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As many of you are no doubt aware, the Tier system works as follows for weapons:
Tier 1 rolls 3d200* and takes the lowest roll.
Tier 2 rolls 3d200 and takes the median roll.
Tier 3 rolls 3d200 and takes the highest roll.
But what does that actually look like? What are the stats underlying these rolls? And is Sspitfire for hire for making pretty graphs? (Yes. Yes, I am.)
Well here are some visualizations of what it looks like:
Probability Density Functions and their Underlying Tables
And here are some stats:
3d200 Stats (and PDF tables for download)
*3d200 means rolling three 200-sided die or randomly generating a number from 1 to 200 three times.
**Ignore the z-axis- it is meaningless.
Goblin Squad Member
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Last winter/spring, Nightdrifter did extensive Monte Carlo simulation, e.g. this thread. Nightdrifter, Nihimon and others (Cheatle, Dazyk, Narad, and other e.g. T7V) have done much of what you have recently rereleased. Consider asking them what they have and give them credit. I expect that you are proud of pulling together what you have, but mush of this was trail blazed. (I understand these analyses are often part of his daily work; I have left that type of work behind, but I do marvel at what he does. Just look at the calculator).
The other main things you could note are these:
Tier 1 attacks are very heavily weighted towards the low end, with roughly 90% of your Tier 1 attack rolls coming up less than 100.
Tier 2 attacks are heavily weighted in the middle, with roughly 90% of attacks being between 28 and 174- and the bulk of that really being between 50 and 150.
Tier 3 attacks are heavily weighted in the upper end, with roughly 90% of your Tier 3 attack rolls coming up above 100.
The expected average of an infinite number of Tier 1 attack rolls comes out to 50.5.
The expected average of an infinite number of Tier 2 attack rolls comes out to 100.5.
The expected average of an infinite number of Tier 3 attack rolls comes out to 150.5.
So if you are wondering, "How much better is Tier 2 than Tier 1?", the answer is "Roughly 2x better."
The 3D graphs are basically just visual representations of everything above.
Sorry for the indecipherability ;) Sometimes I over-estimate my audience's statistical capabilities.
Could someone please post the attack roll and damage equations? I can't find them anywhere.
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Goblin Squad Member
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The other main things you could note are these:
Tier 1 attacks are very heavily weighted towards the low end, with roughly 90% of your Tier 1 attack rolls coming up less than 100.Tier 2 attacks are heavily weighted in the middle, with roughly 90% of attacks being between 28 and 174- and the bulk of that really being between 50 and 150.
Tier 3 attacks are heavily weighted in the upper end, with roughly 90% of your Tier 3 attack rolls coming up above 100.
This! (as was first pointed out before Nightdrifter made his first graph).
T1 attack vs 50 is more likely to miss by a low margin, but occasionally exceeds by a large amount. This is good for T1 vs T3, but unless crits have meaningful effects it slightly favours the defender in T1 vs T1
T2 attack vs 100 (well, 100.5) is symmetric, but compared to a straight d200 is less likely to have extreme high or low numbers.
T3 attack vs 150 is the reverse of T1: more likely to exceed but will occasionally fumble badly.
@Sspitfire:
1) IMO it would make more sense to:
a) plot probability densities (non-cumulative) to show the skewness of the distributions and give a better intuitive comparison of tiers.
b) compute the 'miss factor' (square root and all) vs 50/100/150 and plot that.
c) extended: integrate/average to show the average damage done for various scenarios.
Monte carlo is just one way of doing it. I'd probably prefer the brute force in this case - anyway you already have the probabilities, so moot point.
2) Never overestimate the audience (except when posturing at conferences). Imagine we are all journalists and politicians.
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Goblin Squad Member
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I'd probably prefer the brute force in this case
scratch that. The analytical problem for the first part is simple enough i could do it with pen and paper while supposed to read work emails, so I'd probably try finesse before force (still no monte carlo though).
for 3 dN (with N^3 possible outcomes),
the #outcomes that has a certain dice=x is
#low(x)= (N+1-x)^3 - (N-x)^3
#high(x)= x^3 - (x-1)^3
#middle(x) = 6x (N+1-x) - 3(N+1) +1
The middle one was a bit tricky, so feel free to confirm it independently. Doing it for a d20 in excel gives a checksum of 8000 and the right averages in each column, though.
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Goblin Squad Member
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..and here are the results for 'average damage factor':
vs T1 vs T2 vsT3
T1 82% 59% 39%
T2 96% 82% 61%
T3 100% 96% 85%
Meaning, a T1 roll vs defense 50 on average does 82% of full damage, a T2 roll vs defense 150 on average does 61% of full damage, etc.
Assumptions: +0 attack bonus, +0 reflex bonus (except the 50/100/150 from armor). No crits.
T1vT1 has only 42% chance of a full hit (possible crit), where T3vT3 has a 58% chance of full hit.
Real numbers will be different, but the value of upgrading your armor is very obvious.
Note also that in reality higher tier players will have higher base damage, more hit points and higher protection. Time to kill an equally matched opponent will depend on how different bonuses scale.
(i was surprised to see the same 82% for T1 vs T1 and T2 vs T2, it seems that the square root skewness and the d200 skewness cancel out. Higher number for T3 vs T3 is as predicted).
I speculate that defense buffs like parry/block/evade are slightly better at T3 levels and attack buffs slightly better at T1, but the difference (like the 82% vs 85% avg damage) will be small.
@Sspitfire:
1) IMO it would make more sense to:
a) plot probability densities (non-cumulative) to show the skewness of the distributions and give a better intuitive comparison of tiers.
b) compute the 'miss factor' (square root and all) vs 50/100/150 and plot that.
c) extended: integrate/average to show the average damage done for various scenarios.
Yeah it occurred to me to do the PDF's as well, since that might also be easier for others to understand. I am also going to mess around a little more with the CDF, as well, and see if I can't add some more (meaningful) information to it.
It looks like you have already done b) and c); but maybe there is some graphics I can do with it as well (graphs are fun!).
Goblinworks Executive Founder
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I'd probably prefer the brute force in this case
scratch that. The analytical problem for the first part is simple enough i could do it with pen and paper while supposed to read work emails, so I'd probably try finesse before force (still no monte carlo though).
Spoiler:for 3 dN (with N^3 possible outcomes),
the #outcomes that has a certain dice=x is#low(x)= (N+1-x)^3 - (N-x)^3
#high(x)= x^3 - (x-1)^3
#middle(x) = 6x (N+1-x) - 3(N+1) +1The middle one was a bit tricky, so feel free to confirm it independently. Doing it for a d20 in excel gives a checksum of 8000 and the right averages in each column, though.
Checking your math on "middle":
All permutations of that are: X,X,X; X,X,<X (3 permutations); X,X,>X(3 permutations); X,<X,>X(6 permutations);
The frequency (number of times the outcome happens in N^3 rolls, for each X) of X,X,X is 1 (Every X will come up triples once).
For X,X,<X, it is x-1 (the frequency out of n of rolling less than x; because there are 3 ways to get two the same and one lower, the total frequency is 3x-3
By similar mathX,X,>X has total freqency 3n-3x.
The frequency of X, <X, >X comes out to 1*(x-1)*(n-x). Six permutations contribute -6X^2+6xn+6x-6n
Summing the frequencies, I get 1+3x-3+3n-3x-6xx+6xn+6x-6n
simplify: -6xx+6xn+6x-3n-2
I can factor that to 6x(-x+n+1)-3(n+1)+1
math checks out.