Shout to the Probability Nerds in Here ...

[Eulalios]

... new idea for task resolution mechanic, intended to make odds intuitive but infeasible to calculate for high difficulties / high attributes.

Roll (D+A+B)d20, where "D" = base difficulty of task, "A" = relevant attribute, and "B" = any situational bonus.

For each d20 >= 18, roll another.

Now for each d20 <= 6, remove that one and another that is >= 7.

Now count "N" d20s >= 12. If N > D + b, where "b" = any situational bump, -> success.

[Composer99]

With respect to the 'D' variable in the formula for the number of d20s, does that mean you roll more dice when the task is more difficult? Or is it inversed, and you roll fewer dice?

So, basically:
- you roll a pool of d20s, where each roll of 12 or better is a nominal success
- you get critical successes (natural rolls of 18-20), where you roll an additional die for each critical success
- you get critical failures (natural rolls of 6 or less), where you subtract the critical fail die plus an additional die (presumably of your choice)
- the net number of nominal successes determines whether you pass or fail the skill check, based on the number of successes required

The basic mechanic is the same as the task resolution mechanic in, say, Arkham Horror, only with the critical success/failure add-on.

Now, would it not make sense for your skill (or other relevant) training to determine your die roll numbers, instead of the difficulty of the task (which should certainly determine the number of successes you need to pass the check)?

That is, your formula for the number of dice to roll would be:

Roll (T+A+C)d20, where 'T' = your training in the skill (e.g. number of skill ranks), 'A' = your relevant attribute (or ability modifier), and 'C' is the circumstance modifier (either bonus or penalty)?

[Eulalios]

I'm bashing together a few disparate concepts here. Re the skill setting number of dice to roll, I consciously reject the skills centered paradigm in favor of focusing on character attributes. But skill may lend a situational bonus. Further to that, I also reject a hard threshold for success ... Relying instead on exponentially diminishing chance of rolling all successes, which is what i mean this mechanic to require for a character with mediocre attribute and zero relevant training in a neutral situation.

So the base roll is Dd20 all > 11, you get extra dice for good attribute or situation (including relevance of trained ability). But if it s a (bumpy) situation, then you need D+b d20 > 11, better have good attribute or ability or else a critical success.

[NichG]

So attempting to reduce this for analysis, the only non-trivial part is the fact that low rolls interact with a pool of 'buffer' dice before they start eating into successes.

A single roll has 6/20 'extrabad' results, 5/20 'buffer' results, and 9/20 'success' results of which 3 are plus roll again.

So ignoring the interaction between extrabad and buffer (e.g. imagining no buffer existed) then the average outcome of a single roll would be (-6/20+9/20)*(1+3/20 + (3/20)^2 + ...)

The roll agains amplify the result by about a factor of 1.2.

Now, since there's buffer which occurs at almost the same rate as the extrabad stuff, very roughly speaking changing the -6/20 to -1/20 seems like a reasonable approximation for that interaction. So under that, each die you roll adds about 0.48 to your average result, with the standard deviation going like sqrt(N) as usual for central limit stuff. So you roughly need an attribute+bonus equal to difficulty to have a 50/50 chance, and the amount of deviation you can afford and still have a reasonable chance of success drops like 1/sqrt(D) as the difficulty rises.

So if we were in the very large regime, e.g. D=10000, it seems like basically this becomes a step function (is A+B>D? if so, succeed; otherwise fail).

I guess I'd want to do a simulation run to just make sure there's not some finnicky thing in the interaction between the buffer pool and extrabad pool that changes how the standard deviation grows, but its pretty hard to escape central limit theorem and it doesn't look like that particular mechanism should do it...

[Composer99]

I'm bashing together a few disparate concepts here. Re the skill setting number of dice to roll, I consciously reject the skills centered paradigm in favor of focusing on character attributes. But skill may lend a situational bonus. Further to that, I also reject a hard threshold for success ... Relying instead on exponentially diminishing chance of rolling all successes, which is what i mean this mechanic to require for a character with mediocre attribute and zero relevant training in a neutral situation.

So the base roll is Dd20 all > 11, you get extra dice for good attribute or situation (including relevance of trained ability). But if it s a (bumpy) situation, then you need D+b d20 > 11, better have good attribute or ability or else a critical success.

Okay. I'm still not sure how this justifies, conceptually, the size of your dice pool being set by the difficulty of the task.

So how does this system reflect when a character has (lots of) competence in a given task? How does it model, say, LeBron James making a check to perform a basketball stunt? Or Aragorn making a check to track the passage of Uruk-hai?

[Eulalios]

Okay. I'm still not sure how this justifies, conceptually, the size of your dice pool being set by the difficulty of the task.

So how does this system reflect when a character has (lots of) competence in a given task? How does it model, say, LeBron James making a check to perform a basketball stunt? Or Aragorn making a check to track the passage of Uruk-hai?

The concept is really simple. If you have to get > X on a dY, in order to have a "success," then the likelihood of getting N successes diminishes as ( (Y - X) / Y ) ^ N. Growing the die pool per difficulty just provides this exponential decay of success chance. Adding more dice per attribute and ability, breaks the exponential curve to give a better chance of success.

MAYBE.

I say "maybe" because this mechanic is not simply tractable. For example, in the simple case D = 1 (maybe this is an "edge" case?), NichG's reductive analysis exaggerates success likelihood by 20 %. Oddly, the overestimate is driven by the "critical successes" on 18+.

Re: Aragorn, we add dice for his relevant attribute (maybe "Awareness" if we take the Fictive Hack approach, maybe "Craft" for Adventure Fantasy Game, or of course good ol' "Wisdom" if we want to honor D&D) and for his ranger ability (maybe six extra dice for that?)