Hi Steemians! Today I will treat a simple mathematical model for medieval battles. This post is mostly written as a story. So I think it is easy to read. Technicalities have been put in a separate section at the end. :0)

The mission

Suppose you are an army general in medieval times. An evil Baron has assasinated the King's youngest son. So the King has sent you to invade the Baron's lands to slaugther him and his men. You have a total of 2000 infantry soldiers under your command. Your whole army is armed with spears.

Scouting

The day before the invasion you have sent scouts to your enemy to see how large their army is. When the scouts come back they have good and bad news. The bad news is that the Baron received reinforcements and now has 3000 infantry ready. Furthermore, these soldiers are armed with spears and all seem equally well-trained as yours. The good news is that the Baron's army is divided over three different locations. They hand you a map:

500 men are stationed at the Baron X's gold mines, location (a), 700 men are protecting the Baron's graneries, location (b), and another 1800 men are located at the Baron's palace, location (c). They also mention that these three locations are quite a long distance away from each other. So the Baron does not have enough time to send reinforcement when one location is attacked. The scouts also drew you a nice overview of the situation:

Strategic idea

So what are your strategic options? It makes sense to not split your forces but keep them all together. Which location should you attack first (a),(b),(c)? And where to attack next? Does it even matter? To answer these questions you need information about how many causualties are suffered during each battle. These answers will be given by a modelling the battle process.

Lanchester battle model

Fortunately, Archmaester Lanchester has offered his services to help you with the mathematical model. The Archmaester takes out his quill and a big piece of paper and says "Let us first begin with assumptions".

Source

1. "I will assume that since spears are long weapons. That soldiers on both sides are in striking distance of the enemy"

2. "My recent documentation shows that one of our soldiers kills on average 1/10 enemy soldier per minute. We will assume that is also the case for the Baron's soldiers "

"And now for some notation. Let t denote time per minutes with t=0 corresponding to the start of the battle. Denote by A(t) your army size at time t and by B(t) the Baron's army size at time t. Consequently, A(0) your armies' starting size and B(0) equals the Baron's army starting size."

"So we observe: our casualties per min at time t is B(t)/10, similary, the casualties per min at time t for the Baron is A(t)/10.

Archmaester Lanchester then starts writing down symbols you have never seen before (If you are interested in what he writes down check the technical appendix below). After a few minutes he arrives at the following two formulas:

"The battle is won at a time t=T when the Baron's soldiers are reduced to zero, B(T)=0, and the battle is lost at a time t=T when your soldiers are reduced to zero, A(T)=0. Armed with these formulas you can predict everything "

Results

Recall that 500 (a) 700 (b) 1800 (c). Then using Equation (1) you can write down all the possible battle scenarios and see if you will win and compute how many soldiers will survive:

(a)->(b)->(c) : 141 soldiers survive
(a)->(c)->(b) : 141 soldiers survive
(b)->(a)->(c) : 141 soldiers survive
(b)->(c)->(a) : 141 soldiers survive
(c)->(a)->(b) : 141 soldiers survive
(c)->(b)->(a) : 141 soldiers survive

You make the observation that it is does not matter (the reason for this is given in the Technical Appendix)! You always win! So finally you can rest assured that you will defeat the Baron!

God Speed - Edmund Leighton

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Conclusion/Reflection

In the current setting we computed that independent of where we attack we will always defeat the Baron with the same amount of casualties. Now suppose that the Baron would have rearranged his soldiers in the following way: 300 (a) 800 (b) 1900 (c). Using the formulas in (1) you will see that it is not possible to defeat the Baron in any scenario. Concluding:

The distribution of the Baron's soldiers over the three locations determines if you can defeat him or not

Of course it should be noted that this is a highly simplified model. Battle formation, moral etc. are not included in this model. In addition, we expect that the casualties taken is a discrete process. However, in this model it is continuous since the Equations (1) can be evaluated for any t.

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Sources and Further reading

Source top photo - Battle of Grunwald by Jan Matjeko

The model treated in this post is the most simple version of the Lanchester model. For the wiki click here. I gave my own spin on explaining it.

This model is connected to predator-prey models. More specifically, it is connected to Lotka-Volterra, see this wiki.

Equations written using Quicklatex it is free to use!

Second figure made using Inkscape

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Technical appendix (for those who completed a calculus course)

The equations resulting from Archmaester Lanchester's modelling assumptions give a linear ODE:

Since it is a linear ODE it is easy to compute solution vectors and the flow.

Recall that in the result section we obtained the same results for any battle scenario. By using the equations in (1) you can prove that independently of the distribution of the Baron's soldiers the final result is fixed. In a mathematical setting we say that the underlying operators commute.

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Thank you!

Thanks for being so kind to read my post. You are awesome! Please follow me if you enjoyed it. If you have any questions just post them below and I will answer them. Or if you might have a nice topic you want me to cover also let me know below. :o)

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Spotted Owl - Jessie Eastland