Erik
Moberg:

Models
of International Conflicts and Arms Races

3. Different weapons' effect in attack and defense

Attack
more difficult than defense (k > 1)

Attack
easier than defense (k < 1)

4. The nations' information about their situation

Originally
published in Cooperation and Conflict, Nordic Studies in International
Politics, 1966, nr 2, pp 80-93.

© NISA -
Nordic International Studies Association 1966

In this essay a series of models will be presented
which deal with international conflicts and arms races. The essay is entirely
theoretical; not even operational definitions of essential variables, such as
"arms level", are given. The purpose of the models is twofold. First
it is thought that models of this kind might be useful for clarification of
concepts. In the essay this is illustrated by a discussion of some strategic
concepts, such as "balance of power" and "deterrence", with
reference to the models. Second, there is a heuristic purpose, i. e. it is
thought that models may aid fantasy and that therefore an exploration of models
of this type might result in a better understanding of possible relations
between nations.

In all the models there are only two nations and the
focus of interest is on these nations' decisional situations.

Two nations A and B have the arms levels x and y
respectively. Here "arms level" is thought of as a comprehensive
measure of a nation's total amount of armaments. Each nation can vary its own
arms level. Concerning the course of an eventual war between these nations the
following assumptions are made. First, the nation with the highest arms level
wins the war in the sense that when the war is finished the victor has complete
control over A and B.^{1} This means that the kind of war taking place within
the model is a total war leading to total victory for the one party and
unconditional surrender for the other. As a result of the war the victorious
nation gets, or rather takes, the other nation. However, in spite of the
victory it is not certain that the war is profitable for the winner. This is so
because, in order for the war to be profitable, the winner must also value the
war trophy, i. e. the other nation, higher than the various losses
incurred in the war. So in order to determine whether or not the war is
profitable for the victor an estimation of the war losses is needed.

It is assumed that the total losses in the war,
i. e. the sum of the losses of the two nations, is a function of the arms
levels x and y. The losses can therefore be written f(x, y*). *Regarding
this loss function, the following is assumed. It is defined at all points
(x, y) such that x, y ≥ 0. It is differentiable. It increases monotonously
along the line x = y. Furthermore it is assumed that

(2.1) f(x, y) = f(y, x),

(2.2) f(x, 0) = 0, and

(2.3) ∂f / ∂x > 0, when x < y.

Formula (2.1) is an assumption about symmetry. (2.2)
in combination with (2.1) says that if the arms level of one of the nations is
0 there are no losses in a war between them. Formula (2.3) states in
combination with (2.1) that if the nation with the lowest arms level increases
its level while the arms level of the other nation is kept constant the losses
in an eventual war will increase.

These assumptions seem fairly reasonable. It seems
more problematic, however, to determine what happens to the total losses, when
the nation with the highest arms level increases its level while that of the
other nation is kept constant. Here one could argue in two ways. Either one
could say that because the total amount of weapons used in the war increases,
the losses should also increase. Or, to argue the other way, because the nation
with the highest arms level becomes still more superior the war will be of
shorter duration. The losses should consequently decrease. The two arguments
give the following alternative assumptions concerning the loss function.

(2.4 a) ∂f / ∂x > 0, when x > y, or

(2.4 b) ∂f / ∂x < 0, when x > y.

Now, the loss function f(x, y) gives certain
information about the size of the losses in a war between the two nations. But
to determine whether a war is profitable for the winner or not, a comparison of
its evaluation of the losses and of the war trophy is required.

Consider a point in the sector x > y,
i. e. the sector where A wins a war. Assume that A is considering an
attack on B and therefore tries to evaluate the war. It is assumed that in
making this evaluation A compares the status quo, i. e. the state of the
two-nation world before an attack by A on B, to the postwar state resulting
from A's attack on and war with B. This post-war state is characterized by the
fact that A can consider B a revenue and f(x, y) determined by the arms
levels x and y a cost.^{2} A's comparison of the two states will give an order
of preference of these states. If the post-war state is preferred to status quo
the war is profitable, otherwise not. Assume that when the comparison is made
B's arms level is kept constant at, say, y_{0}. A might then find that
for a certain arms level x_{1} it prefers status quo to the post-war state, while
for another arms level, say x_{2}, it prefers the post-war state to the status quo.
Here x_{1} and x_{2} are both greater than y_{0}, but
nothing is assumed about their relative order.

The following two assumptions, which are relevant when
the nations consider whether to prefer the status quo or the post-war state,
also are made.

(i) If a nation's preference-order of the two states
is reversed at two different points in the x-y-plane, then there is a point
between these two points at which the nation is indifferent toward the two
states.

(ii) If a nation at a certain point in the x-y-plane
is indifferent toward the two states, then the nation is also indifferent
toward the states at all other points at which the losses are of the same size
(that is, at all points at which the value of f(x, y) is the same).

In the situation described, where A compares the two states,
the first assumption implies that there exists a x_{0} such
that x_{1} > x_{0} > x_{2}, or such
that x_{2} > x_{0} > x_{1}, and
furthermore such that A is indifferent toward the two states at the point (x_{0}, y_{0}). The
second assumption implies that A is indifferent toward these states at all
points on the line f(x, y) = f(x_{0}, y_{0}) = K_{B}, where K_{B} is constant. This line is called A's indifference
line. Because the constant K_{B} tells how great the losses are that A is prepared to
suffer in order to conquer B, it is possible to consider K_{B} a
measure of the value of B to A.

Now it is also convenient to define two possible
military policy goals for a nation.

*Military policy goal 1. *A nation has this goal if it tries to get into such a
position that it is profitable for it to attack the other nation. This goal
could be considered aggressive, and a nation with such a goal is also called
aggressive.

*Military policy goal *2. A nation has this goal if it tries to get into such
a position that it is not profitable for the other nation to attack it. This
goal is defensive and a nation with such a goal is called defensive.

In order to get a clear view of the conditions under
which the different goals are realized for the two nations it is convenient to
draw the indifference lines of A and B in the x-y-plane (fig. 1). Only the
first quadrant where f(x, y) is defined is considered. In the sector
x > y the nation A wins a war. Within this sector the line

(2.5) f(x, y) = K_{B}

is relevant, because on the one side of this line it
is profitable for A to attack, while on the other side the opposite is true.
Correspondingly, in the sector y > x, the relevant line is

(2.6) f(x, y) = K_{A}

The slope of these two lines is obtained by
derivation. Derivation of (2.5) gives

dy / dx = – (∂f / ∂x) / (∂f / ∂y)

According to (2.3) the derivative ∂f / ∂y > 0 when x > y. The sign of ∂f / ∂x depends on which of the alternative assumptions (2.4 a) and
(2.4 b) is considered valid. Thus dy / dx is negative when
(2.4 a) holds while (2.4 b) would make dy / dx positive.
The same reasoning holds for the line (2.6). In fig. 1 therefore the two
lines have the slope which follows from assumption (2.4 b). If it is
assumed that K_{A} and K_{B} are both positive, i. e. that each nation
represents a certain positive value to the other, it is also clear that the two
indifference lines will cut the line x = y outside the origin, as in
fig. 1.^{3}

The three areas I, II and III in the figure now have
the following meaning with respect to the previously defined military policy
goals.

Area I: Within this area military policy goal 1
is realized for A and no goal is realized for B.

Area II: In this area goal 2 is realized for both
nations.

Area III: In this area goal 1 is realized for B
and no goal is realized for A.

In connection with this model one can now define the
concept of *balance of power. *Balance of power is said to exist between
two nations when it is not profitable for either of them to attack the other.
This means that in fig. 1 there is a balance of power between A and B in area
II.^{4}

It is now easily seen how the model illustrated in
fig. 1 can be looked upon as a model for arms races. If both nations are
aggressive A will try to reach area I by increasing the arms level and B will
try to reach area III in the same way. Thus an arms race will result. On the
other hand, if one of the nations, say A, is aggressive while the other is
defensive there will be an arms race along the line (2.5) because A is striving
to reach area I and B area II. It is also possible to elaborate the model in
such a way that there will be an arms race even when both nations are
defensive. This particular case, which corresponds to a situation where both
nations have uncertain information concerning the opponent's arms level, will be
discussed later.

If it is assumed that (2.4 a) is valid instead of
(2.4 b) the derivatives of the indifference lines (2.5) and (2.6) will be
negative. This situation, illustrated in fig. 2, obviously is different
with respect to the nations' propensities to arm. For instance, a nation will
not by an increase of arms level reach a point where an attack on the opponent
is profitable. On the contrary such a point would be reached by disarming.
Perhaps this somewhat peculiar fact indicates that (2.4 b) is a more
reasonable assumption than (2.4 a). In any event the different consequences of
(2.4 a) and (2.4 b) make further investigation of the loss function
important. In this essay, however, (2.4 b) will henceforth be considered
valid.

A problem related to what has been said so far affects
the distinction between aggressive and defensive nations. Hitherto it has only
been said that the distinction should be made but nothing much has been said
about how an aggressive and a defensive nation should be defined respectively.
Two possible definitions are suggested here.

*Definition 1. *A
defensive nation is characterized by the fact that there are no arms levels
(i. e. no points in the x-y-plane) at which it prefers the postwar state
to the status quo. This means that the defensive nation contrary to the
aggressive one, lacks an indifference line.

*Definition *2.
The defensive nation, like the aggressive one, has an indifference line. They
are distinguished, however, by the fact that the defensive nation prefers the points
in the x-y-plane where it prefers the status quo to the post-war state, whereas
the opposite is true for the aggressive nation. From the point of view of
behavior this definition implies that a defensive nation, if it finds itself at
a point where it prefers the post-war state to status quo, it will disarm
rather than attack.

For the main part of this essay it is not necessary to
decide between these two definitions, since the discussion is not carried so
far as to be affected by such a decision.^{5} Besides, it is easily
seen that the model might be developed in such a way that the two definitions
are used simultaneously. Apart from the possibility of distinguishing between
aggressive and defensive nations there is also the possibility of using the two
definitions for distinguishing between three groups of nations: nations without
an indifference line and among those with an indifference line, nations which
prefer to be on the one side of it (area II) and those which prefer to be on
the other side (area I or III). It might be interesting to investigate whether
this classification of nations with respect to goals could be fruitfully
applied to real nations.

It should be of great interest to the nations in the
model to be able to decide whether the opponent is aggressive or defensive.
Then, the simplest thing for the nations to do would be to apply the
definitions of "aggressive" and "defensive" utilized in the
model. However, one can easily imagine that the model is constructed in such a
way that this simple application of the definitions is not possible. For
instance this might be the case if the nations have limited information about
each other. In this situation the nations might be forced to determine the
opponent's goal by a more indirect method. For example, changes in the
opponent's arms level could be looked upon as indicating an aggressive goal or
a defensive goal. Conversely, one could imagine that the nations deliberately
change their arms levels in order to communicate to the opponent their military
policy goal. These problems will be discussed later.

When compared to reality it is immediately seen that
the model presented so far suffers from serious oversimplifications. Some of
these are listed here. Later in the essay it is shown how some of them could be
eliminated by introducing further assumptions.

(i) It is a simplification that attack and defense
should require the same amount of armaments, i. e. that the nation with the
higher arms level wins a war independent of whether it is the attacker or the
defender.

(ii) The important problem of what sort of information
the nations have about each other, and how this affects their behavior, is not
explicitly treated.

(iii) A further simplification is that the only type
of war conceivable is war leading to unconditional surrender. Therefore the
possibility of limited war ought to be introduced.

These three simplifications are discussed in the next
pages. There are, however, additional simplifications, unfortunately not dealt with
in this essay, among which are:

(iv) There are only two nations. A more realistic
model should allow for several nations.

(v) Costs of armaments are not considered although it
is obvious that in reality these costs are of great importance to a nation when
determining its arms level.

(vi) No distinction is made between different kinds of
weapons. It appears feasible, however, to give the nations the capacity to
change the very shape of the loss function by introducing a distinction between
offensive and defensive weapons.

It is often the case that the amount of weapons
required for defending a certain position differs considerably from the amount
required for attacking and taking the same position. In military operational
planning it is, for example, often estimated that in ground combat the attacker
needs a superiority of 3:1 in relation to the defender. If this idea is applied
at the national level, the model presented so far could be criticized on the
ground that it is assumed that the nation with the highest arms level wins a
war regardless of whether or not it is attacking. By changing this assumption
the model could be made more realistic.

It is therefore assumed that the ratio between the attacker's
and the defender's arms levels determines which nation wins a war. If this
ratio exceeds a certain minimum value, which is a parameter in the model, the
attacking nation wins, otherwise the defender wins. This parameter is called k*. *If k > 1
attack is said to be more difficult than defense and if k < 1
attack is easier than defense. Thus, the model already presented is the special
case where k = 1. In the following the balance situation in the model
is examined when k > 1 and when k < 1.

In the x-y-plane is drawn the line y = kx
and its reflection in the line x = y (fig. 3).These two lines
divide the first quadrant in three sectors S_{l}, S_{2} and S_{3}. In S_{l} A wins a
war independent of who attacks, and the same is true for B in S_{3}. In S_{2} neither
nation wins a war in which it is the attacker.

In order to know the positions of the indifference
lines it is also necessary to describe the loss functions. As a matter of fact,
two loss functions are utilized here: one which applies when A attacks B and
another when B attacks A. Each of these functions has properties analogous to
the properties of f(x, y). This means that for the function which applies
when A attacks B and which is called f_{A}(x, y) the derivative ∂f_{A} / ∂x is less than 0 at points where x / y > k
(i. e. in the sector S_{1}) and greater than 0 where
x / y < k (i. e. in S_{2} and S_{3}). Except
for the assumption about symmetry (2.1) all other assumptions are the same as
for f(x, y). The correspondence to the assumption of symmetry is that the
function f_{B}(x, y), i. e. the function which applies
when B attacks A, is defined f_{B}(x, y) = f_{A}(y, x).

Now it is possible to draw the indifference lines as
shown in fig. 3. In area I it is profitable for A to attack and in area III
it is profitable for B.

The most striking feature of this model as compared to
the earlier model is that the area in which there is a balance of power, that
is area II, reaches the origin. Hence in this model very low arms levels do not
necessarily mean imbalance.

In this case the line y = kx and its
reflection in y = x divides the quadrant in the sectors S_{1}, S_{2} and S_{3} with the
following properties (fig. 4). In S_{1} nation A wins
irrespective of who attacks whereas in S_{3}, this is true for B. In S_{2} the
attacking nation, which might be either A or B, wins.

Too, in this case the indifference lines give three
areas of interest with respect to profitable attacks. In area I it is
profitable for A to attack, in area III it is profitable for B and in area II,
where there is balance, it is profitable for no nation to attack. It is of
interest, however, that in this model the areas I and III partially overlap.
Consequently, there is in the neighborhood of origin an area in the figure (the
lines cross) in which it is profitable for A as well as for B to attack.
Therefore there is in this area another type of imbalance, different from the
imbalance in the remainder of areas I and III. This new type of imbalance is
quite similar to the imbalance which exists between two nations, when both have
a good first strike capability and lack second strike capability.

The problem concerning what sort of information the
nations have about their situation, and how their behavior is affected thereby,
has not been treated so far. Here are listed some ways in which the nations may
lack such information.

(i) they may be uncertain about the shape of the loss
function.

(ii) they may be uncertain about the position of the
opponent's indifference line.

(iii) they may be uncertain about the opponent's arms
level.

(iv) they may be uncertain about whether the opponent
is aggressive or defensive.

Here will be discussed the situation where both
nations have uncertainties of types (iii) and (iv) but not of types (i) and
(ii). Furthermore it is assumed that the construction of the model is such that
"defensive" and "aggressive nation" are defined by definition 2
above. Because both nations are uncertain about the opponent's goal they have
to infer this by finding out whether the opponent, when he changes his arms
level, strives to get into area II or the one of areas I and III which is
relevant to him. But such observations regarding changes in the opponent's arms
level are difficult to make because it is also assumed that the nations do not
have complete information about this very arms level.

x_{0} = A's actual arms level

y_{0} = B's actual arms level

I_{1} = the interval within which B thinks A's arms level
lies

I_{2} = the interval within which A thinks B's arms level
lies

L_{1} = the line on which B thinks (x_{0}, y_{0}) lies

L_{2} = the line on which A thinks (x_{0}, y_{0}) lies

I_{3} = the interval within which A thinks that B thinks x_{0} lies

I_{4} = the interval within which B thinks that A thinks y_{0} lies

R_{1} = the rectangle within which B thinks that A thinks
(x_{0}, y_{0}) lies

R_{2} = the rectangle within which A thinks that B thinks
(x_{0}, y_{0}) lies

The situation described is illustrated in fig. 5,
where x_{0} and y_{0} signify A's and B's actual arms levels respectively.
It is assumed that the nations' uncertainty about the opponent's arms level is
of the character that they themselves are sure they know in which interval the
arms level lies but admit uncertainty as to where in the interval. In the
figure, I_{1} and I_{2} signify these intervals, i. e. the intervals in
which B and A respectively think the opponent's arms level lies. Because both
nations know their own arms level, A thinks (x_{0}, y_{0}) lies
somewhere on L_{2} and B thinks that that point lies somewhere on L_{l}. In
order to infer the opponent's goal both nations now have to find out towards
which area (I, II or III) the opponent is striving. From A's point of view the
problem looks like this. In order to decide towards which area B tries to move
(x_{0}, y_{0}) nation A must first determine which opinion B has
about the position of the point. At first, since B knows its own arms level, it
is evident that A's opinion about B's opinion about y_{0} is the
same as A's opinion about y_{0}*. *Then, when A's opinion about B's opinion about x_{0} is
considered, it is assumed that A's uncertainty is of the character that he
thinks that B thinks that x_{0} lies within a certain interval, which here is called
I_{3}. To summarize, this means that A thinks that B thinks
that (x_{0}, y_{0}) is somewhere in the rectangle R_{2}.
Correspondingly B thinks that A thinks that the same point lies somewhere in
the rectangle R_{l}.

The following example illustrates how the nations'
behavior is affected by the uncertainties described. Assume that R_{l}
completely, or mainly, lies within area I (or that R_{l} is
moving towards this area if a certain interval of time is considered). B will
then reasonably infer that A has aggressive intentions and will, on the basis
of the information B has, expect an attack from A at any time. The result of
such a war depends on where (x_{0}, y_{0}) lies. If the situation is the one illustrated in the
figure it is clear that B cannot, regardless of whether B is aggressive or
defensive, welcome an attack from A because (x_{0}, y_{0}) is in
the area where war is non-profitable for both nations. B therefore will try to
move R_{l} into area II. One conceivable way for B to do that is
by increasing his own arms level since that could move the interval I_{4} upwards.
But such an arms level increase by B could also move R_{2} towards
III, which in turn would stimulate A to increase his arms level. It is
interesting to note that the described process in its entirety is compatible
with both nations having defensive military policy goals.

The conclusions in the last paragraph are vague and
tentative. This is due to the fact that no explicit assumptions are made about
which factors determine the two nations' opinions about each other in different
respects. However, it seems feasible to build a more complex version of the
model in which such explicit assumptions are included.

In the previous models all wars that could take place
were wars leading to unconditional surrender; this surrender occurred when all
weapons of the losing nation were lost or destroyed. The purpose of the
following model is to eliminate this simplification. Thus in this model limited
wars are permitted. Here a war is called limited when the adversaries neither
try to force each other to surrender unconditionally nor consume all their
weapons in the war. This last fact means that both parties are perfectly
capable of continuing the war, but for some reason they choose not to do so.
The purpose of this model, among other things, is to find an explanation for
why this is so. That is, why should a nation consider it profitable to start a
war, but consider it unprofitable to continue the war until the opponent
surrenders unconditionally? Could such behavior be the result of rational
calculations as was the behavior described in the previous models? It is
possible that such behavior could result from nations not having correct
information about each other's arms level. In this case a war could be started,
but then when, as a result of combat experience, the nations acquire better
information about each other's arms level, they might both prefer to end the
war. In this model, however, it will be shown how a limited war can result from
rational behavior without assuming false information.

Here it is assumed that each nation, A and B, has two
kinds of weapons at its disposal: weapons designed for total war and weapons
designed for limited war. This distinction is inspired by Kissinger, who
writes: "... two basic organizations would be created: the Strategic Force
and the Tactical Force. The Strategic Force would be the units required for
all-out war. ... The Tactical Force would be the Army, Air Force and Navy units
required for limited war."^{6} The distinction between these two kinds of weapons
may be such that nuclear weapons are present only among the weapons for total
war. That is the case in Kissinger's later book *The Necessity for Choice *but
not in the earlier *Nuclear Weapons and Foreign Policy. *The arms levels
of the two nations with respect to armaments for limited war are denoted x and
y respectively. Correspondingly the arms levels with respect to armaments for
total war are denoted X and Y.

Between the two nations of the model limited wars as
well as total wars can be fought. The assumptions concerning the course of both
types of war are similar to the corresponding assumptions in the previous
models.

In a limited war the amount of weapons introduced by A
and B are x and y respectively. If x > y nation A wins the limited
war, whereas B wins if y > x. Victory in a limited war means that
those forces of the one nation designed for limited war beat or destroy all of
the opponent's corresponding forces. In this situation the winning nation would
have complete control over both nations had there not been any weapons designed
for total war.

Now it is assumed that it is possible for the victor
in a limited war to occupy or take only part of the losing nation's territory.
How great a part he will take the victor decides after the following
consideration. If it is assumed that A wins and that the part of B that A takes
is called b, then b has to be great enough compared to A's war losses and at
the same time so small that it is not profitable for B to start a total war. It
is therefore only in cases where such a b exists that A can profitably conduct
a limited war for a limited purpose. The conditions for the existence of such a
b will be discussed later.

The losses that A suffers in the limited war will be
denoted h_{A}(x, y). This function is defined in the sector
x > y of the first quadrant and has properties corresponding to
(2.2), (2.3) and (2.4 b) in the original model. If B wins a limited war
its losses amount to h_{B}(x, y). This function is defined in the sector
y > x and has properties equivalent to those of h_{A}(x, y).
The assumptions concerning h_{A}(x, y) and h_{B}(x, y) consequently
imply that the nation which wins a limited war can decrease its losses in such
a war by increasing its arms level while the losing nation, by increasing its
arms level, will increase the victor's losses.

In a total war the weapons introduced are X and Y
respectively. Here the course of the war is completely analogous to that of a
war in the original model. Who wins will depend on which is the greatest of X
and Y. The total losses are determined by the function g(X, Y) which, with
respect to derivatives and zero-values, has properties corresponding to those
of f(x, y). The correspondence to (2.4 b) is assumed to be valid.

With respect to the relations between the arms levels
of A and B two cases may be distinguished, which will be discussed in the
following. In case 1 x > y and Y > X. In
case 2 x > y and X > Y. (The two remaining cases
where y > x and X > Y and where y > x and
Y > X imply, as will be evident, nothing new compared to the cases
1 and 2.)

It is first seen that in this case there are three
possible final states in the model.

The first of these is the state that will result if
there is no war at all, i. e. status quo.

The second final state will be the result of a limited
war. In this state A has won the war and suffered some losses, namely h_{A}(x, y).
Furthermore A has taken the territory b from B. In this state therefore B will
continue to exist as a nation, although it has lost b. Naturally also B has
suffered some other war losses, but because of the nature of the conclusions
drawn from the model it is possible to disregard these losses.

The third final state will result from a total war. In
this state B has won the war and taken A but also suffered the losses
g(X, Y). A has ceased to exist as a nation.

These three final states are called T_{l}, T_{2} and T_{3}. Now it
is of interest to know the order of preference of these states for A and B
respectively. Suppose, for example, that A prefers T_{2} to T_{1} and that
B prefers T_{2} to T_{3}. In that situation A would start a limited war in
order to take b. But if instead B prefers T_{3} to T_{2} it would
no longer be profitable for A to start a limited war because B would then respond
by initiating total war. Consequently, it is important to see what determines
and affects the nation's orders of preference.

Concerning A it is obvious that T_{3} is
ranked last in every order of preference. The relative order between T_{1} and T_{2} will depend
partly on the size of h_{A}(x, y) and partly on the size of b.

Now assume that in a certain case b has the value b_{l}. Assume
furthermore that there are two points (x_{1}, y_{0}) and (x_{2}, y_{0}) such
that x_{2} > x_{1} > y_{0} and such
that in the first point A prefers T_{1} to T_{2} whereas in the other it prefers T_{2} to T_{1}. Then
there is between x_{1} and x_{2} an x-value x_{i} such that A is indifferent to the two states in (x_{i}, y_{0}).
Therefore A's indifference line is the line h_{A}(x, y) = h_{A}(x_{i}, y_{0}) = C,
which is drawn in fig. 6.

Assume that instead of b_{1} a value
b_{2} such that b_{2} > b_{1} had been considered. The
indifference line in the figure would then move upwards, because the larger
territory b_{2} would be worth greater losses than the territory b_{1}.
Therefore the constant C can be written as the monotonously increasing function
C(b). Hence the complete equation of the indifference line will be

(5.1) h_{A}(x, y) = C(b)

Thus if x and y are given (5.1) will give a value of b
for which A is indifferent between T_{1} and T_{2}. In order to avoid confusion, the variable of (5.1)
is therefore written b_{A}. And the equation is

(5.2) h_{A}(x, y) = C(b_{A}) where dC / db_{A} > 0.

In considering B's orders preference of it is obvious
that T_{1} is always preferred to T_{2}. What is
interesting therefore is to find the conditions under which T_{3} is
preferred to T_{1} and, when this is not the case, the conditions under
which T_{3} is preferred to T_{2}. If the relative order
between T_{1} and T_{3} is considered first an indifference line can be drawn
exactly analogous to the line in the original model (fig. 7). The equation
of the line is g(X, Y) = L, where L is a constant.

In the area where B prefers T_{3} to T_{1} the
state T_{3} is also preferred to T_{2}. But in
the area where T_{1} is preferred to T_{3} the order of T_{3} and T_{2} depends on
the arms levels and the size of the area b, which in state T_{2} is taken
by A.

Assume, in the same way as when A's orders of
preference were considered, that in a certain case b has the value b_{l}. Assume
furthermore that there are two points (X_{0}, Y_{1}) and (X_{0}, Y_{2}) such
that Y_{2} > Y_{1} > X_{0} and such
that in the former B prefers T_{2} to T_{3} and in the latter it prefers T_{3} to T_{2}. Then
there is also a point (X_{0}, Y_{i}) where Y_{1} < Y_{i} < Y_{2} and at
which B is indifferent to T_{2} and T_{3}. The equation of the indifference line will be
g(X, Y) = g(X_{0}, Y_{i}) = K(b). Now, for a territory b_{2} greater
than b_{1} it is clear that B is prepared to suffer heavier
losses in order not to lose the territory. Thus the function K(b) is
monotonously increasing. With arms levels given, the equation
g(X, Y) = K(b) will then determine the value of b which makes B
indifferent to T_{2} and T_{3}. The variable b is therefore written b_{B} and the
equation of the indifference line is

(5.3) g(X, Y) = K(bB), where dK Ú dbB > 0.

Now assume that the four arms levels, that is x, y, X
and Y, are given. (5.2) and (5.3) then determine b_{A} and b_{B}
respectively. If, for these, the inequality (5.4) b_{A} < b_{B} holds,
it is profitable for A to start a limited war against B. This is so because,
for an arbitrary b in the interval between b_{A} and b_{B}, the
state T_{2} is preferred to T_{1} by A and simultaneously T_{2} is
preferred to T_{3} by B. For B it is therefore important to realize the
inequality

(5.5) b_{A} > b_{B}

If A is defensive it is also satisfied with this. But
if A is aggressive it will try to realize (5.4). How A and B can try to affect
the relative order of b_{A} and b_{B} is seen from the signs of the following derivatives,
which are obtained from deriving (5.2) and (5.3).

(i) ∂b_{A} / ∂x
= (∂h_{A} / ∂x) / (dC / db_{A}) < 0,

(ii) ∂b_{A} / ∂y = (∂h_{A} / ∂y)
/ (dC / db_{A})
> 0,

(iii) ∂b_{B} / ∂X = (∂g / ∂X) / (dK / db_{B}) > 0,

(iv) ∂b_{B} / ∂Y = (∂g / ∂Y) / (dK / db_{B}) < 0.

Thus, A can try to achieve (5.4) by increasing x,
which will decrease b_{A}, and/or by increasing X, which increases b_{B}. Nation
B, on the other hand, can try to achieve (5.5) by increasing y, which will
increase b_{A}, and/or by increasing Y, which decreases bB.
Increasing the amount of weapons designed for limited war and increasing those
designed for total war could therefore be equivalent measures when a nation
aims at realizing a certain military policy.

Here some strategic concepts may conveniently be
discussed. Assume that (5.5) holds and that consequently it is not profitable
for A to start a limited war. This means that B, although it is inferior with
respect to weapons for limited war, is able to deter A from starting such a
war. This kind of deterrence will be called *type II deterrence *with
reference to Herman Kahn's similar concept with the same name.^{7}

Because there will be no war, neither limited nor
total, in the situation described, it represents a kind of balance. Now, one
can imagine that one or both nations are badly informed about their situation.
For instance, assume that A, due to ignorance concerning B's arms level or
concerning B's evaluations, falsely believes that (5.4) holds. A will then
start a limited war against B. But because (5.5) in fact holds, B will respond
by starting a total war against A. Here A's starting of a limited war can be
looked upon as a disturbance in the original balance situation and because this
disturbance results in a total war, one can say that in the original situation
there is an *unstable balance of powe*r. The course of the war, starting
with a disturbance and developing into total war, can be considered a case of *escalation
*or *eruption.*

In this model the policy known as "massive
retaliation" can be seen as a special case of type II deterrence. It is
already assumed that x > y and Y > X. If (5.5) is
true and consequently A is deterred from starting a limited war by type II
deterrence, one can say that B has conducted a massive retaliation policy, if
it is also true that y and x are both very small. If A represents the Soviet
Union and B the United States, the model could be compared to the world
situation in the fifties and the early sixties. In the middle of the fifties,
when X was very small, the policy of the United States was massive retaliation.
Then X gradually increased. One result has been the claim for increased forces
for limited war in the United States, i. e. a claim for greater y. In the
model such an increase in y might keep (5.5) true despite the increase in X.
This comparison between the model and reality must not, of course, be taken too
literally.

In this case x > y and X > Y. Obviously no war
is profitable for B. For A a limited war or a total war, or both, may be
profitable. It is clear, however, that A wins a limited as well as a total war.
Consequently B is deterred from starting a limited war as A was in the previous
discussion about type II deterrence. The "structure" of the
deterrence is different in the two cases, however. This is so because in the
previous case the deterred nation, i. e. A, would win a limited war, whereas in
this case the deterred nation, i. e. B, will lose an eventual limited war. In
this case it is convenient, again with reference to Herman Kahn, to speak about
*type III deterrence.*^{8}

This section will now be concluded with a discussion
of two problems related to the last model. With reference to case 1 one
can say that these problems concern the conditions under which B will increase
the level of conflict to a total war. Hitherto it has been assumed that this
will happen only if B prefers T_{3} to T_{2}. One difficulty with this assumption, and this is the
first problem, becomes obvious if one imagines that several consecutive attacks
are possible. In this case A can start by taking a territory from B which is so
small, that according to the assumptions made, it is not profitable for B to
start a total war. Then A could take a new piece of territory of a similar size
from B, and so on. The territories taken by A in this manner could finally add
up to a size which would be more than enough to make it profitable for B to
start a total war, had all the pieces of territory been taken simultaneously.
From that point of view it might have been rational behavior on the part of B
to respond to A's first small attack by initiating total war. Thus, the assumption
that B starts a total war only when it prefers T_{3} to T_{2}, is by
no means self-evident.

In spite of this objection it seems, however, that in
reality nations often do hesitate to increase the war level in the same way as
B, which according to the assumption raises the level of conflict to a total
war only when it prefers T_{3} to T_{2}. Therefore it could also be advantageous to make a
great conquest in several consecutive steps and thereby succeed without
provoking the opponent to retaliate. This has happened in reality.

The second problem is this one. It is conceivable that
B, even if it does not prefer T_{3} to T_{2}, starts a total war in order to take its revenge for
A's limited attack. If B does this, it could be seen as the fulfillment of the
following kind of deterrence policy. B tries to deter A from limited attack by
making A believe that B's response will be total war and by utilizing the fact
that A prefers T_{1} to T_{3}. Here B hopes to be successful in deterring A without
taking into account the relative order of T_{2} and T_{3} in B's
own order of preference. With reference to the US strategic debate, this policy
can be considered a version of the *minimum deterrence *policy.^{9}

^{1} Nothing is assumed about what happens if the two
nations have equal arms levels. Such an assumption would most likely only
complicate the model without adding anything of interest.

^{2} The reason for considering the total losses in the
war, i. e. f(x, y), a cost for A alone is the following: as a result
of the war B becomes A's war-trophy, and all damage done to B can therefore be
considered a cost to A.

^{3} This implies that disarming below a certain level is
not possible if both nations want to realize the defensive military policy goal.

^{4} This definition of balance of power has the reasonable
consequence that, other things being equal, the more valuable a nation is to the
opponent the higher arms level it needs.

^{5} This is not true for the section "The nations'
information about their situation".

^{6} Henry A. Kissinger, *Nuclear Weapons and Foreign
Policy, *1957, p. 419. The quotation is made only to suggest that the
distinction itself is reasonable. The assumptions made about the effects of the
two types of weapons are not from Kissinger.

^{7} See the definition of "Type II Deterrence" in Herman
Kahn, *On Thermonuclear War, *Princeton University Press, 1961,
p. 126. Also see Herman Kahn, *Thinking about the Unthinkable, *Weidenfeld
and Nicolson, 1962. On page 108 is written: "U.S. military policy
currently seeks to achieve at least six broad strategic objectives:

1 Type
I Deterrence - to deter a large attack on the military forces, population, or
wealth of the United States, by threatening a high level of damage to the
attacker in retaliation;

2 Type II
Deterrence - to deter extremely provocative actions short of large attack on
the U.S. (for example, a nuclear or even all-out conventional strike against
Western Europe) by the threat of an all-out U.S. nuclear reprisal against the
Soviet Union;" and so on.

On page
112 Kahn continues: "At some future date, the non-nuclear capability of
NATO may be sufficient to repel a large but conventional attack on Western
Europe. Until this time Western Europe will probably depend, at least in part,
on Type II Deterrence (or Controlled Reprisal) to deter such attacks."

^{8} See On *Thermonuclear Wa*r, p. 126. This
deterrence is also called, by *Kahn and others, *"graduated
deterrence". See, for example, Sir Anthony W. Buzzard, "Massive
Retaliation and Graduated Deterrence", *World Politics, *1956, p.
229. Sir Buzzard writes when pleading for a graduated deterrence: "We
should not cause, or threaten to cause, more destruction than is necessary. By
this criterion, all our fighting should be limited (in weapons, targets, area
and time) to the minimum force necessary to deter and *repel *aggression,
prevent any unnecessary extension of the conflict, and permit a return to
negotiation at the earliest opportunity *- without seeking total victory or
unconditional surrender." *(italics mine)*.*

^{9}
See, for example, *On
Thermonuclear War, *p. 7 ff.