Last time, we added standard dice to exploding dice in order to smooth out the resulting probability distribution. Now we’ll finish up our discussion of exploding dice by looking at opposed rolls.
The Laplace distribution
Opposed geometric rolls result in a discrete version of a Laplace distribution, which is basically two geometric distributions glued back-to-back. The half-life on each side is the same as that of the individual geometric distributions. With an opposed roll both players have a chance to succeed or fail regardless of the disparity in modifiers. This is unlike the non-opposed roll, where the roller may be fail-proof if their modifier is high enough. Here’s the probability mass for a half-life of 3 (AnyDice):
And a corresponding EHP plot:
Notice that the opposed roll improves the quality of the approximation, much like adding a standard die did in the non-opposed case. Adding additional standard dice beyond this pushes some of the probability from the center out into the tails as well as smoothing the tails even more. However, I would consider the opposed exploding dice by themselves to be smooth enough in this case, and it’s fewer dice to deal with.
We don’t necessarily have to use the same half-life or even the same type of distribution for all of the dice. Here are some examples of mixed opposed rolls (AnyDice):
All of these use an exploding d10 for the attacker’s dice, and hence the right tail is similar for all of them. But the choice of the defender’s dice allows us a good deal of control over the behavior of the left tail.
Recall also that, because 1dX is the same as X+1–1dX, a standard non-exploding die can be moved from one side to the other. So the standard dice we added to the exploding dice last time can alternatively be framed as an opposed dice roll between an exploding die on one side and a non-exploding die or dice on the other.
Apart from attacker vs. defender, other options for deciding who rolls what dice include players vs. NPCs, creatures vs. objects, or proficient vs. non-proficient.
Note that controlling the left and right tails independently is only possible because the distribution of an exploding die is not symmetric. If we take the sum or difference of two rolls with symmetric distributions, the distribution of the sum or difference is also symmetric and thus it is impossible to control the left and right tails independently.
Living Myth RPG also came up with the idea of using a Laplace distribution. Read their article here.
That’s all for this series. We looked at effective HP curves for roll + modifier systems with binary succeed/fail outcomes, considering three different types of distributions:
- Single standard dice, which are uniform distributions.
- Sums of standard dice, which are normal-ish distributions.
- Exploding dice, which are geometric-ish distributions.
We paid special attention to the last, which requires some extra work to approach a geometric distribution well, as well as having some special behavior in opposed rolls. The payoff is the less extreme tail behavior in terms of EHP, which may allow designers to more safely increase the scaling of modifiers.
I tried to keep this series focused on effective hit points, leaving out less relevant content. If you want to see some of the odds and ends that were left on the cutting board, you can read the appendices.