British statistician George Box once stated, “All models are wrong, but some are useful.” Recently, several high-profile combat models have been more wrong than useful. For example, a model incorrectly forecasted that the Afghan government could withstand a Taliban takeover for at least 6 months, as opposed to collapsing within 10 days. Another model predicted that Kyiv would fall within 3 days following a Russian invasion. The inaccuracy in these models were due to their attempts to capture the dynamic, complex relationships associated with combat.
Meanwhile, a much simpler model may be able to answer some of the questions about the Russia-Ukraine war, such as why the Russians were so ineffective and what will likely happen as the war continues. This simple combat model is based on the Lanchester equations, which were developed in 1916. The Lanchester equations consist of a series of differential equations that approximate the rate of combat losses for two opposing armies. Although somewhat basic, these equations have been successfully used over the last century to model attrition warfare, similar to the conflict in Ukraine.
The Lanchester equations sets the casualty rate for an army as a function of the size of each army. For modern combat, the Lanchester equations typically take two forms. The first, Lanchester’s Second Linear Law, is used for unaimed fires, where a military indiscriminately shells large swathes of land rather than specifically targeting enemy forces. In this case, the casualty rate scales with the number of firers and the number of targets. The second, Lanchester’s Square Law, is used for modern militaries that are concentrating their forces and targeting specific enemies. For this case, the casualty rate scales with the number of firers.
An analysis of the Russian Army would indicate that they are using primarily unaimed fires. The Russian use of unaimed fires explains the large number of rounds fired by Russian forces despite hitting very few Ukrainian military targets. It also explains the significant amount of collateral damage imposed on the Ukrainian population. As such, Lanchester’s Second Linear Law would be the appropriate for capturing the attrition of Ukrainian forces by the Russians. Meanwhile the Ukrainian Army is concentrating their fires, choosing to target key Russian targets, including command nodes, logistical hubs, and key terrain. In this case, Lanchester’s Square Law best captures the attrition of Russian forces by the Ukrainians.
The different equations for Russian and Ukrainian attrition are primarily due to their respective levels of training. For a military to concentrate their fires, they must be able to coordinate and synchronize their elements. This can only be achieved through training. The Ukrainian military had seven years to prepare for this war and have trained extensively with NATO forces. This training has allowed the Ukrainians to properly target Russian forces. Meanwhile, the Russian use of unaimed fires is indicative of a lack of a training, which was likely accentuated by their use of conscripted soldiers.
Solving the equations requires knowing the starting size of each force. Although there is substantial uncertainty in these values, several sources approximate the Russian invasion force as 190,000 troops. Meanwhile, the Ukrainian military was approximately 145,000 troops at the start of the war. These numbers only encompass their trained military personnel and does not include paramilitary organizations or civilian volunteer groups. The other required information is the attrition coefficients which can be estimated from the casualty rates throughout the war.
When the differential equations are solved, they indicate that the Russians had a strong advantage at the start of the war. Indeed, they destroyed a number of Ukrainian targets during the first few weeks. However, their effectiveness continually diminished as the war progressed at a rate much faster than the Ukrainian forces. This diminishing combat effectiveness is readily apparent from open source sites, such as oryxspioenkop.com, which has shown a continual destruction of Russian equipment, and decreasing numbers of destroyed Ukrainian equipment.
The models further indicate that at approximately six months, the war would change as the Russians would lose their numerical advantage. This coincides well with the shift in the war with the Ukrainians seizing back large amounts of territory. It would also explain the Kremlin’s strong push to recruit 50,000 new soldiers to replace their combat losses.
These models also provide insight into the likely outcome of the Russian plan to field new “volunteer” battalions. Since these troops will be even less trained than the Russians forces currently in Ukraine, they will likely continue the use of unaimed fires. As a result, the surge in soldiers will see a similar trend where they achieve initial successes but long-term failures. Indeed, the models would indicate that the Russians’ best course of action would be to fully withdraw from Ukraine, extensive training, and a reinvasion a later date.
As stated previously, models based on the Lanchester equations are notoriously simple. Regardless, even a simple model can be useful. They clearly indicate the source of the Russian failure is a lack of training, manifesting itself in the use of unaimed fires. Meanwhile, the Ukrainian preparations for the war have allowed them to coordinate their fires, providing them a large advantage over the Russian forces.