Saturday, February 9, 2013

Stupid Dice Tricks: Exploding Dice and the Dice Chain

Oh look, it's "funky dice Friday!" Why not? I mean, we all love dice, right? If we didn't love bizarre polyhedrals, we'd all have a different hobby, no? And so here we all are again, ready for more stupid dice tricks. Last week, we took a look at the concept of central tendency and how important it is in multiple-dice situations. Believe it or not, I do have more to say on the topic, but I'll be saving that for at least another week. Today, I want to talk about a different spot of modern dice logic, exploding dice, its connection to another spot of traditional dice logic, the dice chain, and its origin in a very early RPG dice mechanic, DARO.

Exploding Dice

Exploding dice are a very common feature of a lot of modern RPGs. Sometimes it's just one die (such as WEG's d6 system and its Wild Die), sometimes it's every die that a player rolls (such as in Savage Worlds), but whenever exploding dice are rolled and the maximum value is shown, the die is rolled again and the new value is added to whatever has been rolled thus far. So, if you're rolling 2d6, say, and the dice come up 3 and 6, you would re-roll the 6 and add the result to 9. Okay, you've got that, nothing really ground-breaking there. Standard notation for an exploding die adds an exclamation mark ("!") after any die that explodes (for example, 2d6! means that both dice explode, whereas d6+d6! means that only one of the dice may explode).

Image Search for DARO = Cool Ruins

DARO

Way back in the day, Tunnels & Trolls came up with this spot of dice logic called DARO: Doubles Add and Roll Over. Look at that! D&D's redheaded cousin had an exploding dice mechanism! How's about that? Here's how it would work, say you were rolling 3d6 and the results came up 3, 3 & 4; you'd re-roll 2d6 and add the result of that roll to 10. Fascinatingly enough, getting a DARO result has the exact same probability as getting a die to explode, but has a far greater impact on the re-roll since two dice are being rolled instead of just one.

The Dice Chain & Raises

So, this concept may have a bigger impact on some games (such as DCC & Savage Worlds) than others (like any of those games that just use one boring sort of dice), but all games that mix polyhedrals seem to use some sort of dice chain or another. The dice chain of any game is the progression of dice from fewest number of sides to most. Here are the two most common dice chains I tend to see in games I play:

Standard Dice Chain: (Sometimes d2) => d3 => d4 => d6 => d8 => d10 => d12 => d20
Extended Dice Chain: d2 => d3 => d4 => d5 => d6 => d7 => d8 => d10 => d12 => d14 => d16 => d20

You probably recognize the extended dice chain from DCC and that's where I learned to love it, too. Recently, the concept of raising die type on the dice chain has figured more and more into RPG mechanics, some of which I'm sure has to do with the influence of DCC, but it has pretty solid roots in prior systems as well (usually using the standard dice chain). Basically, raising the dice type just means replacing one particular die with the dice type immediately larger than it on the chain. My invented notation for a dice type raise is to add a capital "R" after the dice type of any die being raised (for example, 1d4R is 1d6 in a standard dice chain but 1d5 in an extended dice chain).

Explode or Raise?

Really, there are two different questions here. One is "given the opportunity to raise a dice type or choose to make a lower die exploding, which should you choose?" The other is "is it worthwhile to or desirable at all to raise the dice type upon explosion of a lower die?" I'll answer these questions in the order that I asked them.

Given the opportunity to raise a dice type or choose to make a lower die exploding, the answer is technically to take the raise. The mean of the raised roll in this case, is 4.5, whereas the mean of exploding die result is 4.18, meaning that technically the better answer is to take the raise. However, the low probability difference being discussed here (is 32% significant? I'll leave that for you to judge), I'd say take whichever result you like more. You were going to do that anyway, but if you happened to like the raise, then now you have some math to back you up.

Right, so, on to the next question. Some folks out in interwebsland have suggested some mechanics where exploding dice also raise the dice type. It started with +Erik Tenkar statting up a magic dagger, with the thought that maybe if max damage would explode and increase the die type. Using the dagger's d4 damage type and several urgings from +Harley Stroh, I started doing some math to figure out which would do more damage, and I ended up figuring out that a straight explosion would average out to do more damage than the raising explosion. And, I'm sure, over a set of equal numerical results, I was right. However, if you factor the results out over an equal number of explosions (that is to say, explode the dice an equal number of times between the d4! results and d4!R results; let's say over three explosions and raising explosions), then the results end up skewed in favor of the raising explosion (d4! = mean of 3.32 and d4!R = mean of 3.458), but very narrowly (again, 0.138 isn't very different).

What does this all mean? Take the raise. Take the raising explosion. Give us a reason to mess around with our funky dice and use all of the weird polyhedrals in our dice bags. We brought them for a reason, so the more cool stuff we get to do with them, not only will we be happier, but it's something close to scientifically proven to be better! Not only is fun on our side, but so is SCIENCE!

Specifically, Gamescience.

Because they're the guys who make all the cool funky dice.