## 2021-05-17

### 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?

1. Funny, my inclination is the opposite of Joshua's - to retain the excitement of a high attack roll, but linearize the benefits. Specifically, to eliminate the damage roll and have damage equal the amount by which the attack roll defeats the target's armor class. Using a two-handed, magical, or otherwise advantageous weapon can simply be boiled down to an attack roll bonus, since this will also implicitly increase the damage dealt. Haven't implemented it at the table yet because I'm still tinkering with how to handle large creatures, whether to just give them better to-hit rolls or to give them some kind of "add-on" damage bonus.

1. Huh, interesting proposal, I wouldn't have thought of that!

2. The root of the idea was from White Wolf games - after counting out "successes" from your dice pool and comparing it to the enemy's defense, you add any "extra successes" to your dice pool for the damage roll. Then stew that together with pondering over the years about armor as hit prevention versus armor as damage ablation, and also ideas on how to make fighters more deadly as they gain levels without using clumsy piecewise functions like the AD&D "Warrior Attacks Per Round" table or 3E iterative attacks.

2. I am working on a exploding d6 damage idea. Where warriors gain the exploding damage. A natural 6 indicates a wound that incurrs a penalty. Cure Light Wounds then literally heals 1 wound. More powerful versions can heal more wounds simultaneously.

To hit rolls are in effect a success at disadvantaging the foe, damage being the obscure disadvantage and physical wounds afflicted on the 6 then need healing.

AC in abstract terms is avoiding casualty or not being disadvantaged by the opponents attack.

I will say fueling combat factors with the difference between rolls indicating the disadvantage if any afflicted has a sweetness to it. This would be more like 2nd edition Tekumel or Arduin I believe.

Great topic. TY

1. Interesting, thanks for saying that!

3. Dueling nor fueling sorry

1. And yet somehow they both work! :-D

4. I dislike nat-20 crits, and don't use them anyway.

I *have* used exploding damage for some games in the past (usually to represent the destructive potential of modern weapons, i.e. guns and explosives). This stat analysis makes me feel much better about it.

I don't *think* I would ever use it for AD&D (damage dice are so variable in that game), but if I went back to OD&D - or even B/X (its variable weapon damage is pretty limited in scale) - I'd consider using it after reading this post. The fact that the expected average doesn't exceed the "normal average" of the next greater die is a BIG PLUS in my book, as I too had feared such a mechanic would create Over-Awesome daggers and slings.

Thanks for this!

1. Right, me too! Initially I had that same expectation (about the, um, expectation). It was Joshua who first pointed out the facts in that same conversation.

5. "(uh... not that you'd need any)" But what if I rolled 2 damage on my dinfinite? ðŸ™ƒ

I never tried to work out the exact formula, but the end result fits with my empirical observation that it provides excitement despite having very little impact on actual results in play.

1. Of course the chance of that is... very small. :-D

6. Fantastic idea. One of the recurring disappointments players experience is rolling a nat-20 and following it up with a miserable damage roll (one that could in theory be smaller than a non-crit). So much for extraordinary success! With this system, the nat-20 could be used for some other boon -- such as 5e-style advantage of the next attack, or a "Luck" point or some kind of other meta-currency, a free stunt of some kind, or maybe even an immediate additional attack.

1. That's actually more-or-less the genesis of this conversation.

2. Great point about "what else does nat-20 trigger, then?". At the moment I kind of have a crush on having it damage the enemy's armor (starting with broken shield, if any).

3. "what else does nat-20 trigger, then?".
Sweep Attack obvs :)

7. I love exploding dice. They are fun at the table! I use them for missile weapons since I don’t allow dmg bonuses for dexterity. We had an epic heavy crossbow shot the other night (2d4+++).
I like the idea of damage being the difference beyond AC. This is basically what the Frostgrave miniatures game does. Opposed attacks. Eliminates all dmg rolls.
As for nat20s, nothing is perfect but I’ve settled on max roll plus the roll. So a d6 crit is 6+d6.

1. Nice! That's a halfway common take on the nat-20 damage. In fact I think Paul may have said he does that?

2. I believe Paul has mentioned doing so in the past. In Ten Dead Rats, however - can't speak for any other games he might be running - but in TDR a 20 is just a hit. He uses a critical hit table when PCs reach 0 hit points to dish out long-term or permanent injuries.

3. How much do they slow down combat?

8. Naturally I think this approach is pretty nifty. When I've tried it at the table I find the players really respond to the excitement of the explosions, even if it changes the expected value very little.

as I see it, it solves both the Yay! Crit! Boo, Crap damage! problem and the If It Hits It Crits problem when the characters can only hit on a nat 20. Neither of those is a game-breaker, but it's nice to sidestep them, particularly when you can do it in a way the players find fun.

1. Yeah, I'm very taken by all those advantages (frankly, top of all, the need-20-so-always-crit corner case).

9. One benefit of exploding dice is that players freaking love them. Savage Worlds uses exploding dice for almost all rolls, and it is the main mechanic from that game my players are always asking if I can add into D&D somehow. Exploding dice are just psychologically fun as hell.

IIRC Lucy Blumire uses exploding dice as the "special benefit" axes and axe-like weapons get in her B/X weapon rebalance. Helps give a reason to use the axes in B/X which are otherwise more or less pointless mechanically (before y'all start writing replies, yes I know that an axe is a more useful noncombat tool than a sword and that combat stats aren't everything).

https://www.kickstarter.com/projects/llblumire/fresh-from-the-forge

1. > yes I know that an axe is a more useful noncombat tool than a sword

Except, that ain't true. Battleaxes have a thin profile to keep the weapon light, swingable, easy to position for parrying, and ideal for hewing flesh rather than splitting wood. Quite unlike the triangular profile and heavy head of a wood-chopping axe. You *can* cut wood with a battleax, but it's not ideal.

And HEMA fighters have found that axes are actually pretty great weapons, at least when held in one hand and paired with a shield. A one-handed axe doesn't offer the hand protection of a sword but it's great for disarming opponents of their weapons or especially their shields by "hooking" them with the axe-head.

Combine that with the fact that a blow from an axe will concentrate force far better than a sword (though not to the extent of a flanged mace or a pick/hammer) and war axes definitely have their niche in combat. But axes designed for combat are secondary or emergency tools at best (and the same is true of certain robust swords like hunting swords, heavy hangers or cutlasses carried by army engineers/sappers in the 19th C., seaxes and messer knives, and so forth).

2. Wedge-headed splitting mauls are actually historically pretty recent, at least in Europe. Among continental Europeans who split their own wood, a significant portion even today use a sharp, thin-profile axe with a flick-of-the-wrist technique. Look up splitting axe vs. splitting maul if you're interested in more info. Gransfors makes a nice splitting axe that's actually pretty similar in size and weight to a historical one-handed battle axe, albeit with a rectangular profile rather than bearded. It's quite likely that a poor viking armed with only a spear, shield, axe, and knife would use the same axe for both combat and for mundane tasks like splitting firewood.

3. Great conversation here. On Matt's note of the use in Savage Worlds: I've played a little bit of Savage Worlds (Paul & Max run that quite a bit at conventions), but never leaped for joy at the exploding dice, I think largely because the often-subjective nature of the Raises usually left me disappointed at the judgement of their effect. To my mind (partly knowing the odds against it), 4 Raises or whatever should make me liege of the kingdom, and they rarely have much extra benefit in practice. If we hard a hard quantified effect (e.g., D&D hp) then maybe I'd find them more interesting.

4. Here's a good breakdown of the competing design priorities of wood-cutting and person-cutting axes, with photos showing the differences.

Obviously heavier axes are better both for chopping wood and hacking through armor, but the weight is substantial. Chopping down a heavy dungeon door of the types described in D&D would be kind of absurd with any weapon-weight blade, and a long, hard job even with a felling or splitting axe.

https://www.arms-n-armor.com/blogs/news/fighting-axes-vs-wood-axes

10. Sorry to bother you on this post, but is it possible to contribute to a document that is only decodable if two people contribute to it? Like, lets say there was source code that we didn't trust if only one person wrote it. Would there be a way to write your bit of the code but it would never be published until the other person submitted their code? The only reason I ask you is that you previously mentioned receipts for handing in work in college, etc.

1. I guess I don't know? Not sure if you may be thinking of another poster -- I don't recall talking about receipts for work in college. Maybe I've gotten hazy.

2. Might have been alexschroeder.ch

11. This is actually great info. I currently give max damage + exploding die on a crit (20 or 10+ roll needed) so average damage on a d6 crit would be 10.2, but could rarely explode a significant increase.

I give all backstabs extra dice and make them all exploding instead of the multiplier as well. So 2d6 up to 5d6.

If I started over I would consider using exploding dice for all damage in lieu of crits, but limit it by class as suggested above - great idea btw, although I would give thieves exploding dice on daggers and ranged weapons and clerics would get it against certain foes.

1. Very nice, that would all be cool to play with, I think!

12. I don't use critical hits, but thinking about the use of exploding dice for crits, instead of having a max roll explode, I think the best use would be to have a 1 explode. That way, when the inevitable nat 20 attack roll is coupled with the inevitable 1 on the damage roll, the player gets to roll again and add the 1. Hardly the massive critical damage players seem to expect in a game like 5E, but it would help lessen the let-down, while keeping most of the benefits of the possibility of your damage die exploding.

1. Wow, interesting reversal!

13. 1) 20 always being a hit and a crit. Using a confirmation roll fixes the problem, but when a crit is not confirmed it feels like the crit was granted, then taken away. I changed it in my game to, on a 20, the attack was so good, it got the opponent so far out of position, that you get an immediate follow up attack. It’s the exact same mechanic, but somehow it feels better. The follow up attack may miss, but it’s not taken away from the player, like the crit is. The players seem to respond better to it.
2) I would also consider making d20’s explode. The difference is minimal, a 20+d20 will hit in most cases, but I don’t like the discontinuity. Also solves the problem of really hard to virtually impossible task all needed a 20 on the die to succeed, maybe a 20 succeeds, maybe you need a 20 then a 10 plus, or 15 plus, or another 20 followed by a 16 etc.
3) An exploding d6, can’t do 6 points of damage. I read on someone’s blog somewhere, that exploding roll should be n-1, so a 1 is a 0, allowing a d6 to do 6 points of damage. It also keeps a d1 from doing infinite damage, I imagine it keeps d2 and d3 from being the same. You are back to excitement, followed by letdown, but I kind of like it.…
4) Level of success determining damage done. I love the idea of these systems, most seem tedious. I have always wanted to design a system using a d30, skill adds to the roll, defenders’ skill armor etc subtracts. Use an exploding d30 roll to hit. The tens die is level of success. 0-9 is a miss, final result is 37, then roll 3 damage dice.
5) Sorry for the long necromantic post.

1. Those are solid ideas. I did gloss over the fact that exploding-d6 can't score a 6, etc. The thing that probably bugs me the most in standard rules is a 20 being both auto-hit and a crit, such that targets hittable only by 20 are (in theory) always hit for crits.

No sweat with the necromancy. I commonly get comments on posts 10 years back now. And I was just thinking about tweaking my nat-1/20 rules, so good timing, thank you!

14. There's been some discussion about critical hits over on Grognardia in recent days, and in the discussion, exploding dice came up, so of course I had to cite this article.

It's led me to a cool little house rule for critical hits that I'm definitely going to have to try out in my next campaign:

>A natural 20 attack roll is a critical threat, the possibility that a critical hit may occur. When it happens, the damage die rolled is an exploding die, with the caveat that each time the die explodes, you subtract 1 from the damage, so that for a die size dN, damage results which are multiples of N aren't impossible. If, for example, you roll a natural 20 when attacking with a dagger, you roll an exploding d4. If it explodes 3 times — let's say that you roll 4–4–4–1 — that's a result of 10 damage.

Now if my math is right, exploding dice that subtract 1 for each explosion all have a uniformly increased expectation value of just 0.5 extra damage, regardless of the die-size. So implementing this rule is, long term, kind of like saying that on a nat 20, the damage dealt is increased by +½ point — far less impactful than a double-damage crit, which means that it ought not to disrupt the OD&D combat system as it's intended to work. (And indeed, for monsters behind the screen, on a "crit," I can just flip a coin and add +0 or +1 damage instead of dealing with exploding dice. Practically the same effect.)

In fact, the primary purpose of even doing this would be a bit of mathematical sleight-of-hand, to make players who *expect* something cool to happen on a nat 20 think that something cool is actually happening when they get to roll exploding damage — even though the actual impact is really quite minor.

1. Yeah, that's an interesting novel take -- you're right, that make a very small difference in play.