Someone Is Wrong On The Internet

Someone recently linked me to an /r/rpg essay that was so completely wrong I felt like I had to tell the world about it. Particularly, if people are still linking this essay two years later, this feels like something I might want a response essay to link to rather than explaining fresh why the essay is wrong every time it comes up. Feel free to use this blog post likewise.

You can read the essay in its entirety at the link above, but the main thrust of it is this:

[T]here is a popular fallacy that I hate to see perpetuated on reddit, that goes something like:

“3d6 is a better resolution mechanic than d20 because it gives a bell curve distribution and the d20 is linear.”

I see a variation on this at least once a week on /r/rpg, and it surprises me that people don’t understand this simple fact: if all you are doing is determining success or failure, it doesn’t matter if the underlying range of dice results is linear or bell-curved. Both methods result in a flat percentage chance to get one or the other.

For example, suppose I need to roll 16 or higher on a d20. That gives me a 75% chance of failure, and a 25% chance of success. If I need to roll 13 or higher on 3d6, that represents almost exactly the same situation (74.27% failure, 25.73% success). If you take a histogram of the results of 10,000 rolls in either of these cases, you will see the same thing: a big lump on one side, and a smaller one on the other side.

This is at best extremely misleading, because 13 and 16 are different numbers. While it is technically true that 13 on a 3d6 and 16 on a d20 have similar odds, both dice sets have the same average of 10.5, which means a 16 is just as far from the average result of a 3d6 as it is from a d20. In other words, saying “you must roll a 16 or higher to succeed” means a task is just as many points away from the average result of your character’s roll no matter which dice you’re using, however the odds of rolling that 16 or higher are 25% on a d20 but under 5% on a 3d6. This is what people are talking about when they refer to bell curves: The d20 is far more likely to produce extreme results than the 3d6. The odds of a natural 20 are 1/20. The odds of a natural 18 are 1/216.

Now, the point of the essay is that the 13+ on the 3d6 and the 16+ on the d20 have about the same odds, so just use whichever number gives the right percentages and it doesn’t matter which is which. There are two problems with this. First of all, the essay is pretty damn vague about whether or not that’s the actual point. That’s the least wrong interpretation of the essay, but nowhere in the essay does anyone actually come out and say that. Secondly and more importantly, that is still wrong. I’ll demonstrate with a quick thought experiment.

Take a series of three locked doors of steadily ascending difficulty and a rogue trying to get past them. According to the essay, it should be possible to get the same probabilities for this scenario out of both a d20 and a 3d6 system. Let’s start with the doors in the d20 system, which are DC 15, DC 20, and DC 25. The rogue picking the lock has a +5 bonus, so the rolls needed are 10, 15, and 20. This is a 55% chance, a 30% chance, and a 5% chance of success.

Now we try to simulate the same thing on 3d6 and everything is bonkers. If we try to port the numbers over directly, that definitely won’t work. A rogue with a +5 bonus has 62.5% odds of hitting that DC 15 and 0% odds of hitting the DC 25. So let’s ignore the exact numbers of the DC and instead try to pick whatever DC most closely matches the percentages of the d20 version (like the 13+ and 16+ example used in the original essay). DC 21 instead of the DC 25, and nothing really lines up with the others at all. The closest to 30% odds is DC 18, which gives 26% odds, and the closest to 55% odds is DC 16, which gives 50% odds. So our DCs are 16, 18, and 21 instead of being DC 15, 20, and 25. We could smooth that out to DC 15, 18, and 21, and indeed, having DCs spaced 3 points apart is much better than 5 points apart for a 3d6 system, but then we’ve somewhat significantly changed the odds from 55% on the d20 to 62.5% on the 3d6. That’s not a huge difference, but it isn’t nothing.

Then it gets worse. Let’s assume that another rogue comes by who’s slightly sneakier and has a +7 bonus. Under the d20 system with its DCs of 15, 20, and 25, the rogue is now succeeding 65% of the time on the first lock, 40% of the time on the second, and 15% of the time on the third. He’s ten percentage points more likely to succeed on everything.

Now let’s give the same +2 bonus to the rogue under the 3d6 system and its DCs of 16, 18, and 21. His results are now 74% against the DC 16 (which gets even further away with an 83% if we use the DC 15 instead), 50% against the DC 18, and 16.2% against the DC 21. This is the worst possible result. The odds for the toughest door about line up, but the odds for the first two doors are way off, especially if we go with the DCs of 15/18/21 that give us a consistent 3 points of difference between the two the way the d20 system has a consistent 5 points of difference.

This is actually an even worse result than if all three of the probabilities failed to line up, because if the +7 bonus were badly off of all three results, you could switch to a +6 bonus and see if that fixes it. Unfortunately, since some of the +7 rogue’s results line up between d20 and 3d6 but others don’t, that means that switching to a +6 will fix the broken results but break the working results (which is almost exactly what happens if you do give the sneakier rogue a +6, although only if you use the 16/18/21 set of DCs).

Bear in mind that even before we looked at two different rogues, we already had a 50% success rate in 3d6 being compared to a 55% success rate in d20, simply because there is no possible number in 3d6 that lines up with 55% at all. The difference between 50% and 55% is the difference of a +1 bonus in d20. Attempting to force a 3d6 system to conform to the probabilities a d20 system produces is, in this way, a lot like giving your players -1 penalties or +1 bonuses pretty much at random, as the probabilities of one distribution fail to line up with the probabilities of the other in one way or another.

Bonuses are in virtually every rollover RPG ever made and they interact radically differently between d20 and 3d6. Even if you’re making an RPG about grey blobs with identical stats it still wouldn’t work, because 16 and 13 happen to line up like this from d20 to 3d6 (respectively), but most results on the 3d6 do not line up that well with results on the d20.

A Redditor named Matt_Sheridan posted a comment on the original essay, with only two lonely karma. It is the second from the bottom when sorting by controversial, so it’s not that he received a lot of upvotes and also a lot of downvotes. He just didn’t get a whole lot of attention for his comment at all. This is too bad, because his comment pretty much completely sums up the reason why the entire essay is junk:

This is kind of a spherical-cows-in-a-vaccuum argument, isn’t it?

Yes. Yes it is. If you are making a game where nobody ever has any bonuses and all actions have a 25% chance of success (or one of the other probabilities where d20 and 3d6 happen to line up), then this essay is a keen insight on the differences between 3d6 and d20. For everyone who isn’t making Spherical Cows: the Vacuum, the essay is white noise.

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