the interrelations of these four variables equilibrium might be examined. Since a syswhich Professor Homans sets forth in Chap- tem that is in equilibrium will not generally
ters 4 and 5 of his book.4
remain there unless the equilibriumis stable,
4See especially the italicized statements in op. we will ordinarily be justified in using the
cit., pp. 102, 111, 112, 118, 120. The reader can conditions of stability in predicting the beperhapsbest test the translationhimselfby reference havior of any system that is observed to
to ProfessorHomans' text. In doing so, he should
take due note of footnotes 2 and 3, above. Pro- remain in or near equilibrium.
fessor Homans has been kind enough to go over
the equations (1.1)-(1.3) with me. He concludes
that the mathematicaltreatment does not do violence to the meaningsof his verbal statements,but
that the equationsdo not capture all of the interrelationshe postulates-that they tell the truth, but
not the whole truth. With this later qualificationI
would concur.
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A FORMAL THEORY OF INTERACTION IN SOCIAL GROUPS
(4) Starting with the assumptionsof equilibrium and stability, we may be interested
in predicting what will happen if the independent variables or the constants of the
system are altered in magnitude-that is,
what will be the new equilibrium position to
which the system will move. This method,
the method of "comparative statics," is one
of the most powerful for deriving properties
of a gross qualitative characterthat might be
testable even with relatively crude data.
Our method, therefore, will be to derive
first the conditions of equilibrium, next the
conditions of stability, and finally the relations that can be obtained by applying the
method of comparative statics.
Equilibrium. An equilibrium position is
one in which the variables remain stationary.
Hence the conditions of equilibrium can be
found by setting dF/dt and dA/dt equal to
zero in equations (1.2) and (1.3), respectively, and solving the three equations for
I, F, and A in terms of E. Designating by
Ion F. and A. the equilibrium values corresponding to E., we find:
(1.4) Io=aFo+a2Ao
(1.5) O=b(I-j3F.)
(1.6) O=c1(F.-,yA.)
+C2(E.-A.)
Eliminating I0 from (1.5) by using (1.4),
we get:
(1.7) Fo=-
a'2A
A0
Substituting this value of F. in (1.6) and
solving for Ao, we get:
(1.8) A,=[
C2)(3-al)
L(Cl'Y+C2) (j3-ai)
-(c,
a2)
J
(ClY+C2)
(3-ai)
-
and
(1.12) cly+c2+b
(3-ai)>O,
(1.13) (#-a,) (clry+c2)-a2cl>O.
Since all constants are assumed positive,
we obtain from (1.13) the requirementthat:
(1.14) 98>ai
If (1.14) holds, (1.12) will, in turn, be
automatically satisfied.
Hence (1.13) and (1.14) together give us
necessary and sufficient conditions for stability. We proceed now to an interpretation of
these conditions.
Stability condition (1.14) may be written:
(1.15) jPF.>a1F.
That is, we require for stability that the
amount of interaction (,BFW)required to generate the equilibrium level of friendliness be
greater than the amount of communication
(a1F.) that would be generated by the equilibrium level of friendliness in the absence of
any group activity. For if this were not so,
(i.e., if al>,8) an initial level of friendliness,
F1, would produce interaction, I1=a1F1,
which would further increase the friendliness
atF1
> F1, and we would get
to F2=Il/,8=
[C2(/3-ai)+C1
whence:
[
It is a well-known property of such dynamical systems that for stability the real
parts of the roots of A must be negative, and
conversely, that if the real parts of the
roots are negative, the system will be stable.
By solving (1.11) for A, this can be shown
to imply:
C2 (-ai)
E.=[
(c a2)]
205
y(j3-al)-a2
~
E,
an ascending spiral such that the amount
of friendliness and the amount of interaction
would increase without limit:
Fi<F2<Fa<...
<F.,
and
Il<I2<IS< ... <In
Stability of Equilibrium
To determine whether the equilibrium is
We can show that the other stability constable, we consider the so-called "character- dition (1.13), is requiredto prevent a similar
istic equation" associated with equations ascending spiral between A and F.
(1.2) and (1.3) after I has been eliminated
Behavior of the System: Comparative
by substitution from (1.1): 5
Statics. The equalities and inequalities we
-b (P-ai) -X I ba2
(1.10)
have derived as conditions for equilibrium
c1
I
and stability of equilibrium enable us to
deduce certain propositions about how the
When expanded, this becomes:
S
=0O
-a2cl
X+b
(coy+c2)
(g-al)
(1.11) V+t jCl'+C2+b(P-ai)
-(ClY+C2)-X
5 The mathematicaltheory involved here is discussed in Paul A. Samuelson,Foundationsof Eco-
nomic Analysis, Cambridge: Harvard University
Press, 1947, p. 271.
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206
AMERICAN SOCIOLOGICAL REVIEW
system will behave when its equilibrium is
disturbed, assuming the equilibrium to be
stable.
Equilibriummay be disturbed by a change
in E, the task imposed on the group, or
by changes in one or more of the coefficients
of the system (e.g., an increase or decrease
in a2). We wish to predict how the variables
of the system will respond to such a shift.
The change in the equilibrium value of
A with a change of E can be determined
from (1.8). Stability requires (by (1.14))
that the numerator of the right-hand side
of (1.8) be positive, and (by (1.13)) that
the denominator be positive. Hence:
(1.16)
dE >0
From (1.7), remembering (1.14), we get
similarly:
dF0
dF0
>0
hence dF
(1.17) dF,>0,
dE.
dA,
Finally, from (1.4), we get:
dA.
dF,
d1L
(1.18)
dE = a
dE
+a2
dE
>0
We conclude that an increase in the activities required of the group by the external
environment will increase (when equilibrium
has been re-established) the amount of
group activity, the amount of friendliness,
and the amount of interaction. As E decreases toward zero, A, F and I will decrease
toward zero. But this is precisely the hypothesis that Homans employs to explain
social disintegration in Hilltown,6 and to
explain the difference in extension between
the primitive and modern family.
We ask next how large Ao will be in relation to Eo. From (1.8), in its second form,
we see that the numerator on the right-hand
side will be larger than the denominator if
and only if:
(1.19)
y(P3-a,)
<a2
If (1.19) holds, then, we will have
AO>EO,otherwise Ao?Eo. We will refer to
a group satisfying condition (1.19) as one
having positive morale. If the condition is
not satisfied, we will say the group has negative morale.
What relations among the coefficients are
conducive to positive morale? From (1.19),
6 Op. cit., pp. 356-362.
7 Op. cit., pp. 263-265.
we see that a2 should be large, relative to
the product of y and (/3-al). But large a2
means high interdependence, i.e., the group
tasks are highly interrelated. From our previous interpretation of y (i.e., that 1/y
measures spontaneity), we see that a high
degree of spontaneity is conducive to positive morale-with large 1/y, or small y,
friendliness will tend to produce a relatively
large amount of activity in addition to that
required by the external environment.
As mentioned above, another condition
conducive to positive morale is that (/3-al)
be small: that there be a strong feedback
from friendliness to more interaction to
more friendliness. But we have seen that an
approach to zero of (/t-al) means an approach to an unstable condition of the
system (see equation (1.13)).
Now, from the stability condition (1.13),
we know that a large value of (yc1+c2) aids
stability, but if we want y small relative to
a2 for positive morale, we must depend on
the ratio c2/c1 for stability. That is, under
conditions of positive morale we require that
the activity level, A, be more strongly influenced by the external demands than by
the level of friendliness.
While we must be careful not to expect
too much from a theory as highly simplified
as this one, it may be interesting to note
that the phenomenon of negative morale
appears to be not unrelated to Durkheim's
concept of anomie. In particular, a division
of labor within a group that would result
in little interrelationshipof tasks (a2 small)
would, in our theory, be conducive to negative morale. This is a prediction that has
received a considerable amount of substantiation from the Hawthorne studies and other
empiricalobservationsin industrial sociology.
We may inquire finally as to the time
path whereby the system readjusts itself
when it is disturbed from an initial equilibrium by a change in E0. It can be shown
that the roots of X in (1.11) are real. This
implies that the system will not oscillate,
but will start out toward the new equilibrium
at a rapid rate, approachingit asymptotically.
GENERALIZATION
TO A NON-LINEAR
SYSTEM
It is time now to relax the assumption of
that the relations
equations (1.1)-(1.3)
among the variables of the system are linear.
The reason for dwelling at length on the
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A FORMAL THEORY OF INTERACTION IN SOCIAL GROUPS
207
linear equations is that they can be regarded
as an approximation to the more general
equations of the non-linear system in the
neighborhoodof points of equilibrium.
Since we really do not have much empirical data as to the exact forms of the
functions relating our variables, we shall
strive in our treatment of the non-linear
(A1,F,)
system to make as few assumptions as possible about these functions. The price we
Y (A1117)=
shall have to pay is to restrict ourselves
largely to a graphical treatment and to the
derivation of gross qualitative results. Nevertheless, in view of the roughness of the emA
pirical observations we might hope to make,
this restriction cannot be regardedas unduly
FIG. I
serious at the present stage of development Through any point (A1, F1), draw a short
of the theory.
d . Then
We will now assume our equations to be: line segment with slope p/+= dA
this segment points along the path on which
(2.1) I =f (A, F)
dF
our system would begin to move if started
g
F)
(2.2)
t- =g(I,
from (A1, F1).
dA
By drawing such a line segment for each
(2.3)
((AF; E)
point of the (A,F)-plane, and connecting
where f, g, + are functions whose properties these into continuous curves, we find the
remain to be specified. If we replace I in paths the system will follow from any initial
(2.2) by its value as given by (2.1) we positions to the subsequent position (and
a new possibly to equilibrium). The collection of
obtain, in place of (2.1)-(2.2)
all such paths is commonly called the "direcequation:
tion field" of the system (see Figure 2).9
dF
(2.4)
dt =g(f(A,F),
F)-0(A,F)
where p is again a function of unspecified F |0
form. Henceforth, we will work with the
system comprised of equations (2.3)-(2.4)
-two differential equations for the determination of F and A.
Our method will be graphical, based on
the "phase diagram" of F and A.8 Let us
regard E, for the present, as a constant-a
given parameter. Equation (2.3) gives us
the time rate of change of A, and (2.4)
the time rate of change of F, both as functions of F and A. Dividing the second by
the first we get
dF/dt
q)(AF)/+(AF;E)
dA/dt=
the rate of change of F relative to A
for each pair of values of F and A. Now
consider a graph (Figure 1) whose x-axis
measures A, and whose y-axis measures F.
df/dA-
8 On the method employed, see Alfred J. Lotka,
Elements of Physical Biology, Baltimore: Williams
and Wilkins, 1925, pp. 77-97, 143-151.
L
M
A
FIG. 2
9 For a more detailed explanation of the construction of the direction field, see Lester R. Ford,
Differential Equations, New York: McGraw-Hill,
1933, pp. 9-11. The direction field corresponding
to the linear system of this paper is discussed and
illustrated by Ford on pp. 48-52. His Figure 14,
page 51, corresponds to the case of stable
equilibrium.
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AMERICAN SOCIOLOGICAL REVIEW
208
Now consider the set of points
(2.5)
dA
Tt
=;(AF;E)
=O
at which A is not changing. Equation (2.5)
will, in general define a curve in the (A,F)plane. At any point on this curve, since
q is zero and hence A is constant, but not F,
the path of the system will be vertical
(either upward or downward as 4)>0 or
4)<O, respectively).
Consider next the sets of points
(2.6)
dF
dt
-
(,)_
at which F is not changing. At all points
on this curve, since 4 is zero, the path of
the system will be horizontal (either to right
or left as b>0 or t<0, respectively).
At the point, or points, where (2.5) and
(2.6) hold simultaneously-that is, where
the two curves intersect-the system will be
in stable or unstable equilibrium. The equilibrium will be stable if any path very close
to the point of equilibrium leads toward it
and unstable if any path very close to it
leads away from it. (This definition of stability can be shown to be equivalent to
a suitable generalization of the analytic
definition we employed in the linear case.)
Figure 2 illustrates the direction field and
the curves frO and 0=0. There are two
points of equilibrium, K and L. Equilibrium
at K is stable, at L unstable.
It should be remarked that if the system
starts off at any point above the lower of
the two broken lines in the figure, it will,
in time, approach the point of stable equilibrium, K; while if the system starts off
below this broken line, F and A will ultimately decline and approach the point Mthe group will, in fact, dissolve.
Now the paths taken by the system from
various initial points will depend on the locations of the curves q '0 and 0) O, and
their points of intersection. The particular
shapes and positions of the curves, as drawn
in Figure 2, represent empirical assumptions
as to the shapes of the functions q and 4.
What can we legitimately assume about these
functions? To answer this question we must
ascertain the empirical significance of the
two curves q= 0 and 4)0.
Equation (3.3) says, in effect, that for
a given amount of external pressure (a given
value of E) the amount of activity under-
taken by the group (A) will tend to adjust
itself to the level of friendliness (F). Our
empirical assumption is that, given E,
greater friendliness will tend to produce
greater activity. If this is so, then the equilibrium value of A must increase as F increases; that is, the curve i/iO must have
a positive slope. We now make the second
empirical assumption: that there is a saturation phenomenon-that as F continues to
increase, A will increase only at a diminishing rate. If this is so, the curve U=O must
be concave upward as we have drawn it.
In the particular case illustrated in Figure 2,
it is assumed that E is sufficiently great
so that there will be some activity even
in the absence of friendliness. This is represented by the fact that the curve cuts the
x-axis to the right of the origin. Later,
we will consider the case also where this
condition does not hold.
Equation (2.4) says that the amount of
friendliness in the group (F) will tend to
adjust itself to the amount of group activity (A). Again we assume that greater
activity will tend to produce greater friendliness; hence that the curve 4=0 must have
a positive slope. If we now assume that
this mechanism is also subject to saturation,
the curve must be concave downward. Finally, we assume that unless the activity is
above a certain minimum value there is no
tendency at all for friendliness to develop
(q 0 cuts the x-axis to the right of the
origin).
In the particular case shown, f0- cuts
the x-axis to the right of q-O. If this were
not so, the point L would disappear and
the system would move toward the stable
equilibrium, K, from any initial point, including the origin. We will consider this case
0O
later. In the particular case shown,
is sufficiently far to the right that it intersects d) O. If this were not so, the system
would move toward the origin from any
initial point. This case also will be considered later.
Finally, it should be mentioned that the
particular assumptions we have made about
the curves do not depend in any essential
way upon the precise indexes used to measure
F and A. For any given scale used to measure
F or A, we can substitute another scale, provided only that the second scale has the
same zero point as the first and does not
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A FORMAL THEORY OF INTERACTION IN SOCIAL GROUPS
reverse the direction of change (i.e., that
we do not have F1> F2 on the first scale
but F'1< F'2 for corresponding situations
measured on the second). To be more precise, our concavity properties may be altered
but not the order or character of the equilibrium points or the presence or absence
of the region below the lower broken line.
Since the conclusions we shall draw depend
only on these properties of the graph, a
change in the index employed cannot affect
our results.
Suppose now that we begin with the system in equilibrium at K, and progressively
reduce E, the external pressure to activity.
A reductionin E may be assumed empirically
to reduce (through the mechanism of equation (2.3)) the equilibrium value of A
associated with each value of F-i.e., to
move the curve V/'O to the left. In the
simplest case (in first approximation) we
may assume that the shape of the curve
is unchanged. Then, as ifrO moves to the
left, its intersection, K, with /=O will move
downward and to the left along p O. We
have shown:
Proposition 2.1. As E is decreased the
equilibrium levels of A and F will be
decreased.
This proposition also held in our linear
system.
As + O continues to move to the left
(continued reduction in E) the two curves
will eventually intersect at a single point
of tangency. Let us call the value of E corresponding to this position of tangency ET.
As E is reduced below ET, the two curves
will no longer intersect and all paths of
the direction field will lead to the x-axis
and, if q-O now intersects with the y-axis,
the system will come to rest at the origin.
We have shown:
Proposition 2.2. As E is decreased below
some critical value, ET, F will go to zero;
and for some sufficiently small value of E
(equal to or less than ET depending on the
location of the intersection of l(A,F;ET)
with the x-axis) A will go to zero.
Here we find, in the non-linear case, a
new phenomenon-a dissolution of the group.
It might be supposed that if a group has
been dissolved by reducing E below ET it
can be restored by again increasing E to
ET. This does not follow. For if the system
is initially at the origin, its path will lead
209
toward K only if q=O intersects the x-axis
to the right of 0==0. But the smallest value
of E for which this condition holds is obviously greater than ET. From this follows:
Proposition 2.3. The level of E required
to bring a group into existence is greater
than the minimum value, ET, required to
prevent the group, once formed, from
dissolution.
To illustrate Proposition 2.3 we show, in
Figure 3, the path that will be followed by
F and A when E is (1) reduced from some
initial value, EK, to ET, (2) then to some
lower value, EL, (3) then increased to EO,
where i-iO intersects the origin, (4) finally
increased to EM where ifrO intersects the
x-axis just to the right of q O. In the descending portion of the path, the decrease in
F lags behind the decrease in A; while in
the ascending portion of the path the increase in F again lags behind the increase
in A. Hence the whole path forms a loop
in the counter-clockwise direction in the
(A,F) -plane.
F
(E;O
S~~~~~~
9(E)=O
._
A
0
FIG. 3
Notice also that the system remains-at
rest at the origin so long as E is below E.
and that A increases, but not F. as E increases from E. to EmS
In the linear model we studied the effects
upon the equilibrium values of A and F
of certain shifts in the parameters,a,, a2,
and ,8 of the system. With E fixed, an
7,
increasein interdependenceof tasks (increase
in a, and a2), an increasein congeniality
(decrease in ,B) and an increase in spontaneity (decrease in -y), within the limits
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AMERICAN SOCIOLOGICAL REVIEW
210
imposed by the stability conditions, all resulted in an increase in the equilibrium
values of A and F.
In the non-linear model an increase in
interdependence of tasks or an increase in
congeniality would be represented by a shift
upward of the curve p0O; an increase in
spontaneity would be representedby a clockwise rotation of the curve t=O about its
intersection with the x-axis. In all cases, if
we began from a position of equilibrium, the
new equilibriumvalues of A and F would be
larger than the initial values.
SOME APPLICATIONS OF THE MODEL
While the model described here was suggested by Homans' analysis of behavior in
The Human Group, we have attempted to
present only part of his system: in particular
we have omitted reference to phenomena of
hostility, and to interpersonal differentiation
(kinship and leadership). On the other hand,
the mathematical model is capable of application to some situations that lie outside
Homans' analysis. In this section we shall
discuss briefly a few of these.
(1) Formation of Cliques. Define variables IL, A1, F1, and E1 to refer to behavior
in a specified group, GI; and I2, A2, F2 and
E2 to refer to behavior in a group, Gii,
which is a subgroup ("clique") within GI.
Then we might postulate equations of the
form:
(3.1)
dA
=4i (Al, F1, A2; E1)
(3.2) dF1/dt=41(A1, F1)
(3.3) dA2/dt-4'2(A2, F2, A1; E2)
(3.4) dF2/dt=42(A2, F2)
These equations are similar in form to
(2.3) and (2.4) except for the presence of
the "coupling" variables: A2 in (3.1) and
A1 in (3.3). The meaning of this coupling
is that activity within the clique (A2) is
assumed to interfere with and depress
activity in the larger group (A1) and activity
within the larger group (A1) is assumed to
interfere with clique activity (A2). We might
also have further complicated the model by
adding coupling terms to (3.2) and (3.4)
("conflict of loyalties").
The behavior of the system (3.1) to (3.4)
can be studied as follows. We take E1 and
E2 as fixed. Then for any given value of
A2, we can set (3.1) and (3.2) equal to
zero and find the corresponding equilibrium
value, A*, of A1. This value, A*, will depend
on A2, and, under our assumptions will decrease as A2 increases. Similarly, from (3.3)
and (3.4) we can find the equilibrium value,
A* of A2 for each value of A1. A position
of equilibrium of the whole system will be
found at the intersection of the two curves
A*=A1(A2) and A* =A*(A1) in the plane
whose x-axis represents Al1and whose y-axis
represents A2. If the two curves do not
intersect, then the clique and the group cannot coexist in equilibrium.Even if the curves
intersect, the equilibrium may be unstable,
but we cannot here go into the exact conditions of stability.
(2) Competition of Groups. Instead of a
clique within a group we might have two
groups competing for the membership of a
single individual. In this case, the variables
A1, F1, Il, E1 would refer to the intensity
of his activity in the first group; A2, F2,
I2, E2 to the intensity of his activity in the
second group. We can then proceed exactly
as in the first case.
(3) Activity of an Individual. The variables in equations (2.3) and (2.4) need not
be interpreted as group activity. Instead,
A might be interpreted as the amount of
time per day an individual devotes to any
particular activity, F as the amount of satisfaction he obtains from the activity, E as
the pressure on him to engage in the activity.
In this case we might want to make different
assumptions as to the shapes of the curves,
c/=O and q(=O, in the phase diagram than
in the previous cases, but the general approach is the same. Similarly the model of
equations (3.1) - (3.4) might be interpreted
to refer to an individual's distribution of
attention between two activities.
(4) Regulatory Enforcement. Still another application of models of this general
class would be to the phenomena associated
with the enforcement of a governmental regulation (e.g., gasoline rationing). Here A
would be interpreted as the actual degree
of conformity to the regulation, F as the
social pressure to conform, E as the effect
of formal enforcement activity. The reader
may find it of some interest to translate
the theoremswe have previously derived into
this new interpretation.
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A FORMAL THEORY OF INTERACTION IN SOCIAL GROUPS
CONCLUSION
In this paper we have constructed a mathematical model that appears to translate with
tolerable accuracy certain propositions asserted by Homans to hold for behavior in
human groups. We have examined at some
length what assumptions the model requires
and what further propositions can be deduced from it. In particular, we have seen
that it offers an explanation for some of
the commonly observed phenomena relating
to the stability and dissolution of groups.
In the last section we have shown that
models of this general class can be applied
to a rather wide range of behavioral phenomena beyond those originally examined.
211
We do not imply from this that the psychological mechanisms involved in all these
situations are identical. The underlying
similarity appears to be of a rather different
character. In all of these situations there
are present: (a) an external (positive or
negative) motivational force toward some
activity, and (b) a secondary "internal"
motivational force induced by the activity
itself. It is the combined effect of two such
motivational forces that produces in each
case phenomena of the sort we have observed. And especially when the relations are
not linear (and the non-linear must be supposed to be the general case), "persistent"
and "gregarious" patterns of behavior can
result.
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