Probably Probability

Oh dear. I’ve been reading about probability and emergence again, and, even worse, pondering the implications for wargaming. This is likely not to end well.

The issue is this. Consider a range of outcomes. A priori, we have no reason to choose one over another, and so the logical thing to do is to assign an equal probability to each of them. Given that probability is defined on a scale of 0 to 1, and that we have A possible outcomes, the probability for each specific outcome will be 1/A. Of course, we hit a slight issue if A becomes infinite, but let us ignore that in the interests of sanity and the fact that we, as human beings, do not really deal in the infinite.

Now, suppose that the system we are considering actually activates, and we get some state of affairs, call it Y. This has happened, of course, with probability 1/A, but is not the actual state of affairs. All the other states of affairs now vanish, as this is the one we have to deal with.

A number of things can now happen. It is possible that the state of affairs Y is not stable, and that it collapses back into the undetermined state we first had. Thus, while momentarily being different, Y has no long term effect on the reality we observe. In terms of, say, thermodynamic theory, this is the equivalent of all the molecules in a room being found in one corner, but then spreading out again before anyone attempts to breathe in. It was a possibly interesting event, but of no lasting consequence.

Another possibility is that the state of affairs Y is stable. This then means that another manifold of possibilities will present itself as outcomes from this state of affairs. These will, of course, be different, at least in principle, from the previous manifold of possibilities, and, potentially, have different weightings of probabilities. Thus, the selection of state of affairs Y has dictated that another set of possible outcomes is available, which is different from the first set.

Yet another possibility is that state of affairs Y is stable, but is then disrupted by event E. Event E might be more or less probably in Y, but is such that instead of a gently evolving system depending of even probability distributions, E resets the system in some way, meaning that the system we are looking at is now in a very different set of circumstances. Let us call this state of affairs X.

We therefore have a set of states of affairs, the original one (call it W), the newly selected one, Y, into which W evolves with probability 1/A, and the disrupted Y, which I have called X, which happens with probability P(E), where P(E) has to be measured on Y, of course.

What we have here, therefore, is a system that shows both stability and the possibility of sudden change. The paradigm example of this would be the solar system, in its current form. The state of affairs W would be the configuration of the planets as measured at some time. Y would be the expected configuration at some later time, given what we know about planetary motion, Kepler’s laws and so on.

In this scenario, event E would be some massive object passing by the solar system and disrupting the planetary system. This may well be a low probability event, but if it happened it would be very noticeable.

So, what has this to do with wargaming?

Well, consider your army as the system W. It will evolve in certain ways, to state of affairs Y, given tis orders, the terrain and so on. You expect it to behave itself, to evolve in a fairly predictable way, just like the planets. You tell a unit to go there, and it goes.

So, what are our events, E?

Probably, we do not actually have too many of them. Of course, a unit can be shot to bits by enemy fire, or fail a morale roll, but I suspect it is disputable as to whether these are not simply another state of affairs Y, just a less desirable one from the point of view of the player.

So what sort of thing could create a truly disruptive event E? How about, for example, the sudden emergence of an enemy force behind your left wing? That, I suspect, could be quite a disruptive thing. And yet, it seems to me that often our rules just allow our troops to raise their eyebrows a little, perhaps sigh theatrically, turn, and face the new foe.

Or how about the above scenario plus the misinterpretation of incoming banners, giving the opportunity to shout ‘Treason’ at least for one side? As happened at Barnet, this can be rather disruptive, too. But is it, can it be, accommodated within our rule sets.

The point is, I think, that our rule sets, and, probably, our views of wargaming are based within the paradigm of gently evolving, logically acceptable events. This phalanx advanced and, after some resistance, the enemy run away leaving it victorious. That tank shoots at this one, and it may or may not disable it, but in the overall flow of events it does not make a huge difference, only that in the next turn this tank is still available to shoot back. The new state of affairs, Y, is similar to the previous one.

It is, I think, more unusual to have an event like E in a wargame. Occasionally you could put something into a scenario, such as refighting Barnet; you would need a treachery rule for that, but on the whole we do not like such things. Our wargames are nicely logical, evolving systems with clear cut probabilities at each stage. We like to be able to give an account of them, even if that account blames the dice, because even those probabilities are accounted for in a slowly evolving system.

Of course, real life is not like that, there are events which have recently been characterised as ‘black swans’. But if we put those into a wargame, the gamer on the receiving end might feel very hard done by. It is not the way we expect the game to go. Wargames, it seems to me, rely on gently evolving probability manifolds, and we do not like the disruptive events.