Monster Math: OD&D Challenge Rating

The rat post was fun! For my next trick, I will reveal the secret "challenge rating" system inside OD&D.

Virtual Hit Points

First, let me explain in greater detail some ground I covered in the last post. Hit dice are not the final word on monster longevity because they do not account for the likelihood that they will get hit by an attack. This means we can get a more accurate picture by calculating a figure's "virtual hit points": how much longer they last in combat as a result of their armor class. I did this years ago, but have a more accurate function now because of a simulation I ran. The results are as accurate for 4 hit points as they are for 35 hit points.

AC	Avg. HP Adj.
9 [10]	100%
8 [11]	111%
7 [12]	125%
6 [13]	142%
5 [14]	168%
4 [15]	198%
3 [16]	251%
2 [17]	334%

As an example, an unarmored character with 10 hit points doesn't get anything extra, but if they were wearing plate armor it would be as if they had 25 hit points instead. Another case: an adult red dragon with 40 hit points and an armor class of 2 has 132 virtual hit points (or 100 if you knock them down by 1, which I like since I like even-numbered ascending AC).

Monster Levels

Whether or not this was systematized in depth, it seems that the dungeon encounter tables tend to relate a lot more to virtual hit points (meaning, both hit points and armor class) than just raw hit dice or points. Here is my raw datasheet, and below is the table from the previous post where I find the average hit points, armor class, and virtual hit points of each monster level.

Monster Level	Avg. HP	Avg. AC	Avg. VHP
I	2.73	6.6	3.5
II	6.06	6.2	8.2
III	13.8	6.0	19
IV	14.6	4.3	29
V	21.5	4.5	43
VI	32.2	3.4	80

You can sort of use this to come up with monsters on the fly, although "monster level" by itself is not helpful compared to how often each monster level occurs on a particular dungeon level (put a pin in it). Still, it helps us notice patterns. A level 1 monster lasts about as long as an unarmored HD 1 figure, and you can only really vary by improving their armor class.

Higher level monsters have greater variance in hit dice because they support a wider range of armor class values. A level 6 monster can have 32 hit points and an armor class of 2, or 47 hit points and an armor class of 4. OD&D, however, tends to prefer scaling armor class rather than hit points. Keeps numbers low! The least powerful level 6 monster is the basilisk (which has petrification; that counts for something), and the most powerful and likely to appear is the balor. (I opted not to find the VHP of individual dragon types at each age level, but theoretically the most powerful could be the ancient gold dragon.)

Dungeon Levels

To reiterate, monster level does not help us much directly because it's a function of a function: first you find the dungeon's level, and then use that to randomly determine the level of the encounter. Why not cut the middleman? We can find the average virtual hit points of an encounter for any individual dungeon level, by taking the weighted average of the possible monster levels' virtual hit points on that dungeon level.

For example, a level 1 dungeon has a ⅓ chance of a level 1 encounter, a ⅓ chance of a level 2 encounter, a ⅙ chance of a level 3 encounter, and a ⅙ chance of a level 4 encounter. If we find the average virtual hit points of these outcomes, we can find the average expected virtual hit points to be found in an encounter on a particular dungeon level. On a level 1 dungeon, then, the average encounter has about 12 virtual hit points.

Here's the leap of faith I'm going to take: if we read this amount as an encounter building budget, the outcome is similar to the "monsters appearing" algorithm that Delta came up with. You know the problem where the "number appearing" stat is only meant for wilderness encounters, and OD&D implies but never explains a method of determining figure numbers in the underworld? Delta's method makes good sense, and this "budget" aligns almost perfectly. For a "party of from 1-3", as per Volume III p. 11, this method would suggest a level 1 encounter of 2 orcs, 3 goblins, or 4 kobolds.

All you have to do is vary the budget, like Delta does by rolling 1d3 per 2 characters, to account for encounters which are more dangerous than this average; or only allow especially dangerous monsters when the party has from 4-6 adventurers (which is intuitive, since most parties I've played in have that many players!). Below is my table based on this, where VHP-2 is based on the smaller party size and VHP-4 is based on the more typical party size.

Dungeon Level	VHP-2	VHP-4
I	12	24
II	20	40
III	36	72
IV to V	42	84
VI to VII	56	112
VIII to IX	74	148

Maybe make it so the budget is 5-6 times the number of players times the dungeon's level (where, for some reason, levels 4-5, 6-7, and 8-9 are combined). Enjoy! Or not. I'm fucking with you after all :) Mwah!