I was chatting with +Peter V. Dell’Orto and +Tim Shorts last night online, walking them through the basis of my Heretical DnD project.

We were talking about the consequences of taking wounds, or having had your Stress Points worn down. Just brainstorming in an idle fashion, tossing out ideas, and someone, I think Peter, mentioned that if things were in a really bad way, that you could pick up Stacked Disadvantage.

Hmm, we said. What would that be? Well, probably taking the lowest result of more than 2d20. But does that look interesting compared to regular Advantage/Disadvantage?

Turns out it does.

The first question is whether stacked disadvantage has enough mechanical utility – does it actually drop the chances of being successful by enough to matter?

Turns out it does.

I’ve created two new categories of being disadvantaged, called “Stacked Disadvantage” and “Tim Shorts.” The first is lowest of 3d20, the second is 4d20.

You can see from the probability of exceeding a given difficulty number that the Stacked Disadvantage is significantly less favorable than picking the lowest of 2d20, and the Tim Shorts level approaches the actual probability of Tim rolling higher than a 1 on any important combat task.

So that’s one way to look at it. Another is to look at the equivalent probabilities of meeting or beating a target number – basically take all the modifiers that go on either side of the equation and make it 1d20>N. So if you normally roll 1d20+5 vs. AC 16, this becomes 1d20 greater than or equal to 11 (which should be, and is, a 50% probability).

That chart shows what simple mental calculation shows too: the odds of rolling a good roll with Stacked Disadvantage are really bad.

Risk Assessment

Finally, another simple way to get a feel for what’s going on here is a risk-based one. What number can you expect to roll under X percent of the time? That’s a way of asking how concentrated towards the bottom of the target number range the dice will be, and as you can see, it’s pretty ugly.

Again, as you expect, with a straight-up roll, half the time you’ll roll 10 or less, and 90% of the time you’ll roll 18 or less. That’s a boring flat distribution, but that’s 1d20 for you.

The rules-as-written disadvantaged mechanic concentrates it more tightly. you’ll be rolling 10 or less 75% of the time, and you have only about a 5% chance to roll 16 or better.

Stacked Disadvantage has 90% of your rolls being 11 or less. And 19 times out of 20 your best roll can only be a 12. Note that rolling 1d20+5 vs AC 16 only really requires an 11, but that means Stacked Disadvantage takes that chance to pretty low – about 12% in fact.

Equivalent Penalties

One last way to look at this is that given Disadvantage, Stacked Disadvantage, or Tim Shorts, what’s the equivalent penalty?

This method of looking at it has its limits. Your odds of rolling (say) a 20 are only 5% with 1d20, and saying that you are at -1 because your success chances go from 5% to 0.012% really, really understates how unlikely the die is to come up 20 on all three (or four, if you’re Tim) dice. But the depth and breadth of the valley of doom is illustrative of how deep in the trouble pool you are.

Parting Shot

Stacked disadvantage may well have its place in the pantheon of elegant but effective mechanics. And that, of course, means that stacked advantage has its place too.

As much fun as it is to bust on Tim for the frequency of his rolling a 1 at exactly the wrong time, in reality, going to the fourth d20 doesn’t really buy you much. Unless the player is attempting a ridiculously easy task – rolling with proficiency and/or expertise and an attribute bonus against a DC 10 or lower target (so net difficulties of 3-5), there’s so little probability of success that the GM might as well say “nope, can’t roll.” Also, the difference between lowest of 3d20 and 4d20 isn’t that much: never more than the equivalent of another -2 tossed on top of everything else.

So I’d leave the “lowest of 4d20 “level behind, but Stacked Advantage and Disadvantage may well find their way into my stable of things to use when GMing 5e.

Oh: I don’t expect I’m the only one to think of this (and in fact, Peter was the one that did), but it seemed cool enough to write up anyway.

Follow-up and Commentary

Well, this looks like it’s going to be a popular post, or perhaps notorious. I’m getting a lot of feedback, especially on Reddit, but it’s good stuff.

Disadvantage pulls in the median by 4 (as you note); stacked yanks it down two more, from 6 to 4. That’s not that big of a deal.
However, what Stacked does do is pull in the tail further. The 90% for Disadvantage relative to a straight roll drops from 18 down to 14, and then Stacked drops it further down to between 10 and 11, which is about the same oomph as Disadvantage.
Beyond Stacking once, I agree: no profit in it.
2. Why not do advantage too?
A good question, and while I address it in my closing, a G+ commenter made a great point:

Also is the stacked dis/adv something that the DM holds onto until times where it is called for(kind of like handing out inspiration) or is it something that is always in play? If it is always in play I think it can really break the game once you start building characters to take advantage of the rule. Right of the top of my head I could build a straight up human variant barbarian who would have stacked advantage on almost, every attack. I think this is the reason WotC decided to not make dis/adv stackable right out of the gates.

Yes, quite – I’d not considered that one fully. That point about a double-stack adv/disadv by character design is a good one. My mental image was that this sort of thing would be only applied based on conditions. So that (for example) if you were, um, grappled AND on ice, yeah, that’s doubly sucky.

But the probabilities invert for advantage, and you’d be rolling 10+ 95% of the time, and a 20 about 19% of the time. The good news is that that sort of skew is still inherently capped at 20, which is the awesome part of the mechanic – no automatically rolling a 40 or something.
3. You’re going to turn the game into 3e/Pathfinder (or GURPS, for that matter)
I think that rolling 3d20 and picking the lowest (or highest) doesn’t run into the Death by Modifier problem that some don’t care for in games that feature a heavy dose of such things. GURPS is a bad one about this, to the point where I wrote a post about trying to cut down on modifier-driven game delay. Pathfinder does have a component of bonus-hunting to it, and furthermore, the bonuses get very large.
I don’ t believe this to be the case for this method, because it still is contained within the core concept of “bounded accuracy” in 5e. The worst you can roll is a 1; the best is a 20. No matter how many stacked ads/disads you do, that’s still true.
It does compress the probability distribution a ton, ’tis true. But modifiers are still there, and if you’re a high-level character with STR 20 and +4 for proficiency, even with stacked disadvantage you roll between 10 and 29 . . . though 90% of your rolls will be from 10-20  (and half of them will be 10-13). So it’s a real crimp in the style of a high-level character, but you can still thump a guy with chain mail and a shield 22% of the time.

## DnD5: The Standard Array

I got to thinking: How did the Standard Array come about?

I figured they did some sort of simulation. Take 4d6, drop lowest, sort them in order, and take the mean, median, or mode of each row.

I wondered, though – what that would look like, and how much variation would there be. I mean, just in my recent character generation forays, I’ve had some really good rolls, and some bad ones. Also, Order of the Stick style, what if 4d6/drop lowest was a metagame universal rule? So that each person that lived basically had that die total. So if you looked at “the upper 20% of adventurers,” you’d see one stat block, “lower 25%” would be the underperforming bandit cannon fodder of the world, etc.

Let’s hit it one thing at a time.

4d6, Drop Lowest: The Simulation

Not being a random number purist, I used RANDBETWEEN(1,6) in Excel to make my die rolls. I took four “dice,” added them together, and subtracted the smallest one.

I did this about 160,000 times. That should force the system to get close to constant values as possible. To the right is what AnyDice says about the distribution from which we’re picking:

So, what does that look like? Interestingly enough, it’s not quite what the book gives you.

 Mode 9 11 12 13 14 16 Median 9 10 12 13 14 16 Average 8.5 10.4 11.8 13 14.2 15.7 DnD 5e 8 10 12 13 14 15
Mode: This gives the number that appears most frequently out of 160,000 rolls. Compared to the 5e norm, it is one point better on the bottom two rolls, and you pick up a 16 instead of a 15 for your best. The mode of the full distribution is 13 (from the graph).
Median: Line up all 160,400 trials, and give me the average of trial 80,200 and 80,201. The median is also the 50% percentile. This matters a bit because 4d6 drop lowest is going to be weighted slightly to the higher numbers. The median of the distribution itself is 12.
Mean (Average): Sum of everything divided by number of trials. The mean of 3d6 is 10.5, the mean of 4d6 is 14. The mean of 4d6 drop lowest is 12.24. You can reproduce the standard array if you assume you round up numbers that are 0.75 or higher, and round down lower than 0.75. Strict truncation would give you 8, 10, 11, 13, 14, 15. Strict rounding would be 9. 10, 12, 13, 14, 16.
That last 16 that appears in all of the above except the actual standard array is important, because in nearly all cases, you can choose your highest stat so that you’re beginning the game with your primary attribute at 18. You may not want to do this, but you can – at least for anything but Wisdom, which for some reason is the only stat you can’t start with a race that gives you a +2.
Percentiles

So, if 4d6 drop lowest were a universal law of some sort, what would our percentiles look like?

First, why do we care? It’s a tool for GMs to tweak their campaigns. If you think the standard array is too stingy or too rich, you can adjust it. If you don’t want the girl who really lucks out on her rolls (it’s no surprise that out of 160,000 rolls, you can find an array like 16, 16, 16, 17, 17, 18) to overshadow completely the poor schlub that rolled 6, 9, 10, 11, 11, 12, then you can give the same array – for good or ill – to everyone.

So, here we go:
 10th Percentile 6 8 10 11 12 14 25th Percentile 7 9 11 12 13 15 40th Percentile 8 10 11 13 14 15 DnD 5e 8 10 12 13 14 15 50th Percentile 9 10 12 13 14 16 60th Percentile 9 11 12 13 15 16 75th Percentile 10 12 13 14 15 17 90th Percentile 11 12 14 15 16 17

As it happens, the D&D standard array can be matched by assuming that you’re taking the 45th percentile of the array I lined up. So 5% below the median is the “tax” you pay for avoiding the possibility of hitting the 10th percentile.
10th and 25th percentile characters would make good hirelings. Those not quite good enough to venture out as superlative adventurers on their own, but honestly not that bad, either. A 4th level Champion Fighter with 10th percentile stats and the right combo of Feat and Archetype is hitting twice per round at 1d20+5 to hit, 1d8+3 damage each, with AC 17 (chain mail and the boost to AC from Dual Wielder) and 36 Hit Points. That’s a credible mook right there.
What does the 90th percentile get you? Mostly, not much – it shores up weaknesses rather than give extra strengths. AC would stay the same (it’s gear-based), throw down half-orc instead of mountain dwarf, and you’re still looking at two attacks, 1d20+6 (only a 5% difference) twice for 1d8+4 (one more point of damage each). You’ll get a slightly higher initiative. Your saving throws will be better across the board. You have 44 HP instead of 36 (that right there might be the biggest difference).
But in a game that works fairly hard to mostly make your best two, maybe three, attributes the only ones that matter reliably, the difference you’re looking at is probably summed up as “three attribues will have +1 or +2 relative to the 10th percentile guy.”
Parting Shot

In many cases part of the fun of character generation is the choices you’re forced to make. Do you suffer with a -1 or -2 penalty in anything you happen to roll in your “dump stat,” or do you shore up your weaknesses by offsetting them with racial modifiers. If you’re a 90th percentile array human with your worst stat of 12, and highest of 18 – is that fun for you? I can see it either way, and there’s no question that with the proper GM and group, either one could be fun.
Overall, I was curious to see how the standard array was calculated. I suspect they picked about the 45th percentile of the 4d6 drop the lowest distribution.
Looking again at the 90th percentile human, it would mean you start with a minimum +1 bonus to everything you do . . . and will max out your primary stat at 20 early in your adventuring career if you choose. Even the variant human would start with 11, 12, 14, 16, 16, 18 plus a Feat and an additional Skill – a very satisfyingly awesome character.
Though again, getting to high level with some great stats, and some still-poor ones seems like a good “you can’t have everything” challenge. Kinda old school, if you would.

Which reminds me . . . what would 3d6 drop-nothing look like by this method?
 10th Percentile 4 6 8 9 10 12 25th Percentile 6 7 9 10 11 13 40th Percentile 6 8 9 11 12 14 45th Percentile 7 8 10 11 12 14 50th Percentile 7 9 10 11 12 14 60th Percentile 7 9 10 12 13 15 75th Percentile 8 10 11 12 14 15 DnD 5e 8 10 12 13 14 15 90th Percentile 9 11 12 13 14 17

So for +Jeffro Johnson and the 3d6 (in order, no less!) crowd, the current attribute array represents top-quartile type die rolls, and a 3d6 array that mimics the 45th percentile perch of the 4d6/drop lowest method has some tough choices to make . . . and classes that have additional Attribute Increase/Feat options will make a larger difference than those that do not.

4d4+2 – Request from a Redditor

Because it’s easy, I was asked to drop in 4d4+2. That one will cluster around 12, with more dice providing slightly more variation, and of course a hard minimum of 6.

 10th Percentile 7 9 10 11 12 13 25th Percentile 8 10 11 12 13 14 40th Percentile 9 10 11 12 13 14 DnD 5e 8 10 12 13 14 15 45th Percentile 9 10 11 12 13 15 50th Percentile 9 11 12 12 13 15 60th Percentile 10 11 12 13 14 15 75th Percentile 10 11 12 13 14 16 90th Percentile 11 12 13 14 15 17