Complexity

Well, once again I have been lurking on the MiniaturesRulesDesign Yahoo! Group as an interesting discussion has developed, this time about wargames, rules, and complex adaptive systems.

Now, I’m no expert on the latter, but they do sound interesting. The idea seems to be that, for example, an army is not just a system of automata, responding in given ways to given stimuli, but a complex system of units and individuals that can adapt to a changing environment. A similar sort of view can exist of other systems, be they governments, organisations, industrial firms or whatever.

A further idea within the complex adaptive system view seems to be something along the lines of non-linear feedback. A given stimulus can have a range of results, and how that range of results and the specific outcome is determined is somewhat obscure to us (in fact, it could be entirely closed to everyone).

I suppose that an example of this is in the classic Horse and Musket era exchange of musketry, which could have outcomes ranging from no casualties but both sides running away, to massive casualties on one side but the other side running away, to both sides standing shooting each other to bits.

Put like that, the problems with this approach start to become clearer. Firstly, how do you determine the range of outcomes? In a clear sense they are context driven: the possible range of outcomes from an exchange of musketry is determinable, both by ‘common sense’ and also by looking at historical precedents, but this range is, obviously, not the same as the range of outcomes for, say, a cavalry clash, or artillery bombardment or even an infantry charge.

The second problem is, of course, how do we decide on the probabilities of each outcome? Assuming that the outcomes are clearly distinguishable from each other, we need to know the probability of outcome A (say, both sides running for it with minimal casualties) against outcome B (say, few casualties and both sides standing for another go). Given the environmental factors that go into this sort of calculation it is very hard to see that we could, in a reasonable time, determine them in any other way than a simple guess.

There is yet another problem, which is contained within the expression ‘non-linear’. We, as human beings, like linear systems. They are comfortable, they are predictable and we can weigh the outcomes. Non-linear systems are not; a slight change in the input can produce wildly differing outcomes. So far as our preconceptions go, they are unpredictable. This sort of process is often called ‘chaos’, although a better term in deterministic chaos’.

It is a well-known, but somewhat surprising fact, that even linear systems can exhibit chaotic behaviour. The driving equations do not need to be themselves badly behaved to lead to unpredictable outcomes. A quick Google on the terms ‘deterministic chaos’ will give you any number of sites which will explain the mathematics and physics of the systems, and also display pretty pictures of fractal systems (which many chaotic systems give rise to). These were all the rage as T-shirt designs a few years ago as mathematicians and physicists thought it made them look cool.

Anyway, to get back to the wargaming discussion, it was posited that we need non-linear systems in order to show the complex adaptive systems behaviour of armies in combat. Wargame rules, it was pointed out, are usually linear. Our units take a few casualties and are degraded in performance; they take a few more and get less effective, and so on, until they collapse. Even adding a bit of a random factor does not detract from the essential linear nature of the rules. Whoever argued this (and I do not recall who it was, but they did seem to know what they were talking about in terms of complex systems), also argued that we need complex adaptive rules to simulate the complex adaptive behaviour.

It is about here that I, and some other parted company from the idea. I think others did because, short of running lengthy computer simulations of our wargames, we are never going to manage to write, still less play, such a game. I think this is true, but that is not my principle problem; possibly it is because of my physics background, but I do not think we need non-linearity to get chaos.

For example, the logistic map is a nice, linear equation, which looks something like this:

X[n+1] = KX[n](1-X[n])

The X here is the variable we are interested in (classically, it is a population of moths), the n is the iteration, and K is a numerical factor between 0 and 4. For K between 0 and 1, the solution falls to zero and stays there. For k between 1 and 3, the solution converges to a stable number.

So far, so linear.

But for K>3, some interesting things start to happen. Firstly, the solution bifurcates, that is, you get two solutions and the system settles for one of them, but you cannot tell by looking at the initial conditions which one it will go for. As you increase K, the bifurcated solutions bifurcate, and then do it again and again until the whole phase space of the solution is filled with possible solutions, but you cannot tell which one you will land up in. you can see the effect on the Wikipaedia article for the logistic map: http://en.wikipedia.org/wiki/Logistic_map

You might think that that is complex enough, but it is just a start with deterministic chaos.

The point is that even a nicely behaved linear (OK, quadratic, but we can actually cope with that) and deterministic system can show unpredictable behaviour. I suppose that it is possible that a full complex adaptive system to battle simulation (I think you would no longer be playing a wargame) might give you some interesting insights into what might have happened, but I suspect that it might be easier, and more fun, to include something which, in effect, gives bifurcations of outcomes to given stimuli.