Dicey Randomness

[Narvius]

While being bored during maths class today, I thought about Angband. And the [XdY] notation for damage/healing. And ToME 3, which prefers [X-XY].

Until then I thought that XdY perfectly equals X-XY, but that's not true.
I'll take 3d3 as example.
3-9 means that there's an equal chance for each number to be chosen randomly (in that case, around 14% per number).
But 3d3 is different:

| # | A |  B  |
|--------------
| 3 | 1 |  4% |
| 4 | 3 | 11% | A = Possible different combinations
| 5 | 6 | 22% | B = Probability
| 6 | 7 | 26% |
| 7 | 6 | 22% |
| 8 | 3 | 11% |
| 9 | 1 |  4% |

As you see, you'll get more average values.

And now the question I wanted to ask:
Does Angband one random() for each dice
{ for (y; y > 0; y--) foo += random(x) + 1 }
or only one big random using-xy?
{ foo +=randomX(x, xy) + 1 }

Not that it would make a big difference, the average stays the same after all, but I'm in my curious-about-Angband-game-mechanics-phase.

[Mangojuice]

While being bored during maths class today, I thought about Angband. And the [XdY] notation for damage/healing. And ToME 3, which prefers [X-XY].

And now the question I wanted to ask:
Does Angband one random() for each dice
{ for (y; y > 0; y--) foo += random(x) + 1 }
or only one big random using-xy?
{ foo +=randomX(x, xy) + 1 }

Not that it would make a big difference, the average stays the same after all, but I'm in my curious-about-Angband-game-mechanics-phase.

In Angband, it's the former: it really rolls 3 separate dice and adds them together. The average is the same, but the overall effect on game strategy is, I think, not the same.

Due to the normal distribution, you can see the 3d3 weapon is more consistent at doing close-to-average damage, but quite unlikely to get especially high damage. For instance, the expected number of hits needed to reach, say, 5 damage is 1.185 for the 3d3 weapon and 1.286 for the 3-9 weapon - about 10% better. Other targets may work out differently. For instance, if we have to deal 9 damage, we expect 2.001 hits for the 3d3 weapon but 1.980 hits for the 3-9 weapon. These numbers get closer and closer together as the goal goes up.

But there is a substantive difference, too: if, for instance, you are fighting something a little hard for you, it's possible that dealing average damage every time will never be sufficient, but above-average damage could do it. In that kind of case, you'd want to have the more variable 3-9 weapon.

Also, if you play a variant with to_d bonuses expressed in percentages, there is an added benefit from having fewer dice with more sides: the real effect of a "+10%" damage rating is to upgrade a 1d10 weapon to a 1d11 one; a +10% bonus may have no effect at all on a 3d3 weapon.

Of course this is all quite minor in the grand scheme of things.

[Narvius]

In Angband, it's the former: it really rolls 3 separate dice and adds them together.

Okay, thanks, that's what I wanted to know.

How the heck did you calculate that expected blows-stuff? oO

[Mangojuice]

Okay, thanks, that's what I wanted to know.

How the heck did you calculate that expected blows-stuff? oO

For reaching 5 damage it's easy: basically, 2 blows would be enough no matter what, so the question is what's the probability that the first blow does 5 or more? It's 5/7 for the 3-9 weapon, and 23/27 for the 3d3 weapon. 1.286 is about 1+2/7, while 1.185 is about 1+4/27.

For more damage, it's a lot of individual cases; you collect the probability to hit each particular amount of damage in each number of turns, and use that for the next higher number of turns. I made a spreadsheet for some more tests because I got curious. For instance, if you want 20 damage, and you have a weapon with mean damage 6, the expected number of turns is:

1d11: 3.886 turns
3-9 (1d7+2): 3.802 turns
2d5: 3.804 turns
3-9 (2d4+1): 3.783 turns
3d3: 3.783 turns
4d2: 3.817 turns
6d1: 4 turns

(3d3 is better by 0.00008 compared to 2d4+1; not an exact tie.)

So you see the differences become pretty small generally.

[Narvius]

Wow, that's interesting.
You could upload that spreadsheet... ^.^

[bpleshek]

3-9 is basically 1d7+2 rather than 3d3 if you're looking for an equivalent to random distribution.

Brian

[Mangojuice]

Also, if you play a variant with to_d bonuses expressed in percentages, there is an added benefit from having fewer dice with more sides: the real effect of a "+10%" damage rating is to upgrade a 1d10 weapon to a 1d11 one; a +10% bonus may have no effect at all on a 3d3 weapon.

Actually, that's wrong. On closer checking, I note that the Zangband/Oangband style damage algorithm actually handles fractional dice. So a d3.5 is a d3 with a 50% chance of an extra +1. Which is actually kind of bad: that .5 adds .5 to the expected result, whereas a whole 1 (d4 in this case) would add only .5. So, for instance, a 3d3 weapon with +30% damage would do an average of 8.7 damage, while a 3d3 weapon with +35% damage would actually do 7.55 damage average.

[Narvius]

@bpleshek
Yeah, I know. But as said, I refer to ToME 3. It has 3-9 as warhammer damage (at least I think so... *cough*).

[Djabanete]

You should forget about the math and go sacrifice a sheep to the RNG.

[Narvius]

You surely meant a yeek