Wednesday, October 5, 2011

Book of War Core Rules Justification Part 2

I usually don't trust myself to do a computation just once. Customarily I try to construct: (1) a procedural computer simulation, and (2) a formal math calculation, and if the two results synch up together, then I can have some confidence in them.

So last time I presented the Java computer-simulation version of the Book of War Core Rules mechanic (the 3/4/5/6 target on d6 to score a hit on a 1:10 scale figure wearing no armor/leather/chain/plate). For me, that's always the more concrete demonstration, but the truth is that we don't really need it: we can also do a direct math calculation.

Let's just do one case as an example. Say you've got normal men attacking normal men in chain mail (no shield: AC 5). Per the OD&D hit chart, those men hit on a score of 14 or more on d20; i.e. (excluding the lower 13), 7 chances in 20, or probability 7/20 = 0.35. From prior work, we know that on average it takes 1.52 successful hits to kill a 1-HD man. (See the two proofs of that here and here). The one attack roll per BOW turn represents 3 D&D rounds, with 5 men along the front line attacking, but it takes 10 kills before a whole mass figure is eliminated. So the expected value (probability) of figure kills each turn is:

0.35 × 1/1.52 × 3 × 5 / 10 = 0.345

Note that all the factors after the initial probability basically cancel out; so, the probability of a hit in D&D is basically the same as the probability for a figure-hit at our chosen BOW scale! So nice. In any event, we can convert this probability to a d6 target value using our standard formula:

7-6*p = 7-6*(0.345) = 4.93 ≈ 5

And of course, that's the same "AH 5" value that we've been claiming all along for targets in chain mail armor. Moreover, we can do this for every possible AC value if we want to be really careful with it. Here's the result:

(Click above for a PDF version. Or click here for an Excel spreadsheet, if you want to tweak or check the calculations.) Okay, so it turns out that the totally correct conversion isn't exactly just a divide-by-3 operation, but it's really close. Leather & shield should technically be AH5 according to this, like basic chain mail. And we can also see what should happen for mega-hardened AC values at the bottom of the chart (clearly unhittable by normal men in OD&D).

This table actually appears in the Book of War Optional Rules section at the back, in case anyone wants to be totally precise with that. But for most purposes, I'm very happy with the 3/4/5/6 per armor-type Core Rule, being exceedingly simple and elegant. The other thing noted in the table is that while in OD&D, normal men can hit AC -1 on a perfect 20, this 1-in-20 chance is negligible in terms of d6 success rates; so, for example, I'm pretty happy with the language in the heroic-level section wherein I summarize this as, "characters with negative ACs are given AH 7" (or greater) [BOW, p. 13].

One final thing to keep in mind: We've now twice-confirmed that the expected values of BOW combat are the same as D&D combat played tens or hundreds of men at a time (i.e., on average, BOW combat produces the same results as D&D combat). But what could not be kept identical with the scale-switching was the variance of the results; that is, BOW combat should be more "swingy" than if you really played out mass combat at the D&D man-to-man scale. (My estimate is that standard deviation has been multiplied by a factor of √15 = 3.87 , since one BOW roll represents 15 D&D attacks; but maybe less than that because damage hit point rolls have been abstracted out of the equation?) Personally I don't mind that, since it says that we're giving the underdog a bit more chance to come out on top, say.

Nevertheless, in practice we've actually found that games at equal point-values (and no enormous mistakes by either player) tend to be almost supernaturally even; we've seen lots of games that come down to the last two opposing figures on the table, with a single hit determining the victor (even if the game swung back-and-forth before that, over the course of play). So our confidence that Book of War is a game balanced unto itself is also quite high. More on why that's the case a bit later.

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