Scrutinizing Roll and Add Mechanics.
One of the big advantages of Roll and Add mechanics is that they’re linear. Meaning a +1 always produces the same relative gain in probability. When the distribution in uniform that +1 always has the same absolute value, when the distribution is normal, the +1 gives the same bonus relative to the unmodified amount.
So to evaluate the different mechanics it’s helpful to discern exactly how much the +1 is going to be worth for each dice and distribution. The way to do this is using Standard Deviation.
Calculating the Average Value of a +1
To figure out what a +1 is worth you simply need to calculate the Standard Deviation within a given distribution. This number will tell you that about 2/3 of the time you will roll a number between the average plus or minus the Standard Deviation. For example, a 1d20 has a Standard Deviation of 5.77, and a Mean of 10.5 – From this we can discern that 2/3 times we roll the dice we are going to land between 5 (4.77) and 16 (16.27). That means in order to feel like you’ve gained enough of a bonus to break free from that range completely, you’re going to need a +6 bonus. Conversely, a 1d4 has a Standard Deviation of 1.12, so 2/3 times it will give between 1 (1.38) and 4 (3.62), but to feel like you’ve broken free of that you need only a +2. In fact you can calculate how much move a +1 is using a 1d4 than a 1d20 by taking the 5.77 and dividing my 1.12. Which makes the +1 just about five times as valuable on a 1d4 scale than on a 1d20 scale.
Of course, if you’ve been paying attention you’ve doubtless already noticed a problem. A 1d4 has a lower standard deviation, but it also has a smaller range. While a +6 is 1/3 of a d20, a +2 is half of a 1d4. A +1 is indeed more valuable in a 1d4, but only because a 1d4 has such a tiny range relative to 1d20.
You of course could measure that by looking at the Standard Deviation relative to the Mean, specifically dividing the Standard Deviation by the mean we derive the Coefficient of Variation. That gives you a uniform metric that can be used to compare how “random” each dice is going to be. Of course the +1 will have to work around that, but the design principle here is to first determine how random you are looking for. The experience I’ve had talking with friends is that 1d20 is too random, and 3d6 is not random enough. That tells me we are looking for a Coefficient of Variance less than 55% and greater than 28%. Targeting around 30% would give us 3d10 or 3d12. 3d10 produces a range between 3 and 30 or 3 and 36, and gives +1’s similar to a d20. I find that level of granularity to be a little much, 2d10 and 3d8 are closer to the level of granularity that seems appropriate with 2d10 producing results between 2 and 10 and 3d8 producing results between 3 and 24. That means a 19 point range or a 21 point range with a similar standard deviation. 3d8 has the advantage of actually being normal, but 8’s are harder to add then 10’s. Finally, biting the ‘too consistent’ bullet and going with 3d6 is tempting because 6 sided dice are easy to gain.
Some Other Options
That analysis was assuming that all the dice are added together, and was working with numbers of identical dice. That’s of course unnecessary. One option is to have one dice subtract from the other instead of adding them together. 2d6-2d6 is a natural fit for this, it has a Standard deviation that indicates an appropriate level of granularity, and produces a nice bell curve, it also produces a very tidy range a -10 to +10. That itself can have an advantage because comparing scores doesn’t require adding 10 to a defending side.
Another choice is mixing dice. There is no rule that says it has to be 3d8, it could be 1d6+1d8+1d10, however without a compelling reason to do that it seems that that only adds complexity.
What’s your preferred dice mechanic? How much variance do you like? Is the 2 positive and 2 negative a good idea?