Curves vs. Linear Rolls

TL;DR: You can just do d6 x 5 instead of 5d6 for damage and get the same kind of game results, but you have to do it that way for both damage and HP for it to work well. See last two paragraphs for ideas about linear HP if that matters to you.

Anybody who has access to a 1E AD&D DMG and hasn’t checked out the part in the front where it talks about probability tables, please do so now. If you don’t have the book, or if that section never made sense, check out Anydice, which does a good job of explaining visually how any combination of dice gives a pattern of results on average.

Here’s what I’ve been thinking about lately, and I don’t understand why I never thought about it before.

1: M-Us cast spells that require lots of dice to roll. Stars Without Number has weapons that deal 5d12 etc.
2: You might have that many dice to roll at once, but you might have to roll over and over and sum as you go.
3: It takes a while and it’s boring except when you get a really high or low result.
4: You generally get an average result because of the curve of the roll probability table.
5: As you add more dice, the result becomes more predictably average.
Conclusion: This doesn’t work as well as it should.

We can try to fix this problem by addressing any of these five points. There are probably more but I didn’t think of them.

Mainly, let’s talk about shifting the roll from “sum many dice” to “one die times multiplier”.

For example, instead of rolling 5d12, which generally gives us something near 32 (and almost always 18-47) we can roll d12 x 5, which is equally likely to give us 5-60 in 5-digit increments. The min, max, and avg are the same.

This is faster and easier, more accurate, requires fewer dice, and changes the kind of result. The first few are simply advantages, but the last is a sticky point because some people might like the curve instead of the linear 5-60 result.

Let’s assume Gygax knew about these methods (since he did write about it) and intentionally used XdY throughout his game instead of dY*Z. Let’s look at a 5th level M-U from 1E AD&D.

He gets a single Fireball, his best damage-dealing spell, which does 5d6 and the victim(s) save for half damage. The average damage is (take one die’s maximum, halve it, add 0.5, multiply by the number of dice) 17.5. More important is what I’ll call a “reasonable range” which I threw in above with the 5d12. It’s the range where 95% of the results will come out. The RR of 5d6 is 10-25.

But let’s look at this from the perspective of a monster of various Hit Dice, using d8 per HD, using the RR of their average HP rolls:

1 HD: 1-8
2 HD: 3-15
3 HD: 6-21
4 HD: 10-26
5 HD: 13-32

Again, remember this is the range that you could generally expect to have happen: a higher or lower result is possible but highly unlikely, effectively half the chance of a natural 1 or 20 on d20.

What we see is that, even if the 1 HD monster saves, he will definitely die. The 2-4 HD monster will generally die unless he saves. But the 5 HD monster will generally survive even if he fails his save.

The saving throw for these monsters is 20% chance for 1-2 HD, 25% for 3-4 HD, 35% for 5-6 HD. If it saves, it takes half damage.

So imagine a 5d6 Fireball hitting a group of 100 (tightly-packed) monsters. Here are how many remain after the blast and saves are rolled:

1 HD: None
2 HD: 20
3-4 HD: 25
5 HD: All 100

Now let’s look at a d6x5 Fireball against creatures with d8xHD HP.

We have a problem. Our old “exclude the 5% outliers” doesn’t work so well because we eliminated those outliers. Let’s just use the straight result pattern possible. This means the Fireball is 5-30 and the monsters’ HPs are as follows:

1 HD: 1-8
2 HD: 2-16
3 HD: 3-24
4 HD: 4-32
5 HD: 5-40

Here it looks like the Fireball is still generally going to fry the 1 HD regardless of save. The 2-4 HD need to save to survive. The 5 HD will probably make it even if the save is failed.

The saving throw chances and results are the same as above.

And of course, remember that the average damage and HP are the same between the two methods, so laying out the numbers and running a comparison of XdY vs. dY*Z using avg is pointless.

Obviously it’s not going to be exactly the same game. When you gain a level, for example, you can’t just roll your hit die and add to your total. That’s going to result in a curved graph for your HP; that is, you are unlikely to get all 1s or all 6s if your hit die is d6. Instead, you will need to roll HP all over again at each level, simply d6 x Level if your HP are d6s.

This may result in grumblecakes from a player who went from L1 (and rolled 6×1=6 HP) to L2 (and rolled 1×2=2 HP). You could let the player take the better of the two HP totals, but that will skew toward higher HP. It’s not a game balance problem if you also give the same benefit to monsters, but it will result in longer fights since damage is relatively lower than HP and will make damage effects less valuable compared to save-or-X effects. That’s not something you’d be worried about if you usually give max HP at 1st level or let people reroll 1s for HP.

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One Response to “Curves vs. Linear Rolls”

  1. Brendan Says:

    Rolling lots of dice is only an inconvenience in games where it is a frequent occurrence. I’ve never played Stars Without Number, but if firing that 5d12 weapon happens pretty much every round, it would indeed be impractical. (White Wolf dice pools are another example that might be overly heavy in actual play.) But in D&D, shooting off that fireball is a big deal, and rolling all of those dice and carefully summing them adds to the importance of the event, I think.

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