# Finding Maths in the Military

As a maths-teacher-turned-infantry-officer, there are some parts of military planning that I’ve particularly enjoyed. For example, I’ve always been able to quickly calculate the ideal support by fire location using distance, bearings, and all the all-too-familiar trigonometry.

But what other maths is there in the planning and operations? More than the mere rates of movement and working out of an even piquet list?

How are the laws of maths and the scientific method used in military planning?

Challenging the Conventional Force Ratio

In 1988, military historian Joshua Epstein conducted studies into every battle in WW1 and WW2, recording the personnel strength of each side, and the outcome. At the time, conventional wisdom followed that the ideal ratio to attack was 3:1. This doctrine has persisted and is still taught within the Australian Army to junior commanders today.

Epstein showed however, that this 3:1 force ratio did not guarantee success. When forces attacked an enemy with half its force ratio, the attacking force only won 63% of the time. When this was increased to the attacking force to having three times the force ratio of the enemy, this only increased to 73%.

This is significant. Even if a commander had a force ratio of 3:1 in their favour, historically, they’d win less than three quarters of the time.

Epstein’s statistics also show that an attacker still has a considerable chance of winning if the ratios aren’t in their favour – attacking with as few as half the force power of the defender would still give you a 46% chance of winning [1].

When you plot Epstein’s calculations on a graph with an appropriate trend line [2], it becomes possible to predict the likelihood of winning for any force ratio. Such estimates could aid a commander in deciding the precise amount of risk they’d be willing to accept before entering into kinetic action.

The Dupuy Method – Predicting Loss of Casualties Per Day

Many things guide mission success more than just sheer numbers. We only have to look at the results of some of our biggest military defeats (Gallipoli, where we possessed 60% more troops than the Ottoman forces) and our greatest victories (Long Tan, where the enemy out numbered us 15:1) to see that there are other factors at play.

In 1996, military historian Trevor Dupuy decided to model all these factors and predict who would win, in his book Attrition: Forecasting Battle Casualties and Equipment Losses in Modern War. Dupuy preferred to use a force’s ‘power’ than its number of personnel, and described it as the following:

P = N x E x T

Where:

• P is the power of the force;
• N is the number of personnel in a force;
• E is the level of effectiveness or training of the force;
• T is the advantage gained or lost from terrain as well as factors such as surprise (where 1 is neither advantage nor disadvantage).

Dupuy derived these models by drawing upon historical data from World Wars I and II, and the Arab-Israeli Wars. The only real subjective measure here is the effectiveness, or level of training of an opposing force, relative to your force.

Michael O’Hanlon built upon this in his 2009 book The Science of War, using some more contemporary examples.  He estimated that a South Korean soldier is three times more ‘effective’ than their North Korean counterpart, whereas American forces had an effectiveness of 10:1 in the 1991 and 2003 wars against Iraq.

Using these figures, we can predict how large forces will fare when pitted against each other.

What if North Korea were to invade South Korea? North Korea has a regular Army of 1 million strong, compared to South Korea’s 500,000. Assuming neither gain nor lose any advantage from terrain, their ‘power’ can be calculated quickly – showing that even though the North Koreans outnumber the South 2:1, South Korea’s Army generates a power ratio that is 3:2.

The second stage of the Dupuy method involves predicting the daily number of casualties each force will experience. For a given day, with Force A pitted against Force B, the number of causalities lost by A will be:

Where:

• F is the ‘normalising factor’. This is the same for each side, as a rough gauge of the intensity of combat [3];
• N is the number of Force A’s personnel;
• P is the power of the relevant forces, A and B;

Since this is an iterative process and the number of forces will change each day depending on the casualties sustained the previous day, we can rewrite the generic formula for calculating the number of a force’s personnel using sequence notation [4].

Where:

• t is the respective day since the kinetic action began;
• N is the number of personnel in Force A;
• M is the number of personnel in Force B;
• F is the normalising factor;
• E is the effectiveness of A and B, respectively.

This can be developed into a single polynomial equation, only reliant upon the starting force numbers, the days since the battle started, and the effectiveness ratio of the two forces. Using this, you’re able to predict how many troops each side will have on any given day, and even the duration of the conflict.

Admittedly, many of these terms are difficult to know ahead of time, especially with a large degree of confidence. Equations such as those featured in Dupuy’s method will always be better for historical analysis than predictive planning. However, this doesn’t negate their role in planning altogether. In his book, The Future of Land Warfare, Michael O’Hanlon extensively draws upon the predictive powers of Dupuy’s method to advocate for a one-million-strong US Army, needed to combat a most dangerous course of action as predicted by the same equations as above.

Additionally, the Dupuy Institute, founded by and in memory of Colonel Trevor N. Dupuy, has even commercialised on its method by selling it’s rights to hopeful tacticians for a cool \$178,000 USD. In the words of the institute, “amass[ing] historical data… [allow us to use] actual battlefield experience to understand all dimensions of combat”. Such methods are “suitable for planning, for analysis, and for examining a variety of combat situations”, merging “operations research with historical trends, actual combat data, and real world perspectives creating applied military history in its most useful sense”.

Conclusion

While not designed to replace the clever and creative tactics that must come with command, applying maths and the scientific method to war can enhance our decision making and planning processes.

Even though quantitative analysis adds rigor to any plan, there still remains much subjectivity. For example, interpreting what force multiplier you’ll be able to generate based upon your assessment of the terrain, or how much risk you want to tolerate before your mission is impacted, is only built upon through experience.

Even with extensive mathematics, the art of war is just as important as the science.

Notes

[1] Epstein’s study also considers the instance where an attacker would take on an enemy with more than 3 times the personnel – ie a force ratio of 1:3. Remarkably, in such instances the attacker won 60% of the time. I’ve excluded this statistic from my analysis as Epstein’s research on such instances only used a total of 5 cases – a relatively small sample size. If anything, it shows that if good tactics and thorough planning are used, a commander can win against a defending force more than triple its size.

[2] The solid line on the graph represents the trendline itself, labelled with its respective formula and R squared value. The broken lines either side of the trendline represent the appropriate uncertainty of the trendline, calculated using the formula:

For both trendlines, this yielded an uncertainty of 4% and a confidence level of 75%.

[3] We can think of ‘F’ as the intensity of the battle, where 0 represents no conflict at all. The value for F consistently used is 0.01, which is considered a relatively high number. Anything much higher, when conflicts stretch more than a few months, the equation begins to fail.

[4] While on the surface these two sequences look arithmetic, it is actually much more complex, since both sequences rely upon each other as inputs in order to be computed. We can however, use these formulas to generate the first few numbers in the sequence (relative to the starting forces), and then use a Vandermonde matrix to interpolate a polynomial of degree 2 for the general formula.

About the author: Lieutenant Brody Hannan is the Adjutant of the 4th/3rd Battalion, the Royal New South Wales Regiment. Views expressed are his own.

Cover Image Credit: Defence Image Gallery, Petty Officer Damian Pawlenko