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

[Esker]

It occurs to me that one reason the centering and scaling doesn't match probabilities even better than it does is that the discrete nature of die rolls prevents us from actually matching the moments. But this could be dealt with by expanding the DCs in the modified d20 system around 11 instead of 10, and requiring a 50/50 "confirmation" roll if your roll exactly ties the DC. That essentially raises the DC by 1/2, as if you had expanded around 10.5.

 

[Ovinomancer]

I'm not sure why you're focusing on the difference in spacing... In 5e's d20 system, the only thing that's relevant is your chances of meeting or exceeding some threshold; it doesn't matter at all how likely you are to roll any specific value, X, except insofar as that represents the difference in difficulty between a DC X and a DC X-1 roll.

I focus on it because it's a key indicator that you aren't comparing the same things. Here's a great example using the OP's first example. I'll repost it for ease:

Level 20 fighter with 24 strength attacking a foe in +3 plate and shield. (note that this isn't contrived -- I just picked out some reasonably extreme examples).

With 3d6, this is +13+3d6 vs AC 26. 25.93% chance of hitting.

Under the "double modifiers", this is 1d20+26 vs AC 42. 25% chance of hitting.

Alright, this is a true statement -- these things have similar probabilities (~1% different). But, what happens if we alter this highly contrived example just a touch? Let's give the Fighter STR 20, and the target a ring of protection. This makes the normal attack bonus +11 against normal AC 27. Need a 16+ on 3d6, or a 4.64% chance of success.

Now, if we do the doubling, it's an attack bonus of +22 (4 less) against an AC of 44 (2 more). You need a 22 on a d20, for a 0% chance of success.

Before you start, sure, 5e has a 20 always hits rule, but that's not part of the distribution -- it's actually breaking the statistical analysis -- so it cannot be used to justify a system based on an incorrect assumption of similarity in statistical behavior.

The difference in spacing is a critical indicator that such breakpoints exist, this one just barely off of the example put forth to show just how similar the systems are. If you, instead, reduce the difference by 2 instead of increase it, the delta is on 3d6 a 50% chance and on the doubled d20 a 45% chance. Reduce another 2 and it's 74% to 65%. You have a non-linear rate of change, which makes it obvious these two things are not, at all, alike.

The OP's suggestion was that using 3d6 is similar to a system where where bonuses are doubled, DCs (and similarly, ACs) are transformed to be DC' = 10 + 2*(DC - 10), and we use a d20 to resolve outcomes.

Note that this is the same in practice as taking the new DC to be 10.5 + 2*(DC-10.5), where we've used the actual expected value of the 1d20 and 3d6 rolls, because this is 0.5 lower than 10+2*(DC-10), and so it yields success on the same integers.


If you need a natural X to succeed in the 3d6 system (that is, you have a +Y bonus and the DC is X+Y), that becomes a +2Y bonus and a DC of 10+2*(X-10)+2Y. So you need a natural 10+2*(X-10), or 2*X - 10 in the modified d20 system. So we could compare success rates for each value of X with the corresponding target natural rolls.

Math tip: that equation simplifies to 2*DC-10. Much easier to write. It's also slightly off from what the OP suggests, but pretty close, for AC. For DC, it's more (DC-10)*2+6, or 2*DC-14. Still not quite what is suggested, but pretty close.

The OP actually suggests using ability-10 as the bonus rather than 2*ability bonus, which increments in possible single steps rather than by 2's all the time. Not sure why they felt the need for that bit of granularity while just doubling everything else, but hey, it's good.

I do find it odd that you think I misunderstood what the OP was suggesting. I didn't. It's just not based on good math.

Alternatively we could leave bonuses and DCs the same and scale and shift the roll instead. Compare 1d20 to 2*3d6-10. Using a target of X on the 2*3d6-10 roll is equivalent to a target of ... 2*X-10 on the modified d20, just the same as if we'd rescaled bonuses and DCs.

This does pretty much nothing for increasing correctness, but it does immediately show how sensitive the analysis is to the arbitrary selection of offset. Your suggestion now only lines up at 2, where the graphs cross, and then almost again at 13 and 14 on 3d6 (which get close to 15 and 17, respectively, on the d20).

So the OP was off by one in their graph in terms of illustrating the impact of their proposed system relative to using 3d6 with regular bonuses and DCs. But actually using -10 vs -11 only affects which system makes for easier rolls, not the sizes of the gaps, since 10 and 11 are equal distances from the mean, and so you're essentially just swapping successes and failures and inverting the labels on the x-axis.

No, the OP wasn't "off by one" as that actually makes the artificial comparison strictly worse. You'd need to choose a different scalar to align at 11 rather than 10. Which, interestingly, while the OP chose 10 as the center for the scaled 3d6, he left 10.5 for the d20 center, meaning the graphs don't even have the same mean value. The warning signs for bad math are just all over this -- and, indeed, it's terrible math.

 

[Ovinomancer]

It occurs to me that one reason the centering and scaling doesn't match probabilities even better than it does is that the discrete nature of die rolls prevents us from actually matching the moments. But this could be dealt with by expanding the DCs in the modified d20 system around 11 instead of 10, and requiring a 50/50 "confirmation" roll if your roll exactly ties the DC. That essentially raises the DC by 1/2, as if you had expanded around 10.5.

Nope. Makes it worse. It might be nice if you at least plugged these things into Anydice or did some actual looking at what monstrosities you're birthing before throwing more bad math at bad math as if that'll fix it.

Using the OP example and the ones above, just reduce the likelihood of the d20 system hitting by 5% for each, making the deltas 5.93% on the OP (3d6 better), 10% on the next to last, and 19% on the last. The second one doesn't change -- 3d6 can still hit and d20 cannot.

And, one more observation on that example that d20 can't even hit -- 3d6 only goes up to 18, so d20 has 2 more numbers in it's upper range and still fails to be able to connect.

 

[Esker]

Nope. Makes it worse. It might be nice if you at least plugged these things into Anydice or did some actual looking at what monstrosities you're birthing before throwing more bad math at bad math as if that'll fix it.

The issue was that 2*3d6-11 has a mean of 10, whereas 1d20 has a mean of 10.5. By introducing a "confirmation" mechanic, we're effectively reducing every roll by 1/2 (if it's easier to see, you can imagine rolling the 1d2 confirmation die after every roll and subtracting 1 on a 1, and leaving the roll alone on a 2, but this has no impact unless our roll exactly meets the DC so in practice you'd only need to roll the confirmation die in that case).

This isn't reducing the likelihood to succeed by 5%; if applied to a d20 it's essentially reducing it by 2.5%, so it's as though we made the d20 have a mean of 10, like the 2*3d6-11.

But that's not actually quite want we want --- I did mess that up in hastily firing off an idea that I hadn't had a chance to sit down and work through yet. What we want is to apply the confirmation die in the 3d6 case, not the d20 case. We either (1) roll 2*3d6-10, applying the confirmation mechanic, which should be comparable to an unaltered d20 system, or (2) just roll 3d6 with the confirmation mechanic, and compare that to a d20 system where bonuses are doubled together with DCs' distance from 10.

Let's look at (1) first. Here are the probabilities of meeting each target natural roll from 1 to 20, comparing vanilla 1d20 (no changes to DCs or modifiers), compared to 2*3d6-10 with the confirmation correction:

Once we correct for the mean of 2*3d6-10 being off by a half, the approximation error is symmetric around the DC/bonus combinations with a 50% success rate. The 3d6 curve is of course nonlinear -- no points for pointing that out; everyone already recognized that -- and so the discrepancy in success rates is not constant, but it is at worst 4%.

And here are the probabilities for a "reduced randomness" scheme: 3d6 with a confirmation correction and no change to DCs or modifiers, compared to 1d20 with doubled modifiers and new DCs set to 10 + 2*(DC - 10) -- that is to say, expanding around 10 by a factor of 2. (The x-axis is the target 3d6 roll; the roll needed on the d20 is different)

As advertised, using 3d6 produces less variation in rolls. But if we adjust modifers and DCs we achieve the same effect with a d20. We do have a bit of an issue on the extremes, in that the scaled d20 approximation makes some rolls guaranteed or impossible, whereas they are around 96% or 4% with 3d6. But again that's something everyone has acknowledged throughout this thread, so there's nothing special in you pointing that out -- it's not a problem with anyone's math, it's a fact that distributional approximations like this have the worst fit in the tails.

Note that in the latter case, the curves cross at 10.5, not at 11. This is due to the fact that by altering our DCs, tasks that previously had DC 11+modifier are no longer 50% success rate tasks; they've gotten a little more difficult, whereas the 50% success rate now sits between old DC 10+modifier (still DC 10+modifier) and DC 11+modifier (now DC 12+modifier). Introducing the confirmation mechanic on the 3d6 roll does the same thing: you no longer necessarily succeed on a DC 11+modifier task if you roll a natural 11; you only have a 50% chance.

 

[Ovinomancer]

The issue was that 2*3d6-11 has a mean of 10, whereas 1d20 has a mean of 10.5. By introducing a "confirmation" mechanic, we're effectively reducing every roll by 1/2 (if it's easier to see, you can imagine rolling the 1d2 confirmation die after every roll and subtracting 1 on a 1, and leaving the roll alone on a 2, but this has no impact unless our roll exactly meets the DC so in practice you'd only need to roll the confirmation die in that case).

Not what you say above, and still a really weird, non-mathematical thing to do to correct at a single point in the distribution because your choice of arbitrary centering (to get, recall, a optical match but not actual match in probabilities) to look better. You're chasing rabbits down the wrong holes and claiming victory because you found an orange plastic cap that you're calling a carrot.

This isn't reducing the likelihood to succeed by 5%; if applied to a d20 it's essentially reducing it by 2.5%, so it's as though we made the d20 have a mean of 10, like the 2*3d6-11.

But that's not actually quite want we want --- I did mess that up in hastily firing off an idea that I hadn't had a chance to sit down and work through yet. What we want is to apply the confirmation die in the 3d6 case, not the d20 case. We either (1) roll 2*3d6-10, applying the confirmation mechanic, which should be comparable to an unaltered d20 system, or (2) just roll 3d6 with the confirmation mechanic, and compare that to a d20 system where bonuses are doubled together with DCs' distance from 10.

Let's look at (1) first. Here are the probabilities of meeting each target natural roll from 1 to 20, comparing vanilla 1d20 (no changes to DCs or modifiers), compared to 2*3d6-10 with the confirmation correction:

Once we correct for the mean of 2*3d6-10 being off by a half, the approximation error is symmetric around the DC/bonus combinations with a 50% success rate. The 3d6 curve is of course nonlinear -- no points for pointing that out; everyone already recognized that -- and so the discrepancy in success rates is not constant, but it is at worst 4%.

I don't know what you're actually graphing, here. Your d20 line has data outside of the possible range, and the value for 1 on d20 is incorrect. The rest are correct (2-20), but these obvious issues make me very leery of what it is you're doing.

Secondly, you've done exactly what I said the OP did -- you've tossed data from the scaled 3d6. Here, you've actually truncated the values form the graph which makes it look even more visually similar. This is bad math.

I think you've set yourself up to fail, here, because your next bit is way, way, way off the rails.

And here are the probabilities for a "reduced randomness" scheme: 3d6 with a confirmation correction and no change to DCs or modifiers, compared to 1d20 with doubled modifiers and new DCs set to 10 + 2*(DC - 10) -- that is to say, expanding around 10 by a factor of 2. (The x-axis is the target 3d6 roll; the roll needed on the d20 is different)

As advertised, using 3d6 produces less variation in rolls. But if we adjust modifers and DCs we achieve the same effect with a d20. We do have a bit of an issue on the extremes, in that the scaled d20 approximation makes some rolls guaranteed or impossible, whereas they are around 96% or 4% with 3d6. But again that's something everyone has acknowledged throughout this thread, so there's nothing special in you pointing that out -- it's not a problem with anyone's math, it's a fact that distributional approximations like this have the worst fit in the tails.

I have no idea what you've graphed here. The d20 in your scheme does not display a different CPDF -- it's still linear. But, here, you have a weird hybrid curve that goes to a 0% probability when you need a 16. Can you not even roll a 16 on a d20? Clearly, you can, you can roll all the way up to a 20 (not featured on your graph), so whatever you've done here, it's not the density function of a d20.

The best I can figure is that you've graphed a d20 scaled by 1/2 and recentered. You've increased the slope of the d20 line, but tossed a bunch of data. Your d20 values go from being able to generate a value of 5 through 16. This is NOT what a d20 can generate.

I also tried to understand your graph as using the expanded d20 roll as graphing likelihood of meeting a DC, but I can't get that math to work, either, even trying out a number of possible mistakes.

I was able to recreate your graph, though. I divided a d20's probability by 2 and recentered it at 10.5, then changed the first and last data points to be an average of the data point before and the data point after (which is why you have those ramps to 100% and 0% on the d20 line). I then recentered the 3d6 line to 10.5. I don't really know why we did this, because both lines were centered on 11, but whatever, it's a bunch of arbitrary decisions to make lines look like each other so it's all the same.

roll	d20	3d6	scaled d20
1​	100​
1.5​	97.5​
2​	95​
2.5​	92.5​	100​
3​	90​	99.76852​
3.5​	87.5​	99.53704​	100​
4​	85​	98.84259​	100​
4.5​	82.5​	98.14815​	100​
5​	80​	96.75926​	100​
5.5​	77.5​	95.37037​	97.5​
6​	75​	93.05556​	95​
6.5​	72.5​	90.74074​	90​
7​	70​	87.26852​	85​
7.5​	67.5​	83.7963​	80​
8​	65​	78.93519​	75​
8.5​	62.5​	74.07407​	70​
9​	60​	68.28704​	65​
9.5​	57.5​	62.5​	60​
10​	55​	56.25​	55​
10.5​	52.5​	50​	50​
11​	50​	43.75​	45​
11.5​	47.5​	37.5​	40​
12​	45​	31.71296​	35​
12.5​	42.5​	25.92593​	30​
13​	40​	21.06481​	25​
13.5​	37.5​	16.2037​	20​
14​	35​	12.73148​	15​
14.5​	32.5​	9.259259​	10​
15​	30​	6.944444​	5​
15.5​	27.5​	4.62963​	2.5​
16​	25​	3.240741​	0​
16.5​	22.5​	1.851852​	0​
17​	20​	1.157407​	0​
17.5​	17.5​	0.462963​
18​	15​
18.5​	12.5​
19​	10​
19.5​	7.5​
20​	5​

NOTE: I did some fast and dirty extrapolation for the 3d6 curve for the .5 values by averaging the preceding and following values. This is because there isn't a probability for rolling, say, 6.5 on 3d6 but it's a pain to get Excel to graph data where values are missing.

Note that in the latter case, the curves cross at 10.5, not at 11. This is due to the fact that by altering our DCs, tasks that previously had DC 11+modifier are no longer 50% success rate tasks; they've gotten a little more difficult, whereas the 50% success rate now sits between old DC 10+modifier (still DC 10+modifier) and DC 11+modifier (now DC 12+modifier). Introducing the confirmation mechanic on the 3d6 roll does the same thing: you no longer necessarily succeed on a DC 11+modifier task if you roll a natural 11; you only have a 50% chance.

Sigh. That makes things easier -- you need a 10.5 to succeed 50% of the time instead of a larger value, 11. This actually increases the likelihood of success by a small amount. That is, if what you've done was remotely correct at all. I point this out because you've not only managed to do horrible things to math, but you've failed to even apply a correct interpretation to the horrible things you've done as if they were okay.

Look, no matter what you do to target numbers, the CPDF of a d20 doesn't change. If you don't get a straight line from 100% to 5% from 1 to 20 with a slope of exactly -1, you've done something horribly wrong.

 

[Ruin Explorer]

Yes, but it pays to consider what you really want out of it. "Limit extremes" is a means, not an end in and of itself. What things actually happening in your game do you not want to happen?

Note that 5e already comes with bounded accuracy.

I concur. Changes like this should always be made with careful reflection on the purpose. Easily the biggest problem with house rules and modifications is that, especially pre-2000, most are/were made without considering the actual goal, instead either being "just because" or reactionary in design (reactionary here meaning "A thing I disliked happened so let's make a rule to prevent it!" without considering how it fits into the greater scheme of things).

I think if 5E used actual fumble rules beyond a 1 simply always missing, then that alone would go a long way towards justifying, say, 2d10 instead of 1d20, because such rules can easily turn a game into a farce. But it doesn't. Of course a lot of groups use very ill-considered "just because" type fumble rules, and I can certainly see that after adopting such, one might be tempted to switch to 2d10 or the like. I saw too many cases in 2E and 3E and other 90s and 00s games where people had whole chains of house rules that were there to compensate for the results of other house rules to think that's a good idea, though.

 

[Coroc]

I still simply do not get what should be achieved with the 3d6 instead of 1d20.
You got to count the faces of three dice, so you waste more time when doing rolls for fast actions.

You make situations which a good DM attaches a threshold to (Bend some bars, read some arcane runes etc.) less likely to occur but you do not reduce the probability of these to 0, which you would, if you just used a threshold!

You make skill checks more reliable, ok I give you that. But what consequences has this for your game?
If you want skilled characters to almost auto-succeed then just give them that on standard tasks and only require a roll on extraordinary displays of their skill.

For saves and attacks which rely on so much else I do not see a point also you destroy the fun of criticals (which in 5e have far less dire consequences than in other editions or systems)

 

[Esker]

Not what you say above, and still a really weird, non-mathematical thing to do to correct at a single point in the distribution because your choice of arbitrary centering (to get, recall, a optical match but not actual match in probabilities) to look better. You're chasing rabbits down the wrong holes and claiming victory because you found an orange plastic cap that you're calling a carrot.

I'm nearing the end of my patience with you, because you're so quick to fire insults and shoot down straw men without actually taking time to consider what anyone else is saying. You keep fixating on the fact that the probabilities don't actually match, and, no of course they don't match exactly. No one was claiming they were an exact match. But they're close, because we've matched the means and standard deviations of the die rolls.

I don't know what you're actually graphing, here. Your d20 line has data outside of the possible range, and the value for 1 on d20 is incorrect. The rest are correct (2-20), but these obvious issues make me very leery of what it is you're doing.

I've only graphed the even target rolls, since 2*3d6-10 can only produce even results before the confirmation correction. Obviously the probability of 1 or higher on a d20 is 1, but that's not on the graph; it's just interpolating between 0 and 2.

If the target is an odd value then the confirmation step doesn't matter because you'll either exceed the target or you'll fall below it, so you can use the probability of hitting one number higher without a confirmation correction. I didn't bother incorporating that because it doesn't change the shape, but since you're clearly intent on fixating on any small missing detail, here's the graph with the odd targets added in.

Secondly, you've done exactly what I said the OP did -- you've tossed data from the scaled 3d6. Here, you've actually truncated the values form the graph which makes it look even more visually similar. This is bad math.

I don't know what you mean by saying that I've tossed data. Unless you mean that I omitted odd-numbered targets. Well they're there now. Happy?

I have no idea what you've graphed here. The d20 in your scheme does not display a different CPDF -- it's still linear. But, here, you have a weird hybrid curve that goes to a 0% probability when you need a 16. Can you not even roll a 16 on a d20? Clearly, you can, you can roll all the way up to a 20 (not featured on your graph), so whatever you've done here, it's not the density function of a d20.

I explained this in my last post: the x-axis shows the target values for the 3d6 scheme. The corresponding targets on a d20 are different because we've changed the bonuses and DCs for that scheme only. If the 3d6 target is X, then the d20 target with rescaled DCs is 10 + 2*(X-10). So the points on the graph at 16 are the probability of getting 16 on 3d6 and the probability of getting 22 on a d20, since these are corresponding rolls. This is a linear transformation and so of course the CDF is still linear. Not sure why you seem to think it wouldn't be.

The best I can figure is that you've graphed a d20 scaled by 1/2 and recentered. You've increased the slope of the d20 line, but tossed a bunch of data. Your d20 values go from being able to generate a value of 5 through 16. This is NOT what a d20 can generate.

Yes, if you prefer you can think of the modified d20 scheme as altering the roll itself instead of the DC: roll the d20, add 10, and divide by 2, rounding down. So the maximum result is 15 and the minimum is 5 (but you only get 5 when the die shows a 1). And so, as I acknowledged, the adjusted d20 is missing a small chance of getting a 3-4, as well as a small chance of getting 16-18: with the confirmation correction, 3d6 has a 96.8% chance of rolling 5 or higher (instead of 100%), and a 3.2% chance of rolling 16 or higher (instead of 0%).

I point this out because you've not only managed to do horrible things to math, but you've failed to even apply a correct interpretation to the horrible things you've done as if they were okay.

I suggest taking a step back and reconsidering your stance that people who produce results that you don't follow must necessarily be mathematically inept, and instead begin from a place of good faith, reading what they have written and trying to make sense of it to the best of your ability before jumping on any perceived missing or incorrect detail as proof that the whole approach is "horrible math". I acknowledge having made some minor errors here and there, and I originally proposed the confirmation correction in a quick post from my phone with what turned out to be some details that were off before I had a chance to sit down and work out the numbers. But I promise you, my grasp of the underlying math is solid (as I believe I've demonstrated in our previous interactions here, so I figured you might extend some benefit of the doubt).

 

[Esker]

I still simply do not get what should be achieved with the 3d6 instead of 1d20.
You got to count the faces of three dice, so you waste more time when doing rolls for fast actions.

The OP's point was that you don't need to use 3d6 to achieve the goals that people have in mind when using 3d6 (namely, making bonuses matter more, and randomness matter less). You can just scale bonuses and adjust DCs, and stick with a d20 roll, to achieve nearly the same success rates that 3d6 gives you, but you've offloaded the mental effort to adjustments you can do away from the table, so that people don't have to add as many dice during the game.

 

[NotAYakk]

The OP's point was that you don't need to use 3d6 to achieve the goals that people have in mind when using 3d6 (namely, making bonuses matter more, and randomness matter less). You can just scale bonuses and adjust DCs, and stick with a d20 roll, to achieve nearly the same success rates that 3d6 gives you, but you've offloaded the mental effort to adjustments you can do away from the table, so that people don't have to add as many dice during the game.

Well, rather that when people describe why they want to use 3d6, they actually don't describe that. Often they talk about liking the "bell curve".

Almost all of the effect of 3d6 using is equivalent to a re-scaling of bonuses/DCs and using a d20. With the addendum that things in the top 5%/bottom 5% (corresponding to natural 20s and natural 1s, which often is played as auto-hit auto-miss even when D&D doesn't say so) aren't as well captured.

If people where rolling 3d6 with the intention of "making modifiers twice as important" and viewed the cost of adding up 3 dice each roll as being acceptable for that, more power to them. But when I people talk about 3d6 that, in my experience isn't how they frame it. Which leads me to suspect they actually think it is doing something else other than what it does.

Now, sometimes slowing down gameplay with "busywork" is worthwhile; D&D with combat reduced to a single weighted coin flip wouldn't be as fun. Rolling 3d6 for skill checks, when skill checks tend to be high impact, might be worth it for that reason.

But I still hold my position that if you want to do something like that, you should make it more interesting; a minigame. You roll 3d6 for a skill check. If the result is under the DC, the DM produces a malus (a cost), and you can say what else you are doing to try to succeed. When you do that, you get to pick up once of your 3d6 and reroll it.

So if you are sneaking into a castle, you roll 3d6+9 against a 20 DC. You roll a 1, 3, 5 and get a total of 18. The DM says as you cross a courtyard, a guard yells out "who goes there" (malus: castle alert level just went up).

You use your mimicry ability to make the sound of a cat yowl. The DM lets you reroll your 1 -- you get a 3. 3+3+5+9=20, so now you have successfully snuck into the castle. Except some of the guards are on a bit higher alert, so after you convince the princess to leave with you it will be a bit harder, which could lead to you having a 10 minute head start instead of a 4 hour one.

But that isn't central to my thesis here.