## Monday, June 6, 2016

### Ballistae and Bell-Shapes

Got in a debate on StackExchange about whether Chainmail's "Fire Optional" rule for catapults (and hence fireballs?) was reasonably realistic or not. Then I realized that the chart that rule generates, so clear in my head, has never appeared on this blog. So let's consider some real-world evidence first.

Strohm, Luke S. "An Introduction to the Sources of Delivery Error for Direct-Fire Ballistic Projectiles". Army Research Laboratory. July 2013. (Link.)

2.2 Normal (Gaussian) Distributions
For direct-fire ballistic projectiles, it is common to assume that error sources and the shot distributions they produce can be characterized by normal (Gaussian) distributions. Normal distributions are defined by a mean (μ) and standard deviation (σ, SD), which produce a bell curve that is unique to the distribution. The mean is the average of the distribution, while the SD quantifies the spread or precision of the distribution. For a one-dimensional normal distribution, approximately 68% of the distribution is within one SD of the mean (+/–) and 95% within two SDs (figure 2).

... In two dimensions, target impact distributions follow a bivariate normal distribution, meaning that the impact locations vary normally in two directions—in this case the horizontal and vertical directions of the target plane.

(Note that the bivariate normal model is the same as I've used in various archery simulations on this site.) Consider also the empirical test of shotgun shell spread presented here: Lowry, Ed. "Properties of Shotshell Patterns". American Rifleman. 1990. (Link.)

Now let's reflect on the rule as written in Chainmail for "Fire Optional"scatter:
Fire Optional: Roll two different colored dice. One color is for an over-shoot and the other is for an under-shoot. To decide which number of use you take the higher of the two. Miss is in inches, shown by dice spots. If they tie then the rock lands at the specified range. This method is simple but effective.
Taking the higher of the two dice biases the scatter towards the high end of the range. This is shown as "Chainmail Fire Option A" below. Note that the resulting probability distribution is distinctly anti-Guassian; it is impossible for a shot to land exactly 1" away from the target; and generally speaking, it's simply total lunacy, some kind of Lovecraftian non-Euclidean physics:

But if we change one critical word to make the rule instead "take the lower of the two" dice, then this mechanic, while still very simple, does in fact generate a quasi-bell-shaped distribution as suggested by the American Rifleman and Army Research Laboratory publications above (shown as "Chainmail Fire Option B" below). It seems patently obvious that this is the better rule:

1. By roll two dice I assume Chainmail used d6. What die do you recommend. Also could you provide a full rule for catapults such as distance, damage, and rate of fire?

1. That's right, they're d6's. The only way I modify that in my games is to roll 2d6 and say every pip below/above 7 indicates 1 inch short/long. I apply that to all fireball and lightning bolt castings (haven't used actual catapults in my games in, like, forever).

2. Thank you for this, I was just going over those rules last week to start working on my houserules for catapults/ballistae for BoW. Would you share the link for the StackExchange discussion?

1. You can see the thread below (search "Fire Optional" for my answer), although weirdly the sequence of comments that spawned this is now gone. I guess on that site the comments are disposable and sometimes they go through and prune them aggressively.

2. It just dawned on me... maybe I'm going about this wrong with the whole catapult thing.

I've always assumed (and it looks like most wargames assume) that catapults and ballistae have pretty devastating results. Chainmail with it's insta-kills, for example

But when I look at S&S, the other go-to for how Gygax, et al, looked at things, I see that catapults become much less effective.

In S&S, a figure of 10 men has their average hit dice computed - on a d6, that would be 3.5, so 35 hit points. A "small catapult" would do 30 points of damage, a large catapult would do 40. So from an S&S perspective, it requires a large catapult to have a chance to kill a figure of normal men. At least as I'm seeing it (and that's not including the myriad of adjustments one can make to the attack.)

So maybe I'm approaching this wrong. Maybe the idea of the catapult is that it isn't necessarily an insta-kill machine, but a softening up machine. Hear me out on this.

In my houserules for BoW, when I've gone from 1:1 to 10:1 and back (especially in my "take 100 men underground" experiment), I came up with a concept of a figure being "diminished" in effectiveness. This came about as the dungeon will kill individuals in a figure (through green slime, or traps), but not the entire figure. Once half the figure had lost their men, I called that figure "diminished" and implemented a -1 to their combat roll. To take into account the loss of combat effectiveness.

This meant I had to track which figures in a unit were diminished, but that was easy enough. Poker chips or some sort of token.

Maybe I can apply the same logic to using catapults or ballistae. If they successfully hit a figure, the figure is diminished or affected (loss of figure cohesion, casualties within the figure) - maybe only for the next turn or two.

It's a consideration and one that I am liking more and more as I chew on it. Especially since a catapult or ballistae is a 1:1 figure in a 10:1 scaled battle.

3. If that works for you, that's pretty cool! Admittedly one of the design principles of BOW was to not have any per-figure record-keeping (just presence of a figure on the table, and maybe hits for a unit with one marker die).

4. @Delta - I can understand that. I guess in that case, I would use a diff colored die for that figure, or put the figure on a poker chip or some other marker. I'm used to doing that with other games, so it wouldn't be too hard to adapt.

I also keep a roster of units with their stat blocks in front of me when I play wargames, and I will track the casualties and/or conditions on that. I did it for the 100 men vs. a dungeon scenario that used BoW and it worked out well for me.

3. There is a slight difference when the target is a vertical field, in your examples, as opposed to a horizontal field.

If the error in vertical height is a gaussian distribution depending on the error in the launch angle alpha, then the error in horizontal distance will not be linearly related but related through the function sin(2.alpha).

So for a tiny error in alpha you will get approximately a gaussian distribution but something a little more complicated for larger errors.

1. ... actually no there is a tan(alpha) relationship between max-height and range so again gaussian for small error in alpha but more complex for larger alpha error. Anyway fun problem!

2. True, but you're still not going to produce a bathtub-shaped (anti-bell-shaped) distribution as in the written Chainmail rule. In choosing between option A or B, B will still be the better fit, by far.

3. Oh yes, you spotted an elegant fix for that alright.

4. I'm pretty sure the "roll 2d6 and take the highest" must have been a typo (or not what the author intended).

1. Right, that's totally what I think, too. Only problem is when it was re-used later in Holmes Basic D&D (under "Giant"), it STILL says "take the greater".

5. Maybe they wanted catapults to mostly miss. Yeah the outcome graph looks weird but anecdotally, individual results would look reasonable.

1. I'll have to disagree on that one.