I came across the Lanchester Equations in Peter Turchin’s book “End Times” where he uses them to explain why the North had to win the American Civil War, and was immediately struck by the apparent applicability of what he said to the Ukraine-Russia war. Given Russia is much larger than Ukraine, it looked as if Ukraine was doomed.

This is what he says:

Remember that the North has a large advantage over the South in terms of manpower. To be precise, it is fourfold (twenty-two million Northerners to five and a half million white Southerners). One might think that giving a twofold advantage to the Southerners because of their skill, and another similar advantage due to their higher morale and motivation, would be just enough to match the numerical preponderance of the North. Getting arms is a problem for the South, but they could (and did) rely on overseas imports. Thus, the South has an even chance of winning the war, right? Wrong. Although the numerical advantage of the North is four, it actually translates into a warfare advantage of four squared, which is sixteen. This mathematical result is known as Lanchester’s square law. It sounds counterintuitive, but once the mathematical result is derived from the Osipov-Lanchester equations, it is easy enough to explain in words. … In real life, the course of the American Civil War went much as the Osipov-Lanchester model would have predicted. The Confederate Army won most of the battles, thanks to better Southern marksmanship and horsemanship, as well as better-trained officers and generals. But the North mobilized 2.1 million soldiers against 880,000 Southerners. The Union Army still had to endure heavy casualties; it lost 360,000 soldiers against 260,000 Confederates. But after four years of bloody, bitter struggle, the North ground up the South and won the war.

Turchin, Peter. “End Times” pp.257-258

He gave this explanation in words in another book like this:

Suppose the enemy has 2,000 archers, while you have only 1,000. When the two armies engage in battle, all archers start shooting at their adversaries as rapidly as they can. The enemy army shoots a volley of 2,000 arrows, so each of your troops is threatened simultaneously by two arrows (2,000 divided by 1,000—this is, of course, the average, because one warrior may be hit by three or four arrows, while another gets lucky and sees not even one). Conversely, each enemy archer is targeted by only 0.5 arrows (1,000 divided by 2,000; in other words, half of them have nothing to fear). Allowing for mishits and complete misses, let’s say that only one arrow in ten actually results in a casualty, wounding or killing a combatant. The first volley, then, costs you 200 casualties and the enemy 100. In the next exchange, the 1,900 remaining enemies hit 190 of your men. At this point you have lost 39 percent of your force. Few armies can sustain such casualties, and yours is no exception. Your surviving warriors run away. You have lost. … This effect is known as Lanchester’s Square Law, because during each round of engagement, the proportion of casualties inflicted by an army on its adversary is the square of its numerical advantage.

Turchin, Peter. “Ultrasociety” p.157

The Lanchester Square Equations are a pair of linear differential equations discovered by Frederick Lanchester in 19161. The solution of these equations gives the Lanchester Square Law that Turchin uses.

The equations are:

\[ \begin{align*} \dfrac{dU}{dt} = -r\,R\\ \dfrac{dR}{dt} = -u\,U\\ \tag{LE} \end{align*} \]

\(U\) and \(R\) in Turchin’s context are armies of archers, American Civil War armies, or even whole populations.

\(\dfrac{dU}{dt}\) is the rate of change of the \(U\) army over time. It gets smaller (the right hand side is negative) and does so at a rate proportional to the size of the opposing \(R\) army. The constant of proportionality is \(r\). The larger \(r\) is the more efficiently \(R\) uses its size and the faster \(U\) depletes. The second equation is understood analogously.

If we solve these equations (see end of this piece if interested) while assuming that at the start \(U\) army has \(U_0\) soldiers and the \(R\) army has \(R_0\) soldiers we obtain the Lanchester Square Law. \[ \begin{align*} r(R_0^2-R^2) &= u(U_0^2-U^2) \\ \tag{1} \end{align*} \] At any time \(U\) and \(R\) are the remaining numbers in each army. In Turchin’s archer armies we have \(R_0=2000\) and \(U_0= 1000\) while the constants are both \(\dfrac{1}{10}\) (one in ten arrows kill an enemy). Plug those values into (1), set \(U=0\) and you find how many soldiers \(R\) has left when it has annihilated it’s opponent. The answer is 1732 so \(R\) has lost more archers but still won the battle because all the enemy are dead.

By re-arranging (1) to express \(R\) as a function of \(U\) we can also determine the condition for the \(U\) army to win which (fairly obviously I suppose) occurs provided \(U>0\) when \(R=0\). This is:

\[ \dfrac{u}{r}> \left(\dfrac{R_0}{U_0}\right)^2 \tag{2} \] This ratio rule tells us the how much more efficient a smaller force has to be to win against a larger one. Unsurprisingly, if the “>” is replaced by “=” then you get the condition for mutual annihilation.

Turchin compared the Union population with the Confederate one to obtain his 16 times war making advantage (in (2) \(\dfrac{u}{r}>16\) for the Confederacy to win)

But the Lanchester Equations thus interpreted don’t really make any sense. For they say that the rate of decline of the Confederate population was proportional to the size of the Union population (and vice-versa). But if we look at the US census records for 1860 and 1870 (pdf) it turns out the the Confederate population did not decline. In fact the population of every Confederate State increased between 1860 and 1870. Certainly populations support armies more or less adequately, but this relation is not, I suggest, sensibly modelled by the Lanchester Square Law.

Confederate States Population
State 1860 1870
Alabama 964 997
Arkansas 435 484
Florida 140 188
Georgia 1057 1184
Louisiana 708 727
Mississipi 791 828
N. Carolina 993 1071
s. Carolina 704 706
Tennessee 1110 1259
Texas 604 819
Virginia 1220 1225

Taking a step down in size Turchin notes that the Union army was much larger (2.1 million) than the Confederate one (880,000). If we apply the ratio rule to those armies we get \(\dfrac{Union}{Confederacy} > \left(\dfrac{2100}{880}\right)^2 \approx 5.7\). Turchin (for the sake of argument) grants the Confederacy a four times advantage in skill, motivation etc. but that is still not enough. Still it’s nearer being enough.

There is a paper by M. Armstrong and S Sodergren, “Refighting Pickett’s Charge” (2015) which uses the Square Law for one part of a phased analysis of Pickett’s Charge at the battle of Gettysburg. It seems that it was possible that the charge could have been successful had a few more soldiers been committed or the artillery used a bit differently. Perhaps if they had won Gettysburg…

The Crimean War lasted from October 1853 to February 1856. It ended after the Russians were driven out of Sevastopol and Austria threatened to join the war on the side of the allies (England, France, Turkey and Sardinia). These allies had 603,000 soldiers, while the Russians had 889,000. According to ratio rule for the allies to win against that disadvantage they needed \(\dfrac{Allies}{Russia} > \left(\dfrac{889}{603}\right)^2 \approx 2.2\), in other words they had to be a bit more than twice as efficient as the Russians to overcome their numerical disadvantage. Were they? It seems so, as the ratio of Russian to Allied casualties was \(\dfrac{450}{165} \approx 2.7\)

We do not know the sizes of the armies fighting in Ukraine. We assume Russian forces are more numerous but actually have no reason to do so. In terms of losses Ukraine puts out figures of massive Russian casualties in men and equipment but conceals its own. The data for applying the Lanchester Square Law is just not available. Yet there are actually a priori reasons to assume that the Square Law does not apply.

Most of us have seen videos of small groups of Ukraine soldiers apparently firing randomly in the general direction of the enemy. This mode of fighting (“Area fire”) is definitely not subject to the Square Law, for a rather subtle reason. If forces shoot at random at each other then the damage they inflict is not constant because each time they luckily kill an enemy there are fewer enemy left to (randomly) hit and so their chances of further success, hence their efficiency, gets less. That means that in the Lanchester Equations the proportionality terms \(u\) and \(r\) cannot be constant and so the equations cannot be solved in the manner that produces the Square law. In short, the Square law does not apply to this form of combat.

Thomas Lucas2 observes (pdf) makes this point as follows:

Some have speculated that the linear law may turn out to be the “new modern conditions.” In particular, if a force can orchestrate the battle (perhaps by using information superiority and agile maneuvers), such that engagements are typically one-on-one, the force may be able to trade quantity for quality. (p. 97)

This is a pretty impressive description of what is going on in Ukraine at the moment especially considering it was written in 2004.

I found some even older notes from 2000 by Alan Washburn (pdf)3. He makes the basic distinction between aimed (Square Law) and unaimed (Linear Law) fire, but also talks briefly about a probabilistic version of Lanchester systems using Markov chains. These systems are discrete and stochastic, at each time step \(U\) loses one unit or \(R\) uses one unitor things stay the same. Each of these outcomes occur with a certain probability. He concludes, “Results can be surprisingly variable, with sometimes one side winning and sometimes the other. The winner of a Square law battle generally does so resoundingly …”. Unfortunately he does not say more.

The equations that produce the Linear Law look more complicated than the Square Law ones but in fact are simpler. I have added the derivations at the end of this piece - but the key step involves dividing one equation by the other which leaves nothing but the constants on the right hand side.

\[ \begin{align*} \dfrac{dU}{dt} = -r\,R\,U\\ \dfrac{dR}{dt} = -u\,U\,R\\ \end{align*} \tag{3} \] Washburn in his notes when talking about more complicated systems of equations speaks of them being “Square Law like” or “Linear Law like” depending on how often they include products like \(R\,U\) on the right hand side. Because we have a product in the first equation above we can say the change in \(U\) is proportional to the product of \(R\) and \(U\) so it avoids the problem with the Square Law I just discussed. The Linear Law equations remind me of the Lottka-Volterra equations for predator/prey relations, though they are simpler.

There has been a lot work trying to see which set of Lanchester equations fits battlefield data best. The Lucas paper, in fact, addresses that problem using data from the battles of Kursk and the Ardennes.

The chart below shows the data he uses for the battle of Kursk. It shows the changes in combat power of the German and Soviet armies during the battle:

The combat power of a force is defined as a weighted sum of combat manpower, APCs, tanks, and artillery, with weights of 1, 5, 20, and 40, respectively (p. 101)

Combat manpower Thomas Lucas "Fitting Lanchester Equations to the Battles of Kursk and Ardennes" Naval Research Logistics, Volume 51, pp. 95 116, 2004

Figure 1: Combat manpower Thomas Lucas “Fitting Lanchester Equations to the Battles of Kursk and Ardennes” Naval Research Logistics, Volume 51, pp. 95 116, 2004

I was surprised to see that the decline in Soviet combat power was much greater, also towards the end of the battle things levelled off (in medieval battles the greatest slaughter occured when one side turned an ran). It also looks like the Germans were winning? Well, perhaps they were Lucas tells us that Hitler ordered the attack to be stopped:

Field Marshal Erich von Manstein, Commander of Army Group South, felt that “[stopping the offensive] at this moment [was] tantamount to throwing victory away” (p. 100)

After that things went badly for the Germans. Very complicated, a fool in Berlin may have lost the battle - a chance event not a modelled militaery fact.

Here, anyway, is the conclusion of the Lucas paper:

Lanchester’s intuitive and parsimonious equations have been, since their introduction the better part of a century ago, the most common tool for modeling aggregate attrition. While we are wary about making too much from two battles (though these are all we have of this type), this research adds to the evidence that Lanchester equations may be too blunt of an instrument for modeling the attrition of highly aggregated forces. Indeed, it is asking a lot to address most of the complexities of combat attrition in a model with only a handful (four or five in this paper) of parameters. The failure to find any good-fitting Lanchester model suggests that it may be beneficial to look for new approaches to model highly aggregated attrition (p. 116)

The message seems to be that the equations are beautifully simple but they don’t work any too well, and probably we should just move on. At least one scholar is quite emphatic about this

I conclude: Lanchester equations have been weighed, they have been measured, and they have been found wanting.

A less damming conclusion comes from Paul Syms in some slides prepared in 2017 for the UK Ministry of Defence. I have taken a snapshot of his last slide:

So if you want to use the Lanchester Laws for Ukraine the Linear one is probably best but none of them are very good.

Elegant and simple equations can mislead and they do so because of the seductiveness of their promise of a lot of knowledge cheaply.

Derivation of Lanchester Laws

The generalised Lanchester Equations from Lucas (2004) are: \[ \begin{align*} \dfrac{dU}{dt} = -r(d\ \text{or}\ \frac{1}{d})\,R^p\,U^q\\ \dfrac{dR}{dt} = -u(\frac{1}{d} \text{or}\ d)\,U^p\,R^q\\ \end{align*} \] The parameter \(d\) represents a difference between attacking and defending. I ignore this. For the Square Law \(p=1\) and \(q=0\) which gives us the equations again:

\[ \begin{align*} \dfrac{dU}{dt} = -r\,R\\ \dfrac{dR}{dt} = -u\,U\\ \tag{A1} \end{align*} \]

To solve these equations first divide the second equation by the first and recalling that \(\dfrac{dR}{dt} = \dfrac{dR}{dU} \cdot \dfrac{dU}{dt}\) obtain:

\[ \begin{align*} \dfrac{dR}{dU} &= \dfrac{-u\,U}{-r\,R}\\ \end{align*} \] This gives a single separable differential equation of \(R\) as a function of \(U\). So separate \(U\) and \(R\) and integrate.:

\[ \begin{align*} r\,R^2 &= u\,U^2 + C \tag{A2}\\ \end{align*} \]

We assume that at the start of the battle \(R=R_0\) and \(U=U_0\) which we incorporate into (A2) using the integration constant \(r\,R_0^2 = u\,U_0^2 + C\), so solving for \(C\) and re-arranging we get:

\[r(R_0^2-R^2) = u(U_0^2-U^2) \tag{A3}\].

To get the ratio rule for \(U\) winning rearrange (A2) as:

\[R^2 = R_0^2 - \dfrac{u}{r}U_0^2 + \dfrac{u}{r}U^2\] When \(R=0\) we have: \[\dfrac{u}{r}U_0^2 - R_0^2 = \dfrac{u}{r}U^2\tag{A3}\] If \[\dfrac{u}{r}U_0^2 > R_0^2 \tag{A4}\] then the RHS of (A3) must also be greater than zero which implies that \(U>0\) and \(U\) army has “won”. But the condition (A4) is the ratio rule.

For the Linear Law we take \(p=q=1\) so: \[ \begin{align*} \dfrac{dU}{dt} = -r\,R\,U\\ \dfrac{dR}{dt} = -u\,U\,R\\ \end{align*} \] Here the changes are proportional to the product of the two fighting strengths. Dividing the equations as before gives:

\[ \begin{align*} \dfrac{dR}{dU} = \dfrac{u}{r} \end{align*} \] and integrating:

\[ \begin{align*} R &= \frac{u}{r}U + C \end{align*} \] from which the Linear Law follows as previously: \[ \begin{align*} r(R_0-R) &= u(U_0 - U) \end{align*} \] The logarithmic law has \(p=0\) and \(q=1\), a sort of self-immolating army that inflicts casualties on itself (though perhaps in Crimean War?). I believe that this model has be fit from time to time with some modest success.

  1. Wikipedia and independently by Mikhail Osipov in 1915. However the first version of the Square law was by a Lieutenent J.V. Chase USN in 1902 in 1902. Lanchester had two brothers who founded the Lanchester car company↩︎

  2. Professor Thomas W. Lucas, Operations Research Department, Naval Postgraduate School, Monterey, California↩︎

  3. Alan R. Washburn. Distinguished Professor Emeritus of Operations Research Naval Postgraduate School, Monterey, California↩︎