Sil: melee to-hit chance table

[BlueFish]

Decided to work this out. Basically as I understand it all melee rolls are between two 20 sided dice, the attacker's roll having a positive or negative modifier based on the difference between the attackers' and defender's melee and evasion. Ties go to the defender.

So this amounts to two rolls, each having a range of 20, with those two ranges offset against one another by some amount. I.e., 5-24 vs 1-20, if melee is 4 more than evasion.

So first you work out the chance that both rolls occur within the overlapping region of the ranges. That's the fraction of pairings where both values are within the overlap, divided by the total number of pairings. Easy enough. Within our example, rolls between 5 and 20 are within the overlap. Range is 16, making the number of pairings within the overlap 16^2. Total number of pairings would be 20^2.

Then you work out the number of possible ties within the overlap. That's just the number of values within the overlap. 5 through 20 in our example, making 16 tied pairings.

Then you work out the total number of wins within the overlap, which is just half the number of non-ties within the overlap (since there will be identical number of non-tied wins and losses, statistically, within any overlap region).

If melee is greater than evasion, pairings outside the overlap are automatic wins. If melee is less than evasion, pairings outside the overlap are automatic losses. So chance of win outside overlap is always either 1 or 0 depending on whether melee is higher than evasion.

So the formula for hit chance becomes "chance of being outside overlap * chance of win outside overlap + chance of being inside overlap * chance of win inside overlap".

Here's the table, indexed by "melee minus evasion". Anything beyond these values are automatic hits or misses.

For hair-pulling fun, you can raise "one minus the appropriate value" by the power of N next time you miss N times in a row and die. That'll be the chance of that happening.

-18	0.0025
-17	0.0075
-16	0.015
-15	0.025
-14	0.0375
-13	0.0525
-12	0.07
-11	0.09
-10	0.1125
-9	0.1375
-8	0.165
-7	0.195
-6	0.2275
-5	0.2625
-4	0.3
-3	0.34
-2	0.3825
-1	0.4275
0	0.475
1	0.525
2	0.5725
3	0.6175
4	0.66
5	0.7
6	0.7375
7	0.7725
8	0.805
9	0.835
10	0.8625
11	0.8875
12	0.91
13	0.93
14	0.9475
15	0.9625
16	0.975
17	0.985
18	0.9925
19	0.9975

[half]

That looks right. It can be easy to think of it as chances in 400 of hitting. Seen that way it starts:

1, 3, 6, 10, 15, 21, ... (i.e. the triangular numbers)

The chances of missing are like this starting from the other end.

These numbers are the same with other opposed skill checks. They are out of 100 now, but still start from each side with:

1, 3, 6, 10, 15, 21, ...

[half]

For the maths nerds, this two dice distribution has a nice s-curve shaped culmulative distribution. The triangular numbers mean it is the piece-wise combination of a parabola and an upside-down parabola. Believe it or not, this was why I chose to use two-dice distributions for all the skill checks in Sil. The properties of this distribution are much nicer than for one die (which has a flat PDF and a linear CDF).

(Wow that was even more mathsy than I intended, but I'm sure Nick will appreciate it...)

[BlueFish]

Originally Posted by half

For the maths nerds, this two dice distribution has a nice s-curve shaped culmulative distribution. The triangular numbers mean it is the piece-wise combination of a parabola and an upside-down parabola. Believe it or not, this was why I chose to use two-dice distributions for all the skill checks in Sil. The properties of this distribution are much nicer than for one die (which has a flat PDF and a linear CDF).

(Wow that was even more mathsy than I intended, but I'm sure Nick will appreciate it...)

I think it makes sense from a game design standpoint. The player is likely to be floating around "equal with his opponents" at any given point in the game, and that point should have some gravity. One more point of evasion or melee has more meaning while you're equal than it has at any other point.

[Patashu]

Yeah, the two roll distribution is great for making just barely getting higher than your opponent's statistic worth it.

[taptap]

Originally Posted by half

Believe it or not, this was why I chose to use two-dice distributions for all the skill checks in Sil.

I believe it. Triangular numbers are all over Sil after all. I like the properties of this, compared to the flat distribution in single rolls, or the many complicated systems which however are flattened by introducing a critical roll that always hits. Having it listed as a table is quite helpful when calculating chances for critical hits (is the 0, 1d7, 1lb or the +1,1d8, 1.3 lb shortsword better?).

@patashu: While percentage-wise the biggest effect seems to be in the middle - the relative value you get out of an additional point is biggest in the margins. Say your evasion is 14 points better than your opponents - now gaining two additional points (say finding a nice +2 shortsword as substitute for your +1 shortsword w/ parry) in evasion more than halves the hits against you. (2 points incidentally is the size of the stun effect.)

[Patashu]

Originally Posted by taptap

@patashu: While percentage-wise the biggest effect seems to be in the middle - the relative value you get out of an additional point is biggest in the margins. Say your evasion is 14 points better than your opponents - now gaining two additional points (say finding a nice +2 shortsword as substitute for your +1 shortsword w/ parry) in evasion more than halves the hits against you. (2 points incidentally is the size of the stun effect.)

The relative value is indeed largest in the margins, but the absolute value is largest near the center. If you outclass opponents by that much in the relevant statistic, they're starting to become lesser and lesser threats - though, it depends on what the result of a failed roll will be (As in, will it be catastrophic if it happens once? twice? more times?) - if you REALLY don't want it to happen, the relative gain at the margins is very important and nice to track.

[Nick]

Originally Posted by half

For the maths nerds, this two dice distribution has a nice s-curve shaped culmulative distribution. The triangular numbers mean it is the piece-wise combination of a parabola and an upside-down parabola. Believe it or not, this was why I chose to use two-dice distributions for all the skill checks in Sil. The properties of this distribution are much nicer than for one die (which has a flat PDF and a linear CDF).

(Wow that was even more mathsy than I intended, but I'm sure Nick will appreciate it...)

That is really cool. For anyone who doesn't believe me, note that a parabola is the graph of y = x², and the sum of any two triangular numbers is a square

[Scatha]

Originally Posted by Nick

... and the sum of any two triangular numbers is a square

Of any two *consecutive* triangular numbers.

[half]

Originally Posted by Patashu

The relative value is indeed largest in the margins, but the absolute value is largest near the center. If you outclass opponents by that much in the relevant statistic, they're starting to become lesser and lesser threats - though, it depends on what the result of a failed roll will be (As in, will it be catastrophic if it happens once? twice? more times?) - if you REALLY don't want it to happen, the relative gain at the margins is very important and nice to track.

Yes, I really like the tension between these two things.

Regarding the relative values, I remember playing Pool of Radiance and getting a character to have such a good AC that I went from being hit 1/10 to 1/20. This effectively doubled my character's HP, and I thought it was very interesting. There are a number of things that go against aiming for this kind of thing though. (1) there are several effects (breath weapons, archery, spells) that ignore evasion. (2) you end up using a lot of time in each battle. (3) different monsters have different skill scores so finessing it against one won't be as good against others. (4) it matters by how much opponents hit you, which makes evasion quite useful in the middle too.

Note that protection works similarly at the top end: if an extra point of prot lets you take 1 damage instead of 2, you last twice as long. However the randomness in damage and protection rolls makes this less striking, and (1)-(3) all still apply.

If any of you have played the board game Eclipse, it has a one die (1d6) distribution, but because the default is to hit only on a 6 and 6s are automatic hits, it is always hovering right on the edge of the relative value thing. If you get better accuracy, you hit twice as often, doubling your combat prowess, but then the incentive for them is to get better shields, halving their chance of hitting etc. It is actually a very neat system. I invented a really tight 1d6 evasion/protection combat system the other day and Scatha pointed out that it was equivalent to the Eclipse one...