"Dice & Statistics" Topic

I recently played a game with an old friend, and gaming partner / opponent of long standing. It was 7YW and we were experimenting with his new rules.

I lost, rather badly, not because of any especially stupid deployments or other tactical mistakes, but because we each started off with a Position Battery. The obvious gamer tactic was to win the artillery duel (as my friend knows, i personally think this is a flaw in his rules, but I digress …). With a gun model standing for two tubes and one D6 per tube, we were throwing, to start with, 6 D6 each, hitting on Sixes only.

All 6 of my guns were wiped out at the cost of just one of my opponent's tubes, and the game, effectively, was over.

While this is, as we say in the trade, statistically an outlier result, it is not going to be uncommon. Many gamers do not understand the difference between mean (or average or typical) results versus the mean SPREAD of results. The key is that choosing whether a result is 1-in-6, 1-in-3, etc. will have a major difference on play.

In layman's terms, if I may, examples work best.

It is true that theoretically six D6 hitting on 6 only should achieve the same mean number of hits (i.e. 1) as three D6 hitting on 5 or 6, or two D6 hitting on 4, 5, or 6. (Before I hear the howls, yes, I know that calculating the mean number of hits from multiple dice thrown at one time is technically much more complex than that, but I seek to keep things relatively simple).

However – and here is the rub – it is also true that the typical RANGE of results halves at each point. In other words, with six D6 hitting on 6 only, the range of results is 0 to 6 hits. With three D6 hitting on 5, 6, the range of results is 0 to 3 hits.

So it is true that, regardless of the mean, the typical range of results experienced is directly proportional to the odds of getting a hit with one die. And remember that in most miniatures games, we do not in reality throw enough dice to achieve a statistically large number – meaning that in practice the distribution of hits will deviate to some degree from the theoretical mean.

Another key ramification is that, because past die throws have no effect on the mean or expected future die results, the lower the likelihood of a hit with a single die, the longer and harder it is to dig yourself out of a hole created by an early bad die run, or to claw back to equality against a foe who had climbed a mountain with an early terrific die run. In fact if using D6 with 6 being the only hit, it will take 4 times as many throws to statistically even out versus having 5, 6 as a hit.

What is the real meaning of all this?

We customarily and traditionally use dice as a means of accounting for all those factors and imponderables that were beyond the control of an historical commander.

This is well and good. But in effect, the smaller the chance of a hit with a single die, the greater the element of chance you are injecting into your game.

Every rulesmaker must decide for themselves the degree to which they want their game to be ruled by the dice, and where their own personal sweet spot lies.

Agreed. It's critical, when designing a mechanism, to look not only at the expected outcome, but the probabilities of all of the possible outcomes. This, however, is not always so easy to do. For the rules with which I fiddle constantly, which use opposed pools of binary dice, I ended up writing a SAS program to examine the probabilities, dumping the resulting data into an Access database, and then pulling that into an Excel pivot table. Not everybody has access to these tools. I probably could have done the whole exercise in Excel, but it would have been enormously more laborious (and I'm a SAS programmer by trade, anyway).

On the plus side, having done the requisite legwork, I'm pretty happy with the results. The mechanism performs pretty much exactly as I had wanted it to.

Doug

I had several rounds of 'you're mechanism is flawed' with a friend. At the start of you turn, you roll for an activation, say 8 or better on a 20-side. Then you roll for an additional activations by adding 2 to the previous trigger point each attempt, all in one turn. As you can see, if you roll poorly (me) and your opponent rolls well, they can get 5 0r 6 activations to your 1 (or none) and by the time your turn rolls around again, you have nothing to command but smoking vehicles and an army of the dead. His defense was always 'Well, it shouldn't happen that often' to which I would reply 'But it CAN and when it does, it destroys the game.' Eventually, after a few games, he decided to make the additional activations a little harder by adding 4 to the trigger point. One must always weigh the likelihood of a rare event by the gravity of its effect on the game.

The OP has at least two good points. I have seen rules where the designer has mods for the chance to hit, or dumber of dice or conditions for re-rolls all stirred into a soup, and I feel certain they haven't really grasped the full consequences of each.

Secondly a game nominally of 2-4 hours play should not be killed off by a single early event or decision by a player – sure one side might have bungled their deployment or orders, but a good game design gives them a chance to fix the problem, albeit the other side might pin them to the ropes and make it difficult. But there is still a game as each side plays it out.

Regards

David F Brown

It's another iteration of the game versus simulation argument. Yes, it could happen that side A gets to reduce side B to smoking wreckage before side B can get its thumb out and react. There are plenty of historical precedents for it, and therefore a simulation should allow it to happen, even if it's improbable. A game, though … well … nobody enjoys playing the role of a punching bag.

Doug

One solution would be to have each player use a deck of playing cards or similar for the random element. You don't know what's coming next but the spread of results is guaranteed, so you won't get outlier results.

Cards would still have the opening stage outlier results mentioned here.

Badgers, I appreciate your comment. Card decks have often been used by wargamers. I would, though, like to point out that if you are drawing randomly from a deck, two things, at least, are true.

(1) Your point about the spread of results being guaranteed is true ONLY if the deck is both entirely and exactly used up, i.e. if somehow the rule system ensures that 52 (or 53 if you use a joker) cards, and only 52 cards – or an exact multiple – are used in the course of play.

(2) In any two card-drawings (i.e. usually, when using playing cards in wargames [unlike, say, poker], players are each drawing a single card, either simultaneously or consecutively) the range of possible results is very large. Because of this, although I haven't taken the time to do the statistical math, the range of the results of 26 pairings during the deck as a whole is also quite large – larger than the D6 hits-only-on-6 used as my original example.

The most effective use of playing cards in wargames that I have seen, is when each player draws some set number of cards from the deck and is then able to play those cards individually as the player sees best fit. This allows the player to set a relative importance upon individual pairings, but it still allows for a broad range of results, including losing every single pairing.

Anyone who designs games should know about stochastic variables such as the mean, but also the variance (or spread), which is basically what the OP is describing.

Though people moan, this is one of the advantage of load of dice – it tends to even out. However I think I prefer 3d6 to d6 x 3, as it aggregate dice tend towards the mean.