## Saturday, 16 November 2013

### 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.

1. Yes it's a strange one that isn't it? It must come down to the competitive instinct - we don't mind losing as much if it's a fair fight. But a "black swan" feels such a piece of bad luck it's "unfair".

But then why should it matter it's just a game with no consequences (unlike the real thing, when you really would be justified in feeling miffed). In the gamers' world you can even claim "I would have won if E hadn't happened". Yet Event E's did happen in real warfare so surely it should be part of the range of possibilities we should be prepared to allow in our games (from time-to-time).

Another instance I think, of wargamers preferring wargames to be evenly balanced, fairly abstract games to being gamable historical recreations.

2. Interesting - I wondered what it is you have been reading, as there are a couple of points in your note with which I would take gentle issue, but then I realise that it would add nothing - my offerings would mostly be on choice of terms and other semantics, and I would show myself as the sort of fringe eccentric with a little knowledge who can't suppress the urge to put in his tuppence worth - a role which I detest when someone does it to me.

I am, however, interested in probability. I once made a living at it. No, I didn't work in a bookie's.

The difficulty with probability, and the mathematical treatment of "scientific" wargames, is exactly this lack of swans. The worst example was the famous (or infamous) Shire Publications rules by Arthur Taylor(?), in which he eliminated all randomness from the game and thus (in his own imagination, at least) elevated wargaming to some loftier, more chess-like status.

My problem is that I regularly read of battles in history which were decided - or at least severely affected - by some occurrence or situation which falls outside the normal game rules. Obvious examples might include the French infantry at Waterloo getting tangled up in the cornfields (do you have a rule for that? - I don't) - there must be hosts of others. If the scenario rules explicitly include such a rule then it's a fiddle - almost an apology.

As ever, I don't have an answer apart from this thought:

In war-games rules, the degree of variability (or randomness, or whatever - the range of possible outcomes, anyway) should tend to be larger than we might think of as realistic - especially if we are Mr Taylor. I do not like to see rules which say "cavalry cannot break a steady square of infantry" - I'd rather see "you can charge a square if you want - it's not a great idea, but there is a very remote chance that you might succeed" and let the rules and experience look after themselves. Let the swans in. If something unlikely happens in a game because of an unlikely dice-roll then that's fair enough. Something unexpected happened - we may not know what it is - history may or may not record what it was - but just today the militia broke the grenadiers.

The universe contains - all the time - countless small events. We can't track them all or measure them all - we can only see the effect. Some of the outcomes will be a surprise. That's OK.

3. You, sir, must be a Bayesian!

Now, to your point regarding special or Black Swan events throwing an unexpected or unplanned chaos element into a wargame. These intrusions into the "gently evolving probability manifolds" are not as uncommon as first assumed. Although not a wargame, the game of Monopoly contains "Chance" cards that insert disruptions into a player's progress around the game board. Featherstone, likewise, included Chance Cards into his scenarios along with his notion of Military Possibilities. These events typically are assessed at the scenario level and not at the rules level. That is, singular events pertaining only to a particular scenario.

Below are a few "E" events that I have seen included in scenarios.
- During the battle of Aspern-Essling, the French bridges across the Danube were destroyed on more than one occasion; often at inopportune times and isolating French troops on the opposite bank.
- Orders may be misinterpreted or completely ignored by subordinates or couriers may become lost or captured.
- Fear of disaster rolls. Many rules have such mechanisms which can lead to an entire army disintegrating in a single cataclysmic event begun by a single unit routing.

That is only a start.

Now, in many wargames, a combat result may be below the level of granularity of the players perception. The outcome may seem preposterous or unlikely because the player isn't privy to that level of detail.
When a surprising result such as this occurs, my first thought is that something very interesting must have happened on the battlefield to which I have no knowledge. In the case of cavalry breaking a steady square, perhaps the square opened its ranks to allow shelter to fleeing gunners or officers. Before the ranks could close the enemy cavalry were upon them?

Very interesting topic!

4. I must confess to having played more game systems where sudden and unexpected or at least unpredictable things were possible or even common than those where they weren't but I have also played games where most things were predictable and.or reactions automatic so it is a matter of player choice and preference that availability.

(for example variable length moves, card sequencing of moves and/or combat or initiative systems to vary sequence, activation systems or control /order tests with personality based results for npc commanders, simultaneous written turn orders with generous movement distances (didn't notice the chance that the light cavalry 32" away might charge that battery in the flank?), game length written order that had to be changed by a courier if you cold convince your opponent that you could see a reason to issue a change, chance cards, combat systems where results were hard to predict or at least had a wide range of possible outcomes, and lots more little thing. Lots of variety out there, a matter of choice and preference as to what sort of games you like.

5. Sometime last week, a man on the radio was talking about a football match - making the point that player S has scored twice in the second half, but should have had a hat-trick, since he missed an open goal in the first half. At this point we all throw our sandwiches at the radio, since the entire game would have been different if the earlier chance had been a goal - most importantly, the second half chances would not have occurred, though there might have been other such chances in their place. A goal would clearly have been an event E - a major change - but even a missed tackle or a doubtful offside decision changes the whole game thereafter - the consequences of not firing a single tank, in your example, are not necessarily trivial in the long term (which begs a whole pile of definitions, but no matter).

The events in a battle or a game are not independent - they are consequent upon each other in ways we probably do not understand. As I understand it, Mr Taleb's main point about black swans was that the things we can predict and estimate are likely to be dwarfed and swamped by things we never thought of and could not put numbers to in any case. The main attribute of events like E is that someone observed them and described them. Maybe it's just a matter of degree? - a major step change like a goal or a collapsed bridge will be an agreed fact, and will become an important part of any subsequent narrative. There may well be a very large number of smaller, less noticeable things which did or did not happen which ultimately also have a large effect.

I believe that I am a Bayesian, but I'm not sure what else there is.

1. The other camp is the one held by "Frequentists."

6. Is it E for 'excuse'? As in "I'd have won if my right flank hadn't collapsed."

Seriously though, when is an E not an E?
An event such as one unit being particularly badly hit due to dodgy dice luck sounds to me like just part of Y, all part of the ebb and flow or (less than) gentle evolution of the game, unless it runs off and takes the whole army with it.

An E as a single event which effects the whole outcome of the battle, or game, could be something like the slope a Waterloo being too slippery to climb, or the fog at Barnet or snowstorm at Towton, but this sort of thing could easily prevent a battle taking place at all. Historically accurate to allow for it to happen on a random factor, but it could spoil the day's gaming if you set up the table only to find that snow has closed the passes between the opposing armies.

I think I'm advocating 'E light' as part of games - random mishaps and confusions which are part of warfare and in large scale games are inherent in the dice throws. (Average dice/card luck x small 'E's = Y?) A string of unfortunate small 'E's could snowball and result in catastrophe, but I think that's ok. A major 'E' which might affect the whole battle on its own should be consigned to a specific scenario.

7. My goodness. Loads of interesting stuff here; I can't possibly comment on it all, so I'll try to abstract, and probably return to some of the points in another post.

There does seem to be something of a consensus that we like the slowly evolving game, not the one that changes suddenly. Despite Ross' counterexamples, these ideas do not seem to be terribly mainstream, although some have been around for a long time. We are all Bayseans underneath.

I suspect that this is to do with the expectation as humans that we have that the world is reasonable and rational. We expect to be able to trace reasons, even, perhaps, logically. For example, a major investigation was launched into recent terrorist outrages. Why? Well, it seems to be that we expect there to be a reason, even if that reason is the mental instability of the perpetrator.

We expect, therefore, cause and effect to be linked. Hume observed that this link couldn't be justified, but admitted that when he thought that a good game of backgammon helped. I guess he hadn't discovered wargaming. The world is an interaction of chance and necessity - random events modified by the smoothing effects of laws of nature. This is why the butterfly wing effect doesn't work - the atmosphere, while technically chaotic, also includes damping which prevents minor events growing exponentially.

I think that we struggle with the lower probability events, to get the probability of the event to be sufficiently low. This is the issue with rules which include such stuff as thirsty troops and the low sun shining in their eyes. These things happened in real battles, granted, but they appear to be low probability events. However, since we only have one stream of events to deal with, we cannot calculate the probabilities. So a rule set which allows a game stopping event 1% of the time is probably going to reproduce too many such events for us to accept as 'historical'.

Anyway, I'm very impressed by this debate, and I need to go and look up what 'frequentists' are.

And the book? Actually, it was Alister McGrath's 'Scientific Theology' volume 1, but it reminded me of some work by Bernard Lonergan in 'Insight' about emergent probability. But you probably didn't really want to know that...