D20

(continuing discussion of game characteristics)

There is a subtle but important element into how you generate random numbers for your game (I'm assuming most games will have a random element).

There actually is a game system called "D20" (where all decisions are made by rolling one twenty sided die). I prefer the GURPS system, which uses (mostly) 3d6.

The two systems are superficially similar. The range is about 20, and the average is around 10.

These are the probability (PDF, blue) and cumulative (CDF, orange) distribution functions for a d20. Totally flat. This means that any number from one to twenty is equally likely, similarly going from N or lower to N+1 or lower is always a 5% increase in probability.

So, a bonus which gives +1 is always worth 5%, whether your an expert or a novice.


These are the PDF and CDF for 3d6. Notice the distinct bell curve for the PDF, and the CDF resembles a sigmoid (which has characteristics similar to an exponential curve at the bottom, and logarithmic at the top).
This captures ideas like "diminishing returns". It also means that an "expert" (someone who could make a roll at 16 or lower) is less affected by difficulties (say, -1 on the roll) - going from 16 to 15 is a drop of ~3%. The same -1 is a 12.5% drop for an average person (10 or lower).

GURPS is limited by the human element. People must roll and count the dice, and increasing the number makes it more likely that one will go off the table, etc. A computerized system can generate any curve we like, with any range we like.

I'm thinking either 1-100, or 1-1000 - with this same bell shape. Something like 17d6.