Aggregation, Disaggregation, and the 3:1 Rule In Ground Combat

Paul K. Davis (pdavis@rand.org)

RAND and the RAND Graduate School of Policy Studies

I. Introduction

The purpose of this paper is to illustrate a number of basic principles about aggregation and disaggregation in combat modeling by working through the mathematics and phenomenology of a concrete example: ground combat taking place in a number of sectors and subsectors within a theater. Even this highly simplified example is sufficient to demonstrate the importance to combat modeling in this era of distributed simulation and model families of approaching aggregation and disaggregation with care (see also Davis, 1995). This implies a strong dose of theory and mathematics rather than the usual dash to programming. It also implies not relying solely upon intuition, because aggregation and disaggregation relationships are often much more complicated than original intuition would have it (Davis and Hillestad, 1993; Horrigan, 1992).

For simplicity and analytic tractability, I assume that combat at some level is dictated by the Lanchester square law.[1] I then discuss whether an aggregate law, Lanchester or otherwise, applies at the next level up. The answer is that "it depends." Discussing these issues for the simple case suggests broader principles involving strategies, command and control, and time scales.

A bonus of the discussion is an explanation of how the famous (or infamous) 3:1 rule does and does not apply at different levels of combat. This is particularly interesting in modern times, because American army forces are likely to be engaged in counteroffensive operations. Will they really need 3:1 force ratios to succeed?

II. The Microscopic Model: Lanchester Square Law On Combat Sectors

A. Definitions and Structure

For the sake of analytic tractability, let us assume that ground combat occurs in independent sectors that may be curved, but which do not cross each other or affect one another directly. The attacker and defender in sector i are characterized by scores Ai and Di. There are N sectors of width L over a total frontage W. The attacker and defender have reserves, which constitute fractions fa and fd of their total capabilities A and D. Figure 1a depicts this schematically for a notional theater with rough terrain and isolated lines of communications (LOCs). Although there are some connections with minor roads, flanking operations can be prevented with small forces and the major battles are conducted on the principal LOCs, the "sectors." Figure 1b shows a more classical piston-model representation of a theater with contiguous sectors. In such a case (e.g., the old NATO Central Region), the simple treatment requires the assumption that both sides fall back as necessary to avoid exposing flanks, thereby keeping the sectors relatively independent.[2]

In what follows we consider a number of special cases of increasing complexity. The initial examples imagine very simple static battles in which the forces in given sectors "slug it out." Later examples consider concentration and intra-battle reinforcement and maneuver. Although we could add a movement model and show forward lines of troops (FLOTs) advancing over time, it would add nothing to the discussion.

B. The Lanchester Square Law

Let us now assume that the "detailed" model of combat is given by the Lanchester square law for a given sector of combat. On each sector i of intense combat, attrition of attacker and defender forces is given by:


where the coefficients Ka and Kd are constants reflecting the lethality of shooters and the vulnerability of targets. For sectors with less than intense combat, we assume that the loss rates are smaller by a factor of m.

It is often more revealing to work with the relative or fractional loss rates defined below:


A particularly useful variable is the ratio of relative or fractional loss rates RLR; in the ith sector. If F is force ratio, A/D, then


The ratio of loss rates is a measure of who is winning the slugfest. At a value of 1, both sides forces are shrinking at the same rate and one can say that the battle is at the break-even point. The attacker, of course, seeks an RLR much less than 1. This expression illustrates why (1) is called the Lanchester square law: RLR varies as the inverse square of the force ratio.

C. Deriving Aggregate Expressions

The challenge in the following examples is to derive closed expressions for the aggregate variables A and D, that is, expressions in which dA/dt and dD/dt are functions of A and D, but not microscopic variables such as the sector-level force strengths. These expressions may also contain various constants that depend on Ka,, Kd, and on various averages over microscopic phenomena.

In general we can write


which define the functions Qd and Qa in terms of the sum over sector-level attritions. For the case of the Lanchester square law, and assuming only two levels of intensity, it follows that


where Mi is 1 for intense sectors and m for other sectors.

E. Special Case: the 3:1 Rule

The famous 3:1 rule in ground combat is represented in Lanchester equations by requiring that the ratio of loss rates be 1 at a force ratio of 3:1. That is, a force ratio of 3 implies a break-even situation. This requires the ratio of kill coefficients to be 9. The basis of the 3:1 law is the notion that the defender has a substantial advantage (roughly a factor of 3)--assuming prepared positions and good defensive terrain, which reduce his vulnerability and increase the vulnerability of the attacker (e.g., by channelization) (Dupuy, 1987). The square law is assuredly not a statement of general truth. For example, it does not apply to meeting engagements or mobile warfare more generally. Nonetheless, it has been so widely discussed that it is worth treating. Later, we shall consider cases where the defender has no such advantage.

III. Deriving Aggregate Models For Simple Battles Without Reinforcement or Intra-Battle Maneuver

A. Case One: Uniform Distributions of Forces

1. General Lanchester Square Law (No 3:1 Rule)

For our first case, let us assume no intra-battle reinforcement or maneuver. In addition, let us assume that the sides have spread their forward forces uniformly across the sectors and fight intensely on all sectors. Now let us develop an equation for the aggregate level. In this first case the battle in each sector is the same and it is rigorously true that

and similarly for defender attrition. To have an aggregate model, however, we need to express the variables on the right side in terms of aggregate variables (i.e., A and D, not sector-level quantities). In this case, we have simply:

Applying (4) and the symmetry between A and D we then have

In this particular case the functional form of the aggregate model happens to be a Lanchester square law also--i.e., it is identical in form to the detailed model. Note, however, that the coefficients of the model depend not only on the coefficients of the detailed model (Kd and Ka), but also some gross features of the microscopic problem: the reserve fractions. If one side has a larger fraction of its forces on line than the other, it has an aggregate advantage, because results depend on the number of shooters.

2. Special Case of the 3:1 Rule

In the special case of the 3:1 rule (8) becomes

B. Case Two: Concentration Effects

1. General Lanchester-Square Assumptions (No 3:1 Rule)

There have been historical instances of uniform attacks across a front, but attackers usually concentrate forces on some sectors while fighting a holding battle on others. Indeed, concentration of force is a central feature of maneuver warfare.

Assume that the attacker concentrates on Nmain of the N sectors, conducting low-intensity feints on the other sectors to tie down defenders without suffering excessive attrition himself. On the main sectors, assume the Lanchester square law as before. That is, Equations (1)-(5) apply again except that on the "other" (non-main) sectors, the intensity is reduced by the factor m (m<1) so that the kill coefficients on those sectors are mKa and mKd, respectively.[3] The assumption that m<1 is crucial; indeed, if it were not true, there would be no advantage to concentrating force.

As before, let us now see about deriving aggregate equations. Since there are two kinds of sectors, main and other, we obtain the equations below.

The first equation in (9) is straightforward except that we have had to reorder the numbering of sectors so that the main sectors are labeled 1, 2,...Nmain whether or not they are contiguous. The next equation introduces the concept of sector concentration as a way to express sector force levels in terms of aggregate force levels. If a fraction (1-fa) of the attacker's forces are forward, then in the no-concentration case, each sector would have (1-fa)A/N attacker forces and (1-fd)D/N defender forces. If we represent the number of forces on the main sectors as multiples pa and pd of the no-concentration levels (which cannot exceed N/Nmin), then the number of attacker and defender forces on the other sectors are multiples [1-(Nmain/N)pa] and [1-(Nmain/N)pd] of the no-concentration levels, respectively.

It follows that we have again derived valid aggregate equations, even though the distribution of forces is not uniform.[4]

Once again, the aggregate equations happen to have the same form as the detailed equations (Lanchester square). The coefficients are defined in terms of the coefficients of the detailed model and some gross features of the microscopic problem--in this case, the reserve fractions, the reduced-intensity factor m, the concentration or loading factors, and the fraction of sectors on which concentration occurs.

Although it is plausible that we could estimate reserve fractions from doctrine, how are we going to estimate the concentration factors and the fraction of sectors on which concentration occurs? We have an aggregate equation, but we do not know how to evaluate or even estimate the coefficients. Furthermore, it is not apparent that there will be good "representative values" of the coefficients because there may be a great deal of variation across battles. Different attacking generals may use different strategies; some defenders may be more clever about anticipating the points of attack; and so on.

From the mathematics alone, then, we can justify an aggregate formulation, but in the absence of more information--and, in particular, a deeper understanding of the military phenomenology--we have no basis for believing that the coefficients are reliably predictable.

To illustrate this point, suppose we derive the corresponding expression for ratio of loss rates. As before RLR is defined as ALR/DLR, where ALR and DLR are the fractional loss rates of the attacker and defender. It follows that

If m<<1, then we have a simpler expression:

from which it follows that

Another expression for RLR in this case follows from physical considerations. If the only attrition is in main sectors, then the ratio of loss rates for the theater is the exchange ratio (dA/dt/dD/dt) in the main sectors divided by F, and the exchange ratio in the main sectors is (Kd/Ka)/Fmain. Thus

Comparing (12) and (14) we see a simple expression for Fmain:

What do the above equations imply about the rule at the aggregate level? Or, to put it differently, what is the force ratio F* at which RLR is 1? Solving for F* by setting F=F* at RLR=1 in (14) we have

It is convenient to write down still other relationships, because in trying to estimate what values of F* are "reasonable" for a range of cases, different people find different representations of the same problem easier to work with. In particular, it is useful to use different combinations of fa, fd, pd, Fother, and Nmain/N as independent variables.

Accordingly, consider that the attacker's force strength is the strength on main sectors plus the force on other sectors plus the force in reserves. However, the force strengths on the main sectors and other sectors can be expressed as main-sector and other-sector force ratios times the defender strengths. With this and some algebra we can derive an expression for Fmain:

If we require that F=F* and remember from (14) that F is approximately (Kd/Ka)/Fmain, we have for m<<1, a quadratic equation for F* that can be solved analytically.

Only the positive root is physically meaningful, since F* must be positive and the negative roots are negative since the square root is always larger than the first term.

From (15), (16), and (19) we have three expressions for F* if m<<1.

The first expression makes it clear that if the main-sector force ratio can be made high, then the break-even point can be made low, as one would expect intuitively. Figure 2 shows the results graphically for three different assumptions about Kd/Ka. The 3:1, 2:1 and 1:1 rules correspond to Kd/Ka=9,4, and 1, respectively. The last of the expressions in (20) is the most useful because it poses the issue in terms of strategy variables. Thus, we can use military reasoning rather than approach the problem as one of pure mathematics.

The way to view the factors in the third expression is perhaps as follows. An attacker must decide how to concentrate his force. To do this he probably estimates pd (it will be 1 if his concentration is a surprise and the defender has not preferentially defended the main sectors) and fd (which might be about 1/4 to 1/3 if the defender has a forward defense). He may establish a minimum value of Fother, taking the view that any lower value would endanger his operation by making counterattacks too feasible.[5] He may specify a minimum value of fa, below which he would be endangering the operation by having too few reserves. Finally, he can derive the value of Nmain/N that will achieve the break-even force ratio (Davis, 1990). He may choose, of course, to concentrate further to win decisively on the main sectors. However, if Nmain/N is too small to be strategically significant--i.e., if a breakthrough on so small a portion of the front would leave too much of the defender's army unscathed and too much of his territory unconquered, then he could reduce further the values of Fother and fa, and reconsider his estimates of pd and fd, which might initially have been conservative. There is no clear-cut optimizing algorithm, because there is no utility function to optimize.[6]

2. Illustrative Results Assuming the 3:1 Rule

Canonical View. Figure 3 shows a relatively canonical view of the problem assuming the 3:1 rule. It assumes the defender has not anticipated the attack and that the defender and attacker have 1/3 and 1/4 in reserve, respectively. We see that the break-even force ratio depends on the fraction of the sectors on which the attacker concentrates (and on the force ratio maintained on other sectors) in a straightforward way.

We see immediately that

Figure 3 is only one of many possible charts that could be drawn varying different combinations of the parameters. Table 1 shows a range of cases taking nominal, defender-conservative, and two attacker-conservative perspectives. The nominal and defender-conservative cases take the view that the main attack might be on as little as 15%-20% of the front. Further, the attacker might tolerate a 2:1 ratio against him in other sectors (Fother=0.5), especially if he could be confident that the defender was not particularly mobile and aggressive. The first attacker-conservative case considers a somewhat larger main frontage, larger reserves, and a less adverse ratio on the other sectors. The last estimate is not unreasonably conservative either; in it the attacker reasons that the defender would surely do some anticipatory counter concentration on the basis of intelligence on massive troop movements. Even a modest counter concentration (a pd value of 1.5) changes the break-even point substantially.
                             Table 1


Representative Bounding Cases Assuming a 3:1 Defender Advantage
------------------------------------------------------------------------
pd      fd      fa      Nmain/N   Fo    F*     Description
------------------------------------------------------------------------
1       0.33    0.33    0.20     0.66   1.6    Nominal
                                   7
1       0,33    0.17    0.15     0.5    1.2    Defense conservative
1       0.33    0.25    0.3      0.67   1.8    Attacker conservative (1)
1.5     0.33    0.25    0.3      0.67   2.1    Attacker conservative (2)
------------------------------------------------------------------------
Summary on the 3:1 Rule and Aggregation. In summary, if the 3:1 rule applies at the sector level, which assumes the defender has major advantages from terrain and preparations, then the defender can tolerate only a much smaller aggregate, theater-level, force ratio--something nominally on the order of 1.5. The attacker will seek a larger number, perhaps out of concern that the defender will observe his large-scale maneuvers and do at least some counter concentration before battle commences. From the attacker's viewpoint, a break-even force ratio of about 2 might seem reasonable. To win decisively, an even larger force ratio might be needed.[7] The requirement would, however, be much lower if the attacker were qualitatively much more capable than the defender (e.g., better morale, training, support forces, air forces, and experience with maneuver). This was the case for the U.S. attacking Iraqi forces in 1991.

This is as far as we can go in the abstract; real-world details matter. For example, in Operation Desert Storm the United States had total information dominance; U.S. generals knew with certainty that the Iraqis had not detected the concentration and mounted a counter concentration (pd=1). Nor could they have done so readily, because of U.S. air supremacy and the lethality of U.S. air forces. Under these circumstances, even a much smaller U.S.-led coalition army could have safely concentrated on a narrow front, broken through, and begun encircling operations to "bag" defender forces.

C. Alternatives to the 3:1 Rule

Most of the equations derived above are expressed in terms of Kd/Ka, rather than assuming the 3:1 rule. What happens, then, if we do not assume the rule? Suppose, for example, that battle was going to be conducted in relatively open terrain and a great deal of tactical maneuver. There might be some advantage to the defender, but not much. Indeed, the attacker might have the advantage by virtue of having the initiative and associated tactical surprise and momentum. In any case, Figure 4 illustrates the consequences of assuming a 1:1 rule rather than a 3:1 rule. This curve is particularly important for the United States in thinking about maneuver warfare.

To illustrate how this can be used, suppose that the attacker wanted to concentrate on at least 30% of the frontage and to maintain a force ratio of .83:1 on "other" sectors. The break-even force ratio would then be about 0.8. To have a decisive victory with a ratio of loss rates of 4:1 favoring the attacker, the attacker would therefore need an overall ratio of about 1.6:1. To have a decisive victory with minimum casualties he might seek a ratio of loss rates of, say, 9:1, which would mean an overall force ratio of about 3:1 or so. Although estimates of force ratio should reflect qualitative differences in fighting capability, not just equipment counts, this conclusion should nonetheless be sobering for those estimating the capabilities the United States might need in future major regional contingencies.

IV. Generalizing: Effects On Aggregation of Reinforcement and Maneuver

A. Reinforcement and Redeployment

A key assumption of the previous cases is that only the forces initially present in the main sectors conduct the battle. But what about reinforcement and redeployment? What if the sides commit their reserves? What if the defender redeploys forces from other sectors? What if the attacker also redeploys forces?

One way to investigate such issues is to develop a simulation model. For current purposes, however, let us instead make some points more qualitatively by looking at the analytics. Repeating (20), we have again that

Now let us account for reinforcement and redeployment as follows. Assume that:

Figure 5 shows the consequences of such assumptions graphically for an aggressive-attacker case. Although real-world changes would be more complex dynamically (e.g., attrition would affect force levels over time, reserves might be initially committed at a higher rate, and the defender might concentrate faster than indicated), this approximate treatment illustrates the basic features. In the example, both sides commit their reserves in time T1 as shown, and the defender proceeds to counter concentrate over time (the x axis only goes to 2 T1, however, so some of the counter concentration is incomplete).

B. Effects of Time Scale

As pd(t), fd(t) and fa(t) change, so also does F*(t). If we use the time-dependent versions of these variables in (20) we can generate the corresponding illustrative plot.

The conclusion here is that if the battle is short relative to T1, then the break-even ratio is essentially the same as in the earlier section that ignored intra-battle reinforcement and maneuver (i.e., around 1.5 or so). However, if the battle is not so intense (lower attrition rates and longer duration), then the average value of the break-even ratio rises sharply, reflecting the fact that much of the battle will be fought under circumstances much less congenial to the attacker than intended. Indeed, if the battle lasts long enough, the sides will concentrate all their forces on the main sectors and the value of the original concentration will be greatly reduced--unless, of course, the attacker reconcentrates and is able always to be ahead of the defender in that respect.

C. Effects On Aggregation Scale: Physical Scale and Command Structure

These observations are reasonable in the abstract, but how do they apply to real-world combat? Roughly speaking, the key point is that forces within isolated sectors with independent generals may be able to call in theater reserves quickly, but they will not be able to draw upon forces from other sectors. That is, in most theater conflicts, T2 is large compared with T1, probably much larger even than shown (e.g., T2 could easily be 10 times greater than T1 because of factors such as terrain, logistics, disagreements between sector commanders, or confused intelligence). As a result, a break-even 3:1 ratio at the sector level translates into something more like half that at the theater level.

By contrast, if one were to try to do the same analysis with subsectors, one would conclude that intra-subsector maneuver would probably happen quickly relative to the duration of the sector's battle. Not necessarily, but plausibly. Thus, if the Lanchester equation and 3:1 rule applied at the subsector level (e.g., battalion-battle), they would probably apply also at the sector level if the higher level defensive command could reallocate forces within its control on a time scale short compared with the duration of the lower-level battle. An important subtlety here is that the relevant duration of lower-level battle includes the times associated with movement, reconnaissance, engagement and disengagement, and full-out battle. A given full-out battle may be remarkably short in modern warfare (e.g., ten minutes). If the defender is good at maneuver, however, and able to engage and disengage readily, he can drag out the duration of battle to improve his opportunity to "reequilibrate" forces. This ability to control tempo and the point of key battles by maneuver has even more leverage than that of prepared defenses, which helps to explain why field officers have long been much less enamored of static defenses than analysts.

The conclusion here is that strategy variables (e.g., Nmain/N) and relative time scales determine the aggregation coefficients. These vary a lot from one level of combat to another.

IV. Implications for Temporary Disaggregation In Simulations, Including Distributed Interactive Simulation

One of the motivations for this paper was to illuminate a problem arising in distributed simulation, the problem of connecting models of different resolution meaningfully. Crossing levels of resolution is notoriously difficult (see Davis and Hillestad, 1992, 1993), but it is even more difficult to do so frequently in the course of a simulation--sometimes disaggregating and then reaggregating as when, for example, an aggregate object must do battle with an item-level object, after which the war proceeds.

Is it reasonable to do such disaggregation and reaggregation? Basing on the forgoing, a key criterion would appear to be whether the real-world aggregate-level object would "reequilibrate" after one of its components had been in battle. If not, there could be important correlations from one battle to the next and the procedure of disaggregating, aggregating, and disaggregating would be improper. But if the reequilibration is realistic, then the procedure may be reasonable--although other kinds of errors can be introduced if, for example, the disaggregation procedure always assumes the same standard formation and tactics.

VI. Conclusions and Summary

This paper has demonstrated many of the challenges in aggregating and disaggregating descriptions of combat processes by working through an analytically tractable model that assumes the Lanchester square law for ground combat in an individual sector such as that controlled by a corps or, in relatively rough terrain, a division. Even though this assumption is certainly not rigorous, it is nonetheless useful for the purposes here and leads to the following conclusions.

A. Aggregating From a Sector Level

Given a Lanchester law at the individual sector level, there may or may not be a valid aggregate model at the theater level. If the attacker and defender apply forces uniformly across sectors and maintain constant reserve fractions, an aggregate model exists and is itself Lanchesterian. If the attacker concentrates forces on a fraction of the sectors, conducting mere holding actions on the others, an aggregate model with constant coefficients is still valid so long as there is no change in the reserve fractions or the allocation of forces across sectors. Again the model is Lanchesterian. In this case, however, the key coefficient governing the ratio of loss rates is a complex function of the attacker's strategy and the defender's anticipation of the attacker's strategy (a function of information and decisionmaking). If the break-even force ratio at the sector level is 3:1, then the force ratio at the theater level is about 1.5, 1.2 or 2.1 for canonical, defense-conservative, and attacker-conservative assumptions, respectively.

B. Effects of Intra-Battle Reinforcement

If in the course of battle the sides commit their reserves and redeploy forces from other sectors to the main sectors, there is no exact aggregate model with constant coefficients except in extreme cases. What matters are the ratios of several time scales: the duration of battle, the time to reinforce with theater reserves, and the time to redeploy from other sectors. The reinforcement and redeployment times depend not only on physical distances, roads, and movement rates, but also on intelligence, decision times, logistics, and the effects of air power. If, for example, the sector-level battle is intense enough or decisions slow enough, then the intra-battle maneuver and reinforcement will be too late and sector outcomes will depend on the initial sector-level force ratios, thereby favoring the attacker. By contrast, if the defender in a main sector can diagnose events quickly and control the pace of events, perhaps by virtue of multiple prepared lines or giving up space for time, then this will not be so and the value of the initial concentration will be less.

C. Break-even Force Ratios At Different Scales (Different Levels of Combat)

While the importance of the relative time scales may seem obvious, these scales are seldom discussed explicitly, even though it has been mysterious to many observers over the years why the 3:1 rule is applied at some levels of combat but not others. As discussed above, if the 3:1 rule is valid at a sector level, the corresponding rule at a super-sector (theater) level may be more like 1.5:1. On the other hand, it is plausible for a 3:1 rule to apply not only at the sector level, but at the subsector level as well. Thus, the same rule might apply to battalion- and division-level battles. The general principle is that if a 3:1 rule applies at a given level, then it will also apply reasonably well at the next higher level if the higher level's defensive resources can be reallocated in a much shorter time than the duration of lower-level battles (or if the attacker is unable to enforce concentration systematically). A subtlety here is that if defending forces can break off battles quickly, this increases the effective duration of the low-level battles, thereby allowing more time for "reequilibration." This gives "active" and "mobile" defense concepts advantages over purely static defenses, although static defenses can often exploit fortifications better.

D. Implications of Mobile Combat

In mobile warfare the defender has less advantage. In this case, the sector level break-even force ratio is 1:1 and the break-even force ratio at the aggregate level may be on the order of 0.8--that is, even an outnumbered side can win. The risks of doing so are considerable, however, because holding actions are more difficult. Battles may be more intense and their durations correspondingly shorter. As a result, concentration of force can be decisive--again, unless defending commanders are deft at breaking off battle when outnumbered and maneuvering quickly to reinforce troubled units. Such maneuver issues are especially important today, because the United States is more likely than not to be engaging in mobile warfare rather than a rigid prepared defense of a fixed line.

E. Disaggregation and Reaggregation Within Combat Simulation Runs

Based on the insights about aggregation relationships, it is possible to draw conclusions about temporary disaggregation in the course of a simulated battle (e.g., in a distributed simulation). By and large, disaggregating from a theater level in which the independent variables are total attacker and defender force levels is arbitrary and unnatural: it amounts to assuming a particular attack strategy for the entire campaign. Such an assumption cannot then be forgotten as one reaggregates, because in the real world theater-level strategies are highly correlated over time (i.e., if the main attack is through the Ardennes on D+1, the Ardennes is probably still a main sector on D+2). By contrast, it is not unreasonable to disaggregate temporarily from a sector level description to a representative subsector-level depiction, and then reaggregate, if the time scales are such that one would expect forces in the sector to "reequilibrate" before the next time period requiring a disaggregated description.

F. Generic Principles.

The purpose of the analysis was more to illustrate methods of aggregation and disaggregation than to work through the implications of the Lanchester square law. Among the more important principles illustrated are the following: A corollary of the last point is that in developing families of models, it may be better to start with more aggregate concepts and develop consistent disaggregated representations and only partial calibrations, than to attempt to work from the bottom up. This may be a radical concept to those wedded to bottom-up approaches. It is contrary to much current discussion, especially by some enthusiasts of distributed interactive simulation who happen to be more acquainted with training and distributed technology than with modeling.

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NOTES

  1. For extensive discussion of Lanchester equations, see Taylor (1980, 1983). Recent work (Hillestad, Owen and Blumenthal, 1993) illustrated how a Lanchester square law can--in simple cases--be a reasonable approximation of events. The authors began with an item-level simulation with individual shooters (e.g., tanks) and kill-per-shot probabilities dependent on range. They assumed flat, featureless, terrain. Even in this case, moving to and understanding the Lanchester representation was nontrivial and, in practice, was informed by theory and experimentation with the higher resolution simulation.
  2. This simple depiction ignores unit structure, treats capability by a scalar score, and does not treat any kind of terrain effects, defensive-preparation effects, movement, or maneuver effects explicitly. It ignores air forces and long-range artillery. Nonetheless, it illustrates important principles.
  3. In this simplified depiction we ignore the fact that the attacker should be able to avoid a high exchange ratio in the non-main sectors, perhaps depending primarily on artillery barrage to tie down forces. We also assume that the intensities and concentrations are the same, sector to sector, within the classes of main- and non-main sectors.
  4. It is common to hear the claim that an aggregate model of a process is only valid if events at the microscopic level are uniform. That is quite wrong, at this example illustrates. However, a sound aggregation must retain information about microscopic configuration. For dramatic examples of configuration effects in aggregation, see Horrigan (1991).
  5. Some of the principal reasons for maintaining a reserve force are "outside the model." At any level of combat, a side with no reserves is exceedingly vulnerable to a random penetration of his line. By constraining fa and fd to be non zero, perhaps on the order of 1/3, we are compensating realistically for inadequacies of the deterministic Lanchester equations.
  6. The Soviet army long used "correlation of force" calculations comparable to those discussed here to make operational decisions about concentration. See, e.g., Hines (1990) and RDA (1990).
  7. In the mid-to-late 1980s there was rancorous debate about the adequacy of NATO's conventional defense posture in the Central Region. The theater force ratio was variously estimated in the range 1.5 to 2.2. Based on the current analysis, one can see why there was a debate. Ultimately, the wartime balance of forces would have depended critically on NATO's response to warning indicators and on whether the Warsaw Pact had improved the readiness of its lowest-quality reserves before beginning mobilization per se (Davis, 1990).