AN ANALYTICAL STUDY OF THE BALANCE
OF POWER THEORIES
By
DINA A. ZINNES*
Department of Government, Indiana University
1. Introduction
Two very different studies of recent
origin suggest and make possible a
reappraisal of one of the oldest concepts
in the literature of international relations:
the balance of power. On the one hand,
Claude’s (1962) excellent analysis of the
various dimensions along which the
balance of power concept has been used
allows us to disregard those discussions
that have been principally polemical,
and to concentrate instead on those which
might explain certain international phenomena. On the other hand, Singer and
Small (1967) have recently proposed
what appears to be a new interpretation
of the balance of power concept and,
further, have collected data and performed a series of analyses that point in
the direction of testing hypotheses concerning the functions of a balance of
power. These two studies suggest that
the time is appropriate to begin enuxnerating the various balance of power hypotheses and devote some concentrated
attention, along the lines initiated by
Singer and Small, to techniques and designs that will test these propositions. These
are the joint purposes of the present paper.
bad in international relations; and as a
label describing a specific historical period,
most notably the 18th and 19th centuries.
More recently, Kaplan’s (1957) analysis
has generalized the label usage by providing abstract characteristics of what can
now be called balance of power systems.
These are the three dimensions along
which we will not venture. If I understand Claude correctly it is the fourth
sense in which I wish to explore the
balance of power concept, what Claude
calls the usage of the balance of power to
denote a situation: ’to indicate the
character of a situation in which the
power relationship between states or
groups of states is one of rough or precise
equality ... Balance of power refers to a
situation in which power is literally
2. The
3. The dependent variable : peace or status quo a
What then are the balance ol power
hypotheses? Following the direction initiated by Haas (1953) some years ago and
updating his extensive bibliography on
the balance of power, I have isolated
what appear to be seven reasonably
distinct hypotheses which predict certain international consequences (the de-
variety of uses of the balance of powder
concept
Claude indicates that there are at
least four quite different ways one might
employ the concept ’balance of power’.
It has been used in a prescriptive sense
to indicate the type of policy a state
is pursuing or ought to pursue; as a
symbol of what is either considered good
or
&dquo;balanced&dquo;
by equivalent power’. (Claude
to a specific
1960, pg. 13). Since it refers
distribution of power Claude feels that
this usage of the term is ’purely descriptive’. For our purposes, however,
the use of the balance of power to denote
a situation becomes more than just a
description, it is a definition of the independent variable in hypotheses about the
relations among nations.
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271
pendent variable) as a function of the
independent variable, a balance of power.
Since there is considerably greater agreement on the definition of the dependent
variable it provides an obvious point of
departure.
The consequences of a balance of
power world appear to be principally
Some writers, though probably a
that a true balance of
leads
to
international
power
peace. While
two.
minority, claim
it is not the only avenue to peace, they
feel that it is one workable solution to
violent international conflict. The majority
of writers, however, contend that the
maintenance of peace is a possible but
not necessary by-product of a balance of
power. These writers argue that the
critical consequence of a balance of
power is that it holds all states in check;
it does not allow any state to become
overwhelmingly strong and therefore in a
position to take over the territory of another state. Consequently, a balance of
power preserves the status quo of the
system and guarantees the independence
of every state, no matter how small. This
may or may not involve war.
Thus we have two possible definitions
of the dependent variable: ( 1 ) the balance
of power preserves peace and/or (2) the
status quo of the system. This discrepancy
between the definitions of the dependent
variable is troublesome but can be bypassed. We will assume that there are
two sets of theories, one for each definition of the dependent variable.
4. The independent variable : six definitions of
a ’balance o, f power’
The major difference in the balance of
power theories, however, lies not in the
definition of the dependent variable but
in the definition of the independent
variable, the concept ’balance of power’.
This is the source of the seven different
theories - one for each definition of the
balance of power. Six of the seven definitions have been abstracted from the
balance of power discussions
and as such share certain characteristics.
We will therefore present them first and as
a unit. The seventh definition, derived
from the recent Singer-Small article,
can then be compared with the six perhaps more traditional definitions.
Most discussions of the balance of
power do not explicitly enumerate the
necessary and sufficient conditions for the
existence of a balance of power world.
They provide instead a description of the
key characteristic of a balance of power
world, i.e. that characteristic without
which a system would not qualify as a
’balance of power’. It is therefore possible
to construct all those worlds (i.e. systems)
which contain this central or key
characteristic. In this sense, then, the
first six definitions do not come directly
from the literature. Rather, they represent
plausible interpretations of balance of
power worlds based on the discussions
concerning the essential ingredient of a
balance of power system.
What is this defining characteristic
of a balance of power world and to what
extent do writers agree on a single characteristic ? Listed below, in as nearly a
parallel fashion as possible, are a series of
quotes in which this key characteristic
is described. This survey covers several
hundred years of writings on the balance
of power and should highlight any similarity of thought among the writers. To
provide parallelism, each quote is structured using the following forxnat :
A balance of power is
numerous
1.
an equal distribution of Power
among the Princes of Europe as makes
it impractical for the one to disturb
the Repose of the other ...’ (Anony...
mous,
1741 )
2. ’The doctrine of counterpoise among
nations, in order to make the declaration of war, by any one, less alluring
...’ (Leckie, 1817)
3. ’... action by
a state to keep its
neighbor from becoming too strong ...
because the aggrandizement of one
nation beyond a certain limit changes
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272
the general system ... (and) may mean
the ruin and subjection of all the other
attention to the mainkind of equality and
between neighboring sta-
neighbors ...
tenance
of
a
equilibrium
tes ...’ (Fenelon, 1835)
4. ’... an arrangement of affairs so
that no State shall be in a position
to have absolute mastery and domin-
the others ...’
(Vattel, 1916)
equilibrium ... between the
members of the family of nations ...
to prevent any member of the family
... from becoming omnipotent ...’
(Oppenheim, 1920)
6. ’... (in the simple case involving
only two units)... two states or
aggregations of power equally balanced... (and in the more complex
are
5. ’...
case
an
of)
five Great
Powers, acting
there was to be
balance between the individual States
when combined ... and the unruly
member against whom they united ...
(Here) there was ... no need of strict
equality and no necessary disturbance
if one State grew stronger than another so long as the growth was not
so great as to threaten the united
strength of the other four ...’ (Pol-
independently ...
...
11. ’... a maneuvering among the
three or several powers with one
main objective in mind, to prevent
any one of them from dominating
the others ... ’ (Kulski, 1964)
If
we
ignore differences
in
expression
particularly evident from one historical
period to another, this list of ’definitions’
illustrates, perhaps surprisingly, almost
complete agreement on the defining
characteristic of a balance of power
world. These writers agree that a ’balance
of power’ involves a particular distribution of power among the states of the
system such that no single state and no
existing alliance has an ’overwhelming’ or
’preponderant’ amount of power. As
Pollard indicates, and others imply, when
there are only two states or alliances of
states in the system the basic requirement (no preponderance) can only be met
if there is an equality of power between the
two. But as you move away from this
simplest case, a variety of distributions of
ance
power are permissable. In effect, any
distribution is permissable as long as the
power of each unit - state or alliance of
states
in the system is less than the
combined power of all the remaining
units.
Clearly, the defining characteristic of
a balance of power world, according to
all these writers, rests on two critical
variables: (1) the distribution of power
among the states of the system and (2)
the alliance configuration. If we assume
that the power of an alliance is a linear
function of the powers of the individual
members (e.g. the power of an alliance
composed of A and B = the power of
A + the power of B), it is possible to
construct the following six balance of
power systems based on a system of five
states which meet the requirement set
forth by the above writers, viz. that there
be no all-powerful, omnipotent unit in the
a
system.
lard, 1923)
‘
7. ’
such a &dquo;just equilibrium&dquo;
in power among the members of the
family of nations as will prevent any
one of them from becoming sufficiently
strong to enforce its will upon others
...
...’ (Fay, 1937)
&dquo;equilibrium&dquo; of the type
represented by a pair of scales.
When the scales are equal, balance
8. ’...
results ... the balance of power
that through shifting alassumes
liances and countervailing pressures
no one
power or combination of
powers will be allowed to grow so
strong as to threaten the security of
the rest’. (Palmer and Perkins, 1953)
9. ’... the system of maintaining an
equilibrium among
(Herz, 1959)
the
powers ...’
10. ’... the quest for safety expressed
in a search for allies. The
safety of all was assured only if no
one nation or group of nations was
... chiefly
to achieve a preponderof power - if, in other words,
rough balance was achieved’. (Stoes-
permitted
singer, 1961 )
-
A balance of power is
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a
world in which:
273
(1)
There
are
no
states have
alliances and all
B
equal power. (A
= E).
states belong to one
=
C = D
of
(2) All
alliances and the power of the
alliances is equal. (A + B = C +
=
two
two
D +
E).
(3)
There are two alliances equal in
power and one non-aligned state.
(A + B = C + D; E).
(4) There are two alliances and a
third non-aligned state, such that the
power of either alliance plus the nonaligned state is greater than the
power of the other alliance. (A + By
C + D, E; A + B + E > C + D and
A+B<C-~D-I-E).
(5) There are no alliances, and the
power of each state is less than the
summed total power of all the remain-
ing states.
( ~ Xi > X~ for j
i*jJ
=1, ..., N,
total number of nation
where N
states in the world).
(6) There is one state or alliance
which is more powerful than any other
unit in the system but such that
condition (5) above is still met,
=
( ~ X~ > X~
i*i
J
N
=
i
for j = 1,
total number of
...,
N, where
states
in the
worldandA>B>C>D>E).
This set of six definitions clearly does
exhaust the possible permutations
and combinations that one might generate
using the two variables and meeting the
basic requirement. These definitions are
however, the more interesting ones and
the ones encountered most frequently
in the balance of power literature. The
not
first two definitions are immediately
derivable from the quotes given earlier.
The third and fourth represent two
plausible interpretations of those balance
of power discussions that point to Britain’s
performance in the 18th and 19th centuries as a ’balancer’. The fifth definition most clearly resembles the notion
of a collective security as attempted in the
concert of Europe, the League, and United
Nations; it is also, as discussed earlier,
easily derivable from the quotes given
above. The last definition has been in-
cluded because of the number of discussions of the balance of power in which the
writer indicates that a balance of power
really means, on the order of a check
account balance, a positive or a balance
in favor of a particular state.
While the above six are different types
of balances of power, note that they all
meet the basic requirement of a balance of
power world. This is perhaps where some
of the confusion in the balance of power
literature lies. One writer describes a
balance of power via definition { 1 ),
another via definition (4), and there
appears to be a lack of agreement on the
use of the term. As can now be seen, the
balance of power definitions have a
common denominator, but they can be
expressed in a variety of forms.
The differences and similarities between
the six definitions can be graphically
shown by examining a hypothetical world
composed of five nations. If we construct
all possible alliance configurations (designated by ABC/DE where the slash indicates an alliance between those nations
on either side) and power distributions
(i.e. rank ordering) for this world, we can
demonstrate which alliance configuration and power distribution combination
each definition chooses as a ’balance of
power’ world. Figure I is such a matrix.
This matrix has been entitled a ’general’
opposed to a ’specific’ matrix because
representation of an alliance configuration or a power distribution refers
to a pattern and not to a specific alliance
configuration or a specific power distribution. For example, the following alliance
configurations and power distributions
are here equal:
as
each
In other
are
are
words, the labels of the nations
The relevant factors
the number of alliances of various
unimportant.
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274
Definitions : *
A balance of power is a world in which:
(1) There are no alliances and all states have
(2)
(3)
(4)
Mt!t~
equal power.
of
two
alliances
and
the power of the two alliances is
belong
There are two alliances equal in power and one non-aligned state.
All states
to
one
There
are
two alliances and
a
third
um
non-aligned state, such that the power of either alliance
plus the non-aligned state is greater than the power of the other alliance.
alliances, and the
(5)
all the remaining states. M
There
(6)
equal.
Im
are no
There is
one
state
or
such that condition
§0l
alliance which is
(5)
[is]
power of each state is less than the summed total power of
above is still
more
met.
powerful than
any other unit in the system but
-
* These definitions follow those given in the text,
**
the most perfect balance
Numbering indicates the rank ordering of the systems from ’1’
of power system - to ’5’ the poorest representation of a balance of power system, under
the Singer-Small definition.
-
sizes and the
placing and number of
equalities and inequalities
Definitions (1) and (6), which are only
power
function of the distribution of power,
pose no problems and are easily located
in the matrix. Definition ( 1 ) is clearly
more
restrictive than definition (6);
the latter definition, which only requires
that there be one all powerful nation,
allows for eight times as many systems
(where each matrix cell is now considered
to be a system) .
a
Definitions
(2), (3), (4), and (5) are
easily located in the matrix. Even
assuming, as noted earlier, that there is a
linear relationship between the power of
individual nations and the composite
power of an alliance of those nations,
without knowledge of the specific power
not
as
of each nation we cannot, for these definitions, determine exactly whether a
system is balanced. Does the following
power distribution and alliance configuration represent a balance of power system
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275
definition (2) (where the power of
alliances must be equal): A>B>
and AC/BDE ? This can only
C> D > E,
be answered by knowing the actual
amount of power possessed by each of the
nations, or, what here amounts to the
same thing, knowing the magnitude of
the difference in power between the nations.
Thus, definitions (2), (3), (4), and (5)
sometimes identify three, but more frequently only two, sets of systems. First,
by a process of elimination, all four
definitions identify those systems (i.e.
matrix cells) that could not qualify as
balance of power systems regardless of
the specific power of each nation (within
the restriction imposed by the particular
2
power distribution under consideration) .2
Second, all four definitions permit the
designation of those systems that could be
but are not necessarily balance of power
systems.3 Only definition (5) designates
the third set of systems, namely those
which would be balance of power systems
regardless of how one assigned power to
the nations within the given power distribution restriction.
Since definition (5) is the easiest to
locate in the matrix, the analysis can best
begin with it. The requirement that there
be no alliances immediately indicates that
balance of power systems under this
definition must fall somewhere in the
second row of the matrix. Hence, by a
process of elimination the first set of
systems - those which could not qualify
under any circumstances - are pinpointed as all remaining cells in the matrix.
The third set of systems designated by
this definition - those systems which
would be balance of power systems
regardless of specific power differences are all those systems (in the second row)
in which the first, or most powerful,
position is tied. Whenever there are two
or more nations having equal superior
power (and no alliances) a balance of
power system results. This is shown in
using
the
two
’
the matrix. The second set of systems those which could be balance of power
systems given specific power differences consists of the eight remaining cells in
the no-alliance row. These cells, where
there exists one nation more powerful
than each of the others, could be balance
of power systems if the difference between
the most powerful nation and the other
nations is not too great, in particular,
not greater than the sum of the remaining
nations.
Definitions (2), (3), and (4) designate
only the first two types of systems: by
implication, those that are definitely not
balance of power systems and those that
could be balance of power systems.
Recall that we are concerned here only
with the patterns of alliance and power
configuration and not with the specifics
of who belongs to what alliance. Note also
that the three definitions depend heavily
on the number of nations in the system:
specifically, whether the number of nations in the system is odd or even.
Definition (2)
there exist two and
two
alliances
only
equal in power chooses
rows 7, 8, 9, 10, and
immediately
12. The remaining rows of the matrix
therefore belong to the first class of systems : those which could under no circumstances qualify as balance of power
systems. With three exceptions, however,
it is not possible further to delineate
balance of power matrix cells. For each
matrix cell in rows 7, 8, 9, 10, and 12,
with three exceptions, there exists at
least one (and frequently more) specific
alliance configuration and one specific
power distribution which could meet
definition (2) criteria. These cells therefore belong to the second class of systems
those that could be but are not always
balance of power systems - and are shown
in Fig. 1. The exceptions concern the
power distribution pattern where all the
nations are equal. Here the three alliance
-
-
configuration patterns ABC/DE, ABCD/
CDE are not acceptable under definition
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276
(2) and therefore belong to the first class
of systems indicated above.
Since definitions (3) and (4) differ
only in the magnitude of the discrepancy
between the power of the two alliances
and the non-aligned nation, it is not
surprising to find that these definitions
coincide in the general (though obviously
not in the specific) matrix. They both
choose rows 6 and 11. Once again, for
reasons just discussed we cannot delineate
beyond this.
The matrix comparisons of the six
definitions, then, show where the differences and congruencies occur. Note that
definitions (1) and (6) do not overlap
nor do definitions (2), (3), (4), and (5),
but that the two sets of definitions do
overlap. Definitions (1) and (6) share at
least one common matrix cell with definitions (2), (3), (4), (5)
if we consider the
matrix cells which could be but are not
necessarily balance of power systems by
definition (5). Hence there are a number
of systems which meet the criteria of two
definitions. The matrix also demonstrates
what systems are not considered balance
of power systems under any traditional
definition.
-
5. The
independent variable : the seventh
definition of a balance of power
Singer’s and Small’s (1967) definition
of a balance of power world provides an
interesting contrast to the six described
above. As
the
with the first six
and Small do not
explicitly enumerate the defining characteristics of a balance of power system.
However, on the basis of their discussion
these features can be abstracted. The
following excerpt should provide the
rationale for the derived definition given
below:
was
case
definitions, Singer
going into the quagmire of
terminological and normative dispute
Without
which has characterized much of the
theoretical literature on the balance
of power, we can nevertheless note that
its defense or justification clearly
the assumption that the
of the international system
can be maintained without reliance
on superordinate political institutions
Central to this notion is the understanding that the invisible or unseen
hand will function only to the extent
that all nations are free to deal with
all others as their national interests
dictate. Thus, it is assumed that every
dyadic relationship will be a mixture
of the co-operative and the conflictful,
with political, economic, ideological,
and other issues all producing different
interest configurations for each possible pair of nations. The net effect, it
is believed, is such a welter of crosscutting ties and such a shifting of
friendships and hostilities that no
single set of interests can create a
self-aggravating and self-reinforcing
division of cleavage among the nations ; A and B may well have competitive economic interests in the
Middle East, but harmonious strategic interests in the Caribbean, while
B and C’s political interests may
coincide in regard to West Africa
and clash in an international organization setting. It follows from this sort
of a model that anything which
restrains or inhibits free or vigorous
pursuit of the separate national interests will limit the efficiency of the
stabilizing mechanism. And among
those arrangements seen as most
likely to so inhibit that pursuit are
formal alliances. Nations in the same
alliance are less free to compete with
their allies in such spheres of incompatibility, and less free to co-operate
with outsiders in areas of overlapping
interests.
rests
on
stability
...
Thus, a perfect (or at least precondition for) balance of power system for
these authors appears to be one in which
every state in the system has equal
positive and negative sentiments towards
every other state in the system. Since
alliances are defined as obstacles which
impede this balance between positive and
negative ties, the operational measure of a
perfect balance of power system is one in
which no state belongs to an alliance.
How does this definition compare with
those discussed above? First, the Singer-
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277
Small definition is not dichotomous as are
the above definitions. The traditional
definitions only allow for balance of
power and non-balance of power systems.
Using the Singer-Small definition one
can gradually move away from the perfect system by increasing the number of
nations having alliance commitments.
Second, unlike the first six, the SingerSmall definition is not a function of
power; it is solely a function of alliances.
Third, there is a difference between the
alliance configuration variable found in
the first six definitions and the alliance
variable in the Singer-Small definition.
The initial six definitions are a function of
the number of alliances in the system (two
or none). The Singer-Small definition
counts the number of nations having at least
one alliance commitment. Only at the
extreme end are the two formulations
equivalent: if there are no alliances then
there can be no nations having an alliance
commitment, and vice-versa. Otherwise,
there is no obvious relationship between
the alliance configuration variable of the
first six definitions and the number of
alliances of this definition. Although
the number of alliances in the system will
always be less than the number of nations
in the system having alliances, since a
given nation may belong to more than one
alliance, there is no simple way to determine the relationship between the number
of alliances in the system and the number
of nations having at least one alliance
commitment.
Let us now attempt to place the SingerSmall definition in the matrix of Fig. l.
The perfect Singer-Small balance of
power system clearly falls in the second row
of the matrix. But since this definition
allows for
degrees of ’balance-ofwe
have indicated, along the
powerness’,
alliance column margin, the rank ordering of the systems from the ’best’ (perfect
balance of power) to the ’worst’ balance
of power system, using the Singer-Small
operational definition (number of nations
having at least one commitment). The
large number of ties is due to the fact that
the Singer-Small formulation does not separate overlapping alliance membership
systems from non-overlapping ones.
How does the Singer-Small definition
compare with the traditional definitions
of the basis of the matrix? It is interesting
that the perfect balance of power system
for Singer and Small coincides with
definition (5) in the matrix. But this
overlap is due to the reason described
earlier: if there are no alliance commitments then there can be no alliances.
Thus, while the two definitions agree on
this one point they are basically of a
different character; one counts numbers of
alliances and the other counts the number
of nations having at least one alliance
commitment.
The difference between the SingerSmall formulation and the traditional
balance of power definitions is emphasized by the fact that systems which,
according to Singer and Small, depart
farthest from a balance of power frequently
qualify under the traditional definitions.
Furthermore, the second and third ’best’
systems under the Singer-Small definition
do not overlap any of the four definitions
which consider alliance configuration.
These second and third choices do overlap definitions (7) and (10), but this is
because these definitions are solely a
function of power, whereas the SingerSmall definition is solely a function of
alliances. If the Singer-Small definition
does not restrict systems on the basis of
power distributions then it can easily
permit those power distributions
definitions (7) and (10).
We conclude then that the SingerSmall definition is of a considerably
different character than any of the
traditional definitions. Systems which
qualify under traditional definitions are
considered by Singer-Small standards to
be poor representations of a balance of
power, while the better Singer-Small rep-
designated by
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278
resentations
are
power systems
considered balance of
the traditional writers.
not
by
6.
Testing a balance of power theory
Combining the two dependent variables
with each of the possible definitions of a
they argue, the superimposition of friendly
and hostile relations reduces the prospects
for war. Alliances are then defined as
obstacles which prevent the successful
operation of crosspressures. The theory
can be outlined as follows:
free, uncommitted nations-3 cross-pressures-~.
(i.e. non-aligned nations)
balance of power we arrive at a potential
of fourteen hypotheses. Clearly the
next step is to subject at least some of
these propositions to empirical tests.
Singer and Small (1967) have begun the
pioneering work in this area. They derive
an hypothesis from their balance of power
model and test it for the 130 year period
between 1815 and 1945.4 Undoubtedly
this is the first attempt empirically to
verify any of the balance of power propositions. Unfortunately, due to difficulties in the operationalization of the
variables and the execution of the analyses,
the results do not permit us to draw any
conclusions concerning the validity of
their balance of power model. However,
their design does suggest modifications
which would allow us to test directly their
interpretation of the balance of power.
Since they have begun the difficult
task of testing the balance of power
theories, it seems wise to pursue their
initial attempts.
There are three basic problems inherent
in the Singer and Small study: the operationalization of the independent and
dependent variables and the assumptions
involved in testing the relationship
between the variables. We will analyze
the study in this sequence. From our
earlier discussion we recall that for
Singer and Small a perfect balance of
power world is one in which every
state has equivalent positive and negative
ties with every other state in the system.
The crosspressures on each state, under
this condition, are at a maximum; hence,
set
peace.
On the basis of the theory the writers
derive their central hypothesis:
’The greater the number of alliance
committments in the system, the more
war the system will experience ...’
A comparison of the hypothesis with
the original theory shows that what was
originally a one-way implication has now
become an equivalence: the theory maintains that free and uncommitted nations
implies cross-pressures, but the hypothesis
indicates that the non-existence of alliances
is equivalent to the existence of crosspressures.
The validity of
an
equation
between
free, uncommitted nations and the exisof cross-pressures in the system
on the index used to
measure the freedom of the nations. The
tence
depends heavily
’interaction-opportunity’ ratio, designed
inversely the amount of free-
to measure
dom or ‘uncommittedness’ of nations in the
system, is formed by counting the number
of nations having at least one alliance,
and dividing this number by the total
number of nations in the system. The
ratio necessarily varies between zero and
one, where zero indicates the existence
of no commitments and hence the maximum operation of crosspressures, and one
indicates that all nations are in some
alliance and crosspressures are nonexistent. To what extent is the interactionopportunity ratio an inverse measure of
the amount of crosspressures existing in
the systema
Suppose the world contained only five
nations, A, B, C, D, and E, and consider
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279
the following
tions :
possible
alliance
configura-
(1) ABC/DE
(2) ABC/BCD/CDE/DEA
(3) AB/AC/AD/AE
(4) ABCDE
the interaction-opportunity ratio,
the amount of crosspressures existing in
each of the four international systems is
1. Accordingly,
the same, namely 5/5
for each of these systems crosspressures are
non-existent. In the first system this result
is intuitively appealing. With all nations
committed to one of two sides there is
little likelihood that all pairs of nations
have equal positive and negative ties.
But what happens in systems two, three,
and four? Could not one argue that in
system two, A’s, B’s, and E’s membership
in two alliances and C’s and D’s membership in three alliances produces crosspressures ? Are not some of these alliances
contributing rather than diminishing crosspressures ? In other words, the interactionopportunity ratio is a valid measure of
crosspressures if it can be assumed that
nations belong to one and only one
alliance. When multiple memberships
occur, an index is required which not
only considers the decrease in crosspressures as a consequence of committed
nations but also accounts for the increase
in crosspressures as the result of overlapping alliance membership.55 Thus, the
principal difficulty with the interactionopportunity ratio as an operational measure of the independent variable is that
it assumes nations belong to only one
alliance.
To what extent is the one-allianceper-nation assumption met in the SingerSmall data? The number of alliance
memberships held by each nation per
year was determined and an ’overlap’
index constructed by summing the number of multiple memberships held by
each nation. For example, in 1833 Austria,
Prussia, and Russia each belonged to
alliances. The amount of overlap
for that year would be 1 + I + 1 = 3.
If a nation belongs to four alliances in
the same year it contributes three to the
’overlap’ index. In general, the overlap
index for a given year can be computed
from the formula:
two
Using
=
where n = the number of alliances in
which nation i holds membership, and
m
the number of nations in the system
belonging to any alliance.6 Note that the
index is only concerned with the set of
nations having some alliance member=
ship?7
The results of the
analysis indicate that
one-alliance-per-nation assumption is
only valid for 22 of the 130 years: from
the
1824 to 1832, 1849, 1860 to 1862, 1865,
1871 to 1876, 1879 to 1880. Only for
these years is the interaction-opportunity
ratio an adequate measure of the amount
of crosspressures existing in the system.
There are, however, indications in the
text that the authors were aware of the
overlap problem, though perhaps from a
different viewpoint.8 In an attempt to
measure structural characteristics of the
alliance system they construct an additional index: a ’bipolarity’ index. This is
calculated for each year roughly as
follows:
1. Determine all alliance memberships.
2. Determine the targets of these
alliances.
3. Eliminate these two sets of nations
(allied and targets) and count the
number of nations remaining in the
system: (the free, non-aligned nations).
number of free nations is
since each nation can ally
with all other free nations, and this
result is added to the product of the
number of free nations times the size
of the largest alliance or the sum of the
targets, whichever is bigger. The
resulting figure is divided by the
number of pairs theoretically pos4. The
squared,
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280
sible
as
determined
by n ~ 2 l~
where
the number of nations in the
The bipolarity indices can
therefore be described by the two
ratios:
n
=
system.9
The bipolarity index for all three systems
is 3/21. Again, only the first would appear
to be truly bipolar.
The bipolarity index, then, contains
the same difficulty encountered in the
These ratios
again vary between zero indicating complete bipolar cleavage in
the system - and one - indicating the
total absence of such cleavage.
The first significant feature of the two
ratios is the important role played by the
interaction-opportunity index: systems
having no overlapping membership are
indistinguishable from systems in which
considerable criss-crossing of alliance
membership exists. Since this new index
is specifically designed to measure the
number of free nations in the system. One
extreme, when there are no alliances and
extent to
bipolarity is non-existent, is intuitively
appealing. The other extreme, however,
produces difficulties. If there are no free
nations in the system then bipolarity, or
cleavage, according to the index, is
maximum. But compare the following
two
systems of five nations:
(1)
(2)
ABC DE, where the target of one
alliance is the other; and
ABC, BCD, CDE, where the targets in order are D, E, and A.
Both systems,
according to the bipolarity index,
maximally bipolar. Yet
the
first
would
only
appear to be truly
are
’bipolar’ as that term is commonly used.
difficulty does not occur only at the
extreme. Compare the following three
systems, each containing seven nations:
The
(3)
(4)
(5)
ABC DEF G, where the target of
one alliance is the other, and
ABC BCD EF G, where the targets
of the alliances in order are D, E,
and A, and
ABC BCD CDE DEF G, where the
targets of the alliance in order are
D, E, F, A.
which a system contains cleavthe
defect
here would appear to be
ages,
serious.l°
particularly
Having examined the operational measure of the independent variable, we turn
now to a consideration of the dependent
variable. To restate the original hypothesis : ’The greater the number of
alliance commitments in the system the
more war the system will experience ... ,’
(italics mine). What is meant by ’more
war?’ The crosspressures balance of
power theory would seem to dictate that
’more war’ be defined as the probability
that war breaks out. The fewer the alliance commitments, the greater the crosspressures in the system pulling each nation
towards divergent goals, the less the
probability that strong animosities will
develop between nations, the less probable is war.ll
But how does one measure the probability of war? Singer and Small propose
three different operational measures for
the dependent variable. Of the three,
the one that would seem to be the best
measure of the probability of war is the
frequency or number of wars that began
each year. This, however ,is the index
which the authors feel is the least in-
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281
structive of the three, since it does not
discriminate between large and small
wars nor between long and short wars:
’Thus the correlation of alliance aggregation with mere war frequency would be of
limited interest (if not downright misleading) and further graduation is clearly
necessary’. An examination of the war
data, however, discloses a perhaps more
serious defect with the use of this index:
there are few instances in which more
than one war breaks out in any given year.
Necessarily, a correlation between this
index and any measure of the independent
variable will be, and in subsequent
analyses is found to be, small. The low
frequency of wars per year is partially
alleviated by using three and five year
periods. A count is made of the number of
wars that began not in just the following
year but in the following three and five
years to obtain three different frequency
measures (wars that broke out in one year,
within three years, within five years).
While this increases the frequency count
somewhat, it adds a new problem for
testing the hypothesis (to be discussed
below).
It would seem that the difficulty here
is that the dependent variable is actually
dichotomous. Since wars can be of any
size involving from two to all system
nations, a simple count of the number of
wars which began in each year would be
inappropriate. If all nations were involved in a particular war, clearly only
one war could begin that year. And, should
the war last several years no wars could
begin for as many years.
Two alternative measures might be
suggested. The simplest expedient would
be to treat war as a dichotomous variable:
either wars began or did not begin in the
system for any given year. A somewhat
more complicated measure, but one which
would discriminate between wars involving only two nations and wars involving all nations, would be to calculate
along the lines of the Singer-Small
-
interaction-opportunity index - the number of dyads at war. The number of
dyads at war divided by the number of
possible dyads in the system could be
interpreted as the probability of war for
that year.
The other two measures used by
Singer and Small were designed for the
purposes of discriminating between wars of
different size and severity. As they explain it:
identified those international
which interest the student of
international relations as he examines
the effects of alliance patterns, the
next point to note is that these wars
differ markedly in their duration,
magnitude, and intensity. Thus, the
correlation of alliance aggregation
with mere war frequency would be of
limited interest (if not downright
misleading) and further gradation is
clearly necessary. This gradation is
achieved in the first instance by use
of the nations-months-of-war measure,
so that the simple magnitude of each
war is the sum of the months which all
nations individually experienced as a
participant in the war ... In addition
to nation-months as an index of the
magnitude of war, at least one other
factor seems to justify consideration.
That fact we will call severity and it
will be measured by the number of
battle-connected deaths of military
personnel sustained by all participants
in any given war.
Having
wars
The purpose of these
ably
measures
is presum-
of different
provide probabilities
war - wars of varying magniand severity. Unfortunately, how-
to
types of
tidues
ever, the
of these measures alters the
being tested. Compare the
use
hypothesis
following two
statements:
( 1 ) Crosspressures lead to a lower
probability of war,
(2) Crosspressures lead to smaller and
less
severe wars.
Only the first hypothesis can be deduced
from the crosspressures balance of power
theory. Yet the use of the magnitude and
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282
and
three, and five
severity measures changes the first hypoth-
tions)
esis into the second, and it is the second
that is actually being tested. What is
wanted is a technique which will hold
constant the varying magnitudes and
severity of war. Do alliance commitments and the probability of wars of
a certain magnitude and severity correlate significantly?
We come now to an examination of the
third aspect of the Singer-Small research
design: testing the hypothesis. As the
authors describe the analyses: ’the method
to be employed is a trend analysis. After
developing several different measures of
alliance aggregation and several measures
of the onset of war, we will examine the
extent to which the two sets of variables
rise and fall together, over time’. Correlations were computed using five variations
of the operational measure for the dependent variable (’more war’) and seven variations of the operational measure for the
independent variable (’alliance aggrega-
year lags.13
The last technique - the use of three
and five year lags - warrants special
consideration. Here the measures of the
two variables were summed and/or averaged, thus making the data non-independent. If we sum the nation-months
for wars beginning in 1815, 1816, and
1817 (or average over the three years for
alliance aggregation), then sum over
1816, 1817, 1818, etc. the data become
highly interdependent. As such, a Pearson
product moment correlation (i.e. any
significance test thereof) is inappropriate.14
That this lack of independence is an
important contributor to the results can
be shown by a comparison of the one year
lag condition with the three year lag
condition for the severity and magnitude
tion’), namely:
variable
in which
always assigned
a war began.12 The correlations were
computed within three time periods,
1815 -1945, 1815 -1900, and 1900 1945, for the entire international system
and for only part of that system, the
central system (nations either located in
Europe or deeply involved in relatively
durable relationships with European na-
The
were
measures
for the
dependent
to the year
using
one,
war measures.
Table I (next page) shows that in all cases
the three year lag condition is higher.15
A second difficulty in the conduct of
the analyses was hinted at in the previous
section. Since the magnitude and severity
alter the hypothesis being
tested the correlation design should be
appropriately amended. The revised hypothesis given above - crosspressures leads
to smaller and less severe wars - should
in fact more accurately read: the greater
the crosspressures, given that war has
occurred, the smaller and less severe the
wars will be. A correlation should then be
war measures
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283
Table 1. A
comparison of correlations between percent of alliance committed nations and
severity and magnitude of war for two lags
Taken from J. David Singer and Melvin Small, ’Alliance Aggregation and the Onset of War,
1815-1945’, in Insights and Indicators in World Politics: Ten Quantitative Studies, ed. J. David Singer
(Glencoe, Ill.: Glencoe Free Press, in press).
*
p < .05, i.e. the null hypothesis can be rejected at the .05 level.
computed using only
creases, the
began.
years in which wars
Since there are 55 years during the
total period in which no wars began, this
difference could be significant. On purely
intuitive grounds it would seem that this
restriction would increase the correlations found since it will eliminate many
years in which the dependent variable
has a zero value, years which, by inspection, appear to have varying values
for the independent measures.
abilities for
7. A retest
of the Singer-Small theory
What modifications could be made in
design to afford a
theory? A number of
were
suggestions
implied in the foreand
our
going pages,
purpose here will be
to summarize these points.
With respect to the independent variable, two approaches can be suggested.
First, since the interaction-opportunity
index is appropriate for years in which
no overlapping alliance membership occurs, a correlation could be computed
using only the 22 years which meet the
one-alliance-per-nation assumption. Second, during the years in which overlapping occurs a multiple correlation
could be computed using both the interaction-opportunity index and the overlap
index as joint predictors of war. Since,
as the interaction-opportunity index inthe Singer-Small
closer test of their
expectation
is that the
prob-
increase; whereas when
overlapping increases, one would expect
war probabilities to decrease, the interwar
action index
must
thus be modified. By
the interaction-oppor-
simply reversing
tunity index, i.e. by determining the percentage of nations having no commitments, both independent variables will
predict in the same direction.
Two alternative
measures
for the de-
pendent variable, probability of
have already been described: (1)
war
as
a
dichotomous variable
war,
or
treat
(2)
of war’ index by
nation pairs at
the
of
number
dividing
war by the number of possible pairs in
the system. The use of the second index
is straightforward, and the usual correlation can be computed against the alliance
measures. If war is used as a dichotomous
variable, a measure of association could
be computed between it and a dichotomized or trichotomized interaction-opportunity index using simple chi-square
formulae. Or, an appropriately modified
correlation formula might be employed
to compute correlations between the
essentially continuous interaction index
and the dichotomized war variable.
The combination of modified measures
for the independent and dependent variables should provide results on the cross-
compute
a
’probability
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284
pressures balance of power theory. While
it is beyond the scope of this ’paper to
carry out these various tests, the results
of one small exploratory test might be
mentioned. Fischer’s Exact Test was
applied to the data for the 22 years which
met the one-alliance-per-nation assumption (i.e. years having no alliance overlap). The number of alliance commitments for those years were classified into
two categories: low number of commitments and high number of commitments.16
The cut-off point was determined by the
mean number of commitments for those
years (which was 8). War was treated as a
dichotomous variable and of the 22 nooverlap years we determined how many
years fell in each of the four categories:
Table 2. A test
of
the
Singer-Small hypothesis
Small
hypothesis can be accepted: some
relationship exists between the level of commitments in the system and the outbreak of war.
A similar type of analysis was attempted
for the overlap years, by holding the
level of overlap constant. Separate tests
were made for years in which there were
few overlapping alliances and years in
which there were many overlapping
alliances. However, strictly speaking, this
analysis is incorrect, since the overlap
index and the alliance commitment
index are not independent: if there are
few alliance commitments there can only
be a few overlaps, and if there are a
large number of overlaps (particularly
if the number of overlaps exceeds the
number of nations in the system) there will
probably be many commitments. Whether
for this reason or others, the chi-square
computed for these data were not signifi-
cant.
Summary and conclusion
There are, then, at least seven distinct
balance of power theories. Six of these
share a common concern, the seventh is of
a considerably different character. Each
of these should be subjected to empirical
tests, and this is obviously the next stage
in an analysis of the meaning and significance of the balance of power. An initial
attempt in this direction has been made
by Singer and Small. Taking their interpretation of the balance of power theory,
they have derived and tested an hypothesis. We have examined this study and
discovered that certain aspects of the
8.
Following the Singer-Small argument it
felt that low commitment years
should coincide more frequently with nowar years, while high commitment years
should tend to contain more war years.
No lags were used in this calculation, since
it was observed that alliances tended to
endure for a number of years, and it
seemed unnecessary to introduce lags to
measure their effect on the outbreak of
war (furthermore, an inspection of the
data showed that there was little difference
between one and no year lag conditions).
The Fischer Exact Test is comparable
to a chi-square analysis, except that
exact probabilities are directly obtained.
For the above analysis p
.039. In other
words, we can reject the null hypothesis
that the two variables are independent
at the .039 significance level. Thus,
for the no-overlap years the Singerwas
=
research
design mitigate against drawing
any conclusions concerning the viability
of their balance of power theory. However, these writers have presented us
with some fascinating data and the first
thrust towards testing any of the theories.
Hopefully, the stimulus of their research
and some of the suggestions made herein
will bring forth a series of studies designed
to evaluate the viability of the balance of
power model.
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285
NOTES
particularly wishes to thank J. David Singer, Rudolph Rummel, Dean Pruitt,
Harold Guetzkow, Charles Hermann and Harold Sprout for their many useful comments and
suggestions on an earlier draft of this paper, and Klaus Knorr and the Center for International
Studies at Princeton for providing the facilities which made possible the writing of this article.
1
A ’specific’ matrix can easily be constructed by considering all permutations for each alliance
2=
configuration and each power distribution. This would increase the matrix by a factor of (5!)
14,400, and make it completely unwieldly. Note also that we have included all possible overlapping alliance configurations. It should be mentioned that no general formula exists for determining the number of alliance configurations for nations when one ignores labels and allows for
overlapping. A solution, for example, does exist for finding the number of r-termed splittings of a
given integer M. But in our case we are also interested in determining the number of ways in
which m can be split.
2
For example, using definition (2) regardless of the specific power of each nation the power
configuration of A = B = C = D = E would not be acceptable; whether the nations each had
ten units of power, twenty, or thirty would be irrelevant since it is not possible to divide these
nations into two equal alliances.
3
and AC/BDE could
The example given earlier illustrates this point :A > B > C > D > E
be a balance of power system for definition (2) if A’s power is not that much greater than B’s.
In other words, according to definition (2) AC must equal BDE, and this is not ruled out by the
power distribution of A > B > C > D > E.
4 Half of the data on which this
study is based can be found in Singer, J. David & Small,
Melvin, ’Formal Alliances, 1815-1939’, Journal of Peace Research, 1966, Vol. I, pp. 1-32.
5 The
independent variable contains a second, though less serious, difficulty. The independent
variable of the theory is not necessarily equivalent to the one stated in the derived hypothesis.
The number of free, uncommitted nations is not always equivalent to the number of alliance
commitments; the same nation may have more than one commitment. This becomes particularly
obvious in the use of the interaction-opportunity ratio as a measure of the independent variable.
Since the ratio ignores multiple memberships it does not count the ’number of alliance commitments’
but the number of nations which have any alliance commitment. If no multiple memberships occur
then there is a perfect correlation between the two variables, and they can be used interchangeably.
Overlapping alliance memberships, however, destroy this correlation, as can be seen with
reference to the above four hypothetical systems. The interaction-opportunity ratio for all four
systems is one; yet, regardless of how the variable "alliance commitment’ is operationalized, the
four systems yield different values. If we assume every nation in an alliance has a commitment
to all other nations in that alliance and we count the number of commitments from the point of
then
view of each nation &mdash; i.e. if we count all possible pairs of nations within each alliance
the number of alliance commitments for the four systems would be: (1) 8, (2) 24, (3) 8, (4) 12.
If we exclude repetitions across alliances, i.e. if we count BC once in system two even though it
appears twice, then the results reduce to: (1) 8, (2) 21, (3) 8, (4) 12. This can be further reduced
if we do not compute the number of commitments from the viewpoint of each nation, i.e. if we
count only the number of different alliance commitments: (1) 4, (2) 9, (3) 4, (4) 10. In any case,
however computed, when nations belong to more than one alliance a one-to-one correspondence
does not exist between the number of free nations in the system (or the percentage thereof, as
given by the ratio) and the number of alliance commitments.
This difference between the theory and the hypothesis is not, however, particularly serious.
The main concern is to test the theory, and the interaction-opportunity ratio is a direct measure
of the number of free, uncommitted nations in the system described in the theory. While the stated
hypothesis is not directly derived from the theory, the hypothesis actually being tested can be
deduced from the theory.
6
Numerous other indices could be constructed. Two others considered were (1) count the
this was felt to be too conservative,
number of nations belonging to more than one alliance
it would mean that system three (above) would have only slightly more crosspressures than
system one; (2) count the number of times in a given year all alliances overlapped &mdash; this produces
a somewhat inflated figure: given alliances ABC, ADE, AFG this index would compare the first
and second alliance, the first and third alliance and then the second and third alliance producing
an overlap of three. The overlap index described in the text above seemed to be a reasonable
compromise between these two extremes.
7
A table giving the results of these tabulations can be obtained upon request. It indicates the
*
The author
&mdash;
&mdash;
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286
number of alliances each nation belonged to in each year. The lower portion of the table gives
the total number of nations having some alliance commitment, the number of nations in the
system, and the amount of overlap for each year as measured by the index described above.
8 For
example: ’... if the alliance configurations show less and less partial overlap, and they instead increasingly reinforce a tendency toward a very few (but large sized) coalitions, the system’s loss
of interaction opportunities becomes even more severe...’; and again: ’... there should be a difference between the effects of a structure which reflects a crazy-quilt pattern of all sorts of overlapping
alliance memberships and one in which only a very few easily distinguishable coalitions emerge’.
9 There
appears to be an error in the Singer-Small test. As they describe the index, the theoretical number of pairs would fall in the numerator: ’Once the number of feasible remaining defensive alliance links was ascertained, that number was divided into the original number that would
have been possible in the absence of any such alliances, to give the percent of major power defensive alliance ties exhausted’, (italics are mine). Presumably they mean that the remaining
number of possible alliance links was divided
by the number of theoretical pairs. Otherwise
the resulting index would not give the ’
percent... of alliance ties exhausted.’
10 The
bipolarity index contains two additional problems. As pointed out to this author in
correspondence with Dean Pruitt, the construction of the index assumes that already allied nations
cannot form additional alliances. In other words, to discover how many alliances could still be
concluded in the system after eliminating existing alliances, one only counts the number of possible alliances between free nations and others in the system. Alliances which cut across existing
alliances are not considered; in system one above, for example A cannot ally with D or E.
The second difficulty with this index is that it does not measure a bi-polar cleavage in the system.
As shown in the examples above (especially systems three and five), if two systems have the same
number of free nations in the largest alliance (or set of targets) then the actual number of alliances
in the system is irrelevant. System three above has two alliances and system five contains four
alliances
yet the two systems have equal bipolarity indices.
11 There are indications in the
Singer-Small text that this was originally their meaning : ’ ....
our dependent variable: the incidence of war ...’; ’Alliance Aggregation as a Predictor to War
...’; ’as the system’s interaction opportunities diminish wars will increase in frequency ...’;
and finally ’alliance aggregation should correlate with the onset of war ...’
12 For
example, if war broke out between two nations in 1827 and lasted for 18 months, the
36 nation-months to the year 1827. The prefirst measure would assign 2 nations &times; 18 months
vious discussions analyzed only measures one, three and five for the dependent variable and
measures one and six for the independent variable because the other measures produced consistently lower (and non-significant) correlations and did not appear to have any greater validity
than the measures examined.
13 A one
year lag correlation was computed between the alliance aggregation measures and
any of the ’more war’ indicators for the following year. Thus, if war broke out in 1827, 1829, and
1834 the number of months all nations participated in each war would be calculated and assigned
to these years; these measures would then be compared with the alliance aggregation measure
for years 1826, 1828 and 1833 respectively. The three and five year lags were computed in a somewhat different fashion. A three year lag for the ’more war’ variable involved adding the specific
indicator over three year periods. In other words, the number of nation-months for all wars
beginning in a three year period are summed; for the above example, the number of nationmonths for 1827, (1828
0), and 1829 are summed. Similarly, the five year lag means that the
measures are summed over five years. The three and five year lags for the independent variable
are computed slightly differently. Averages, rather than totals, are determined for the periods. The
results reported in the body of the text are based on the three year lag for both variables; tables
in their Appendix give results for all combinations of lags for both variables.
14 A similar
problem exists in the correlation reported between the various measures of the
dependent variable and the various measures of the independent variable. A correlation, for
example, between measures one and two for the dependent variable will necessarily be high since
one measure is a subset of the other. A part-whole correlation is required here.
15 The authors seem to
recognize this lack of independence in their data but their solution,
which involves the Fischer exact test, does not remedy the difficulty. The solution would seem to
lie in the elimination of the three and five year lags.
16 Number of alliance commitments rather
than percentage of commitments (over number of
system nations) were used for ease of analysis. A separate test, however, was conducted using
percentages and found to be almost identical.
&mdash;
=
=
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287
REFERENCES
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SUMMARY
This study examines the literature concerning the ’balance of power’ concept, and
abstracts from these discussions seven hypotheses which postulate the international
consequences of a balance of power system. The dependent variable of these hypotheses
is generally agreed to be either peace or the preservation of the system. Considerable
variability, however, is found in the possible interpretations of the independent variable,
a ’balance of power’. Seven different definitions of a balance of power are constructed
from the literature and then compared. One of the hypotheses, proposed by Singer and
Small, has been subjected to empirical tests. However, due to certain difficulties in their
research design, the results of their study are not yet applicable to a further understanding of the validity of any of the balance of power theories. The present author examines
some of the difficulties of this analysis and suggests where the problems lie and several
alternative solutions, with the hope that this initial study will be further pursued to
provide the needed tests of at least one of the balance of power theories.
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