2009-11-24

Are All Hit Dice Created Equal?

Here's a D&D math puzzle. Consider a 4HD creature versus a 1HD creature -- say, an OD&D Hero versus a Veteran. (For this discussion we assume that all hit dice and damage are uniformly d6's.) On average, will the 4HD creature take 4 times as many hits to kill as the 1HD creature?

You might assume so -- I know I did, and that assumption is more-or-less built into the bridge that connects Chainmail to D&D. But somewhat surprisingly, that turns out not to be the case. Consider the following table (PDF): www.superdan.net/download/CompareHD1.pdf

What you'll see is that on average, any creature takes about HD + 0.5 successful hits before being eliminated. That is, we expect about half of the last hit to be "wasted" damage, perhaps as the creature is reduced to 1 or 2 hp, and still requires another full hit before being struck down. And what this means is that, as a proportion of overall HD and hit points, the 1-HD creature types will be "wasting" more of the attacker's hits and damage than higher-HD types.

In the second and third columns of the table, you'll see things like this: Whereas a 1HD creature takes an average 1.5 hits per HD, a 4HD creatures only takes 1.1 hits per HD. In short, a 4HD creatures actually only takes 3 times as many hits as a 1HD creature (on average). And this grows progressively more severe: an 8HD creature only takes 6 times the hits of a 1HD creature, and a 15HD creature really only takes 10 times more total hits than a 1HD creature!

This might be merely a mathematical curiosity. Or, it might be something we have to make decision about if (to pick a random example) we wish to construct a set of mass-warfare rules which replicate D&D results with high statistical fidelity. Should we honor the actual hits-to-kill-over-1HD (as in D&D above), or should we more simply use the HD as hits-to-kill (as in Chainmail)?

I've come to call this the "Packing Problem" in regards to hit values in a classic RPG or wargame -- the fact that very weak targets are wasting more of the enemy's damage on overkill hits, while those with higher hit-point scores are suffering full damage from most attacks, effectively reducing the true value of those higher hit points.

Follow up -- More Hit Dice Stats (checked by closed probability formulas).

1. An interesting post. I'd like to reply tangentially by observing that it made me think of a cool "partial damage" mechanism for D&D.

Instead of pooling all the hit points into a single hit point total, characters should retain the list of points they gained at each level. So a fourth level fighter's hit points might be recorded as 6+3+4+3 rather than 16. Each "level" must be killed individually.

For example, if he takes a hit of 5 points of damage, that would be deemed to have "killed" his fourth level. Any excess killage is wasted.

His hit points are now 6+3+4, and furthermore he now fights and saves as a 3rd level fighter until he is healed.

Not only does this create a nice model for partial damage, it also would ensure that Chainmail and D&D damage models are parallel as originally intended, because now an Xth level creature will take as many hits to kill it as X single-hit-die creatures.

2. Interesting numbers but they completely ignore other aspects of level gain such as improved chance to hit, better saves, and so forth.

3. I had to deal with this issue several months ago when thinking about "wasted damage" for a prospective War & Battle supplement for AD&D. With units of 200 individuals in ranks of 20 and files of 10, and variably sized damage dice with unpredictable modifiers, it became clear it would be excessively difficult to model mathematically.

Another point to bear in mind is that combatants with more hit points have a greater window of opportunity to withdraw from the front line before being outright killed (depending on what you consider the "retreat" number to be with regard to hit points).

4. Well observed. You assume fights are to the death which may or may not be valid. You don't consider, and I understand why, that in many cases fighting ends in at most wounding followed by healing.
So a soldier might have four such fights before a terminal combat, recovering between each bout. Considering all hits absorbed up to death in this way pushes proportions of HTK towards HD again.

5. > a set of mass-warfare rules which replicate D&D results with high statistical fidelity.

I believe basing a mass combat system off the results of a man-to-man combat system is the wrong way to go about it.

Mass combat is different. Formations matter, numbers matter (ability to absorb losses and maintain formation) and large creatures tend to disrupt formations irrespective to their HD (think charging horses and elephants historically). Morale, training, leadership are much larger factors.