Friday, July 2, 2021

Fun with ability scores, modifiers, and data analysis; or, when is a player character unplayable?

The 3rd edition of Dungeons & Dragons has a caveat that after you roll your ability scores (4d6k3, arrange to taste, the same default method as 1st Edition AD&D), you can discard the scores and re-roll if the sum of all the ability score modifiers is 0 or less, or if you didn't roll at least one score of 14 (a +2 bonus) or greater.

The D&D Rules Cylcopedia (where scores are still rolled on 3d6 in order) also includes guidelines for when to discard a character as "hopeless": on pg. 145 (in the section of Chapter 13: Dungeon Master Procedures on "Creating Characters"), the book recommends that a character with all scores below 9 (i.e. all penalties) or two or more scores below 6 (so multiple −2 or −3 penalties) should be discarded unless the player wants to keep the character anyway

Since I'm currently running a Cyclopedia campaign, it's this latter standard that I'm using at the moment. But it also got me thinking: whenever I next get a chance to run Engines & Empires, which uses only four ability scores and a modifier table that runs from −2 to +2, what should the standard be? I used to allow mulligans for any sets of ability scores that add up to less than 42 (because 10.5 × 4 = 42, and because it's an easy number to remember thanks to Douglas Adams). But then it occurred to me that fully half of all rolled sets of ability scores can be discarded by that standard, and if you weed out all the below-average sets, that's really no better than the requirement from 3rd edition that the sum of all your modifiers should be at least +1. At that point, you might as well be rolling 4d6k3 instead of 3d6.

So I've decided that I need a new standard for minimum playable ability scores in E&E, one that accounts for the reduced number of abilities from six to four, and also the fact that the modifier table looks like this:

3 … −2
4–7 … −1
8–13 … no modifier
14–17 … +1
18 … +2

My feeling is, in a game derived from Classic D&D, a character whose modifiers add up to 0 or even −1 is perfectly playable. It's when the total modifiers start coming out to −2 or lower that things get sketchy. Just going by my intuition, I'd say that a character whose total modifiers add up to −3 or less ought to always be discarded, and that a character with modifiers adding up to −2 exactly should be discarded unless the player wants to play that character in spite of subpar stats.

That's just my bare intuition talking, but whenever I come to this point in a deliberation, I crave data. And for that, I turn to something that always makes my nerdy heart sing: Python coding. So I wrote a little script that generated half a million Engines & Empires characters using 3d6 in order and totaled up their modifiers, binning the data into a distribution as follows:

-8:  0/500K ...  0.0 %
-7:  0/500K ...  0.0 %
-6:  1/500K ...  0.0002 %
-5:  52/500K ...  0.0104 %
-4:  786/500K ...  0.1572 %
-3:  7431/500K ...  1.4862 %
-2:  38843/500K ...  7.7686 %
-1:  115510/500K ...  23.102 %
±0:  174884/500K ...  34.9768 %
+1:  115586/500K ...  23.1172 %
+2:  38728/500K ...  7.7456 %
+3:  7288/500K ...  1.4576 %
+4:  837/500K ...  0.1674 %
+5:  53/500K ...  0.0106 %
+6:  1/500K ...  0.0002 %
+7:  0/500K ...  0.0 %
+8:  0/500K ...  0.0 %

Now that is just a lovely normal distribution, isn't it? That part is as expected. But as to what we can glean from the data… that's just beautiful. The first thing we notice right away is that characters with all 3s or all 18s (or nearly so: three 3s and a 4–7 or three 18s and a 14–17) do not occur in half a million characters rolled. There was precisely one instance each of all the modifiers adding up to +6 and −6 (possible with three 18s or three 3s and an average score of 8–13, or possible with two 18s and two +1 bonuses or two 3s and two −1 penalties). But these extreme outliers are essentially negligible. In fact, all of the characters whose modifiers add up to less than −3 or greater than +3 add up to less than 3.3% of all characters rolled! So, really, if we want to clean up the data, we can present it in a form that's a little easier to parse:

-3 or less  ...  2%
-2:         ...  8%
-1:         ...  23%
±0:          ...  35%
+1:         ...  23%
+2:         ...  8%
+3 or more: ...  2%

Looked at this way, it becomes clear that characters whose modifiers add up to −1 or more account for about 90% of all possible score sets. If we say that characters whose modifiers add up to −3 or less must be discarded and characters whose modifiers add up to −2 may be discarded, that's only 10% of all rolled characters, and that feels pretty good to me! It basically confirms my intuition: characters with really bad stats—stats so bad that the character is practically hopeless—are actually pretty rare, only about 2% of cases, and that one character in ten is either a "definitely discard" or a "maybe discard" (with the latter being significantly likelier—fewer than one character in twelve vs. fewer than one character in fifty). I think that just feels right.

The code that I wrote is easily modified to expand the range back up to six ability scores again (or five or, hell, seven for you Comeliness fans out there, or more). And it's no big deal to change the modifier spread to match Swords & Wizardry (−1 for scores below 9, + 1 for scores above 12) or the standard ±3 range shared by Classic D&D and Castles & Crusades or even the ±4 range employed by WotC's d20 System games. I'll definitely need to play around with this and find out what happens when you change the number of scores and the frequency and magnitude of modifiers! 

ADDENDUM: I just ran the code for Swords & Wizardry (six scores, ±1 modifier range), and here's how it plays out!

-6:  ...  0.0 %
-5:  ...  0.0 %
-4:  ...  0.4566 %
-3:  ...  3.3562 %
-2:  ...  11.2074 %
-1:  ...  21.7054 %
±0:  ...  26.7484 %
+1:  ...  21.6188 %
+2:  ...  11.1392 %
+3:  ...  3.3384 %
+4:  ...  0.4296 %
+5:  ...  0.0 %
+6:  ...  0.0 %

What's interesting to me is that the overall picture isn't that different, but in Swords & Wizardry, with six ability scores in play, a character with a pair of −1 penalties and no bonuses still feels totally playable. For S&W (or any of its many, many derivatives), my feeling here is that a character with penalties in three scores and average values in the other three may be discarded at the player's option, and that a character with four or more penalties but no bonuses is best deemed hopeless and re-rolled.


  1. It really depends on where the negatives are placed, compared to the class the player wants to play. If I'm (assuming Classic D&D) playing a Fighter with an Int 3 and Cha 5, but Str 14 and Con 13, I've got a total -3, but I've still got a competent Fighter. I could still play that.

    However, I think your instincts on where to draw the line on the "hopeless" character is good.

    1. To no small degree, it's a matter of intuition when all is said and done (though I do feel better about seeing data and letting that inform my intuition).

      As for character class having an impact on overall playability, I'd agree that it's true to some extent in classic D&D, where Int has little to no direct impact on a mage's ability to cast, Wisdom is similar for clerics, and Dexterity doesn't adjust a thief's skill percentages at all. In fact, only Strength has a direct impact on a fighter's ability to fight in melee, so fighters can't really afford to have low Strength. (But I also wouldn't want to play a fighter with a low Dex, Con, or Cha either if I could avoid it.)

      So I guess this brings me back to a topic that I'll address in my next post: I don't actually like for the ability scores to have that much of an impact on characters in the first place, and I really don't like the idea of low scores imposing such hefty penalties that they cripple characters. The way Swords & Wizardry limits modifiers to ±1 is really admirable, and something that I'm going to try and adapt to BECMI very shortly here.

    2. (Also, of course, the whole matter is considerably mitigated by the fact that scores in classic D&D aren't "placed" at all. You roll 3d6 in order and you get you what you get! But that's a whole separate kettle of fish.)