--Bryan
For the purposes of baselining scenarios, I was trying to figure out how big a skill difference creates how much chance of the higher skill winning. Even confining it to a simple contest (or single roll in an extended contest) there ends up being rather a lot of maths, however I'll point you around those parts if you don't fancy them, and just give you some sample numbers For example, the next bit is all maths, so skip over it if you can't bear that horror, and go to the ****
In a contest between equal abilities, the lower roll wins (380 possibilities), and equal rolls tie (20 possibilities). Each side has equal chance of winning (190/400)
In a contest between non-equal abilities with no mastery difference,
and neither ability equal to 20 or 1, the lower roll wins, and equal
rolls tie, with the following exceptions:
• If the superior ability rolls a success greater than the
inferior ability, and the inferior ability rolls a failure less than
or equal to the superior ability and less than its roll (i.e in 17 vs
14, 17 rolls 17, 14 rolls a 15 or 16, or 17 rolls a 15, 14 rolls a
15) then the lower roll wins. The number of such combinations is a
triangular function, given by n*(n-1)/2, where n = superior ability –
inferior ability.
• If both sides make the same roll, and it is greater than the
inferior ability but less than or equal to the superior ability (i.e.
in 17 vs 14, they both roll 15, 16 or 17), then equal rolls result in
a victory for the superior ability. The number of such combinations
is n, where n = superior score – inferior score.
This implies that:
• the superior ability wins in 190 + n*(n-1)/2 + n
combinations,
• the inferior ability wins in 190 –n*(n-1)/2 combinations,
• they tie in 20-n combinations.
Therefore the superior ability will win in 2*(n*(n-1)/2) + n
combinations. Simplifying, this is n*(n-1) + n, = n^2.
To use a few common examples, the % more contests won (combinations
out of 400 in brackets)
17 versus 14, 17 wins 2.25% more contests. (196 versus 187, 17 ties)
In 14 versus 13, 14 wins 0.25% more contests. (191 versus 190, 19
ties)
In 14 versus 6, 14 wins 16% more contests. (226 versus 162, 12 ties)
In 17 versus 6, 17 wins 30.25% more contests. (256 versus 135, 9
ties).
You may notice that for small ability gaps, the difference in winning percentage is very small. For example, in an extended contest between someone with an ability of 17 and someone with an ability of 14, on average you'd have to go through 22 rounds to hit one where the skill gap made a difference..
The math doesn't work so neatly over a mastery step. With a mastery your chance of winning `big' (minor or better victory, instead of just a marginal one) go up, but how about your chance of success? To calculate this we'll need to do some more maths, so feel free to skip along again, to the ####.
If you plot out on a grid the results, it looks very different with a
mastery compared to without. Without a mastery, the area where the
low role doesn't win is somewhere in the middle of the chart, around
the gap between the two abilities. With a mastery difference, the
areas where the low role doesn't win happen at the top left and
bottom right corners, the areas where the mastery bump turns "success
versus success, low roll wins" into "critical beats success",
or "fail versus fail, low roll wins" into "success beats fail."
For convenience define `a' as the superior ability minus its mastery
(i.e. for 5W, a = 5), and `b' as twenty minus the inferior ability
(i.e. for an inferior skill of 17, b = 3), then we get
two `triangular' functions, one in terms of a and one in terms of b.
When they don't overlap (a+b<=20) and neither skill is 1 or 20, then
the difference in the number of winning combinations of die rolls is:
• (a-1)*(a-2)+a + (b-1)*(b-2)+b.
If a+b>20, then it gets even more complex as you have to subtract a
correction factor of .
• ((a+b-20)(a+b-19)/2 to account for the overlap of the `a'
region and the `b' region.
Percent difference in winning is this equation divided by four. Obviously this is a more difficult calculation to handle on the fly! However what is interesting, when you compare it to the non-mastery gap case is that it is not constant for the difference between the abilities. This is to say that applying the same modifier to both contestants could actually change their odds of winning!
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Some examples are:
• 5W versus 14, 5W wins 10.75% more contests ( 217 versus 174,
with 9 ties) (for the same ability gap without a mastery jump, the
superior skill won 30.25% more contests).
• 6W versus 6, 5W wins 49% more contests (298 versus 102, with
0 tie)
• 17W versus 17: 17W wins 65.5% more contests (331 versus 69,
with 0 tie) (for the same ability gap as the previous example, the
odds of winning go up quite a bit)
To provide a more direct comparison, for an 18 point ability gap,
here are the percentage differences in the chance of winning:
• 19 versus 2: 81% (90.5% vs 9.5%)
• 2W versus 4: 57% (78.5% vs 21.5%)
• 9W versus 11: 32.5% (66.25% vs 33.75%)
• 17W versus 19: 64.5% (82.25% vs 17.75%)
Clearly what this says is that it is not adequate to merely know the gap between skills, you have to have some idea of the actual skill values, as it can make a substantial difference in the odds of winning. `
Of course, I programmed the equations into a spreadsheet to help generate all of these numbers, and I've uploaded that to the files section of the group.
Regards;
-Bryan
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