Lanchester Systems and the Lanchester Laws of Combat

F.W. Lanchester was among the first theorists to apply higher mathematics to warfare. In 1916, he began to analyze the aerial war that was taking place over Europe. His goal was nothing less than the Holy Grail of military analysts: a means of reliably and reasonably quickly predicting the outcome of military encounters (battles to you and me) given some basic information about the forces' status avant de combat.

To do this, he applied a set of methods which had been created (and, until then, mostly been used) to describe the growth of biological populations given certain environmental factors. He recognized that by reversing the state vector, i.e. making the rate of change of the population negative, the methods which offered insight into growing populations could be molded into a system of examining relative casualties in combat. Of the two sides (represented as populations), the first side whose population reached zero would be the loser (as long as the other side had any population - soldiers - left).

The 'Lanchester Equations', as they are commonly known today, are a system of Ordinary Differential Equations. They bear upon the problem of which side, of two, will remain standing after the end of a battle. Warning: I am a lousy mathematician, so I will be parroting some of the more conceptual stuff in this writeup. I came to the Lanchester Systems from the applied side of the fence, as a military geek. If you are curious about the math itself, avail yourself of some of the excellent mathematical folk here on E2; or if I have said something stupid below, /msg me and I can have it checked out.

There are two basic rules that the Lanchester systems produce that are useful to military analysts. They are known as the Lanchester Square Law and the Lanchester Linear Law. They each offer some insight into the outcome of a different type of combat activity. The Square law models an 'aimed fire' encounter, whereas the Linear law models an 'unaimed fire' situation.

'Aimed fire' describes a situation where the shooter is directly aiming at an enemy. If/when that enemy is destroyed, the shooter moves his fire to a new target. Thus, as targets are eliminated, the firepower of the shooters becomes more and more concentrated on those targets that remain.

'Unaimed fire' is (natch) the opposite. When undertaking unaimed fire, a shooter is firing upon an area in which the enemy is presumed to be. The basic difference between this and aimed fire is that even if a shooter destroys an enemy, his fire remains directed at the area as a whole; there is no corresponding concentration of firepower as enemy targets are hit. In fact, in many cases, the shooter won't even know when they've hit a target.

By way of further explaining the relevance of these concepts, I'll jump into the equations themselves.

Lanchester's Square Law (Aimed Fire)

The basic state system looks somewhat like this. Consider the following two equations:

Δx = -βy; x > 0
Δy = -αx; y > 0

Here we have stated that, for each 'iteration' of combat, the attrition of force x is equal to the size of force y at the beginning of the iteration, multiplied by force y's lethality coefficient (represented here as (negative)β). This assumes that force x has soldiers remaining at the start of this iteration. The second equation states the same for force y, with α used to represent the lethality coefficient of force x's units. This version assumes that neither side receives reinforcements.

The lethality coefficient is the probability that a single unit of a force will succeed in incapacitating (killing, wounding, etc.) a unit of the enemy. For example, if we take a day to be the iteration period (typical) then if force x's lethality coefficient is 0.01, then each unit of force x has a one percent (probability 0.01) of removing an enemy unit from the battle. In some circumstances (say, if we are describing weapon-on-weapon attacks like tanks) this is equivalent to the Pk.

While we can solve these two equations for time (the point at which one side defeats the other) we're not really concerned with that; rather, we'd like to know which side wins and how many units the winner retains at the finish. (We're gun geeks, remember?) So, we can remove the time component by finding y as a function of x, from which we get:

dy / dx = (α / β) (x / y)


β(y2 − y02) = α(x2 − x02)

…assuming non-negative values for x and y.

It is assumed that combat continues until one side's unit count reaches zero. If this is true, then at the end of combat (defining the number of units remaining as xf and yf respectively) then we can shortcut the iterations and declare that:

If:    βy02 ≤ αx02 then yf = 0     and     xf = √ (x02 − (β / α) y02)


If:    αx02 ≤ βy02 then xf = 0     and     yf = √ (y02 − (α / β) x02)

So, Custy old chap, I can hear you grumbling, what on earth does that awful chickenscrabble mean?

Ah. Okay. That's a good question. Let's look at the final pair of equations. Those first terms (immediately following the If: statements) apparently determine which way the battle will go! Thus, these terms are called the 'fighting strengths' of the two sides. If one side's fighting strength is greater at the start of combat, it will emerge victorious (yes, this does betray the overly deterministic nature of this whole exercise, but bite down and bear it).

Look at those terms carefully. In each one, we have a lethality coefficient, which can be considered raised to the first power (it doesn't have an exponent). The second variable, however, which describes the starting numbers, is squared!

This is our clue that as far as the Lanchester aimed-fire models go, numbers count more than effectiveness. Thus, it's always easier for a combatant to make up a deficiency in effectiveness with numbers than it is to compensate for lack of numbers with increased lethality.

This makes intuitive sense, as well. The square law means that at each iteration of combat, the side that began with a lower fighting strength (say, y) will lose more units than the other (x). This means that the number of units in y will drop more quickly; as each round finishes, there are more x guns on fewer y targets. Since each unit present has the same chance of eliminating a y target, if there are more x units than y units, some y units are therefore double-targeted (or a few y units are multiply-targeted) raising the chance that they will be eliminated even higher. In response, moreover, y's forces can only engage y numbers of x's forces - meaning that as the iterations continue, their return salvos will only become less and less effective.

This hypothetical model has been exhaustively tested against real-world historical warfare. One well-known model (by J. Engel in Operations Research, 1954) tested the battle of Iwo Jima for its compliance to the Lanchester Square Laws. Engel discovered that after determining the lethality coefficients for each side by mining the historical records of casualties, a hypothetical graph of casualties and remaining troops for each side over time matched the actual historical numbers fairly precisely. While interesting, this doesn't mean as much as it might; while the model fit nicely, in most cases we won't have the actual lethality coefficients available to us ahead of time! It is, of course, possible to arrive at decent approximations of such parameters through careful historical study and consistent analysis. Therefore, the model retains predictive power.

One caveat: the model only works on scales large enough to suppress variation due to generalship, terrain, luck, weather, etc. In other words, local variations of the model are quite likely; if, for example, one sector of the battlefield has an unusually competent officer corps on force x, then that area's casualty rate may vary wildly from the overall model. In such cases, the analyst must either widen the scope of the model until such disturbances can be homogenized, or separately iterate that local area's model. The danger of the latter is that extrapolating the impact the outcome of one local skirmish will have on the battle as a whole is a very iffy business. We can control for this by simply assuming that the forces in that sector are fighting a separate battle, and at the end the survivors either fight the larger remaining enemy or join their surviving allies; however, at that point, our model begins to diverge from reality rather nastily.

The base point here is that deterministic, statistical models break down quickly when applied to problem spaces smaller than their optimum resolution. When the universe of the problem contains fewer points, there is much more of a chance that inequal components of forces x and y will meet, since the organizations aren't homogenous (of course, no human grouping is).

Hmm. What now? Ah yes. Examples and then moving on. The applicability of the Square Law is, as I stated earlier, limited to aimed fire situations. This includes any combat where the units involved are firing at enemy units, and correcting their fire to avoid 'killed' targets. Rifle duels, tank combat, air to air combat, all of these apply. There is another mode of combat, however, that this does not address; that is the model of unaimed fire onto an area (as opposed to unit) target. In this case, while each iteration does of course destroy a certain fraction of targets, the now-proportionally-greater firepower of the winning side is not re-aimed; 'dead' targets are just as likely as live ones to draw fire in the second interation. Thus, neither side gains a 'square' advantage as their numerical preponderance increases. Some examples include mass anti-aircraft barrages, or artillery bombardment of enemy positions.

Lanchester's Linear Law (Un-directed Fire)

Let's keep artillery as our example. Each unit of force x fires at the (estimated) position force y. Specifically, each unit is firing into an area of size A. Each shot will kill any enemy unit in a (smaller) area a inside A. The chance of actually killing a unit, therefore, depends on the number of units y has inside area A. Let's say that x's units can fire at rate r, then we can model the attrition suffered by y (call it δy) as follows:

Δy = −(ra/A)xy

So, in the equation above, x's lethality coefficient is ra/A (let's call it α), and y's (β) is equivalent using the other values of r, a and A. We can thus set up our two initial equations as:

Δx = βxy, x > 0


Δy = αxy, y > 0

If we eliminate time by performing the same mental/pencil gymnastics as we did above (I'll spare you the obfuscated HTML) then we can arrive at the following end state:

yf = 0    and    xf = xo −(β/α)yo    if    αxo ≥ βyo

…and …

xf = 0    and    yf = yo −(α/β)xo    if    βyo ≥ αxo


Now, in this world, the world of the Linear law, we find that fighting strength is equal to the product of the original force levels non-exponential and the lethality coefficient. In this case, there is no gradual increase in advantage as combat continues - the decline is slower. This means there is no advantage to pursuing numbers over lethality.

Where things really get interesting is when one begins to consider (while riding into combat) which world he or she is riding into. Ideally, we find, one would like to be more numerous in the Square law world of aimed fire, and more lethal in the Linear world of unaimed (or, to use a better term, un-directed) fire. So while jouncing about in your MBT, you realize that the ideal situation is for you to live in a square law situation while forcing the enemy into a linear law position! This is perfectly possible; to continue our example, imagine that you have sneakily and cleverly emplaced your tanks in hull-down defilade at the edge of a forest. Your opponent must advance their forces across a large open plain to reach you. In this case, while each of your well-hidden tanks has a square law shot (aimed) at the enemy, he is denied the same!

After the first crashing thunder of guns, not only are your tanks hidden in the forest, but you (clever chap) have ordered your units to employ their smoke generators, thermal pots and infrared targeting systems. Now the enemy's tanks out on the plain are suffering from a number of problems:

  • They are in disarray from the initial volley (we hope)
  • They are short some number that were destroyed
  • They have no aimed fire targets, because:
    • Your tanks are hidden
    • Your turrets are lowered
    • You've put up smoke that they can possibly see through with infrared, but your tanks are too well emplaced to be giving off much visible infrared except when they fire.
All in all, then, you will be taking aimed shots, after each volley of which your tanks can scoot back a couple of meters and become invisible behind ground cover before moving slightly or just waiting and popping up to fire again. The enemy is forced to fire blindly at the forest's edge. That, of course, puts his force squarely (ha ha) into Linear world, while your noble lads are blazing away in square law country. Guess who wins?

The lessons don't stop there. If you rely heavily on indirect fire, as the American military does, then you know that you can balance lethality improvements equally with numerical increases. If you have to carry all your forces several thousand miles as well as keep them supplied with ammunition, toilet paper, beans, etc., then it will pay rich dividends to research increased lethality per indirect fire unit (artillery piece, competent munition, carpet bomber, etc.) As for your aimed fire units, it behooves you to maneuver for local numerical superiority, because even if your forces are twice as lethal as your enemy, if he has twice your forces he still enjoys a two to one advantage (lethality * 2 != numerical difference ^ 2).

In any case, it is also important to remember that numbers and models don't make the world happen, they merely describe it - and even then, not in all cases. The main utilities of the Lanchester models of combat are not that they will tell you in advance who will win; rather, they describe a set of actions and their affect on the outcome. This gives you the box you must strive to think outside of!

If numbers matter, and your opponent (the natives) outnumber your expeditionary forces, then avoid direct fire confrontation and go for imbalanced fields; more, get outside the box altogether. Start looking for massive force multipliers - you're going to need to be clever to make this work. Find their fuel sources; find their supply lines. If you're the defending natives, then realize that you should only join combat when you can mass forces, because you'll need a numerical advantage to make up for your unit's individual weakness. You'll have to be wary of the enemy maneuvering. If the invaders are better equipped than you, try not to get into artillery duels, because his indirect fire strength is likely much, much better than yours. Use your indirect fire against his direct fire units so that they cannot respond; mass your own aimed fire units (tanks) rather than your artillery. Disperse that!

And so it goes. The most important thing to recall about combat models is the old adage: 'No plan ever survives contact with the enemy.'

On the other hand, if you're a fatassed civilian analyst working in sunny Santa Monica, then these models offer you a means of putting your mind-numbingly complete knowledge and historical research to quantitative use.

Some folks have asked where this data comes from. To be honest, it comes from a whole bunch of sources; the math was done by yors utrly from a set of starting equations given to me as an assignment for class at MIT. The remainder was assembled (again by me) for use while I was teaching section for a course in the field from lectures, various books and papers.

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