Book of War: Variance

The Book of War mass-warfare game (see sidebar) is designed to quickly model standard D&D combat in as statistically high-fidelity a fashion as possible. Each d6 roll represents the attacks of 10 men or creatures (or more specifically: attacks by 5 men in the front row, over 3 rounds of combat, i.e., 15 individual attacks). The average casualties from the system are known to be very close to that of actual D&D combat played out en masse.

An outstanding question for me, however, has been: what of the variance? One fairly obvious observation would be that reducing the overall number of die rolls would make the game more "swingy" (i.e., reduced sample size causes higher variation). But on the other hand, by eliminating any damage rolls, that's actually one source of variation removed, so perhaps it pulls things back in the other direction. For a long time my assumption was that BOW combat would be about 3 times more swingy (in standard deviation) than normal D&D combat, but I didn't take the time to find out exactly.

Here are more exact comparisons, made possible by an edit to the BOWCore Rule program seen previously in the link above; revised version here). Numbers below are in terms of individual men killed per turn (3 rounds) by a single mass figure (5 across the front).

Variance in D&D & BOW

AC Mean Stdev Mean Stdev
10 5.9 3.3 6.7 4.7
7 4.4 3.1 5.0 5.0
4 3.0 2.7 3.3 4.7
1 1.5 2.0 1.7 3.7

Conclusion: The mean number killed is, as designed, as close as we could make it using a simple d6 roll. The BOW mechanic is indeed a bit more "swingy" (higher standard deviation/variance; more chance for an underdog to surprise a powerful opponent), but only on the order of about 1.5 times that of regular D&D (not triple the standard deviation as previously presumed).

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