A Blog by Jonathan Low


Mar 20, 2022

How Mathematical Models Predict Russia's Time In Ukraine Is Running Out

Math makes it seem simpler than it probably is, but the underlying dynamics suggest a relentlessly quantitative reality which is difficult to argue with, let alone overcome. JL 

Tim Andersen reports in Medium:

Retired Lt. Gen. Ben Hodges, former commander of the U.S. Army in Europe, recently stated Russia was running out of time to win in Ukraine and might have to make peace within 10 days (because in) a war of attrition, "they don’t have time, they don’t have the manpower and I don’t think they have the ammunition." Lanchester’s power laws describe a set of differential equations which a force will inflict over time given a certain rate. With the influx of weapons from NATO, the Ukrainian advantage is increasing while the Russian advantage is declining. As long as NATO continues to supply Ukraine, and they keep their resolve, Russia will eventually fail by pure attrition.

Retired Lt. Gen. Ben Hodges, the former commander of U.S. Army forces in Europe, recently stated that Russia was running out of time to win the war in Ukraine before it would be forced to make peace.

Russia’s decision to transition to a war of attrition, where they’re smashing cities, putting civilians on the road for fear of being murdered, …they need three things to do this [win], and they don’t have those three things. They don’t have the time, they don’t have the manpower and I don’t think they have the ammunition.

In fact, he suggested that it might happen within 10 days. Hodges went on to clarify that this was dependent on a continuous or even accelerated flow of equipment (supplies and weapons) into Ukraine from NATO nations to destroy long range artillery, rocket launcher, and cruise missile sites. This latter point is most important to what I want to talk about in this article.

His statement may seem confusing given that many nations carry out protracted wars over years, even decades. Vietnam and Afghanistan come to mind. His intuition however has mathematical backing, going back to the War to End All Wars.

World War I was an offensive war of attrition. In other words, two sets of combatants were duking it out against one another, each attempting to dominate the other in a series of offensive and defensive maneuvers, but ultimately each was just waiting for the other to run out of ammo or supplies. The immobility of WWI trenches was legendary. There was no way they were going to move. The Allies and Central Powers were fairly closely matched for most of the war. It was only the entry of America into the war that tipped the scales.

During the war, Frederick Lanchester, best known as one of the fathers of the British auto industry, contributed to the development of the first tanks. He was also interested in aerial battles and considered how to devise a set of laws to determine who would win them.

His laws became known as Lanchester’s power laws. They describe a set of differential equations. The linear laws apply to ancient combat, where combatants fight each other one at a time. His square laws apply to modern combat. (In practice, we use a combination of the linear and square law since modern combat isn’t always unlike ancient combat.)

The laws describe the causalities that a force will inflict over time given a certain rate. You can see why this might be important. As a force takes on casualties, its fighting strength is diminished, which causes it to inflict fewer casualties on the enemy. This means that a weaker or disadvantaged enemy will lose strength almost linearly while the stronger enemy’s losses taper off with time.

The law also tells us how armies can defeat one another when they have an advantage versus when they have numerical superiority.

Below are some example scenarios. We have two Armies: Army 1 and 2. Each engages in a war of attrition. We want to know how their forces decline over time. The top box depicts the case where Army 2 outnumbers Army 1 by 40% and they are evenly matched. The second box depicts when Army 1 outnumbers Army 2 by 30% but Army 2 has double the damage rate. Box three depicts a draw where Army 2 has twice the numbers but Army 1 has four times the damage rate.

Public Domain. https://commons.wikimedia.org/wiki/File:Damagerace.JPG

Looking at the Ukraine-Russia war, Ukrainians vastly outnumber the invading force in terms of warm bodies. Out of a population of 44 million about 3 million have left as refugees. That leaves about 41 million still in Ukraine. Of course many Ukrainians are too old, too young, too frail, or otherwise needed elsewhere and can’t fight, but that still leaves several million able bodied recruits. They also have an influx of well trained veterans from foreign parts who provide not only additional fighting force but improve the forces that they train. That is versus a Russian army that is in the hundreds of thousands and pretty much maxed out in human power. Given its attempts to bring in foreign fighters (with limited success) Russia itself does not seem interested in throwing its population against Ukraine WW2 style. No re-enactments of the Battle of Stalingrad seem likely here.

The Russians have clear advantages in weaponry. They have many more cruise missiles, a much larger air force, bombers, more armor, and air defense systems.

Yet, with the constant influx of weapons to the Ukrainian army from NATO borders, the Ukrainian advantage is increasing while the Russian advantage is declining. (Not unlike the relative strengths of Germany and Britain in 1941 under lend lease.) They only have so many cruise missiles they can use. Based on calculations from their Syrian campaign, each cruise missile costs $1.2 million. About 900 of various types have been used meaning Russia has spent about $1 billion on long range missiles alone. Warplanes cost $12,000 per hour to fly. Helicopters cost $3000 per hour. A T-14 tank costs $3.7 million. Pentagon estimates are that Russia is losing about 50 vehicles a day and has lost about 184 tanks by time of writing (March 15). This from a nation with an annual defense budget about 1/10th that of the US.

In a war of attrition, left to itself Ukraine would certainly lose. They would simply run out of weapons and supplies, but with a virtually inexhaustible supply coming in from the West and Russia losing considerable cash to sanctions, the reverse is true. Ukraine has the people to fight, and it has the will; meanwhile, Russia has neither in great abundance.

Thus, while early estimates suggested that Russia would eventually win the war because of its extra military capacity, that is now only true if Ukraine chooses to avoid a war of attrition.

This means that the longer the war goes on the less favorable peace terms will look for Russia since they will be in imminent risk of outright defeat. The losses to Ukraine will be horrific nonetheless; hence, both have a strong interest in bringing the war to a close soon.

This may be even more apparent when Russia’s finite supplies of helicopters, warplanes, and cruise missiles are taken into account regardless of cost. While Lanchester’s power laws don’t model these very powerful but finite supplies, newer models such as the Salvo combat equations, do. With these equations, developed at the Naval Postgraduate School in 1995, weapons exchanges involving small numbers of weapons (salvos) that are highly accurate change the dynamics of warfare. Naval battles often work like this where missiles, torpedoes, and shells are fired infrequently and are devastatingly effective.

In this case, the effectiveness of each salvo of a weapon must be evaluated and an accumulation determines the overall outcome of the war. This applies well to tank battles, modern air battles, ballistic missiles, and naval battles where targeting tends to be accurate.

The war in Ukraine, modeled this way, is more complex but gets to the heart of Hodges intuition which is that Ukraine destroying Russian long range salvo capability is critical to winning. While the Ukrainians have done well so far, unless they receive influxes of artillery, tanks, drones, and missiles and are able to attrit Russian long range capability, they will suffer far longer. They have done remarkably well capturing Russian equipment, which, ironically, is what they are trained to use thanks to decades of Soviet rule. That many of them read Russian is a bonus since Russian tanks and other combat vehicles are labeled in that language.

An example of the salvo model below we have two forces, A and B. A has an offensive firepower of α and B of β. Each also has a defensive firepower (given by, say, anti-missile missiles) of y and z respectively. Each has a staying power (the inverse of loss), w and x, and a targeted area ratio of n and m (for area fire that affects multiple units).

The box indicates the number of units in each force. The more units the less total damage in general. The square of number of units in A vs. B has various regions that indicate how much damage a salvo can do to each force depending on the defined parameters and which is hurt more by an exchange.

In the Russian case, area fire is particularly true of thermobaric “vacuum” bombs that can take out multiple fighters, but cruise missiles have an area too.

Example of the outcome of a single salvo exchange. Credit: Michael J. Armstrong, “The salvo combat model with area fire.” Naval Research Logistics (NRL) 60.8 (2013): 652–660.

NATO’s approach so far has been to supply the Ukrainians with anti-tank and anti-aircraft weapons as well as small arms that a single person can fire. This is intended to attrit the Russian supplies of tanks, armored vehicles, as well as helicopters and low, slow aircraft. This reduces the targets available to the Russians but also requires the Ukrainians to rely on more guerilla style tactics like ambushes. Ultimately, NATO needs to get the capability to Ukraine to take out artillery and missile sites that are behind the Russian lines.

As long as NATO continues to get supplies to Ukraine, and they keep their resolve to fight, Russia will eventually fail by pure attrition, and Russia is far from willing, it seems, to fight to the last man.

You might ask why, if Russia is so doomed, some wars can seem to go on and on. Partly, it has to do with how wealthy the aggressor is and partly how effective they are minimizing deaths and other losses. The United States and especially a NATO coalition can wage war for a long time. The Afghan war cost about $2.3 trillion and lost about 2500 uniformed coalition forces in 20 years. Russia has lost as many people in days and doesn’t have trillions to spend.

Let’s not forget that Russia, despite its nuclear arsenal and size, is not a wealthy nation. It has a GDP somewhere between Spain and Italy at about 1.5 trillion (before the latest sanctions). The USA is at about 20 trillion while China is at about 14 trillion, ten times the size of Russia’s economy. Canada, another giant arctic nation, even has a larger economy than Russia.

This may put the war in perspective: Russia is trying to play superpower while wielding little more than an exaggerated sense of its own importance. The US and EU can easily outspend and out supply them. All that Ukraine needs is the will to outlast them.


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