## 2014-08-07

### Arena – Ability Distributions

Ability Distributions for 4th-Level NPCs; And a Simple System for Rolling Corresponding Abilities for Like-Level PCs

I'm still working on the Arena program to simulate a population of battling fighters, and using it to assess how high the ability scores are for the characters who manage to achieve higher levels. In fact, as I prepared for my Jul-4 weekend marathon game, I wanted to apply those results. But the evening before I realized I hadn't yet inspected the distribution of ability scores for higher levels.

So here's a closer look at the scores for 4th-level characters in that simulation (the level at which we were playing that weekend). In this case this is the result of a population of 10,000 fighters battling pairwise for 1,000 cycles (with half dying each cycle, the other half gaining prizes and XP). For this run, I've got 42 gladiators who achieved the 4th level. Here is a frequency table of their ability scores:

You can see that the most common Strength score is 16; Constitution around 13-15; Dexterity around 13-14. The scores are approximately bell-shaped around these peaks. Other scores are purely random 3d6, since they give no benefit in the arena combat. Also, here are the hit points:

Most common is to have 26 hit points for these 4th-level fighters, and again roughly bell-shaped around this modal value. That's significantly higher than the average 22 you'd get from a naive dice expectation (4d8+4 [for average Con bonus +1] =  4 × (5.5) = 22).

So then my problem was: What kind of dice-rolling process (hopefully a simple one) can we give to the players that will roughly recreate these distributions of scores? Last time, I suggested that a process like roll-dice-but-apply-minimum (say: 3d6, but minimum "3" on any die; seen in many systems like 1E UA starting hit points) would give on okay average. But that gives a distribution that clusters up at a particular minimum value (i.e., is right-skewed), whereas looking at the distributions above, they're really more properly bell-shaped (as tends to be the result from a natural selection process like we have here). So that prior suggestion simply won't do.

Therefore, I started a different spreadsheet and found myself playing around with several different hypothetical dice-rolling procedures to find something that would work. I went through a few iterations before finding something that satisfied me; both (a) simple at the table, and (b) generates approximately the same distributions as above. The solution, in short, is this: You can't apply a per-dice minimum. What works best is to roll some reduced dice and add fixed bonus (so you do get a bell-like distribution shape, and an inflated average, without potentially blowing past the natural maximum 18 value). Here's what I had my players do that weekend to generate 4th-level characters, and it worked pretty well:
• Roll for Abilities. Pick one ability and roll 2d4+10 (suggest your prime requisite). Pick two other abilities and roll 2d6+6, in order. For your other three abilities, roll 3d6 in order. For resulting ability modifiers see OED (same as B/X except max is ±2).

• Roll for Hit Points. For each level, roll a die of one size lower than normal and add 2. That is: Fighters roll d6+2 each level, Thieves d4+2, and Wizards d2+2. Add Constitution bonus as usual. For multiclass characters (Elves), roll separately for each class in this way, and use the higher total.

So that's where we are at the moment. Notes: This is a work-in-progress (subject to later refinements of the Arena; like actually battling standard OD&D monster types). It's only true to the 4th level. But it does give you something to work with at the table (as I needed for my game Jul-4), and I expect that the basic idea will be retained in the future (roll some reduced dice and add a fixed bonus).

More detail in ODS spreadsheets: