Monday, May 4, 2015

Dungeons & Demographics, Pt. 3

In the last installment, we noted that OD&D text descriptions for leader distributions (read: NPC levels) are totally different than what emerges from a simulation of actually playing out games against the dungeon encounter tables (read: PC levels). But: We did see that if we alter rolls on the OD&D encounter tables in the simulation to the reduced level of 1d6-4, then we do get a population that is passingly similar to the described NPC distribution. Here I'd like to use those simulations to come up with an evidence-based system for generating high-level character ability scores and hit points.

One important thing I noticed in an earlier, partial analysis of ability scores and hit points (link) is that these improved high-level distributions are still bell-shaped. That is: You would not want to use a method like "roll d6's, minimum 3", because that causes a pileup of scores on the low end (e.g.: 3), which is to say, generates a right-skewed distribution. (For example, the "minimum score" method that Gygax uses for hit points in AD&D Unearthed Arcana, p. 74, is contra-indicated.) 

What I came up for a simple dice-rolling method for such heightened, generally bell-shaped scores is to roll two dice of some type and add a fixed bonus, such that the maximum result is still 18. Consider these dice to be our toolset for this purpose: 3d6 (Expected value 10.5), 2d6+6 (E = 13), 2d4+10 (E = 15), and 2d3+12 (E = 16). Now we look at the average scores for our population generated by the Arena program (with an encounter roll modifier of -4), and try to find the best fit at each level:

I've batched up the levels into groups of about 3 at a time for convenience. At higher levels, the picture is pretty clear; scores in each of Strength, Dexterity, and Constitution trend upward as only the fittest characters survive to higher levels, as we would expect. In the 2nd-4th level tier, characters have an average of 13 across each of these abilities, arguing for a roll of 2d6+6 in three abilities of the player's choice. For levels 5th-7th, it looks like one ability is about 15 (2d4+10), and the two others 13 (2d6+6). For levels of 8th and above you have at least two scores at 15; in this case I'll take a bit of artistic license and use one 2d3+12, one 2d4+10, and two 2d6+6 scores. Not too bad.

But now let's focus on the slightly troublesome 1st level. Whereas the 0-level clearly depicts the straight 3d6-roll for newly generated normal men (average about 10.5), characters surviving to 1st level do evidence about a 1-point advantage across each of Strength, Dexterity and Constitution (average about 11.5). The problem is, using the dice "tools" above, none of the abilities quite reach halfway to the 2d6+6 roll method (mean of 10.5 and 13 = 11.75). But if we pool the overall improvement in all three scores, then we get a value a bit over 3 (3.2). So to reflect that basic trend, I'm going to give 1st-level characters in my games a boost of about 3 points by selecting one single ability -- likely their prime requisite -- and rolling 2d6+6. (This replaces my old house rule of rolling all 3d6 and then swapping two of the player's choice.)

In summary, here's my new evidence-based house rule for rolling ability scores:
  • Level 0: Roll all abilities 3d6 in order. 
  • Level 1: Roll one selected ability 2d6+6, others 3d6.
  • Levels 2-4: Roll three selected abilities 2d6+6, others 3d6.
  • Levels 5-7: Roll one ability 2d4+10, two 2d6+6, others 3d6.
  • Levels 8+: Roll one 2d3+12, one 2d4+10, two 2d6+6, and two 3d6.
Hit Points: Note that in the table above, average hit points at each level are truncated, not rounded (average hit points at 0-level are really 3.5, at 1st level about 6.5, etc.) The best method to generate higher-level hit points is to simply re-roll any dice that come up 1 or 2. (Or equivalently: roll a die of one size less and add 2.) From a purely technically standpoint you wouldn't want to do this for the very last level (the character has not proven survivability through that level yet), but for practical purposes I don't want to split that hair in my house rules, nor have to remember about it.

Final Thoughts: While we've developed this system looking at the "NPC" population at the -4 encounter die level, the truth is, any other close modifier generates about the same ability scores and hit points upon surviving to higher levels. So I'm comfortable using this system for generating scores for both high-level PCs and NPCs. It's conceivable that future refinements of the Arena simulator program might change things here, but my expectation is that the method will remain pretty stable going into the future.


  1. I have found your three part analysis very interesting. I would like to offer you two methods to generate ability scores that I use. The first method I came up with for Dragon Warriors. The second method I came up with Shannara Campaign using Castles and Crusades.

    Roll 3d6 and refer to below chart:

    3 or 18 = 18
    4 or 17 = 17
    5 or 16 = 16
    6 or 15 = 15
    7 or 14 = 14
    8 or 13 = 13
    9 or 12 = 12
    10 or 11 = 11

    The second method is 4d3 + 6

    1. Well, the problem with the first method is that it creates a right-skewed distribution; 54 ways to get 11, 50 ways to get 12, 42 ways to get 13, and so forth in monotonically decreasing fashion. The variable ability scores we get from the simulation are basically bell-shaped which argues against that.

      The second method would be bell-shaped, so it's at least in the running. Personally I try to use fewer dice instead of more.

  2. So does this mean that the ability scores of your player's characters will increase as they level up?

    1. No, it's only done the one time when the character is generated. It's just that if generated at a higher level, then the character is presumed to have higher scores which allowed survival to that point.

  3. Someone on Facebook pointed out an interesting coincidence about the reroll 1 or 2 hit points at starting level: it's the same as Moldvay suggests in Basic D&D, bottom of p. B6:

    (First level characters may easily be killed in battle. As an option, the DM may allow a player character to roll again if the player has rolled a 1 or 2 for the number of hit points at first level only.)

  4. Here's an odd thought, a year later: you could abstract this to just bonuses from stats for NPCs. First level NPC gets a +1 from his presumed prime requisite of 13. At second level, he gets +1 from another attribute, another at fifth, and another at eighth. Also at eighth level his presumed prime requisite is 16, raising that bonus to +2. So we can assume that tenth-level magic-abuser with the buccaneers has Int 16, three other stats at 13 (I'm going with Wis, Dex, Con), and the last two at 10 or so, but only worrying about the bonuses since we're not likely to use anything else in play.