If you don’t expect it, you will not find the unexpected . . . 

Uncertainty and expectation are thoroughly familiar elements of ordinary human life. Also deception and concealment are rather familiar. However, their systematic and studious exploitation in the form of stratagems is more typical for one special region of life: military strategy and tactics.

The theory of games is an important step in elevating the art of stratagems from the darkness of pure intuition to the light of systematic, though inductive, inferences. The light which the theory sheds on the function of such elements as uncertainty, deception, and concealment illuminates at the same time one of the oldest and most important countermeasures: preparedness and flexibility of preparedness.

One of the most efficient means of achieving flexibility of preparedness is multiple-purpose weapons. It is not difficult to appreciate, and the history of war confirms, that the frequency and potential of multiple-purpose weapons increase with the complexity of weapon systems. The German dual-purpose 88-mm antiaircraft and antitank gun of World War II is an instructive example.

The definition of a multiple-purpose weapon is simple: it is a weapon system that can serve more than one purpose. It is not to be confused with commonality, which presupposes at least two single-purpose weapons for which common parts are sought This may or may not result in monetary savings.

Any advantage of a multiple-purpose weapon (relative to its single-purpose competitors) which stems exclusively from its multiplicity of purposes is called “leverage.” To be sure, there are also disadvantages. They should not be belittled. If they were flatly negligible, we would have only multiple-purpose weapons and tools. The disadvantages stem from the fact that multiple-purpose weapons are, generally, more complex than their single-purpose competitors. The consequences are weight, volume, and cost penalties to multiple-purpose weapons if they match the performance of their single-purpose competitors or, conversely, performance penalties if weight, volume, and cost are kept comparable.

The subject of this discussion is not a complete utility analysis of multiple-purpose weapons but only the first step to such an analysis, which is to introduce, to define, to understand, to apply, and to analyze qualitatively and quantitatively the concept of the leverage of multiple-purpose weapons. The assessment of the disadvantages, though indispensable for any complete utility analysis, would only becloud the issue. One of the oldest and most successful tools of scientific methodology is to divide complex problems into subproblems and, when studying a certain effect, to exclude as much as possible the perturbing influences of other effects.

The leverage of multiple-purpose missiles will be discussed as applied to two strategic examples: the dual-purpose missile for strategic aircraft and the area ballistic missile defense. In each case the leverage is first identified, explained, and analyzed in qualitative terms. Then we present a simple, mathematical analysis, which is dispensable for the reader who is not so mathematically inclined.

The Dual-Purpose
Missile

The dual-purpose missile is carried by strategic aircraft and can function in two modes: air-to-air and air-to-ground. In the former mode it serves as bomber defense missile; in the latter, as attack missile. Its two single-purpose competitors are the air-to-air bomber defense missile and the air-to-ground attack missile. The aforementioned disregard of any disadvantages, stemming exclusively from the multiplicity of purposes, implies that the dual-purpose missile can achieve in all its purposes the same performance as its single-purpose competitors for the same weight and volume (costs are not considered at all). This, then, implies that the bomber can load as many dual- as single-purpose missiles.

Some economic leverages of the dual-purpose missile are immediately recognizable, though probably of minor practical importance. These are the learning-curve effect (the unit cost of an item is a monotonically decreasing function of the number of items produced) and certain simplifications in operations, maintenance, and logistics. These are, incidentally, the major and probably the only advantages of commonality. They are, however, of no further interest to the present considerations.

The two most important leverages of the dual-purpose missile are its loading leverage and its stockpile leverage. The latter is a consequence of the former but is, nevertheless, an additional effect that pays additional dividends.

For discussion of the loading leverage, let us assume that the bomber is to be loaded with single-purpose missiles. Hence, a decision has to be made about the mix of bomber defense and attack missiles. They shall be so mixed that the mission effectiveness (measured in terms of weapons delivered to ground targets) is maximized. This is the “optimal mix.” Clearly, this optimal mix needs to be determined on the basis of an expected combat situation, for the actual (future) combat situation is not known when the bomber is being loaded. To be sure, there is generally more than one expected combat situation, since there is generally more than one maker of decision information.

However, at this point it is not yet important that there is more than one expected combat situation. Since they are described by certain distribution functions (for the number of threat interceptors per bomber and for other constituents of the scenario), the various expected combat situations can always be unified by applying the principle of superposition to the various distribution functions. This allows for normalized weighting factors which take into account the relative authority of the advisers to the decision-maker. So it may suffice at this time to assume that one expected combat situation can be generated. The mix of single-purpose missiles can then be optimized on the basis of this expected combat situation.

Now, when the bomber enters the real combat situation, its loading with single-purpose missiles will be optimal to the degree to which the real conforms to the expected combat situation. For example, if the bomber encounters fewer threat interceptors than expected, it has a surplus of bomber defense missiles and a corresponding deficit of attack missiles. This will result in a penalty to the mission effectiveness. On the other hand, if the bomber encounters more threat interceptors than expected, its survivability will be less than optimal with respect to the aforementioned maximization of the mission effectiveness. In other words, the inevitable uncertainty about the future combat situation causes effectiveness penalties to single-purpose missiles.

It is precisely this effectiveness penalty which is avoided by dual-purpose missiles. This is the loading leverage.

The loading leverage is typical for all leverages. In the last analysis, all leverages stem from the uncertainty that is an inevitable and ubiquitous ingredient of any military scenario. Multiple-purpose weapons avoid the penalties that are incurred from these uncertainties by single-purpose weapons.

The stockpile leverage is a consequence of the loading leverage. To describe it, let us again assume that the bomber is to be loaded with single-purpose missiles. For each expected combat situation there is a corresponding optimal mix. For each optimal mix there is a corresponding stockpile of bomber defense and attack missiles. It follows that for each expected combat situation there is a corresponding stockpile of single-purpose missiles which will permit optimization of the bomber loadings. Therefore, if the expected combat situation changes, the bolder of single-purpose missiles has only two alternatives: to change or not to change his stockpile. In the first case, he will adjust his stockpile so that he will be able to optimize loadings on the basis of the new expected combat situation. This means, however, that he will have to procure additional missiles of one kind (most likely, but not necessarily, bomber defense missiles) and that he will have to retire a corresponding number of missiles of the other kind. In the second case, when he refuses to adjust his stockpile to the new expected combat situation, he will have to send the bombers into combat with suboptimal loadings. That is, he will have to accept a corresponding penalty to the overall mission effectiveness (apart from the highly probable reduction of bomber survivability). Therefore, if the expected combat situation changes, the holder of single-purpose missiles has to choose between two evils: either to procure additional missiles of one kind and to retire a corresponding number of missiles of the other kind, or to accept a penalty to the overall mission effectiveness.

Again, this disadvantage is avoided by the dual-purpose missile. This is the stockpile leverage.

It is evident that the stockpile leverage, like the loading leverage, sterns from uncertainty. But it is now necessary to take a closer look at the nature of this uncertainty. It is helpful to distinguish between two classes of circumstances that cause multiplicity of expected combat situations. The first class has already been alluded to: the fact that there is, generally, more than one adviser to the decision-maker, more than one source of information and intelligence. This circumstance may be described as “disagreement between different persons at the same time.” The second class of circumstances is concerned with “disagreement of one person with himself at different times.” It is, in this perpetually changing world, merely the result of honesty and courage; it should not be confused with inconsistency. In contrast, inconsistency requires neither honesty nor courage; it is the “disagreement with oneself at the same time.”

This second class of causes (disagreement of one person with himself at different times) is probably more important than the first. It means that, even if there is only one decisionmaker, his expected combat situation is most likely to change in the course of time. The causes for this change are changes in intelligence information, in offense and defense inventories, in technology, strategy, tactics, objectives, and various beliefs. When these changes occur, the holder of single-purpose missiles has to take the aforementioned choice between two evils, while the holder of dual-purpose missiles enjoys the loading and stockpile leverages.

For a brief mathematical description of the loading leverage, let n denote the number of threat interceptor encounters per bomber. This number n is an integer random function of the individual bomber. This means that it is neither precisely predictable nor entirely unknown but is covered with a probability distribution function P (n). This probability distribution function is the essential ingredient of the expected combat situation about which disagreement is principally conceivable and generally the case. In many practical cases it will suffice to select a relatively simple distribution function such as the binomial distribution, which has only two (semi-) independently selectable parameters, say, the expectation or mean value <n> of n and the variance or dispersion o².

Let now S stand for “survival,” and let P (S|n) denote the (conditional) survival probability, given that the bomber encounters exactly n threat interceptors and that he has sufficient bomber defense or dual-purpose missiles to engage each of the n interceptors. The probabilities P (S|n) can all be generated by means of the simple formula

P(S|n) =P(S|1)ⁿ. (1)

Let then P_s (S|n) and P_d (S|n) denote the conditional bomber survival probabilities for single- and dual-purpose missiles, respectively. For simplicity, it is assumed that these survival probabilities are zero (or negligibly small) if the bomber runs out of lethal defenses while still being engaged by interceptors. To formulate this assumption analytically, let  m₁, m₂, and m₃ denote the numbers of bomber defense, attack, and dual-purpose missiles per bomber. The aforementioned assumption is then expressed by

P_s (S|n) = 0 for n> m₁     (2a)

P_d (S|n) = 0 for n> m₃     (2b)

This is, however, the only difference between the two sets of survival probabilities, for the earlier stated assumption that the dual-purpose missile matches the performances of its single-purpose competitors at equal weight and volume implies that

P_s (S|n) = P_d (S|n) for n < m₁ < m₃            (3a)

and that

m₁+ m₂  =  m₃  (3b)

The overall (unconditional) bomber survival probabilities for single- and dual-purpose missiles are now, respectively,

                  m₁

P_s (S) = Σ P(n) P₈(S/n)                             (4a)
               n=0

                 m₃
P_d (S) = Σ P(n) P_d(S|n).                            (4b)
              n=0  

For simplicity it is also assumed that the bombers encounter the interceptors prior to the delivery of their attack or remaining dual-purpose missiles to ground targets. In other words, all attack missiles and all the remaining dual-purpose missiles are delivered to ground targets if, and only if, the bomber survives all interceptor encounters.

Figure 1. Effectiveness of dual and single-purpose missiles as a function of uncertainty

If, then, the bomber is loaded with single-purpose missiles, the expected number <m₂> of attack missiles deliverable to ground targets equals the product of the number m₂ of attack missiles loaded and the overall bomber survival probability for single-purpose missiles. This is expressed by

                        m₁

<m₂> =m₂ Σ P(n) P_s (S|n).                        (5a)

          n=0

If the bomber is loaded with dual-purpose missiles and if it encounters exactly n interceptors, then it will deliver exactly (m₃-n) dual-purpose missiles to ground targets. The probability that this will happen is P (n) P_d (S|n). Thence the expected number <m₃> of dual-purpose missiles deliverable to ground targets is

                   m₃

<m₃> = Σ (m₃ –n) P (n) P_d (S|n).               (5b)

              n=0

It is now asserted that

<m₂> < <m₃>                       (6a)

for all interceptor distribution functions P ( n) and that

<m₂> =<m₃>                        (6b)

if, and only if, the interceptor distribution function has “δ-character,” i.e., if

P (n) = {1 for n = m₂

            {0 for all other n.          (7)

Of course, the δ-shaped distribution (7) is logically equivalent with certainty; and it should not be surprising that, under this condition and conditions (3a)and (3b) single-and dual-purpose missiles have equal effectiveness.

Figure 2. Effectiveness of dual and single-purpose missiles as a
function of real and expected combat situation

Assertion (6a) is expressed, in more detail, by

    m₁

m₂ Σ P(n) P_s (S|n) <

    n=0

    m₃

    Σ (m₃ – n) P (n) P_d (S|n).     (8)

n=0

The proof of this assertion follows the line of reasoning that was applied in explaining the loading leverage. First, it is assumed that the bomber meets fewer interceptors than expected, that is: n < m₁.

It then follows from equation (3b) that n < m₃ - m_2,  which is m₂ < m₃ - n. If this latter relation is heeded in relation (8), one can see that, for n < m₁, the terms on the left side are smaller than their corresponding terms on the right side. Next, the transitional case n = m₁ is considered. Clearly, this means that the bomber encounters as many interceptors as expected. It then follows from equation (3b) that m_{2 =} m₃  -n. If this is heeded in relation (8), one can see that the term for n = m₁ on the left side equals the corresponding term on the right side. Finally, it is assumed that the bomber encounters more interceptors than expected. This means that n > m₁. For these values, the survival probabilities on the left side of relation (8) vanish, whereas the terms on the right side are still positive until n > m₃ from which point on these terms also vanish.

It follows that the left side of relation (8) is always smaller than the right side except for the δ-shaped distribution function (7). This proves assertions (6a) and (6b). The reader’s attention is called to tile fact that this roof is independent of the interceptor distribution function P(n).

Figures 1, 2a, and 2b show some quantitative results on the loading leverage. They refer to one bomber that can load 20 single- or dual-purpose missiles, a binomial distribution function P(n), and a bomber survival probability for the single engagement,

P(S|1) = 0.9.

This probability combines some important parameters, such as single-shot kill probabilities of bomber and interceptor, the first-shot probability of the bomber, and others. The ordinate of Figures 1, 2a, and 2b displays the mission effectiveness measured by the expected number of attack or dual-purpose missiles deliverable to ground targets per bomber.

In Figure 1, the mean value <n> = 4 is fixed, whereas the standard deviation σ is varied. The value σ = 0 refers to a δ-shaped distribution, which amounts to certainty. For this value of σ, dual- and single-purpose systems deliver an equal number of missiles to ground targets. From here on, the effectiveness of dual-purpose missiles stays almost constant, whereas the effectiveness of single-purpose missiles decreases monotonically. This reflects the aforementioned fact that the dual-purpose missile avoids the disadvantages incurred by single-purpose missiles from the uncertainty about the future combat situation.

Figures 2a and 2b show the effects of variations of the expected number <n> of interceptors per bomber. Loadings with single-purpose missiles have been optimized in Figure 2a for <n> = 2 (low threat) and in Figure 2b for <n> = 6 (high threat). These are the points of closest approach between the curves for dual- and single-purpose missiles. For <n> = 0 (no threat at all), the delivery of attack missiles to ground targets has in Figures 2a and 2b the values < m₂> = 15 and and <m₂> = 12. From these numbers, the optimal mixes can be inferred. They are shown in the accompanying table.

From this table the stockpile leverage can be assessed. As has been mentioned, the holder of single-purpose missiles has two options: to change or not to change his stockpile. Consider the former option first. Suppose the threat has changed from low (<n> = 2) to high (<n> = 6). Under the low threat, 5 bomber defense and 15 attack missiles have to be stockpiled per bomber. Under the high threat, these numbers are 8 and 12, meaning that, per bomber, three additional bomber defense missiles are to be procured and that three attack missiles are to be retired.

Consider, then, the second option. Figure 2a refers to the low threat. If loadings with single-purpose missiles are optimized for the low threat <n> = 2 or if the stockpile of single-purpose missiles permits optimization of loadings for this threat only and if, then, the high threat <n> = 6 materializes, the effectiveness of single-purpose missiles is penalized so strongly that dual-purpose missiles are now twice as effective as single-purpose missiles.

If weight or volume effectiveness penalties for dual-purpose missiles were incorporated, the curves for dual-purpose missiles would be lowered relative to the curves for single-purpose missiles. Consider, for example, a 10 percent volume penalty to the dual-purpose missile. This means the bomber can load only 18 dual-purpose missiles as opposed to 20 single-purpose missiles. The curves for the dual-purpose missile in Figures 2a and 2b are then lowered so that they touch the curves for single-purpose missiles just at the points of closest approach, that is, at <n> = <n>_opt. But for all other values of <n>, the dual-purpose missile would still be superior to single-purpose missiles.

Area Ballistic
Missile Defense

The purpose of missile defense is to defend certain installations that are potential targets for ICBM attack. If an interceptor-sensor pair can defend more than one target, it is a multiple-purpose system. Strictly speaking, there is no single-purpose system, for there is no point defense either. However, the area which a point defense interceptor can defend is so small that it may be considered as one target and, when compared with the size of the United States, as a point. On the other hand, an area defense interceptor can defend an area that contains more than one target. The larger this area, the larger the “degree of area coverage.” A more suitable, though inverse, measure of the degree of area coverage is the minimum number b of interceptor bases required for complete coverage of the United States. The highest degree of area coverage corresponds to b =1 which implies complete coverage of the United States by one single interceptor base. If T denotes the number of targets to be defended, the lowest degree of area coverage corresponds to b=T which implies that each target needs its own interceptor base.

The concept of “multiple-purpose system” is here used in an extended meaning. The previously considered dual-purpose missile is a dual-purpose weapon by virtue of its capability to perform two different functions, viz., air-to-air and air-to-ground. The area defense interceptor can perform only one function, but it serves multiple purposes by virtue of its capability to defend a multiplicity of targets. It has this capability in proportion to its degree of area coverage.

The multiplicity of purposes of area defense results in at least two leverages: “numerical interceptor leverage” and “weapon exchange leverage.” Let us consider single-purpose or point-defense systems first.

A point-defense interceptor can defend one target only. If, by the end of the war, the target was not attacked, then the interceptor was wasted. By the same token, if one target is attacked by fewer and another by more warheads than were “expected,” then the first will have a surplus and the second will have a deficit of defenders.

This effect should not be viewed as matter of mere coincidence. In fact, it will be strategically planned and optimally exploited by the offense. The means of generating the effect of interceptor surpluses and deficits at the various targets is to attack the targets deliberately with different degrees of intensity, studiously selected. Offense strategies that are, for this purpose, deliberately heterogeneous with respect to the attack of the targets are called “selective.” This “selectivity” of offense strategies and the selectivity of defense strategies (to be discussed shortly) should not be confused with “preferential” attack or defense. Preferential offense or defense strategies are also heterogeneous with respect to the attack or defense of the targets, but for entirely different reasons. Preferential strategies are heterogeneous because the targets themselves are heterogeneous. For example, the targets may differ in value, vulnerability, and accessibility.

Selectivity is a strategic means of purposely creating uncertainty in the opponent without regard to the diversity of the targets, whereas preference is dictated by and a consequence of the diversity of the targets. Selectivity is concealed; preference is predictable.

The disadvantage to the defense which stems from the concealed selectivity of the attack is avoided by area defense in proportion to its degree of area coverage. This is one root of the leverage of area defense.

Another root is that area defense can counter the selectivity of offense strategies with its own weapons, that is, with “selective area defense.” To explain this concept, it is first necessary to distinguish between two classes of defense strategies: gross strategies and detail strategies. The gross strategy determines the distribution of interceptors over the interceptor bases. Since this distribution is open to offense intelligence (for instance, reconnaissance satellites), it follows that gross strategies are nonconcealed.

In contrast to gross strategies, detail strategies do not refer to the interceptor force as a whole but only to the interceptors of one particular base. There are two classes of detail strategies: those of the first class are preallocative, selective, and defensive; those of the second class are postallocative, nonselective, and offensive. The strategies of the first and second classes are, in brief, called “selective” and “primitive,” respectively.

A selective strategy preallocates the interceptors of a particular interceptor base to the defense of the targets which are to be defended from that base. This preallocation is selective, i.e., deliberately and studiously heterogeneous.

Under a primitive strategy a warhead is attacked whenever it comes within reach of the first interceptor, regardless of the target at which the warhead is aimed. A primitive strategy focuses on the attack of the attackers, without regard to the targets to be defended. It is inherently offensive.

A selective strategy focuses on the defense of the targets and attacks attackers if, and only if, the preallocation of interceptors to the defense of the targets calls for the attack. Otherwise, the warhead is not attacked. Hence, selective strategies are essentially defensive. As has already been pointed out, the deliberate and studious heterogeneity of the defense of the targets under selective strategies has nothing to do with the possible heterogeneity of the targets themselves but is a stratagem to exploit the concealment of detail strategies with the express purpose of creating in the enemy as much uncertainty as possible.

Point defense has the disadvantage that it commands only gross strategies, which are always nonconcealable and therefore nonselective and offensive. Area defense has the advantage that it commands both gross strategies and detail strategies. The latter may be selective or primitive. Whether and when to employ selective or primitive strategies depends on the “level of protection.” This, in turn, depends on the “defense job,” the “strength of the offense,” and the “strength of the defense.”

Let the defense job be characterized by the number of targets T to be defended, the strength of the offense by the number M of deliverable warheads in the offense inventory, and the strength of the defense by the number N of interceptors. For simplicity, assume unity single-shot kill probabilities for the interceptors and unity target-destruction probabilities for the warheads. Consider then two levels of protection as follows:

    Level of Protection I: N > M

Level of Protection II: N < M - T.

In the first case, there are more interceptors than warheads. In the second case, the number N of interceptors is not only smaller than the number M of warheads but even smaller than the number M reduced by the number T of targets. Clearly, defense level I is relatively “high,” and defense level II is relatively “low.”

Apply now primitive and selective defense strategies to both defense levels. Consider first defense level I, i.e., the high defense level. If defense applies primitive, i.e., offensive, strategies, it will attack and destroy all M warheads so that all targets will be saved. If, at the same high defense level I, defense applies selective, i.e., defensive, strategies, it will defend the various targets to various degrees. Therefore, some targets may be defended by fewer interceptors than the number of warheads attacking. These targets will be destroyed. Hence, it the defense level is high, offensive strategies are the best defense.

Consider now defense level II. If defense applies primitive, i.e., offensive, strategies, it will attack and destroy exactly N warheads. Hence, M — N warheads will penetrate. But M — N>T. Hence, the number of penetrators is more than sufficient to destroy all targets. On the other hand, if defense applies selective, i.e., defensive, strategies, it defends the various targets to various degrees. Some targets may not be defended at all. Assume that the number of targets that will be defended is A (A < T). The average number of interceptors per target “selected” is then N/A. At some targets, this number will be larger than the number of warheads attacking these targets. These targets will be saved. Hence, if the defense level is low, defensive strategies are the best defense.

The two preceding conclusions have an implication with respect to the popular strategic rule “Offense is the best defense.” In this unqualified form, the rule is false. Whether and when offense is the best defense depends on the defense job and the relative strengths of the opponents.

A mathematically rigorous criterion of when to employ defensive or offensive strategies is provided by the concept “assured defense level.” To define this concept, let M denote the number of deliverable warheads in the offense inventory (for simplicity they are assumed to have equal yields, circular error probable, etc.). This number M constitutes the highest possible attack size and is therefore called “maximal attack size” or “threat size.” It must be assumed that the defense has a fair estimate of the maximal attack size but does not know the actual attack size before the attack is completed (with the exception that it cannot exceed the maximal attack size) This is an element of uncertainty in missile warfare that is often overlooked, particularly in critiqueless applications of the theory of games.

Let T denote the total number of targets to be defended and T₈ the expected number of targets that will be saved under maximal attack. The assured defense level is then defined as

t = T₈ /T                                                    (9)

which is a variable with variability from zero to one.

The particular assured defense level where defense has to switch from defensive (selective) to offensive (primitive) strategies is called “critical defense level” and denoted by t^¤. To define t^¤, let b denote the minimum number of interceptor bases required for complete coverage (by radars and interceptors) of all T targets to be defended. Of course, b and T can only assume integer values. For purely mathematical reasons it is advantageous to impose the further restriction that

b=T/n for 2 < n < T                                (10)

so that both b and n are integers. The critical defense level is then defined as

t¤ = 1 - ½b.                                             (11)

Since the value b = T, which applies to point defense, is excluded, it follows that the concept “critical defense level” is not applicable to point defense. This is necessary, for the critical defense level is that level where defense, however, commands only nonselective, to primitive (offensive) strategies. Point defense, however, commands only nonselective, i.e., primitive strategies. Area defense has to apply selective (defensive) strategies if t < t^¤ and primitive (offensive) strategies if t > t^¤ .

With the aid of the critical defense level, it is now possible to describe quantitatively the numerical interceptor requirements. As before, it is assumed that the single-shot kill probabilities of the interceptors and the target-destruction probabilities of the warheads are unity. For point defense, the critical defense level is not needed (and is not applicable). For purely mathematical and therefore unimportant reasons (integer problem), the interval 0 < t < 1 is split into two parts, allowing representation of the number N_PD( t) of point defense interceptors required for providing the defense level t as follows:

               {M – T(1 –  t) (t <1 – 1/T)

N_PD(t) = {         1 – t                                     (12)

              {TM                 (t = 1)

which is, for given M and T, a function of t.

In representing the number N_AD (t, b) of area defense interceptors required for providing the defense level t, essential use is made of the critical defense level:

                  {(M – T(1 – t) (0 <  t  < t^¤)

N_AD(t,b) = {    2(1 – t)                                  (13)

                 {b[M – T (1 – t)] (t^¤) < t).

Here, the splitting the interval into two parts operational or strategic and therefore important reasons, because the point of separation, t  = t^¤, is the point where area defense has to switch strategies. No switch of strategies is required for point defense.

It is now possible to describe the numerical interceptor leverage by the “leverage factor”

               __N_PD(t)    .

p(b,t) =     N_AD(t, b)                                      (14)

This function is easily generated from equations (12) and (13) and is illustrated by Figure 3. This figure shows p as a function of t with b = 1, 2, 3 as parameter. The following observations can be made: First, p is never smaller than two, which implies that the number of point defense interceptors is at least twice as high as the number of area defense interceptors, regardless of the defense level and the degree of area coverage (measured inversely by b). Second, for constant b, p increases monotonically with t and becomes very large for high values of t. This implies that, for high defense levels, many more point defense than area defense interceptors are needed. Third, for constant t, p increases monotonically with the degree of area coverage. Of course, the unit costs for radars and interceptors increase also monotonically with the degree of area coverage. It can therefore be surmised that an optimal degree of area coverage exists for which the total system cost effectiveness is maximal.

Figure 3. Numerical interceptor leverage

The weapons exchange leverage is a consequence of the numerical interceptor leverage. To describe it, let ΔM denote an increment of the number M of deliverable warheads in the offensive inventory, and let ΔN denote that increment of the number N of interceptors required for compensating for the increment ΔM, that is, for restoring that assured defense level which prevailed prior to the increment ΔM. The weapons exchange ratio is then defined as

E = (ΔN/ ΔM)_t                                 (15)

where the subscript t indicates the constant assured defense level.

Figure 4. The weapon exchange ratio

It is typical for the inherent unsymmetry of the offense-defense relation in missile defense that E is, for all practical purposes, larger than one. In Figure 4, E is shown as a function of t with b as parameter. The following observations can be made: First, the weapons exchange ratio for point defense is always greater than one. That is, the number of point defense interceptors required for restoring the defense level is always greater than the number of warheads required to offset it. Second, the weapons exchange ratio for point defense increases monotonically with t and becomes eventually prohibitively large. That is, even if higher defense levels could be obtained by point defense, they could certainly not be maintained against a determined threat in a missile race. Third, the weapons exchange ratio for area defense is smaller than one if t < ½. This is a decisive advantage of area defense at low defense levels. Fourth, the weapons exchange ratio for area defense does not, at higher defense levels, increase monotonically with t but is truncated so that it will never exceed the value E = b.

It is now abundantly clear that leverage stems from uncertainty. The uncertainty may be the ubiquitous, inescapable uncertainty of ordinary life, as is the case in strategic bomber penetration, or it may be studiously generated by a cunning opponent, as is the case in missile defense.

It is sometimes naïvely complained that rational (the fashionable word is “meaningful”) decisions are not possible because too much is unknown. If this were true, we could as well stop all contingency planning, even to the carrying of a raincoat when the weather looks stormy. For the significant things that are known with certainty are not many, and the body of the unknown will always exceed the body of the known.

The art is not so much to remove uncertainties—though this can be quite useful—or to create the illusion of removing them but to live with them. In the words of George Eliot: “No great deed is done by falterers who ask for certainty.”

Office of Research Analyses, OAR  

Contributors

Dr. Bruno J. Manz  (Ph.D., Technishce Hochschule, Aachen, Germany) is Chief, Operations Analysis Division, Office of Research Analyses, OAR, Holloman AFB, New Mexico. He began his study of physics in 1946 at the University of Mainz, Germany, and came to the United States, 1957, to accept a position as physicist, Aeroballistics Laboratory, Army Ballistic Missile Agency, Redstone Arsenal, Alabama. Dr. Mantz has been at Holloman since 1959 and in his present position since 1965.

 Richard H. Anderson (M.S., New Mexico State University) is an Operations Research Analyst, Office of Research Analyses, Holloman AFB. After military service in the Air Force, 1953-57, he joined the Research Division at Holloman and transferred to Redstone Arsenal as mathematical adviser to the Director of the Aeroballistics Laboratory, 1958. Mr. Anderson returned to Holloman in 1959.

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