Elegance and Complexity

One of the joys of having written a set of wargame rules is, of course, the opportunity for people to criticise them (I may be being a tiny bit sarcastic, here). Recently, the order mechanics of Polemos: SPQR were praised by someone, who then went on to say that the combat mechanics were far too complex because they had 20 factors which needed to be added up.

Now, there are several responses, more or less grumpy, which I could make to those sorts of comments, along the lines of ‘you might think this is complex, but look at those rules over there’, or possibly some insulting reference to the fact that adding and subtracting numbers is not actually complex per se, and I’m sure that given some thought I could come up with other, more or less crushing, responses.

On the other hand, I’m not sure that any of these sorts of responses would help and, after all, my critic did describe the order and movement rule as elegant (although, of course, someone else described them as a ponderous game mechanic – I guess you cannot please all the people all the time).

I have written before about the problems of complexity in battles, and I think it is generally agreed that a battle is a complex thing which cannot be handed by probabilistic mathematics, and least within any reasonable time frame. So I will focus on the other factor that is mentioned in this context, that of elegance.

Now, in my chequered past, I was a physicist and, as such, encountered, as regular readers of this blog are probably painfully aware of by now, mathematical models. The interesting thing about these things, in the context here, is that some models were described as elegant, and even as a struggling student with a reasonably low mathematical ability, I could see that some were, indeed, aesthetically pleasing.

The most obvious place where this occurs in modern physics is in the theories of quantum mechanics (QM). It is not a widely known fact that there are three basic approaches to QM. These are the Schrodinger equation, Heisenberg’s uncertainty principle and Dirac’s spinor.

I will not go into detail, but I will say this. Of the three, I used Schrodinger’s equation most, because it gave tractable answers that could be compared with observations (which I what I did as an active researcher). However, the most elegant of the theories was that of Dirac. I recall sitting in a tutorial while my tutor expounded Dirac’s theory and came up, inevitably with a 4 vector (a single column of items, with four entries). The top two ever electrons, and the question was ‘what are the bottom two?’.

The answer, of course, was “anti-electrons”, and that piece of work both predicted anti-matter and won Dirac a Nobel prize. But how did anyone know at the time it was right? Why would anyone go and look for anti-matter, which is a pretty counter-intuitive sort of thing?

The answer, of course, is in the elegance of the theory. Nothing is forced; anti-matter arises quite naturally in the mathematics. Dirac knew it was correct because it was an elegant theory.  

Now, of course, we have to let complexity have a bit of a play. In some senses, particle physics is easy, because the objects at its heart are simple. Bosons, electrons, leptons and so on are, at core, fairly simple, (so far as we know) unstructured objects. They are tractable by mathematics, conceptulisable (if not imaginable) by our minds, and, of course can be observed (or strictly speaking, their effects can be observed) in the world.

On the other hand, someone once said ‘Understanding atomic physics is child’s play compared with understanding child’s play’. Human behaviour, even in relatively benign environments, is not simple, and cannot be predicted by mathematics, no matter how elegant.

That being the case, how can a set of wargame rules ever be described as elegant? Not only are we dealing with humans on mass, under very stressful circumstances, but we cannot even start a mathematical model of the environment. There are too many factors to take into account.

Let me take an example, that of DBA. There are 17 basic troop types, with a factor against foot and one against mounted. There are 6 tactical factors. However, by my counting, there are 78 different combat outcomes. This is generally accepted to be a fairly elegant example of a set of wargame rules, but it has to be accepted that the complexity is hidden in the combat outcomes; as I recall, these are the bits that people find most confusing.

Secondly, let me have a look at PM: SPQR. There are 11 troop types, each with 4 factors, and 22 tactical factors (not all of which apply, of course). However, there are 5 outcomes to combat.

If you add on to this analysis the fact that PM: SPQR has provision for different grades of troops (so is more like DBM than DBA in this respect), then, I submit, that it is no more complex than DBA. The difference, of course, is where the complexity is hidden.

Now, I am not claiming that PM: SPQR is the simplest, most elegant system that is possible. I do think that some aspects could be tidies up, but I do claim that, by comparison with a widely used fast play rule set, it is no more complex. After all, the close combat factors are clearly set out in two tables, and a little experience means that you can glance at the table top situation and work out the factors very quickly. In DBA, I’ve never managed that, I always have to read the outcomes (it helps, I discovered, if you read them out loud).

So where are we left?

I think the answer is that a battle is inherently complex, and that complexity has to be represented somewhere. In SPQR it is in the initial factors, in DBA it is in the combat outcomes. There just does not seem to be an elegant solution to this, so, to return to my QM analogy, we will just have to stick with Schrodinger, as Dirac seems to be beyond our reach.