Saturday 9 June 2012

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.


  1. Hi,

    I could not agree more that starting from overall results and stepping down, in a way guessing partial input of factors, is a good way to go.

    There is one major reason for this. We have got, from historical accounts, two kind of data which may be countable. This is casaulties and army strength. Although many historical accounts are of dubious credibility, comparing them we can get some range of numbers, which may probably cover the real number. As we can also try to extract and correct the extreme results by comparing different accounts, our guessing may be reasonably close to real in many cases.

    Another thing we know* about is tactical results. For example we know* there are no accounts of archers winning over hoplites without support; we also know that lightly armed javelin-throwers could not stand an attack from trained heavy infantry in closed formation and so on. Collecting tactical results and numbers of casaulties as compared to army strength gives us some data which may be used in guessing probabilities in our games. It is equation with many unknowns, but at least we have got some knowns also. And the happy thing is, that overall results should be satisfactory - we start from the base treated as "established" and end comparing with this base. If we do our homework right, it is a self-realizing wish.

    *Of course, our knowledge is faulty for many reasons, but we have to base on something or go crazy. I think you were writing about this before.

    On the other hand, we can try to go from bottom to top, but this is hard and unrewarding. In the Red Army they liked statistics a lot. Soviet historical studies are full of numbers. Apart from their usually very low credibility, they present different pieces of interesting data. For example (based on extracts from an article of colonel W.Pruncow, 11 June 1944) a regular infantry battalion of 600 soliders could fire away 7150 bullets and 31500 pieces of shrapnel per minute, not counting hand grenades. Probability of hitting a man-sized target firing a rifle at 300m is close to 1 (from theory and practice -all this is true). Col.Pruncow conclusions go in another direction, but one could guess "if this is so, any foot enemy should be stopped and slaughtered in several minutes". Seems like everything is correct. Of course the enemy is cunning and will try to disable defences before attacking. Yet still, if we deduct 90% of rifle firepower and 100% of mortars and hand grandes (I think this is reduction one can be happy with) around 715 enemies should be hit by the remaining defenders in just one minute. This is like twice the number of actual loses of a German division attacking strong positions for a whole day during Kursk battle. And they were assaulting much stronger units than an infantry battalion. :) So theory and practice is like lightyears away.

    I suppose this gives a nice example of problems which we meet when trying to built rules from the bottom side.

    Best regards,

  2. Hi,

    I think, most rules have big fudge factors. For example, many older rule sets started from the premise of how far a man could walk in a minute, or 5 minutes and worked out their ground scale from that, but then had to fudge it because bodies of troops have never moved that fast.

    I suspect that the real point lying behind this is that our preconceptions, our a priori concepts of how stuff should work just do not apply to a real battle, and so switching back from them to the real world requires a lot of blurring at the edges.

    The overall impact of this seems to be that we can only hope to simulate an overall outcome (within a range of possible outcomes) not every step of how we got there. Keeping track of each bullet is theoretically possible, but if we do not know when they were fired in 'real life' then it does become rather pointless.

    I think your bottom up example is a good case in point. Theoretically the German divisions should have been mincemeat, in reality they were not. Why? Well, I don't know, but i guess for starters we can assume that not all the bullets were actually aimed at anyone.