1. Introduction

When President Reagan was at loggerheads with Congress over arms control and the MX missile, he appointed the Scowcroft Commission to mediate a compromise. After their 1983 report, the hot political topic of strategic missiles overshadowed their arms control suggestions, so their final 1984 version stressed particular recommendations about arms control. One was the importance of devising a measure of weapons' effectiveness (President's Commission on Strategic Forces, 1984, 5). The goal was to "index each type of weapon through some measure of it such as 'destructive potential' . . . ". Such a measure, they said, would promote an agreement on equivalent reductions between adversaries possessing different kinds of forces, and would help gain acceptance of an arms agreement by the domestic constituency.

Such indices have arisen in arms control proposals before and since. One framework has been to agree on a rate of tradeoff among weapons within which each side would maximize its own goals, as if it were solving a linear program. In 1976 Andrei Gromyko proposed that each B‑1 bomber count three units against a ceiling of 1320 units while a B‑52 or a ballistic missile launcher would count only one (Talbott, 1979). Kent (1986) and Drell (1982) proposed exact numerical schemes to govern an arms build‑down agreement. This approach was implemented in a limited way in the START I Treaty which came into force in 1994. The nuclear warhead of an intercontinental ballistic missile effectively counts for twice as much against the quota, as that of a cruise missile carried on a heavy bomber.

One significant use of an effectiveness index involved not weapons' numbers but features. The so-called Nitze index was the product of the emitted power of a radar installation multiplied by area of its antenna. Paul Nitze, on Nixon's negotiation team for the first SALT Treaty, argued persistently that there should be a mathematical limitation on the anti-ballistic-missile capabilities of either side's radar. In the signed treaty, the measure was known as the potential of a radar site, and could not exceed 3,000,000 watt-m². Talbott (1987) suggests that setting a limit with precision has helped deter violations and accusations of violations, and has preserved the ABM Treaty over the years.

A measure of importance might also be valuable for of conventional arms control. The discussions on Conventional Forces in Europe (CFE) deal with weapons of many types. Whereas START negotiators handle ten types in the category of ICBMs, for example, tanks and armoured combat vehicles alone come in over 90 types (Leavitt, 1990). The talks have dealt with other classes of weapons, and subdivided them according to their location in several zones depending on their proximity to Central Europe. Although no formal methods have evidently been used so far, a measure of value for groups might simplify the negotiations.

The notion of importance arises also in military planning. An early application of military worth measures was investigated at the RAND Corporation in the late 1940's, when Ed Paxson, the founder of the systems analysis division, used a formal index to guide the design of strategic offense and defense (Digby, 1990). Another use assesses the "military balance," where analysts judge the value of individual items and add up the holdings in each side's arsenal. A large literature developed indices comparing NATO and the Warsaw Pact, for example. Many automated war games use numerical rules to decide how to allocate forces, how much one side should sacrifice to eliminate some adversary units, or how to tabulate scores at the end of the session.

A potential use of measures in military analysis involves the tenet of a defender's innate advantage, that the attacker typically needs three times the defender's strength to win. Does the historical record support this? The claim presumes some way to calculate the force ratio. If the adversaries had only one type of weapon in two different amounts, calculating the ratio would be easy, but if they possess various sorts in different proportions, an analyst must score individual items and add.

This paper presents a theoretical way to calculate the "importances" of weapons, and illustrates its behaviour through examples. The measure takes as given the benefits of having a set of weapons, and tells how to allocate the credit among them. It is very general, applicable to any problem where the values for the outcomes of the conflict can be specified. If one accepts certain postulates as compelling, it is the only way to measure importance. In some applications it can also measure the importance of weapon qualities, such as firepower versus vulnerability.

The method requires a great amount of information on the benefits of the weapons and of many possible subsets of the weapons. This limits its applicability, but the purpose of the paper is not to suggest a measure for use in some of the arms control and warplanning applications mentioned above. The contribution it hopes to make is to clarify intuitive patterns of thought. The definition should show what a logically correct concept of importance would be like, bringing out any inconsistencies in more practical measures.

Section 2 develops the definition of importance for discrete weapons and Section 3 illustrates it with examples. Section 4 extends it to continuous quantities which can apply often to assessing the features of a single weapon, and Section 5 adds some examples. Section 6 compares the measures with some categories of operations research techniques for assessing weapons values. The problems with present formal methods and informal metaphors turn out to be the same: they ignore synergy and redundancy among weapons, and they assign values to weapons based on their individual characteristics without considering the benefits these weapons confer on the possessor.

A final aim is to understand the images people use in their thinking, which are probably more influential in determining actions than any analysis. We hear about who is "ahead," whether the other is "catching up," whether there is a "military balance" or a "gap," what is each side's "strength" compared to the other's, but these metaphors are misleading, in my view. They are especially inappropriate in the context of nuclear weapons, which in no sense balance off against each other. The paper will try to clarify the notion of importance of weapons in way that can be compared with people's natural thought processes.

2. The method for discrete weapons

The method is based on Shapley's value of coalitional games (1953), supplemented by Young's more recent discoveries about the Shapley value (1985). There is no game theory involved here, however, since there is no strategizing or choosing moves -- Shapley's mathematical structure is borrowed without its original content. This application resembles the use of the Shapley value to study cost allocation problems in accounting. In that context, an activity that serves several parties' needs might incur a certain cost, and the question is how much of the total cost should be attributed to each party. (Shubik first applied the Shapley value for this purpose in 1962, and Young (1995) reviews the literature since that time. Biddle and Weinberg (1984) compare the approach with other methods of cost allocation.)

The three kinds of problems -- coalitional games, cost accounting, and measures of military effectiveness -- are similar but not at all identical. Somewhat different mathematical questions arise in each one. One central observation applies to both the cost allocation and the military worth case. It is that worth cannot be measured simply by the marginal contribution of a weapon -- how much it adds to the benefit from one's current holdings. One might, for example, have two weapons at hand, where exactly one is necessary and sufficient for provide some benefit b. A straight marginal worth approach would assign value 0 to both, whereas the only plausible assignment is b/2 to each.

As often happens, an intuition that seems right is at bottom correct, as long as it is formalized in the proper way. Identifying importance with marginal contribution led to a problem because the added value of each was 0 given the other weapon was in the arsenal. But each weapon is in fact worth something because it would make an important marginal contribution if the other were not there. This suggests the rule: Do not look at the marginal contribution only to the complete arsenal, consider what the weapon adds to all subsets. This will be formalized as Principle 2.

Suppose I hold a finite set W of weapons. Suppose that I can measure my benefit from this set, and also measure my benefit from having any subset. The latter is a hard requirement in practice but the idea is simple: If I possess 20 ships and 100 tanks, how well would I do with 19 and 100, with 20 and 99, and so on, for all lower amounts? If S is some subset of the total arsenal W, the number b(S) will designate my benefit if I hold only the weapons in S. The function b will be called the benefit function.

The importance of the weapon x, given that the benefit function is b, will be written I_b(x), and the "right" or "appropriate" function I_b is what we are trying to find. The following three principles restrict I_b

and, taken together, they determine it uniquely in the discrete case.

Principle 1: 3_x₀_W I_b(x) = b(W) - b(N).

This principle says that the sum of the importances of one's weapons equals the benefit derived from having them. It seems most natural to talk of importances as additive, and it is hard to think of any simple alternative. The principle seems intuitively compelling.

The next principle involves the concept of equivalent weapons: two weapons x and y are equivalent if replacing one by the other in any set S leaves b(S) unchanged. That is, equivalence means that no matter what other weapons are held, replacing one by the other does not change the benefit you receive.

Principle 2: If x and y are equivalent for b, then I_b(x) = I_b(y).

The third principle is more subtle. It stems from the vague notion that the importance I_b(x) of a weapon x ought to reflect the added advantage you gained from acquiring x. As noted above, defining importance simply as the marginal contribution b(W) ‑ b(W ‑ {x}) will not work. It would clash with either Principle 1 or 2. The two weapons in that hypothetical arsenal must be equal in importance -- we cannot arbitrarily specify one or the other as the redundant one -- and their importance must sum to 1.

The refinement of the marginal contribution idea is to determine importance by the set of all possible marginal contributions the weapon might make. For each x we will define a function on the subsets of W (where the notion of subset includes W itself),

b_xN(S) = b(S) ‑ b(S ‑ {x}) if x is in S,

b(Sc{x}) ‑ b(S) if x is not in S.

This is the marginal contribution function, b_xN(S), a series of numbers that shows what benefit x is adding to or could add to S. As the notation suggests, it is like a derivative of S with respect to x. For the two‑weapon situation above where either was necessary and sufficient, b_xN( N) = b_xN(x) = 1, where N is the empty set containing no weapons, and b_xN(y) = b_xN(xy) = 0. (We are using the simpler notation of "b_xN(x)" for "b_xN({x}).") Principle 3 states that b N is sufficient to determine the importances. Even comparing two different benefit functions, identical marginal benefits mean identical importances. Young (1988) calls this the marginality principle.

Principle 3 (marginality): For a given weapon x, I_b(x) is determined by the marginal contributions b_xN.

Young proved that only the following function satisfies the three principles:

I_b(x) = 3_S_f_W b_x!(x) (w - s)! (s - 1)!/s!

Here w is the number of weapons in W and s as the number of weapons in S. So the importance of x is a weighted average of the marginal contributions of the weapon x, the weights depending on the number of weapons in the subset.

3. The discrete case: some examples

Example 1. A command centre and two units.

Suppose that we have a command/supply centre C and two units J and K that attack the enemy directly -- C is the "tail" and J and K are the "tooth." The command centre plus either unit will succeed, but with no engagement units you will lose and with no command centre you will lose. With values 0 and 1 for losing and winning, the benefit function for this game is: b(CJK) = b(CJ) = b(CK) = 1, and b(S) = 0 for all other sets S.

The formula of the theorem gives I_b(C) = 2/3 for the command centre and I_b(J) = I_b(K) = 1/6 for each engagement unit. (For n units the command centre has value n/(n+1).)

Why not 1/2, 1/4, 1/4 or something else? Applying the logic of Young's proof to this particular example shows why not. Call the original benefit function b₁ and define a second scenario: b₂(CJ) = 1, b₂(S) = 0 for other S. In this alternative situation only C and J succeed, and adding K nullifies them -- b₂(CJK) = 0 as if K were an effective counter-weapon held by the adversary. Unit J's contributions in b₁ and b₂ are the same, since in both it contributes 1 to the set C, 1 to CJ, and 0 to others. Principle 3 then requires that J's worth must be the same in the two. Define a third situation: b₃(CJK) = ‑1, b₃(S) = 0 for other S. (Here the existence of K turns the situation to your loss.) In b₂ and b₃ K's contributions are the same: ‑1 to CJ and CJK and 0 to other subgroups, and thus K's worth is the same. But worths in b₃ can be assigned by Principles 1 and 2 alone since the C, J and K are equivalent weapons in b₃, and the values must sum to ‑1. Therefore K gets ‑1/3 in b₃ and thus also in b₂. Since C and J are equivalent in b₂ and the sum of the three worths must be b₂(CJK) ‑ b₂( N) = 0 by Principle 1, then J must be assigned 1/6 in b₂ and hence also in b₁. The values of the other two follow since J is equivalent to K and the three sum to 1.

Young showed that such a chain of b-functions can always be constructed, leading from the original to a situation solvable by its symmetry alone.

Example 2. Deterrence with triads.

If Example 1 is modified to include three units instead of two, it resembles US strategic nuclear forces, the units being the so‑called legs of the triad: intercontinental missiles, submarine‑launched missiles and bombers. The commonly used metaphor of legs suggests that with only two legs the system would topple. (This is misleading, of course, since the explicit goal of having a triad is that each unit should stand by itself, even if the others have failed.) Example 2 involves triads on both sides. Suppose there are two sides, each with a command centre and three engagement forces. A side is defined to have the ability to attack if it has the command centre plus one or more of the three forces. A side loses if it cannot maintain deterrence, if it does not have the ability to attack and the other does. Otherwise it wins.

Here weapons are held by both sides. The adversary's weapons are measured from your viewpoint, so will receive negative values. Another new aspect of this example is the suggestion of common interests. You win if both you and the other have the ability to attack, or if neither does, implying that if one were to define payoffs for the other, both could win at once. It is reasonable that a definition of military worths should not suggest that the situation is zerosum.

A simple computer program could be used to calculate the formula above. It would involve determining the outcome of all 2⁸ subgroups of weapons, and adding the weighted contributions. In fact a faster method was devised by Hart and Mas‑Collel (1988) to give the results presented in Figure 1.

According to Figure 1 your weapons' scores total 1/2 and the other's total ‑1/2. By Principle 1 the sum must be zero since a world with all the weapons is as beneficial to you as one with none. Your command centre turns out to be more valuable to you than the other's is threatening to you (.418 versus .332). I have found no intuition behind this, but on the other hand there is no inherent reason they should be equal

2. The method with continuous weapons or features

Here we consider systems that can vary continuously, from zero to their extant value. One might model a large arsenal in this way, but a straightforward application is to the qualities of an individual weapon -- firepower, armour, speed, etc. -- which are normally continuous. One generalization of the Shapley value to this case is called the Aumann‑Shapley value. Owen (1983) gives an introduction with some simple calculations, and Tauman (1988) has written a survey.

The definition of the Aumann‑Shapley value in this context is as follows. Suppose a weapon possesses n continuous features to degrees given by the vector (a₁, a₂, . . . a_n), where its benefit from possessing each to degree (t₁a₁, t₂a₂, . . ., t_na_n) is given by b(t₁, t₂, . . ., t_n). We will assume that the features are measured such that b(0,0, . . . 0) = 0, and that f has continuous partial derivatives in the set [0,a₁] x . . . x [0,a_n]. The Aumann‑Shapley value of the i'th item is defined to be

₁

I b_i(t,t,. . .t) dt

^t=0

where b_i(x₁, x₂, . . . x_n) is the derivative of b(x₁, x₂, . . . x_n) with respect to its i'th variable.

Intuitively one could regard the derivatives b_i as measuring the marginal contribution to an arbitrary set of preexisting features. The integral takes an average of i's marginal contributions. Whereas the Shapley value considered all smaller subsets of the discrete arsenal, this measure looks only at those "weaker" weapons possessing all features in same proportions. The average is taken over the ray connecting the zeropoint to the true values. This follows the intuition that each feature is made up by an infinity of small bits. A random sample of all bits will yield the same proportions as in the whole population -- the infinite sample will eliminate random variations.

This is an intuitive justification of course. The real support for the procedure depends on the axioms that lead to it uniquely. Unfortunately the three principles stated in Section 1 are not enough -- many other methods satisfy them in the continuous case, such as treating each quantity as all-or-none and calculating the Shapley value. However modifying Principle 3 above, and adding one further reasonable axiom yields the Aumann-Shapley value uniquely (Billera, Heath and Verrechia, 1981; Young, 1988).

Principle 3 is to be changed to Principle 3' (marginality and strong monotonicity), that the importance measures are not only determined by the marginal contributions but they are determined by it in a monotonic way: when an element in the set decreases the importance measure does not increase.

The additional principle says that when two features have similar effects on benefits, their importance assignments should be connected in a specific way. Consider a cargo plane that can carry 160,000 pounds of supplies. We might (rather perversely) divide its payload capability into two features instead of one, and assert that it can carry 80,000 pounds and 40 tons. The principle requires that our measure not be misled by the arbitrary distinction, that the importance of carrying a certain amount of tons should be 2000 times the amount for pounds.

Principle 4. (aggregation invariance). Suppose there exists a function b* and real vector (8₁, 8₂, . . . 8_n) such that f(x₁, x₂, . . . x_n) = f*(E8_ix_i). Then the importance assigned to a weapon (a₁, a₂, . . . a_n) following benefit function b, should equal 8_i that assigned to a weapon with a single feature E8_ix_i, following b*.

Together, Principles 1, 2, 3' and 4 imply the Aumann-Shapley method (Young, 1994).

4. The Continuous Case: Examples

Example 3: Orbiting lasers for boost-phase intercept of ICBMs.

During the debate on strategic missile defenses, models were devised to estimate the size and technical requirements of a space-based defensive system. One by Garwin (1985) will provide an example of an importance calculation that involves continuously varying features. For simplicity, we will use a formula Garwin develops along the way to his full model, but the application to his complete model is straightforward.

ICBMs will be launched from a single point; above are satellites with laser weapons. The density of the satellites over the site is s and they are deployed at altitude h. The laser on each has brightness B, as measured in energy delivered per steradian of solid angle. A satellite learns immediately when a booster has been destroyed and can retarget its laser instantaneously. The booster is vulnerable only during its burn time of T seconds, and its hardness J is measured in the energy per area of its surface required to destroy it. The Earth is flat here, so that the number of satellites that can participate in the engagement is limited by the distance where the laser beam becomes ineffective, rather than by the horizon.

The number of ICBMs that are destroyed can be calculated to be roughly

N(s,B) = (BsBT/J) log(BT/Jh²).

We will consider the components whose importance is to be measured to be s and B. These are the two variables under the control of the side that deployed the satellites.

The difference with past applications here is that brightness and density are not dichotomous. If they were dichotomous, the answer would be straightforward and trivial: nothing would happen if either one took the value zero, so both features should receive equal weights. The Aumann-Shapley value, however, considers the marginal contribution that a small increment in brightness makes over a continuous range of densities, and likewise the contribution that an increase in density makes over a continuous range of brightnesses.

To calculate the importance of brightness at the pair of actual values s* and B*, first, we calculate its marginal contribution at various levels of the two variables. We calculate N_s(s,B), the partial derivative with respect to s, averaged over all situations in which s and B are in proportion to their actual values, uniformly weighted:

I_s(s*,B*) = I N_s (ts*, tB*) dt

This is the per unit importance of density at the values s* and B*. The total importance is found by multiplying this by the actual density s*. A similar calculation is made for B.

Garwin chooses B* = 269 megawatt-megameter² per meter², J = 200 megajoules per meter², T = 100 seconds and h = .5 megameters. A density s* = 1 satellite per megameter implies that 164 satellites will be within the range of engagement. Then N(s*,B*) = 2657 -- the system can destroy 2657 boosters. We calculate

N_s(s,B) = (BBT/J) log(BT/h²J)

N_B(s,B) = (BsT/J) (log(BT/h²J) + 1),

and

I_s(s*, B*) = (BBT/4J) (2 log(BT/h²J) - 1) = 1223 boosters/(satellite/megameter²)

I_B(s*,B*) = (BsT/4J) (2 log(BT/h²J) + 1) = 5.33 boosters per (megawatt-megameter² per meter²)

Thus the importance of density of satellites overhead in this system is 1223, and with 269 units of brightness available the total importance of laser brightness in this system is 5.33 x 269 = 1434. This allocates the number of boosters destroyed between the two ingredients, density and brightness. At these parameters, brightness is slightly more important than the total number, but this ratio would change if B or s changed.

Product-of-powers laws of weapon effectiveness

The importances derived in the next examples are instances of a general proposition, easy to derive from the Aumann‑Shapley formula. If the benefit function has the form of a product of powers

a₁^r1 a₂^r2 . . . a_n^rn, then factor a_i's total importance is proportional to r_i. This is a simplification not only for the calculation but for the conclusion, since the relative importances do not depend on the features' current values (in contrast to the orbiting laser example where they depended on the extant s and B.)

Many product-of-powers functions arise in military operations research. Sometimes they result from simple laws of physics, which are often products of powers because of their basis in concatenation operations of measurement (Krantz, Luce, Suppes and Tversky, 1971). Example 4 on the destruction of missile silos fits this group, as does the potential of a phased array radar site mentioned in Example 5. However such laws can arise in other ways. Anderson and Miercort (1995) suggest a great number of them as effectiveness measures, as a consequence of their differential equation models of the interaction of military forces.

Example 4. Warhead yield and accuracy

A products of powers forms the basic model of the countersilo effectiveness of a ballistic missile. Considering the two factors of weapon yield y and accuracy a (defined as the inverse of the circular error probable), we note that these fit the requirement of the Aumann-Shapley method that b(0,0) = 0. The likelihood of destruction of a targeted silo is often modelled as proportional to y^2/3a² (Bennett, 1981). During Cold War debates and since, one often hears that the "destructive power" of strategic weapons has been decreasing in recent years, and the claim has sometimes been accompanied by a graph showing the total megatons falling with time (e.g., United States Department of Defense, 1988, p.110; President's Commission on Integrated Long‑term Strategy, 1988, p.39). This ignores the development of missile accuracy, which rose rapidly (MacKenzie, 1990), and is more important than accuracy when warheads are aimed at hardened military installations. We can now say with mathematical precision (at least if we buy the assumptions of the measure) just how much more important it is: accuracy is more important than warhead yield by a factor of three.

Example 5: The potential of an ABM phased-array radar.

The potential of a phased array radar, its Nitze index, is measured by PA, the product of its power and aperture (antenna area).[1] Nitze's formula would be justified by arguing that the payoff to the radar's possessor increases monotonically with R_max according to the logic that the further out the object is detected the longer one has to shoot it down. Any subset of the variables in the equation could form the basis for an assignment of importance according to the Aumann-Shapley value, as long as (a) they are under human control; and (b) they satisfy the requirement f(0,0, . . . 0) = 0. They first requirement eliminates using Boltzmann's constant or the temperature, and the second eliminates the use of the receiver noise F_n, or the radar cross-section F. This could be circumvented, of course, by re-measuring these two as their reciprocals, as was done early in measuring accuracy as 1/CEP. Without this maneuver, however, this leaves P, A and t_s. The variable t_s is determined by many factors including the wavelength. This is not based on a full-fledged model of how many ICBMs such a radar could stop -- that would be quite complicated, involving the amount of sky to search, the most likely directions of the attack, the qualities of the interceptors, etc. However PA is a reasonable measure for its level of simplicity. The signal strength received from a target depends linearly on power the radar transmitted and the area of its antenna that collects the signal.

Our method gives the obvious result here, that power and aperture are of equal importance. A more interesting aspect of the example is how the practical problem guides us to a choice of particular variables to measure. The features P and A are relatively monitorable, and a choice of A, in particular, is revealed to the other party years in advance of the radar becoming operational. As happened with the Soviet Krasnoyarsk radar, the adversary has time to raise questions and objections under the treaty. Other variables, like the velocity or cross-section of the target, should not be included because they are less monitorable and not under the signator's control.

Other indices have been proposed to measure the effectiveness of a radar/interceptor combination. Dern et al. (1976) presented an analysis supporting an index for a radar-interceptor-missile system, PAV³ where V is the average velocity of a single interceptor.

Importances in the Lanchester theory of combat

The next three examples involve the Lanchester theory of combat. Lanchester models typically use differential equations and so assume a continuously variable quantity of forces on both sides.

Example 6. Numbers versus quality in aimed fire

Suppose two sides X and Y have homogeneous forces x(t) and y(t) that destroy each other according to the equations:

dx(t)/dt = ‑ ay(t), dy(t)/dt = ‑ bx(t).

These would be appropriate for aimed fire, that is, for a context where the two sides target specific forces on the other side rather than send a barrage. The constants a and b are the fire-effectiveness per weapon of X and Y respectively. This involves the rate of fire and proportion of shots that destroy the target.

Lanchester's original 1914 paper proved that the quantity bx²(t) ‑ ay²(t) is invariant as combat progresses, a proposition known as "Lanchester's Square Law." The Square Law has been cited in the non‑mathematical literature on procurement as support for buying quantity instead of high technology (Canby, 1984; Perry, 1984). It suggests that numbers are more important than quality, since it would take a fourfold increase in quality to match a doubling of the numbers. What happens when we compare quality and numbers using the Aumann‑Shapley value?

The first step is to decide on a sensible benefit function. How should we gauge the benefit of having certain forces left after the battle? Should we take simply the size of the surviving force, or should we include their quality as well? If we assume that the forces have no value in themselves, are inanimate objects, we might use for X some increasing function of ax² (the quality and size-squared of X's weapons). If remaining weapons are employed elsewhere as the only weapons in a further battle, this function is the one that determines X's success. Somewhat arbitrarily, we will then take simply ax². Thus given the Square Law, the benefit to X when the two sides possess x(0) and y(0), is bx²(0) ‑ ay²(0) if this is positive (if X would win) and zero otherwise.

Applying the Aumann‑Shapley method, we calculate that the total importance to X of X's firepower effectiveness is ax²(0)/3, and of X's numbers is 2ax²(0)/3. The importance (to X) of Y's firepower effectiveness is ‑by²(0)/3, and of Y's numbers ‑2by²(0). Thus, taken in total, numbers are twice as important as quality both for one's own and for the adversary's weapons.

4. Other Approaches to Measuring Importance of Weapons

Many existing models define a measure, and can be compared with the one developed here. They can be grouped into several categories.

1) The Additive Features Approach

The additive features approach rates each weapon based on its physical attributes, then adds the numbers to get a total score for each side. The two sides' scores are typically compared to see if there is an overall "balance," or for a specific type of weapon to warn of a "gap."

A sophisticated example is the Armored Division Equivalents (ADE) method of the U.S. Army Concepts Analysis Agency, 1974, 1979; Mako, 1983), also called the WEI/WUV ("wee‑wuv") scoring system. The basic method was used by US government agencies to report the military state of affairs vis-a-vis the Warsaw Pact (e.g., Department of Defense, 1988, p.31). The original system, WEI/WUV I, divides conventional US Army weapons into nine categories: tanks, armed helicopters, mortars, etc. Each category score depends on several "performance factors" that such a weapon should have, such as firepower, mobility, or survivability (Figure 1). Each of these is analyzed into objective physical features, or "performance characteristics," which are measured, normalized for a standard weapon of that type, weighted for their importance as a contributor to that performance factor, then added to get the factor score. For undesirable features, like presented-target area, the inverse is used. The scores on the factors are then weighted for importance and added to derive the Weapons Effectiveness Index (WEI) for the particular weapon. The various weights are estimated by a panel of experts, and differ depending whether the weapon is used offensively or defensively. The WEI values are multiplied by further weights portraying the importance of the category (of tanks, helicopters, etc.) and totalled for all forces to get a Weighted Unit Value (WUV). The WUV is normalized by dividing by 48,743 (in the defensive case), the nominal total for an American armoured division. This gives the Armored Division Equivalent (ADE) score, a measure of overall strength.

This approach has the advantage that the scores are based partly on objective technical data, but it can be criticized for the subjectivity involved in judging the weights and in selecting the features to include. There is also a deeper structural question, however, involving whether one should combine values by addition. Adding assumes independence. It assumes that an additional weapon makes the same contribution whatever was there before, but weapons or weapons' features are generally not independent. Some weapons are synergistic, such as the command centre and one leg of the triad of Example 2, and others are largely redundant, such as one leg in relation to another. If a weapon's features are combined by addition, a high enough value on one feature can appear to compensate for poor scores on the others, in contrast to reality where both features are absolutely necessary. The author estimated the WEI score of his all‑terrain bicycle, regarding it as an Armoured Personnel Carrier (APC). (Although the normalizing values are kept secret, some can be estimated from known data about the standard weapon, the M113A1, and most others are irrelevant since the bicycle usually scored near zero.) On two features it excelled: presented‑target area and gross weight, and in fact these contributed so much (since when normalized by the measure of the standard APC the bike's normalized value of 1/(gross weight) is about 60 instead of the typical 1) that they compensated for low troop capacity, no armour or firepower, etc., and the bicycle's Weapons Effectiveness Index was double that of the Swiss MOWAG 3M1, the best APC in the report.

The source of the problem is that weapons or their features usually are synergistic or partly redundant. A bicycle lacks certain features necessary for the task, and no degree of superiority on the others can compensate. A defender of the WEI/WUV method might counter that every approach has its range of valid application, that a bicycle is not an APC and should not be scored as one. A report on the "conventional balance" in Europe by the Congressional Budget Office (1988) tries to answer this criticism in the larger context of a composition of an overall forces rather than the design of a single weapon. The charge was that the method may implicitly call for a very unbalanced mix of weapons,

Finally the WEI/WUV method assumes that the added benefit of additional weapons is linear ‑‑ that is more weapons of any kind continue to provide the same additional capability as the first such weapon. This assumption is called 'constant marginal utility' in economic jargon and ignores the fact that, beyond a certain point, additional weapons of one kind might be redundant and therefore of no added utility. For this reason WEI/WUV scores should not be used by themselves to determine the optimal mix of weapons in a division. . . . Rather, the scores should be used to suggest how one mix of weapons deemed plausible by military experts might perform against another plausible mix.

This response has considerable validity, but the writers do not confront the circularity that remains: military experts have chosen to mix weapons in certain proportions, whereas WEI/WUV scores suggest that other mixes would do better. The ideal mix according to WEI/WUV may be recognized as grossly implausible, but smaller shifts in that direction would still be rated as better. Should we then believe the WEI/WUV score?

A later version, WEI/WUV III, prevents one source of the problem of the bicycle, the domination of the WEI score by a few characteristics. It limits the maximum contribution any characteristic can make. This mitigates the counterexample but does not eliminate it: the bicycle scores 56% to 74% of current APCs indicating that the same job could be done with a few more bicycles. The revision does not correct the source of the problem and weapons may well be ranked incorrectly even though they are too similar to make the error blatant.

One approach that deals with synergy is the IDAGAM and COMBAT computer models of battle, devised at the Institute for Defense Analyses (Anderson and Miercort, 1989). The programs group weapons into classes, according to whether weapons of one class require weapons of another class for their protection. In case a side's forces are unbalanced, in the sense that the necessary protecting weapons are lacking, the program COMBAT regards some of the weapons as ineligible to participate in the battle, and so reduces its estimate of the side's effectiveness. Although this approach appears somewhat too dichotomous in the way it portrays protection, it seems an interesting practical solution.

Example 7. Balancing weights versus importance measures.

These criticisms apply to other methodologies that add up values, e.g., measures of the strategic "balance" of nuclear weapons such as Equivalent Weapons discussed by Bennett (1980.) A general problem with military balance analyses is that they neglect the context and dynamics in which the war might be fought. There may be a first-strike advantage as with strategic missiles, or the opposite, a relative advantage to being the second mover, if each side has adopted defensive positions and is hoping the other will leave its trenches. (The WEI‑WUV methodology compensates for this by calculating both an offensive and a defensive value.) Another difficulty is that during the course of a war one side may switch from defense to counterattack, a question is also addressed by Anderson and Miercort.

The metaphor of a balance is innately misleading. Nuclear systems, for example, do not balance off against each other: matching the other side's first-strike capable weapons by adding similar weapons of one's own can make the situation more unstable, not more balanced. In our method, these contextual factors affect the importance through the benefit function b. A mutual first‑strike advantage, for example, would induce a higher probability of war and be included in the model through lower values of the benefit function. The next example asks the conceptual question: If the notion of a balance were true in a hypothetical situation -- if that was how the war outcome would be determined -- how would the numbers that determine the balance relate to the importances? Would they be identical? The answer is generally not.

Suppose side X has m weapons with associated weights x₁, . . . x_m and Y has n weapons with weights y₁, . . ., y_n. The meaning of the "weights" here is that they determine whether the system is balanced. Looking at the situation from X's viewpoint we will assign b(S) = 1 if the sum of the weights of X's weapons in S is at least as great as Y's. Then X's weapons "balance" Y's weapons. Otherwise b(S) = 0. To that balancing weights and importance values can be different, suppose that X has three weapons with weights 2, 1, and 1, and Y has an identical set of three. The importances to X are calculated to be 29/60, 18/60, 18/60, ‑29/60, ‑18/60 and ‑18/60. Thus X's weapon 1's importance (29/60) does not cancel y₂ and y₃'s combined importances (36/60), even though by assigned weight x₁ is enough to balance those two weapons.

2) The Antipotential‑potential Approach:

Several writers (see Spudich, n.d.; Howes and Thrall, 1973; Dare and James, 1971; Holter, 1971; Anderson, 1979, and the thorough summary by Taylor, 1983) have defined weapons' lethality recursively by relating X's values to the rate at which its weapons destroy Y's valued systems, the latter being defined in turn by the rate at which they destroy X's value. Anderson aptly named the idea the antipotential potential (APP). It leads to calculating eigenvalues of matrices involving the attrition rates of each weapon versus each of the opponent's.

In a clever and sensible way this approach makes values dependent on other existing weapons, both one's own and the adversary's, as the diagram shows. However, we see two problems, the first being that it can be used in only in contexts when attrition is linear and when the allocations of weapons to types of targets are fixed throughout the combat. The second difficulty is that APP values of the weapons are defined without reference to the value of the benefits to the possessors. In essence the approach sidesteps Principles 1 and 3 by not defining benefits. One consequence is that the APP values are not completely determined. The values can be found up to comparison of ratios within one side, but the ratio of weapons between sides is arbitrary. That is, if v₁, v₂, are one side's values and v₃, v₄ the other's, then for any positive k, v₁, v₂, kv₃, kv₄ will satisfy the conditions as well. The constant k then has to be chosen using some argument different than the notion that value is the ability to destroy value. The choice of k is critical for computer simulations such as IDAGAM and TACWAR (Hoeber, 1981) that define a side's total force as numbers times corresponding APP scores, and use this for force ratio comparisons. Different authors have suggested various ways of determining the parameter k, but no convincing argument has distinguished itself, and some have encountered paradoxes. Farrell (1975) notes that increasing the firepower of a Y‑side weapon sometimes causes the force ratio to move in X's favour, and Anderson (1975) has shown that dividing a weapon into two subtypes that are effectively identical, can greatly increase one's rated force ratio within the method of Howes and Thrall. Clearly neither of these changes reflects the benefits for a side in a conflict, but APP scores are not defined by benefits, and so this questionable behaviour can arise. For the importance measure presented here, Principle 3 guarantees that a desirable improvement makes a weapon more important.

An important step toward removing some of these objections is Robinson's analyses (1993), in which

3) The Optimal Allocation RuleApproach

Here one sets up a model of combat and finds the rule for optimally allocating weapons to the opponent's forces. Sometimes it will involve an index for the opponent's weapons that states which one should be given priority, and this constitutes a measure of their importance. Concepts like this often arise in models of strategic exchanges (Congressional Research Service, 1985) or missile defense (Karr, 1981), but the following example is based in Lanchester theory.

Example 8. The Isbell‑Marlow fire allocation problem.

Side X has two systems X₁ and X₂ in quantities x₁(t) and x₂(t), and Y has one system of amount y(t). X's systems attack Y's system, and Y can divide its fire between X₁ and X₂. It allocates its fire in proportions c and 1‑c, and unlike Example 6, Y controls the allocation can change it over time. The equations of attrition are:

dx₁/dt = ‑ca₁y, dx₂/dt = ‑(1‑c)a₂y, dy/dt = ‑b₁x₁ ‑ b₂x₂.

The combat ends when one side has no forces left. Which side will win depends on the initial conditions and the parameters. Here we will look only at the case in which Y can win, partly because the other is more complicated and also because we prefer to avoid making assumptions about Y's preference for different mixes of X₁ and X₂ remaining after a defeat of Y. (The condition for Y to win is a₁b₁y(0)² > (b₁x₁(0)+b₂x₂(0))² + (a₁b₁/a₂b₂ ‑ 1) (b₂x₂(0))². Since the Aumann‑Shapley measure looks at only subsets with the same proportions as the whole, the condition holds for all the subsets that it considers.) Taylor (1973) showed that to maximize Y's remaining weapons, Y should fire first at whichever of X's weapons has a higher value of a_ib_i. When that weapon is gone, X switch to the other. He suggested measuring the value of X_i to Y by a_ib_i, on these grounds, that it was the decision variable. It is a combination of the threat from X_i and Y's ability to counter it, and the rule is the simple one: fire at the more valuable weapon first.

To apply the present method, we calculate Y's payoff, based on Taylor's generalized form of Lanchester's square law which states that if c is constant from time t₁ to t₂, then

u²(t₁) ‑ u²(t₂) = [ca₁b₁ + (1‑c)a₂b₂] [y²(t₁)‑y²(t₂)], where u(t) = b₁x₁(t) + b₂x₂(t).

For simplicity and consistency with Example 3 we will define the benefit b(S) of an initial set S of weapons as the square of the number of the weapons Y has left. Effectiveness is not an issue here since we are not evaluating its importance and in any case it is different for the two targets. Assuming from now on that a₁b₁ > a₂b₂ (meaning that by definition, X₁ are the weapons Y shoots at first), we get the outcome of the battle to yield benefit

b[y(0), x₁(0), x₂(0)] = y(0)² ‑ x₁(0)² b₁/a₁ ‑ 2x₁(0)x₂(0)b₂/a₁ ‑ x₂(0)² b₂/a₂.

The resulting Aumann‑Shapley importances for the totals of each type of weapon are

Y: y(0)²

X₁: ‑x₁(0)² b₁/a₁ ‑ x₁(0)x₂(0)b₂/a₁

X₂: ‑x₂(0)² b₂/a₂ ‑ x₁(0)x₂(0)b₂/a₁

or, for importance measured per weapon,

Y: y(0)

X₁: ‑x₁(0) b₁/a₁ ‑ x₂(0)b₂/a₁

X₂: ‑x₂(0) b₂/a₂ ‑ x₁(0)b₂/a₁.

The formulae above are too complicated to comprehend fully but we can extract some of their meaning by noting which factors each includes. Weapon X₁, for example, has a (negative) importance to Y due to x₁, a₁, b₁, x₂, and b₂, but a₂ or y have no effect on its importance. One can see why each factor included should have an influence. Importance of an individual X₁ weapon to Y is aggravated by the number x₁ of such weapons in all, since under the aimed-fire hypothesis, the other X₁ weapons divert Y's fire from that particular item. Naturally, an X₁ weapon is worse for Y as a consequence of its firepower b₁ and its invulnerability 1/a₁. The number x₂ and effectiveness b₂ of X₂ weapons matter because they are firing on Y while Y is attacking X₁ and are lowering y, thus promoting the survival of an X₁ item and consequently harming Y through X₁ weapons. However X₂'s invulnerability 1/a₂ should not matter because Y is not firing on X₂ while under attack by X₁, so this invulnerability play a role only when Y turns to attack X₂. Weapons in X₂ are in a different position, however, and indeed according to the formulae, the invulnerability of X₁ weapons affects the importance to Y of X₂.

Taylor's measures, a₁b₁ and a₂b₂, are different from ours. They should be since he is talking about values to Y for attacking the weapons, not the latters' values per se. His measure should involve only the threat they pose, whereas ours includes not how much Y is able to counter it.

The strong point of the allocation rule approach is that it ties into a well‑defined decision. A shortcoming is the narrow context in which it applies. In Example 5, Y had only one weapon. However if two weapons systems faced two others, and one were especially good against one and the other especially good against the other, no single set of numbers assigned to the adversary's weapons could represent their value.

4) The credit apportioning method

Here one must know the outcomes and the approach tells how to allocate the credit among weapons. The theory in this paper belongs here. It would be more satisfying if we could state in a phrase what these importance numbers mean, beyond that they are the ones that satisfy the axioms. Certain interpretations of the numbers that were available in the original application of the axioms are implausible in the weapons context. Shapley's theory assigned values to people, players of a game, and the values showed players' a priori expectation from participating in the game. When we turn from evaluating people to non‑goal‑seeking objects, the axioms of the Shapley value still seem to apply to the undefined idea of "importance," but the interpretation of the numbers give us nonsense: which type of weapon an individual weapon should prefer to be. Fortunately, Roth and Verrecchia (1979) have suggested an alternative interpretation of Shapley values when they are applied to cost allocation problems, that the value gives the expected benefits to an item's program manager who is about to enter bargaining over a budget. This is plausible for weapons all owned by one side, but it is difficult to accept when the adversary's weapons are involved as well as in Example 2.

In the end the importance of the theory is the structure it sets up, its potential for clarifying concepts. The numbers are interesting but not the goal. Most other approaches to weapons importances come up with numbers that guide a specific decision or expectation, what to procure, how to allocate, who will win. However the present theory has suggested reasons to be sceptical about their internal validity. Its aim is to straighten out our thinking, to avoid logical fallacies, and clarify the meaning of other theories.

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ADE	6 Armoured Personnel Carriers	.30 Firepower	.85 Weapon firepower	.42 Cyclic rate x Max eff rate fire
				.35 Max eff rnge x Muzzle energy
				.23 Dependability
			.15 Portholes
		.30 Mobility	.20 1/Ground pressure .12 Step traversing .12 Trench spanning .12 Water crossing .12 Road speed .30 Mobility .08 Ground clearance .08 Slope climbing .04 1/Gross weight .04 Cruising range .04 Horsepower per ton .04 1/Length per track
		.15 Survivability	.35 Front armour thickness .22 Side armour thickness .15 Survivability .20 1/Presented‑target area .17 Overhd armour thickness .06 Belly armour thickness
		.25 Troop capacity
	1.2 Portable Small Arms...
	55 Tanks...
	36 Armoured Recon Vehicles...
	46 Antitank Weapons...
	85 Cannons/ Rockets...
	47 Mortars...
	44 Armed Helicopters ...
	28 Anti‑ground support Air Defense

Figure 2. The calculation of Armored Division Equivalents according to WEI/WUV I (defensive). Detail is given only for APC's. Scores are assigned to the rightmost cells, and accumulated leftward using linear combinations. Features are normalized so that a typical weapon has value 1, then added according to the weights and numbers in the force. The final result in the leftmost cell would be the Weighted Unit Value.

[1]His index was a specification of any of the general equations that determine the effective range of a radar installation, and we can also look to the models behind these general formulas to measure its justification.

R_max = (P F A (t_s/S)/[4BkT₀F_n(E/N₀)] )^1/4

This equation applies to a radar that searches a volume of space by directing its beam at various angles. Here R_max is the maximum range in which the oncoming object is detected; t_s is the total scan time across volume of solid angle S (which will be set by the minimum dwell time necessary in each direction); P is the transmitting power output, F is the radar cross section of the approaching object, k is Boltzmann's constant, T₀ is the temperature and F_n is the receiver noise figure; A is the aperture size of the radar; E/N₀ is the minimum signal-to-noise energy ratio for satisfactory detection. All is taken to the fourth root since the square root law of distance applies as the radar signal moves to the object, and again as a small amount of it is reflected back.