Aggregation, Disaggregation, and the 3:1 Rule In Ground Combat
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
- Even if the Lanchester square law and 3:1 rule are exact at the sector
level, there is no unique analog at the aggregate level. The coefficients
(e.g., F*) depend on issues of choice, particularly Nmain/N.
- On the other hand, the attacker will probably need an overall force ratio
of at least 1.5--if one can argue that a successful attack will need to cover
at least, say, 15% of the frontage.
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:
- The defender commits his reserves to the main sectors at an even rate over
a period T1.
- The defender counter concentrates his entire force at an even rate over a
period T2 (T2>T1). That is, over a time T2 he increases the concentration
factor on main sectors at a constant rate until all his forces are on main
sectors.
- The attacker follows the defender, maintaining a constant force ratio on
non-main sectors. Thus, the attacker also commits his reserves to main sectors
over a period T1 and redeploys additional forces, eventually all of his
forces, to the main sectors over a period T2.
- We express T2 as a multiple of T1
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:
- Even approximate mathematical analysis can clarify aggregation and
disaggregation issues by suggesting functional forms and likely sensitivities.
- However, aggregation typically depends sensitively on assumptions outside
the detailed model, notably assumptions about higher level strategy,
command-control, maneuver, and time scales. These cannot generally be
determined in advance, making uncertainty analysis necessary at the aggregate
level.
- The often dominating role of these higher level factors is the reason that
aggregate models (even board games) can often be quite respectable without
being derived in detail from, or calibrated against, detailed models.
- Aggregation may also depend sensitively on other assumptions outside the
detailed model, assumptions so implicit as to be largely forgotten. The
"detailed" models may, for example, be deterministic because of implicitly
assuming tactics such as maintaining reserves that hedge against the
consequences of random events. These assumptions must be reflected as
constraints when aggregating or using automated methods such as neural nets or
mathematical programming to find "optimal" tactics.
- Temporary disaggregation within simulated campaigns may or may not be
reasonable, depending on the objectives of the simulation and, importantly,
the time scales involved. By and large, temporary disaggregation is defensible
if, in the real world, forces would "reequilibrate" at the aggregate level
between periods in which the simulation disaggregates. The reequilibration
concept is general, not restricted to ground-force maneuver. The
"reequilibration" may involve, e.g., alertness, allocation of fires,
redeployment of command and control assets, or maneuver of aircraft and ships.
- Validation of aggregation/disaggregation relationships should focus on the
treatment of strategy, command-control, constraints, time scales, and
uncertainties. It should not pivot around whether the aggregate model has been
fully calibrated against a detailed model, because in many cases such
calibration is impossible without mischievous assumptions. On the other hand,
experiments with detailed models can often reveal issues and sensitivities
that would be missed in even a moderately careful mathematical analysis.
Further, they may be a good basis for calibrating some parameters of the
aggregate model, even though other parameters are outside the model.
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.
Bibliography
Davis, Paul K. and
Richard Hillestad (eds.)(1992) Proceedings of Conference on Variable-Resolution
Modeling, Washington, D.C., 5-6 May 1992, CF-103-DARPA.
Davis, Paul K. and Richard J. Hillestad, "Families of Models That Cross
Levels of Resolution: Issues for Design, Calibration, and Management," in
Proceedings of the 1993 Winter Simulation Conference, December, 1993, Society
for Computer Simulation, San Diego, CA.
Davis, Paul K. (1990) "Central Region Stability in a Deep-Cuts Regime," in
Reiner Huber (ed.), Military Stability: Prerequisites and Analysis Requirements
for Conventional Stability in Europe, NOMOS-Verlagsgesselschaft, Baden-Baden,
Germany.
Davis, Paul K. (1995), "Distributed Interactive Simulation in the Evolution
of Warfare Modeling," to be published.
Dupuy, Trevor N. (1987), Understanding War, Paragon House
Publishers, New York.
Hillestad, Richard J, John G. Owen
and Donald Blumenthal (1993), Experiments in Variable-Resolution Combat
Modeling, RAND N-3631-DARPA, included in Davis and Hillestad (1992).
Hines, John (1990), "The Soviet Correlation of Forces
Method," in Huber (1990).
Horrigan, Timothy (1992), "Configurational Effects in
Attempts to Aggregate Combat Models," in Davis and Hillestad (1992).
Huber, Reiner (ed.) (1990), Military Stability:
Prerequisites and Analysis Requirements for Conventional Stability in Europe,
NOMOS-Verlagsgesselschaft, Baden-Baden, Germany.
RDA (1990), Battle Command Training Program (BCTP) Opposing
Force (OPFOR) Command and Staff Handbook, Book 2, Chapter 5, R&D
Associates/Logicon, Los Angeles, CA.
Taylor, James G. (1980), Force on Force Attrition
Modeling, Military Operations Research Society of America, Alexandria, VA
(originally published for the society by Ketron Inc.).
Taylor, James G. (1983), Lanchester Models of Warfare, Ketron Inc.,
Arlington, VA.
NOTES
- 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.
- 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.
- 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.
- 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).
- 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.
- 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).
- 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).
![]()
![]()
![]()
![]()