Telescoping Bell Rolls

So, we've all heard about the "telescoping" d20 rule presented in the Epic Level Handbook. For those cases where automatic success just isn't good enough. It's a wonky patch IMHO. Why? Well, it doesn't take into consideration new game mechanics, like Action Points.

On the other hand, Bell Curve rolling is nice -- it works very nicely with Action Points, since an extra d6 makes your roll 4d6 instead of 3d6. However, the low chance of critical success sticks in my craw. If dice depict the "cinematic"-ness of a game, neither the uniform d20 nor the normal 3d6 really make me excited.

So, what's the solution? I offer for your analytical abuse the following method: roll 3d6, and you get to re-roll any natural "6", including "6"s that come from re-rolls. Add all re-rolls to original result.

The expected value of a telescoped d6 is 4.2 (vs. 3.5 for uniform), but a single die doesn't generate a very nice distribution -- for example, it's impossible to get a 6, 12, 18, etc. because any multiple of 6 is *always* re-rolled. The PDF is a step function with some gaps -- this is also the issue with telescoped d20, it's just a wider step in the step function.

The behavior of telescoped 3d6 is much nicer.

I've got the variance and such worked out at work, but not here at home. Any math people have the ability to make pretty graphs of the "telescoped 3d6" function's PDF? And for 4d6 (Action Dice!)?

Thanks, -- N

Nifft said:

The behavior of telescoped 3d6 is much nicer.

I've got the variance and such worked out at work, but not here at home. Any math people have the ability to make pretty graphs of the "telescoped 3d6" function's PDF? And for 4d6 (Action Dice!)?

Click to expand...

I'm not sure that it is particularly nice, but I can graph these two distributions for you. Here's the 3d6. This doesn't cover all possibilities (obviously, since they're infinite), but it's accurate in what it does cover. There's about a 0.4% chance of leaving this neighborhood on a good roll.

Here's the 4d6 distribution, with about 98% of possible cases displayed. Again, everything shown is accurate.

How would the GM accomadate for players that have an ability score over 18? If Billy rolls 2 sixes, and a five, then rerolls his two sixes and both come up threes, then the final result would be 23 correct? How is a GM supposed to have this character be on par with other character that have abscores under 18?

Does my previous post make sense or did I miss the point entirely?

Frukathka said:

Does my previous post make sense or did I miss the point entirely?

Click to expand...

Yes, you missed the point entirely. I think the idea behind this method is to replace 1d20 rolls with exploding 3d6 rolls.

D'OH! My bad.

It would be kind of interesting for ability scores, though that's not my intent.

-- N

Here's some results from my test program:

>>> dicetest.trial(1000000,3,6)
Trials: 1000000 Dice: 3d6
Avg: 14.000667 Var: 32.6841112393 StDev: 5.71700194501
Max: 67

>>> dicetest.trial(1000000,4,6)
Trials: 1000000 Dice: 4d6
Avg: 18.663042 Var: 43.5712548775 StDev: 6.60085258716
Max: 75

And for comparison:

>>> dicetest.trial(1000000,1,20)
Trials: 1000000 Dice: 1d20
Avg: 11.560508 Var: 55.3982201802 StDev: 7.44299806397
Max: 116

-- N

Nifft said:

Here's some results from my test program:

>>> dicetest.trial(1000000,3,6)
Trials: 1000000 Dice: 3d6
Avg: 14.000667 Var: 32.6841112393 StDev: 5.71700194501
Max: 67

>>> dicetest.trial(1000000,4,6)
Trials: 1000000 Dice: 4d6
Avg: 18.663042 Var: 43.5712548775 StDev: 6.60085258716
Max: 75

And for comparison:

>>> dicetest.trial(1000000,1,20)
Trials: 1000000 Dice: 1d20
Avg: 11.560508 Var: 55.3982201802 StDev: 7.44299806397
Max: 116

Click to expand...

The exact averages for open-ended rolls (reroll max and keep diewise):
1d20: 210/19 ~= 11.053
3d6: 12.6
4d6: 16.8

I'm working out a good way to get closed-form standard deviations, but at least I have an accurate estimate for the exploding 1d6: 3.26190128606002.