# The mathematics of war

War! Hngh!

What is it good for? Well, the development of some interesting mathematics, if nothing else. And raised eyebrows. And scheming/strategising.

Last Monday morning (yes yes, I know – been busy, ‘k?!) I successfully managed to hie myself off to Dr Sean Gourley’s speech about, you guessed it, the mathematics of war.

Or, to be particular, the mathematics behind insurgencies’ ability to stave off defeat by much larger, more well-equipped forces. Asymmetric warfare, in other words.

I am now going to attempt to share with you the (hopefully not too garbled) notes and learnings which I took therefrom.

*Nash equilibria, open source intelligence, and oranges*

Firstly, a brief explanation. Up until relatively recently – the last coupla decades, really – warfare has been a more traditional affair. Two sides, lined up against each other, having it out. Clear knowledge of who one’s enemy is, where they are, and at least to some extent, what they’re up to.

And game theory (more specifically, Nash equilibria) was able to adequately help model how such conflicts might go.

Things are now different. The concept of ‘your enemy’ has become far more complicated. There are many conflicts all over the world at the moment, each featuring its own cast of insurgents/guerillas/terrorists/organised criminals and, um, often at least one ‘major’ force.

Gourley was interested in collecting data on these sorts of things (attack size, when, which conflict, deaths, injuries etc), but of course didn’t have official access to such data. Because govts like to keep them to themselves. So, how to get the data he wanted? The answer: open source intelligence.

Yes, once again, ‘open source’ raises its beautiful shiny head above the parapet and grins charmingly at us.

Open source intelligence is, simply, the information that one can gather from citizen reporting, the news, NGO stats and so forth. Yes, there’s a lot of noise, but there’s also some signal in there. Data, in other words. Even looking for second order effects can help one determine if something’s going on. Perhaps the classic story about open source intelligence and second order effects is about the Alliance. Unable to be directly sure whether they had successfully bombed bridges in Germany, they looked at the price of oranges in cities, which had to be imported. Spikes in the price of this acidic, vitamin C-containing fruit corresponded to bridges going down.

*We haz data – now what?*

So yes. Gourley and co. collected a bunch of data for different conflicts around the world (Iraq, Colombia, etc), and then set about analysing it. What they found was interesting – when deaths were plotted against cumulative frequency for a number of conflicts on a log-log graph, the resulting line looked an awful lot like something adhering to the power law, with an alpha (slope) hovering around 2.5.

Or, to put it more simply: the data suggested that insurgent conflicts (fought in different places, for different reasons) around the world might cluster around this value.

The next step was to try to explain this phenomenon.

[Where P (the probability of an attack killing x no. people in a time window t) = a constant multiplied by x (the size of the attack), raised to the power of negative alpha (which is, roughly speaking, the slope of the line when plotted on a log-log graph)]

Lost yet?

For those of you not terribly comfortable with the equation, not to worry. Of more interest is what it means.

Basically, it looks like the number of people killed in an attack is correlated with the strength of the attacking group. And it’s worth being clear here – that’s *strength*, not size. A smaller group of people with oodles of moolah and weaponry is going to do more damage than a larger group with less moolah and weapons. So one could look at alpha as the distribution of attack strengths, which leads us towards an organisational structure.

How?

Well, ask yourself: how does one organise one’s forces to best fight the opposition? Now, bearing in mind that insurgencies aren’t centrally controlled but rather self-organising, how does the insurgency organise itself to take on a much stronger, conventional armed force?

There are a couple of possibilities – one might be taking the whole force, and dividing it up equally. But the attacks that come out of that look more like a Gaussian distribution, not a power law distribution. Which means that’s not how insurgencies are organising.

Instead, the organisational structure which best fits what Gourley et al observed, was that each group would have a small number of groups which killed lots of people, lots of groups which killed very few people (per attack), and a bunch of groups in the middle. Indeed, the maths suggests that it’s 316 times more difficult to kill ten people than it is one person, and 316 times more difficult again to kill 100 rather than 10 people. And that multiplying exponent seems to stay as one looks at different conflicts.

Now, here’s where biology had some lessons for the mathematicians (hah!*). Each group is subjected to forces of coalescence and fragmentation.

With coalescence, there can be a formation bias towards the formation of large groups or towards the formation of small groups. And the actions can be geographic (i.e. dominated by people one is near) in nature, or non-geographic. (hello mobile phones, internet etc).

Similarly, with fragmentation, groups can either split into two and factionalise, or they can split into many parts/shatter.

The interaction between these factors gives rise to different distributions – an understanding of how allowed Gourley et al to start looking at which structures best fit insurgencies.

*How do insurgent organisational structures behave?*

They found that the formation bias was towards large groups , with connections that aren’t geographical in nature. Of course, the stronger they get, the more liable they’d be to find themselves on the radar of whichever the larger, conventional force is. Who would then attack them.

Classic tall poppy syndrome stuff.

Said insurgent group would proceed to shatter (rather than factionalise). However, it wouldn’t shatter randomly – instead, the next most successful group starts to accrete members. So there’s this fluid system which allows a great deal of learning and innovation, as opposed to the conventional forces which have static, rigidly defined operating procedures.

All nicely explained in the picture below.

*Alpha – higher or lower?*

So interesting patterns around alpha – 2.5 appears to be the value at which an insurgency is stably/sustainably fighting against the larger/stronger force. That is, the conflict won’t end with an alpha of 2.5. But what happens when alpha is higher or lower?

If one can drive alpha higher, then one drives the insurgency towards fragmented, fluid groups, and more groups (basically, more towards the guerilla feel). These tend to peter out eventually.

If one can drive alpha lower, one has an insurgency made up of stronger, more robust groups, but fewer of them (more like a conventional war). There is the possibility of an actual win/defeat here.

How does one decide whether to drive alpha higher or lower? Well, look at where alpha is currently, and has been over time, and from there make a decision about what’s achievable.

Fascinatingly, the strategy that the maths suggests is counterintuitive – attack the weak groups, not the strong ones.

*Applications*

Now, what are the applications of this? Well, it’s certainly got the US NSA and other such organisations interested, because of what it might be able to teach them about how to defeat the insurgencies with which they’re involved.

The model’s powerful – it can suggest how many insurgent groups are active, and how to deal with them. Because insurgent wars which drag on kill an awful lot of people.

Which helps answer the question: why did Gourley start this research? Well, he’s of the belief that the more we understand war, and how people die, the more we can stop it happening. Hear hear 🙂

Oh yes, and there are other applications – the model can probably be drawn out to look more generally at how small groups successfully fight large groups. As with companies (how small tech startups beat large tech giants), or even ,medicine (how drugs attack tumours in the body).

Brilliant stuff!

I’m hoping to be able to podcast Gourley’s talk (and the questions afterwards!), but while I wait for permission, here’s a brief TED talk he gave.

**UPDATE: Sean’s given me permission to podcast his talk. /celebrates**

**You can find it here**.
**

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Oh yeah: and I highly recommend the BBC’s “The Story of Maths”, hosted by Oxford professor Marcus du Sautoy.

* You see? Biology’s not just, ahem, “stamp collecting”…

** Full props to the Internet Archive for letting people upload files, for free 🙂

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*References:*

Bohorquez, J., Gourley, S., Dixon, A., Spagat, M., & Johnson, N. (2009). Common ecology quantifies human insurgency Nature, 462 (7275), 911-914 DOI: 10.1038/nature08631

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