[RPG] the average roll of a die with an expanded explosion threshold as compared to one with an expanded success threshold

dicestatistics

This is going to be a bit of a weird question, but I'm figuring dice pool probabilities for a homebrew system and I want to know how certain abilities should be priced regarding their use. The basic roll of this system is roll a pool of d6's, count the dice that read 4, 5, or 6 as a "plus" (similar to Burning Wheel or Shadowrun), and explode dice that show 6. Pretty standard so far, but where I'm running into issues is regarding two abilities.

The first ability expands the threshold of success to threes on the d6, but otherwise the explosion threshold remains the same.

The second ability is the more difficult one; it expands the explosion threshold to fives and sixes on the initial roll but not any rerolls from exploding dice; the success threshold remains the same.

My question is this: What do the statistics for all three of these look like? I've attempted to use Anydice to answer this question but it doesn't appear to have a way of looking at the side of the dice rather than the number printed on it, so I don't think that can answer my question. I'm just looking for the average of total "hits" or "successes" per roll here, not the actual total number. Any detailed answer would be much appreciated.

Best Answer

Here's an AnyDice program to simulate it:

function: test N:n against X:n reroll Y:n later Z:n {
    if N >= Y { result: 1 + [test d6 against X reroll Z later Z] }
    if N >= X { result: 1 }
    result: 0
}
output [test d6 against 4 reroll 6 later 6] named "success on 4-6, reroll on 6"
output [test d6 against 3 reroll 6 later 6] named "success on 3-6, reroll on 6"
output [test d6 against 4 reroll 5 later 6] named "success on 4-6, reroll on 5-6 (re-reroll on 6)"

Looking at the summary output, we can see that the baseline roll (success on 4+, explode on 6) yields on average 0.6 successes per die. Lowering the success threshold to 3+ increases this to an average of 0.8 successes per die, whereas lowering the explode threshold to 5+ gives 0.7 successes per die. (I was a bit surprised that the mean success rates would work out to such nice round numbers, but it seems they do.)

Thus, lowering the success threshold is a significantly better investment than lowering the explosion threshold.

In fact, this is easy to see intuitively: lowering the success threshold by one gives you a 1/6 chance of an extra success; lowering the explosion threshold by one gives you a 1/6 chance of an extra die roll. Given that the expected number of successes per die is less than one, it's pretty obvious that, given the choice, you should choose an extra success over an extra die.

Ps. I did some more testing, and it sems that, if you modify the second ability to also allow successive rerolls on 5+, the average number of successes per die increases to 0.75. That's still less than for the first ability, but closer.

Allowing a reroll on any initial success (i.e. on an initial roll of 4+), but only on a 6 for subsequent rolls, turns out to give the same average number of successes, 0.8, as lowering the success threshold to 3. Thus, if you really want the two abilities to be equally good, that might be a good option to consider.

(Or you could keep the improved reroll ability as originally suggested, i.e. only improving the first reroll chance by 1/6, and make it cost half as much as the improved success ability; if you let the player take the improved reroll ability twice, that would be, on average, equivalent to the first ability.)

Related Solutions

[RPG] How to model Burning Wheel probabilities in AnyDice

AnyDice handles exploding dice in a peculiar fashion that's poorly-suited for roll-and-count dice pools. I have cobbled together a somewhat inelegant solution, but it appears to work correctly.

Here's the link to the program.

Instead of using dice in the usual manner, I've created several functions that, given parameters of a die, return its output, i.e. the number of successes it has produced.

function: roll ROLL:n threshold T {
  if ROLL >=T {result: 1}
  result: 0
}

function: rollexploding ROLL:n threshold T{
  if ROLL =6 {result: 1+[rollexploding 1d6 threshold T]}
  result: [roll ROLL threshold T]
}

function: rerollfailures ROLL:n threshold T{
  if ROLL < T {result: [roll 1d6 threshold T]}
  result: [roll ROLL threshold T]
}

function: rerollthenexplode ROLL: n threshold T {
  if ROLL < T {result: [rollexploding 1d6 threshold T]}
  result: [rollexploding ROLL threshold T]
}

Then I made a wrapper function that would figure out which of these functions to call, and handle requests for multiple dice being rolled:

function: wrapper DICE:n threshold T explode E reroll R{
  RES:0
  loop N over {1..DICE} {
    if E & R {RES: RES+[rerollthenexplode 1d6 threshold T]}
    else {if E {RES:RES+[rollexploding 1d6 threshold T]}
      else {if R {RES:RES+[rerollfailures 1d6 threshold T]}
        else {RES:RES+[roll 1d6 threshold T]}
      }
    }
  }
  result:RES
}

Finally, I made a bunch of functions that simplify input by providing pre-determined combinations of parameters. I'll provide only 3 of them here:

function: b DICE:n{
  result:[wrapper DICE threshold 4 explode 0 reroll 0]
}

function: be DICE:n{
  result:[wrapper DICE threshold 4 explode 1 reroll 0]
}

function: gr DICE:n{
  result:[wrapper DICE threshold 3 explode 0 reroll 1]
}

To use it, type things like output [b 4]

[RPG] Odds for dice pool with exploding dice

There's a built-in explode function, but you have to use it before doing your other processing. Specifically, use FOOdBAR to have BAR evaluated as a sequence and used to generate FOO "dice" with arbitrary side counts and values. Here, DICEd([explode d6]) does the job, turning an exploded d6 into the base for the actual roll. (explode doesn't separate dice that you give it — you have to do that, or you'll get a 5d6 roll that only explodes, once, on 30.)

You can't compare sequences to a threshold to get a list or sum of values against that threshold — you'll just get the sequence summed and checked against the threshold. Instead, use [count VALUES in SEQUENCE] for each of the valid values (3-6). And as it turns out, since every 6 triggers a new die and all those dice count, you can just double-count 6s once exploded. So really that should be [count {3..6, 6} in ROLL].

The results, as best I can manage, aren't pretty. Each explosion will either fumble harder (1), make no progress against fumbling (2-5), or break even and get a new chance to do the same thing (6), and the more dice you start with, the more chances for explosions. There is no provision for fumbles to peter out in explosions, as one would expect from a great success; the only question is whether the chain ends with a 1 or not.

Best Answer

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[RPG] How to model Burning Wheel probabilities in AnyDice

[RPG] Odds for dice pool with exploding dice

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