War in Ukraine II: The Model

Peter Turchin


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The first part of this series gave an introduction to the Osipov-Lanchester (OL) model and illustrated the ideas with the example of the American Civil War. In this second part, my aim to extract a prediction from the OL model for the War in Ukraine. A few general points/reminders:

  • I am not taking sides and I am not discussing the rights and wrongs.
  • War is evil, but needs to be studied.
  • I am interested in a scientific prediction, not prophecy.
  • The model is very simple and its prediction is not a statement of what will be, but a way to find out how reality deviates from its prediction.
  • I am not basing this prediction on a direct analogy with the American Civil War (ACW); I use it only as an illustrative example.

In fact, and as some people pointed out in the comments to the previous post, the ACW may not be a good example of the Square Law (while being a good illustration of a more general OL approach). More generally, a few studies, I am aware of, that attempted to test the OL model came to the conclusion that the actual exponent relating numerical advantage to war advantage is almost never 2, but is typically between 1 and 2. But this is not going to affect what follows.

The core of the OL approach is to model the dynamics of casualty rates inflicted by each army shooting projectiles at the enemy. It’s a model of attrition warfare (although the model is usually applied to a single battle, I use it to model the course of an entire war). Since over 80 percent of casualties in the Ukrainian conflict are inflicted by artillery, to a first degree of approximation we need to know how many shells are fired by each side. There is a general agreement by all sides that the Russians expend many more munitions than the Ukrainians. Specific numbers might be something like 5,000 shells per day fired by the Ukrainians as compared to 20,000 shells fired by the Russians. These numbers primarily refer to heavy guns, shooting 152 mm ammunition (USSR/Russian standard) or 155 mm (NATO standard). The numbers represent averages. For example, the Russian ammunition expenditure has varied between 10,000 and 50,000 shells per day, or even more. The overall conclusion is that Russians have roughly a 4:1 advantage in artillery, although it could be easily 3:1, 5:1, and even 2:1 or 10:1 (I’ve seen all such ratios mentioned by different sources).

The main assumption I am going to make is that the rate of casualties inflicted on the enemy is proportional to the number of projectiles expended. In the numerical example worked out in Ultrasociety (p. 157) I assumed that the ratio of arrows shot to casualties inflicted (killed or wounded) is 10:1. In the Ukrainian war it is more like 20-30, or even 40 shells per casualty (it also depends on whether casualties are defined as KIA or also include wounded), but the general principle of proportionality is the same. Thus, according to this model, the prediction is that Ukrainian casualties must be roughly 4 times that of Russians (but the range could be 3–5, or even 2–10). Once we have better data on the ammunition expended by each side during the course of the war (after the war ends), we will be able to tighten up this prediction.

This prediction comes from a very simple model, and reality can deviate from it due to a number of factors (remember, this is not a prophecy). Let’s consider some such complicating factors. Note, depending on how important these factors are, the prediction could be substantially different. In other words, we have alternative predictions resulting from different assumptions—finding out which of these alternatives matches the data best is what the scientific method is about.

  • The prediction assumes that neither side has a serious technological edge. There are diverse opinions on whose guns/tanks/airplanes are better. A post-war assessment will show how accurate this simplifying assumption is.
  • Skill. Because the war actually started in 2014, by February 2022 the Ukrainian artillerists had an advantage in skill as they have been fighting against the Donetsk and Lugansk militia for eight years. But as the conflict lengthened, they lost this advantage as Russian gunners increasingly gained experience.
  • Morale. There are conflicting reports on this aspect. Furthermore, in an attrition warfare morale plays less important role than in mobile warfare. In the absence of a compelling argument one way or the other, we stick with the simplifying assumption of no advantage for either side.
  • Defense/offense. Both sides have conducted defensive and offensive operations. Furthermore, in attrition warfare, there is no clear difference between offense and defense, as the same piece of territory may change back and forth due to attacks and counter-attacks.
  • Although artillery dominates, it is not the only force that inflicts casualties. Air force, guided missiles, land mines, and drones are also important. It’s an open question how adding them to the equation may change the prediction.
  • Variations in time and space. Unlike a mathematical model, with its smooth curves, actual casualty rates will fluctuate in time with the intensity of conflict. There are “battles” when fighting is intense, and lulls between them when casualty rates are low. Space can also be important. One side achieving a local numerical advantage could shift the relative casualty ratio in its favor.
  • Logistics. The OL model doesn’t include space, but in order to be shot, munitions need to be delivered to the front lines.
  • Production. This is the most important factor affecting the long-term course of the conflict, and it requires more detailed discussion (which I defer to a future installment).
  • War goals of each side and their determination to achieve them. Again, this is very important, and I will devote a separate post to it.

To summarize, what we have here is a clear and specific prediction coming from a simple model. And there are many ways in which it can go off track. The goal of this exercise is to determine how close the prediction is to the actual outcome, once the war ends; and, even more importantly, which of the factors listed above will have a significant effect on this outcome.

In the next installment I plan to provide an interim assessment of the state of this war; at least, as much as we know about it.

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One more factor that is difficult or impossible to measure or include in this model is battlefield intelligence including satellite imagery; the US support to Ukraine probably gives their side a significant advantage. This likely manifests as a higher kill rate for Ukrainian long range weaponry, ceteris parabis.


Then based on the number of shells ukranian casualties must be four times that of russians. Why did ukranians regain territory?


Production. This is the most important factor affecting the long-term course of the conflict…”
Production of gun, shells, tanks, and drones. And soldiers.

Paolo Ghirri

one assumption is not even that no allied enter in the war?
i find very difficult to think that if Ukraine eventually start to show exhaustion Poland or other east european country don’t came to help directly

or to say in another way, this prediction is valid if and only if the scenario is “Russia against ukraine(+nato material help)” correct?

Air force: one thing that i don’t understand is where is Russian AF? maybe is only FOW, but it seem to me that the intensity of use of AF is at very low level: why? in reserve for a long attrition war (as France in 1940)? lack of ammunitions (as Germany at the end of france campaign)? corruption (big on paper but only on paper as Italy 1940)?

also: (maybe this is yet planned for future installment) if food and energy price stay high we don’t have to expect instability events inside west europe and US that limit the capacity to support Ukraine?

and: most of west analyst see the Prigožin affair as sign of russia internal instability: i think this is the opposite, it seem to me similar to Marcus Aemilius Lepidus and Octavianus in 36 ac, or for some extent to The Xi’an Incident in 1936


Af isn’t used by either side due to the effectiveness of ‘modern’ AA systems. I believe that we are at a tactical stalemate in warfare doctrine and air dominance isn’t possible anymore between near peer enemies.

It might be different with 100s of stealth fighters but even this is questionable and neither side has them or will have them in the near future


The Russians are holding back on airplanes and helicopters because they have a limited supply and they take a long time to build so these assets can’t be used in the same fashion. Russia has lots of old tanks, old artillery and men to waste but they can’t afford to waste their airpower.

Michael Moser

i think there are additional factors, it’s also about technological capability, Additional factors would be:

  • Who has air superiority? Capability and quality of air defense systems?
  • Precision of the artillery? Quality of mil. intelligence and ability to make use of it?
  • Capability to disrupt supply lines by deep strikes?

In the American Civil War both sides had equipment of the same type. I am not sure you have this situation now.


My laypersons opinion was that if two sides of a conflict are in a stalemate or a war of attrition, and one side is reliant on a foreign benefactor than that side is at a disadvantage.

What historical events relate to that I’m not sure, but Vietnam was a candidate in my mind. The south Vietnamese relied on the support of a foreign benefactor but it might be more complicated. The North Vietnamese had benefactors too but they were not based in a distant country.

Last edited 2 months ago by Ollie

Artillery has changed with the introduction of smart shells. Today it would be more relevant to measure the ratio of smart munitions each side possesses rather than the number of artillery tubes.

Peter Mott

I learned about the Lanchester laws from “End Times” regarding the American Civil. I was immediately struck about the application to Ukraine-Russia war – which dismayed me – because Russia is much bigger than Ukraine.

I did some investigation, started writing something, then wondered whether I was taking his (PT’s) remarks too seriously so I posted a tweet (https://twitter.com/peter_mott/status/1678446249085739008) and left it.

Almost immediately after that PT posted the first part this blog so I finished my piece and have posted it at https://rpubs.com/peter2108/lanchester (this is a free site for posting Rstudio stuff).

It’s about the Lanchester laws referencing what PT says in “End Times” (not this blog). My conclusions are:
(1) PT is unclear about the scale at which these laws can be applied. Battles seem best.
(2) There are two laws (Square and Linear) and it is controversial which should be used in any given case.
(3) Using them is difficult and the results are not very satisfactory, but the Linear Law seems best in Ukraine.
(4) There are scholars who think the laws are discredited (of course there usually are 🙂 ).

John Strate

The model and the argument seem sound. Other authorities and knowledgeable persons (John Mearsheimer, Alexander Mercouris) appear to agree. There would seem to be uncertainty regarding Putin’s goals and whether they have expanded over time to include control of four additional oblasts and regime change.


How do you account for quantity, quality and conviction of allies in your models? These are very hard to judge but they have been determinant in many conflicts.

RL Green

Regarding the main assumption that the rate of casualties inflicted on the enemy is proportional to the number of projectiles expended, that is generally true. A presentation I just watched does give this some relevant nuance in the context for the war in Ukraine. See “Cluster Munitions & Artillery in Ukraine – Attrition, Ammunition & Adaptation in 2023″ at https://youtu.be/1zcUe47xerQ.

Alan Patrick

Most wargame models I’ve seen have an attack and a defence algorithm. The Osipov/Lanchester model only has an Attack function, ie. how many potential hits are caused.There may be differences between Russian and Ukrainian Defence functions, i.e. ability to avoid getting hit/hurt. Even a small difference causes a big variances over time. Alos, I read that Ukrainian ability to get soldiers off the battlefield fast is better, so they may have a better casualty return function too.

Peter Mott

There is a more complex version of the Lanchester law used by Thomas Lucas that does. https://core.ac.uk/download/pdf/36736335.pdf


I’d argue there are a couple of other points that will impact the accuracy of this model:

  • where each army chooses to expand its munitions – Russia seems to be expanding a large proportion of its firepower on attacking civilians, presumably in order to destroy Ukrainian morale. If significant enough numbers of shells are being used in this way, then the casualty estimates for Ukraine per shell will be skewed by civilian casualties who never would have fought anyway.
  • The availability of cheap smart artillery in the form of drones – Ukraine has the advantage here, although Russia is trying to catch up. Cheap abundant drone warfare seems on the surface to be enabling Ukraine to deliver much higher shell/casualty ratios than this model might suggest.

Finally, is there an argument for including China as a supporting partner for Russia? Admittedly they have not gone all-in to the degree some may have expected/feared (yet) and appear reluctant to openly aid Russia, but there are definite hints they are supporting Russia on the DL.

Steven Johnson

Attacking civilians? The world has experience of a major power targeting civilians: The Korean war, the Vietnam war, the Serbian war, the Libyan war, the Afghanistan war. Watching the US spokesmen assuring the world they meant to a deep-penetrating bomb to kill pretty much everybody in the air raid shelter in Baghdad on the grounds it was “for” command staff families, thus a CCCI installation or the declaration the Baghdad water and sewer was as valid a military target as the power grid (still not fully repaired!) has given me an entirely different impression. There are no surgical strikes in war, that’s nonsense. But the notion Russia targets the civilian population the way the Ukrainians have been targeting civilian targets in Donetsk and Lugansk for years clearly needs much stronger documentation from reliable sources.

And that doesn’t even begin to approach the targeting of civilian populations by sanctions. I believe economic warfare is still warfare. It may be part of hybrid warfare, but it’s still WWIII. Nobody can demand the current re-division of the world has to look like the previous exercises.


I have seen no convincing evidence of Russia targetting civilians. The Ukrainian army has been doing so since 2014. Western media report Ukrainian attacks on civilians as Russian ones.

For instance, Patrick Lancaster, an American reporter living in Donetsk, attended the aftermath of a Ukrainian Tochka-U strike that landed a couple of blocks from his home in the retail area of his purely civilian neighbourhood. His video of the dead and dying and the desperate attempts to save the wounded were very grueling. They were too much to resist for Western media, which purloined them as illustration of Russian war crimes in attacking civilians. Several of Lancaster’s reports have been mis-used in this way.

Similarly, the missile attack on Kramatorsk railway station that killed civilians waiting to be evacuated was used by the media for a couple of days as proof positive of Russian barbarity. The story suddenly disappeared when footage from an Italian TV crew revealed the missile’s serial number, which was in the middle of a sequence known to be used by the Ukrainian army.

Anyone depending on the Western media for assessment of military behaviour, and indeed of morale on the two sides, risks being seriously misled.

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