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

[NotAYakk]

It turns out that, in a roll over/under system, replacing 1d20 with 3d6 doesn't do much of what people want.

Mostly what it actually does is double bonuses and DCs, outside of crit hit/miss zones.

Change attribute bonuses to be (stat - 10), halved for damage. Make AC delta from 10 double -- so mage armor is 16+dex, plate is 26, shields are +4. Enchanted weapons go up to +6 to hit. Proficiency starts at +4 and scales up to +12 (expertise twice that).

Saves are 6+proficiency+attribute bonus. Bless is +1d8 instead of +1d4 (or 1d4*2). Bard inspiration dice are doubled. Etc.

You can see this graphically -- see AnyDice select "graph" and "at least" to see two curves right on top of each other -- but a concrete example might help.

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.

I got lucky -- I don't expect that to usually happen. So, I can do this again. A level 10 20 dex fighter with a +2 bow, +1 arrows and archery style using sharpshooter attacking a suit of animated armor (AC 18).

3d6 +5 (dex) +4(prof) +2 (bow) +1 (arrows) +2 (style) -5 (SS) = 3d6+9 vs AC 18. 74.07% chance of hitting.
1d20 + 10(dex) +8(prof) +4(bow) +2 (arrows) +4(style) -10 (SS)= 1d20+18 vs AC 26. 65% chance of hitting.

This one is about the peak of the difference you get.

Level 1 warlock with 16 charisma doing an EB on a dex 14 leather-clad bounty hunter.

3d6 + 3 (cha) +2 (prof) vs AC 13 = 90.74%
1d20 + 6 (cha) + 4 (prof) vs AC 16 = 85%

---

Why this works is because the mean of 3d6 is 10.5, just like 1d20. The standard deviation of 1d20 is sqrt( 399/12 ) and the standard deviation of 3d6 is sqrt( 3 * 35/12 ) = sqrt( 105/12 ). Which means SD(1d20) = 2 * SD(3d6).

When we double the modifiers on 1d20 rolls, we in effect halve its standard distribution. The resulting distributions have the same first and second moments. The difference -- the third moment -- is far smaller than you'd naively expect from looking at the "roll exactly" curves.

D&D is a roll over/under system, not a roll exactly. Looking at "roll exactly" curves is looking at the derivative, when you should be looking at the values -- which is the cumulative distribution function. On anydice, this is "roll at least" or "roll at most" graphs.

Note that 2d10 is about half way, and represents about x1.5 modifiers.

 

[toucanbuzz]

What triggered this analysis?

 

[Maestrino]

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.

The other major difference in the two approaches I can see is the change in outcomes. With 3d6 you get a range of 3-18 distributed in a bell curve, with a significantly higher chance to get a result in the middle of the range. On a d20 the range is 1-20 with a flat 5% chance of any particular result occurring.

So this 3d6 system gives a slightly higher chance to hit as constructed, but eliminates the critical hit/miss mechanic (or if you redefine it to be rolling three ones or sixes, then it makes such an occurrence much less likely.

 

[dave2008]

The reason to use 2d10, or 3d6 over 1d20 is to limit the extremes. Simple example:

A crit with a 1d20 happens 5% of the time. (or 1 in 20 attacks)
A crit with a 2d10 happens 1% of the time. (or 1 in 100 attacks)

The extremes are much less likely in a dice pool system.

 

[Umbran]

The extremes are much less likely in a dice pool system.

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.

 

[NotAYakk]

The other major difference in the two approaches I can see is the change in outcomes. With 3d6 you get a range of 3-18 distributed in a bell curve, with a significantly higher chance to get a result in the middle of the range. On a d20 the range is 1-20 with a flat 5% chance of any particular result occurring.

The "middle range higher chance" is 90%+ covered by the change in standard distribution, which is basically making bonuses bigger.

The bell curve almost completely evaporates when you look at the "roll at least". It is only significant when looking at "roll exactly", and D&D isn't based on "roll exactly".

What triggered this analysis?

Seeing yet another "use 3d6 instead of d20" comment that talked about bell curves.

The reason to use 2d10, or 3d6 over 1d20 is to limit the extremes. Simple example:

A crit with a 1d20 happens 5% of the time. (or 1 in 20 attacks)
A crit with a 2d10 happens 1% of the time. (or 1 in 100 attacks)

The extremes are much less likely in a dice pool system.

It depends what you define as a crit. The outer extreme "5%" events mechanics can differ between them if you want.

Other than the crits, the distance to the extremes is can again be mostly reflected by a change in modifiers (make them bigger) and DCs (make them further from 10.5).

Please look at AnyDice then click on "at least" and "graph". The "eliminated extremes" is almost entirely an artifact of the change in standard distributions. After doubling modifiers and distance from 10 of DCs, 3d6 has larger extremes than d20 does (without crit/miss), and all of that occurs in the extreme 5% probabilities.

If you want to eliminate extremes, double modifiers to your d20, and double distance from 10.5 on DCs, and you'll do the same job.

 

[dave2008]

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.

Agreed, I wasn't trying to advocate one way or other. Just pointing out a reason to do it.

 

[dave2008]

Other than the crits,...

Crits are the only reason to go with a dice pool system IMO. There is no "other than the crits." Crits are everything in this discussion IMO, everything else is just gravy.

 

[dave2008]

The "middle range higher chance" is 90%+ covered by the change in standard distribution, which is basically making bonuses bigger.

What if you don't want to change bonuses?

 

[Don Durito]

If you want to use 3D6 you should only really be using it for skill roles outside of combat.

And of course you need to change the DCs.

Combat already works on an effective bell curve, because it is resolved using a whole load of D20 rolls, not just one.

3D6 for skills only is a relatively small change. 3D6 for combat requires a complete rebuild (for one thing it messes with bounded accuracy at outlier results are going to become a lot less likely).