Probability and Wargaming

A few posts ago, Adam was worrying about being able to compute probabilities for wargames, and then the discussion turned on whether this was a particularly useful thing to be able to do. I suspect that opinions may vary on this, so I would like to have an explore of the space which the question opens. Unfortunately, the question turns on probability, which is notoriously difficult to understand (humans are not very good at calculating probabilities) and mathematical, but I will try to talk in concepts rather than maths.

To start with, consider something which is called the grand canonical ensemble. This is a statistical physics term. To see what it means consider a room at normal atmospheric pressure and temperature.  Now this will have a fixed number of particles in it, and they will be arranged in a particular way. The grand canonical ensemble is a assembly of all the possible ways that the particles can be arranged in, from the one where they all clump in a corner, to the one where they are all exactly evenly spaced throughout the room.

Now, the grand canonical ensemble is a concept which we never find in real life, but mathematically it allows us to make links from the distributions of the particles to the normal situation we find in the room, that is the most likely distribution. All we need to do is to count the different ways that the particles can be distributed. Of course all the particles look the same, and so some different distributions will be effectively the same. If we count all the similar looking states as the same, we find that the normal, more or less even distribution is overwhelmingly the most probable one.

Which is just as well, as otherwise, every once in a while, we would walk into a room and suffocate.

Now, it has to be admitted that battles are not of this type, but it does, I think, give us a way in to considering the next thing, which is a philosophical item, arising from what is called modal logic.

Consider an action, and the whole range of possible outcomes, suppose the action is firing a shell from a tank gun. Now, there is a range of possible outcomes to that action – say we hit the target, or miss by 5 meters or the shell bounces off the target. For each outcome, there is a possible world. In one world, the target has a hole in it, in another the shell has left a crater 5 meters away and so on. Each of these is a possible outcome, and so, by analogy with the grand canonical ensemble, we have an ensemble of possible worlds.

Now, to some extent, we can calculate the probability of the outcomes, of each individual possible world, at least in principle. We might make our gunners fire 100 shells, and find that they hit the target with 25, miss by 5 m with 25, by 10 m with 25 and by 20 m with 25. Pretty poos shooting, you might think, but I said I would try to keep the maths simple.

So we can say that in a quarter of our possible worlds the target has been hit. Of course, the complications then start: how badly has it been hit? Do we have a mission kill or simply a glancing shot and so on. We can add probabilities, and hence possible worlds to this, but I’m sure you can spot the difficulty into which we are madly rushing.

The possible worlds which we are considering are not only multiplying at an alarming rate, but they are also becoming conditional on some previous possible world being the case. So if we have a 25 per cent chance of hitting the target, we then have a (say) 25 per cent chance of disabling it, we have a 12.5 per cent chance of landing up in a mission kill final possible world.

Now, while this is fairly all right to consider as a single event, even though for a single event the contingency is starting to get on top of us, as the number of possible worlds multiply, we do, inevitably, have another problem.

My manifold of possible worlds currently relate to a single instance of a shell being fired. I do not know that statistics, but my understanding of a tank battle is that several hundred, or thousand, or even more, tank shells would be fired. Clearly, the manifold of possible world outcomes becomes too difficult to handle in any realistic sense. The complexity and inter-relatedness of our possible worlds is going to cripple any attempt to treat in a sensible way.

For example, in one possible world tank A may fire at tank B and miss, and then tank B fires at tank C and hits. But if tank A had actually hit and disabled tank B, tank B would not be available to take its pot shot at C. Our possible worlds are not only running wild, but tripping over each other.

Clearly, while this sort of probabilistic approach is very tempting to the wargamer, the sheer complexity of the outcomes, in even a simple scenario, becomes overwhelming. Once we start to try to account for crew training, fatigue and so on it becomes impossibly complex, particularly as we can, in fact, only guess at the relevant probabilities.

So, aside from despairing and going fishing instead of wargaming, what can we do? I think this is where we deploy our secret weapon, abstraction. We cannot trace the probability of each shell in a battle, but we can look at the overall results and, as it were, work from the top (the outcome) down to some sort of probability of a given action.

But it is only ever an abstraction, I think.  I cannot see that it provides any particular insight into the detail of what has happened, only into the outcome as opposed to other outcomes. That is, our need to abstract means that any insight accrued is limited by the abstraction itself. A wargame is only ever going to give a highly granular insight into why stuff happened.