D&D General Replacing 1d20 with 3d6 is nearly pointless

[Ovinomancer]

Have you looked at the graphs I've been linking to, or not?

How math works is you do steps and results come out.

Do I have to take a screen shot? I have to take a screen shot. naughty word.
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So here we have the CDF (cumulative distribution) of 1d20 and the CDF (cumulative distribution) of 3d6 with different averages and standard deviations normalized.

The 1d20 curve is a line. The 3d6 curve is the set of black points. Notice how the 3d6 curve is close to, but not exactly on, the 1d20 line. It only differs significantly at the 5% "critical hit/miss" cases that correspond to 1 and 20 on the d20 roll.

I horizontally scaled 3d6 by a factor of 2, which corresponds to "bonuses and penalties are twice as large, conceptually, in a 3d6 based situation".

So yes, that is how that works. The distributions are similar in CDF, because you can see it. Yes, one is a flat distribution and the other is a normal(ish) one, but we aren't playing "can you roll a 7", we are playing "can you roll a 7+" when we play D&D. And "can you roll a 7+" corresponds to the CDF (the integral) of the distribution.

And when you integrate things, the differences between a flat distribution and a curved one fade away pretty fast.

This isn't "mathturbation", because I actually checked my results. I even shared links to those results being checked. I am not sure why I expected people to actually click on those results before saying "this is naughty word".

Anyhow, here is the results inline.

Quite possibly a slightly different value than "2" would be more correct once we neglect tails -- a different value than "2" would correspond to a change in the slope of the 3d6 part of the graph, and making it slightly less steep might improve the match (except for the tails). But 2 is so close I really don't care.

Sigh. The screenshot you presented should have clued you into a problem with what you did. The scaked and recentered 3d6 curve has values from -5 to 25 while the d20 is 1 to 20. All you've done is stretch a normal cumulative dustribution and note that, if you strech it enough, the middle part looks straight(ish). You're ignoring ~1/3 of the data points to do this.

This is like blowing up a circle to a large enough circumference that a close look at a tiny part of the arc looks like a straight line. But, despite doing this, a circle is not a straight line. This is why when you do math, you really need to understand what your doing -- what assumptions are necessary. Just doing math doesn't mean you'll get the right answer. Especially with stats.

 

[NotAYakk]

Sigh. The screenshot you presented should have clued you into a problem with what you did. The scaked and recentered 3d6 curve has values from -5 to 25 while the d20 is 1 to 20.

The total probability of values from -5 to 1 on the 3d6 curve is under 5%.
The total probability of the values from 20 to 25 on the 3d6 curve is under 5%.

All you've done is stretch a normal cumulative dustribution and note that, if you strech it enough, the middle part looks straight(ish). You're ignoring ~1/3 of the data points to do this.

I'm ignoring the 5% most extreme values on both ends, and looking at the middle 90%.

With a d20 on an attack, a natural 1 already misses and a natural 20 already hits; in a sense, it corresponds to a -infinity and +infinity.

I talked about the outlying 5% cases already, and now you bring it up as if it was some big gotcha. Those are the crit/auto hit/miss mechanics.

This is like blowing up a circle to a large enough circumference that a close look at a tiny part of the arc looks like a straight line. But, despite doing this, a circle is not a straight line. This is why when you do math, you really need to understand what your doing -- what assumptions are necessary. Just doing math doesn't mean you'll get the right answer. Especially with stats.

I assume we care about "does this attack hit/miss" experience at the table.

Given a game played with double-modifiers and d20, and another played with 3d6, I claim distinguishing between those games with a log of hit/misses (and not the rolls) will be a herculean task.

You'd basically have to find some creature whose chance of being hit is right on the edge of possible for the 3d6 case (in the "long tail" of 16-18) and tease out if the chance is different than 5%.

Suppose we want a 2 SD error bar on the sample. We have some random variable H. Its true value is either 1/20 or 1/216. How many samples do we need to distinguish that?

Quick napkin math (I think the right answer involves using student's T? It is basically a polling problem.) gives me about that it would take about 100 samples of "creature we know needs to be hit on an 18 on 3d6" to see a significant (2 SD, or 0.03 P-test) difference between the d20 with double modifiers and auto-hit on a 20 and the 3d6 with normal modifiers.

Or, tl;dr, we really don't care about events with really low probability, as they don't happen often enough to care about them. And the entire "tail" you are pointing at adds up to a low probability event.

 

[TheCosmicKid]

I did an extensive analysis of 3d6 vs d20 a while ago when we considered going to 3d6. Like a lot of people, it works best for skill checks and even saving throws because they are simple "all-or-nothing" rolls. Using 3d6 increases the likelihood of "typical results" which is what I'd expect from a single-roll test.

This is sort of true, but you have to be careful how you understand it. A 50% probability check on a d20 -- say, a +4 roll against a DC of 15 -- is still a 50% probability check on 3d6. Your odds of rolling an 11 are higher, but your odds of rolling an 11 or above are still the same. What the normal distribution on 3d6 does is make the probability of success/failure "fall away" from 50% faster as your bonus or the DC changes.

I'm not attributing any fallacy to you in particular. Maybe you already know this and I'm preaching to the choir -- awesome! But in my experience a lot of people hear "increased likelihood of typical results" and think that it means they're more likely to hit these DC 15s on a +4 because they'll roll more 11s. So I'm just clarifying that the math doesn't work that way.

 

[Blue]

Indeed. Not sure where this came from - the only place I've seen this argument is stat generation. 3d6 vs. 4d6 drop lowest vs. a straight d20 roll, etc.

A couple times a year someone announces that they want to move from d20 to 3d6 to "reduce swinginess". When really what it does is make that even the smallest modifier swings the success/failure chance a lot around the middle of the range, which is where bounded accuracy often puts us.

It's usually because people confuse large ranges of numbers as swinginess, when really it's boolean so success/failure is the only swinginess and this exasperates how much modifiers change that.

 

[Flexor the Mighty!]

If I ever go back to 5e I may tinker with a 3d6 system. Crits IME are the driving force of most encounters and made combat crazy "swingy".

 

[BrokenTwin]

If you're using 3d6 to make crits rarer, why not just make any triple a crit? That's a 1/36 chance, which is less than max on 1d20, but still significantly higher than the odds of rolling 18 on 3d6. It's a lot easier to remember than a range of "crit numbers", and requires zero math to realise at the table.

Plus, it makes reading the dice a bit more dynamic, and adds a bit of suspense for those buggers who insist on rolling one die at a time.

 

[Ovinomancer]

The total probability of values from -5 to 1 on the 3d6 curve is under 5%.
The total probability of the values from 20 to 25 on the 3d6 curve is under 5%.


I'm ignoring the 5% most extreme values on both ends, and looking at the middle 90%.

With a d20 on an attack, a natural 1 already misses and a natural 20 already hits; in a sense, it corresponds to a -infinity and +infinity.

I talked about the outlying 5% cases already, and now you bring it up as if it was some big gotcha. Those are the crit/auto hit/miss mechanics.

I assume we care about "does this attack hit/miss" experience at the table.

Given a game played with double-modifiers and d20, and another played with 3d6, I claim distinguishing between those games with a log of hit/misses (and not the rolls) will be a herculean task.

You'd basically have to find some creature whose chance of being hit is right on the edge of possible for the 3d6 case (in the "long tail" of 16-18) and tease out if the chance is different than 5%.

Suppose we want a 2 SD error bar on the sample. We have some random variable H. Its true value is either 1/20 or 1/216. How many samples do we need to distinguish that?

Quick napkin math (I think the right answer involves using student's T? It is basically a polling problem.) gives me about that it would take about 100 samples of "creature we know needs to be hit on an 18 on 3d6" to see a significant (2 SD, or 0.03 P-test) difference between the d20 with double modifiers and auto-hit on a 20 and the 3d6 with normal modifiers.

Or, tl;dr, we really don't care about events with really low probability, as they don't happen often enough to care about them. And the entire "tail" you are pointing at adds up to a low probability event.

Well, we have progress, as now you're not saying the math is correct, but that your argument is correct despite you discarding data. That's good.

As for your argument that we can discard data because it's low probability, you're tossing 10% of the possible rolls. That means that you're discounting 1 out of every 10 rolls. That's not a negligible amount.

Now, if your argument is that you can move the target numbers and bonuses to adjust the needed rolls on a d20 so that it looks more like the center of the 3d6, then you've done a little bit of moving things around to prove something that's largely true without the effort -- the center of the 3d6 is pretty darned close already to the most common needed d20 numbers for most combat efforts. There's plenty of ways to discover this without abusing stats.

 

[NotAYakk]

I said what math I did, not that it was "correct" because of the math. You seem to be projecting.

I scaled them by the standard deviation. This, observably, had the effect I described. It wasn't "correct" because of the math I did, I did the math then I described the results. I then described why those results aren't all that surprising; that dividing by the ratio 2nd moment and subtracting the difference in the first leaves only higher order components, and those components are bounded in effect (small).

We can formalize that if you want, but my argument has not and never did rely on that formilization. It relied on the actual graphs which I posted and the probabilities on those graphs and what those probabilities mean. (to sketch the formilzation argument using high school calc concepts: you'd basicaly mirror arguments like using the low order terms of a Taylor series, and how the tail of the series has a bounded contribution, so can be neglected if you are willing to accept a known error. Except with statistical moments instead of polynomials. I know this argument is plausible, but I am not claiming it is sufficient or nessicary.)

You grabbed onto my description of why the results aren't surprising and started complaining, seemingly without even looking at the graphs, based on you changing your position once I posted screenshots.

I hope this clears things up for you. Getting things backwards can be confusing, and maybe a reread would help.

Have a nice day.

 

[DND_Reborn]

This is sort of true, but you have to be careful how you understand it. A 50% probability check on a d20 -- say, a +4 roll against a DC of 15 -- is still a 50% probability check on 3d6. Your odds of rolling an 11 are higher, but your odds of rolling an 11 or above are still the same. What the normal distribution on 3d6 does is make the probability of success/failure "fall away" from 50% faster as your bonus or the DC changes.

I'm not attributing any fallacy to you in particular. Maybe you already know this and I'm preaching to the choir -- awesome! But in my experience a lot of people hear "increased likelihood of typical results" and think that it means they're more likely to hit these DC 15s on a +4 because they'll roll more 11s. So I'm just clarifying that the math doesn't work that way.

Yeah, know I understand what you mean, but that wasn't really the issue I was talking about. My point was more about how the 3d6 vs d20 concept affects single-roll outcomes versus extended "contests" such as combat.

A skill check is (most often) a single roll, as are many saving throws. This means with the linear d20 the "swinginess" makes your normal efforts as likely as your best effort and your worst effort. That isn't how most peoples' efforts are. The bell curve of the 3d6 more models the likelihood of "typical" results compared to worst and best results.

Combat becomes non-linear because there is a series of rolls involved to determine the outcome (at least in most cases). In an extreme case, you hit on every roll, representing your best effort. However the likelihood of that happening is pretty small (depending on your opponent's AC of course). More commonly, sometimes you will hit and other times you are going to miss. If you look at a particular distribution for a bonus vs. an AC, you see how it is a bell curve. So, you don't need to use 3d6 for combat to make it non-linear.

 

[Ovinomancer]

I said what math I did, not that it was "correct" because of the math. You seem to be projecting.

I scaled them by the standard deviation. This, observably, had the effect I described. It wasn't "correct" because of the math I did, I did the math then I described the results. I then described why those results aren't all that surprising; that dividing by the ratio 2nd moment and subtracting the difference in the first leaves only higher order components, and those components are bounded in effect (small).

We can formalize that if you want, but my argument has not and never did rely on that formilization. It relied on the actual graphs which I posted and the probabilities on those graphs and what those probabilities mean. (to sketch the formilzation argument using high school calc concepts: you'd basicaly mirror arguments like using the low order terms of a Taylor series, and how the tail of the series has a bounded contribution, so can be neglected if you are willing to accept a known error. Except with statistical moments instead of polynomials. I know this argument is plausible, but I am not claiming it is sufficient or nessicary.)

You grabbed onto my description of why the results aren't surprising and started complaining, seemingly without even looking at the graphs, based on you changing your position once I posted screenshots.

I hope this clears things up for you. Getting things backwards can be confusing, and maybe a reread would help.

Have a nice day.

Sigh. Okay, when I said you did mathturbation, you got mad, but that's when you do the wrong math and get confident you did something cool because of the wrong math. What you did -- scaling standard deviations and then thinking that made distributions similar? That's wrong math. It's bogus, utterly. That you saw graphs line up was coincidence -- it had nothing to do with what you did but the fact that you kept picking numbers until you managed to make the center 10 data points of the normal cumulative distribution of 3d6 look like a line with a slope of -1. Making post hoc choices with stats is always dangerous, because you're altering the assumptions that go into a statistical model but not altering the model to account. It leads to your assumption that you found something, when you did not.

If you look at the PDFs for d20 vs 3d6, you might not that you have 18% less chance of rolling a 14 or higher on 3d6, a 20.8% less chance of a 15, and a 20.4% less chance of a 16. Those numbers don't show up much, but that's a pretty big delta. In your streched and recentered 2*3d6-11, that same point is rolling a 12 on the 3d6 part. That's what lines up with a 15 on the d20. The 15 on the 3d6 is over 20 on the d20. I have no idea why you thought these were even comparable. Lines on a graph don't matter much if one "line" is a zoomed in circle and the other is an actual line -- they aren't the same at all.