Exploding Dice Statistics

In January, after our Wandering DMs show on critical hits, someone in the patron's after-party chat (Joshua, but you knew that) suggested replacing Nat-20-triggered criticals in D&D with exploding damage dice all the time. It's not something I ever considered for D&D, but the statistical niceties of that immediately got stuck in my head. Let's look at that a little more closely. (Someone's probably already done that in the past, but as they say: "This is my blog; there are many like it, but this one is mine," or something.) 

A Formula for Exploding Dice

I wanted a simple, closed formula for the expected (average) value from rolling any variety of exploding die. Here you get some math so I can show my work. 

The essential trick/insight/tactic is this: Say we want the expected value E of an exploding d6. There's a 1/6 chance of getting any of the values 1, 2, 3, 4, 5, and then at the end, a like chance to get a value of (6 + E); that is, 6 plus the average result of the same process all over again (recursively, if you will). More generally, for an n-sided die, you have a 1/n chance for each value 1, 2, 3, ... , n − 1, (n + E). Some basic algebra solves for that E:

$$E = \frac 1 n (1) + \frac 1 n (2) + ... + \frac 1 n (n+ E)$$

$$E = \frac{1 + 2 + ... + n} n + \frac 1 n E$$

$$\frac{n - 1} n E = \frac {1 + 2 + ... + n} n$$

$$E = \frac {1 + 2 + ... +n} {n - 1}$$

Note in the 3rd step we subtracted \( \frac 1 n E \) from both sides, generating on the left-hand side a value of \( E - \frac 1 n E = \frac n n E - \frac 1 n E = \frac {n - 1} n E \).

What other ways of writing this are there? That top sum 1 + 2 + ... + n has a name, specifically the (nth) triangle number, which could be denoted simply \(T_n\). It's like the factorial function, but with adding instead of multiplying; and it's sequence A000217 in the Online Encyclopedia of Integer Sequences. So we could just write:

$$E = \frac {T_n} {n-1}$$

If you don't like that (and how dare you), we could use what I refer to as the Gaussian formula to replace it with more primitive operations; \(T_n = \frac {n(n+1)} 2\). Substituting that into our formula for E, we get this version of our exploding-die expected value formula:

$$E = \frac {n(n+1)} {2(n - 1)}$$

Note that the expected value for multiple dice is this number times the number of dice; e.g., for 3d6, just compute E for n = 6 and then triple it (expectation always being a linear operator).

A Table of Exploding Dice Values

Here you have a table of expected values for our standard mostly-Platonic dice, in both the normal-roll and exploding-roll forms. Consider the ratio column at the end, which shows the effective multiplier that exploding is giving you on average. The benefit is best for smaller dice; e.g., for a d4 you're multiplying your expectation by 1⅓, and then it goes down for larger dice. And yet this benefit doesn't flip the ordering of any dice; the benefit on d4 (which I kind of like, say, making daggers a bit more fearful) doesn't even reach the expectation for a standard (non-exploding) d6.

That said, watch out that things get weird in the edge-cases if you go off the top or bottom of this table. If you have 2- or 3-sided dice mechanics, those exploding expectations are actually the same: E(d2exp) = E(d3exp) = 3. And theoretically an exploding 1-sided would generate infinite damage! (E.g., consider the "ordinary rat" by Gygax in the AD&D MMII with that damage specifier; the standard 1st-level adventure just got a lot more dangerous.) On the other end, larger die values have a benefit ratio decreasing towards 1; in the limit for an infinite-sided die, there would be no benefit at all (uh... not that you'd need any).

Note that the ratio of exploding-to-normal damage expectation has a nice, short formula for it. (Thanks to Drew on Twitter for noticing this from the decimal values above.) This is:

$${n(n + 1) \over 2(n - 1)} \div {n+1 \over 2} = {n(n+1) \over 2(n-1)} \times {2 \over n+1}$$

$$ = {n \over n-1} = {n - 1 + 1 \over n-1} = 1 + {1 \over n-1}$$

Graphs of Exploding Dice Values

Here's the other thing that I think is really nifty about the exploding-die mechanic; the distribution becomes right-skewed (instead of uniform, or bell-shaped for multiple dice). Here's the probability distribution for an exploding 1d6 (from Troll Dice, turned on its side for more familiar presentation):

And possibly more illustrative, here's the chart for exploding 2d6 (like the damage for giants in Original D&D): 

Note how that slopes off gently on the extended right-hand side. My understanding is that right-skewed distributions like this are far more common (really, the only thing) for natural or biological processes -- you have a hard lower limit (often 0 or 1), but in theory no hard upper limit (on the right), and therefore the population/sample-space spreads out in exactly this way.

In fact: The famed evolutionary biologist, Stephen Jay Gould, wrote an entire book dedicated to exactly this observation: it's called Full House: The Spread of Excellence from Plato to Darwin.

Are those compelling results? (And did you already know that?)  Would you consider using always-exploding weapon damage dice in your D&D games, instead of a Nat-20 trigger?