1d20 vs 2d10: A Dissection

Forgive me, this is not going to be a flowery blog post. It will be all text and math. Today, I'm going to dissect the application of dice in D&D and explain the inherent flaws in the d20 system as it was inherited by 5th edition.

The Problem: Swing

The #1 complaint against the d20 system is called "swing". The problem comes from a flaw in the phraseology and valuing used in the 3.5e corebooks. In those books, they repeatedly and frequently emphasize that 10 is the average result on a d20, and they value a single +1 as being equivalent to many hours of play and thousands of gold coins. They also repeatedly harp on the idea that a score of 10 (with a modifier of 0) is "average" for a common person, and that a DC of 10 is "easy". 5th edition inherited much of this idiocy.

The problem is that they are using the word "average" in both it's formal and informal uses. Mathematically, the average on a d20 isn't actually 10, it's 10.5. (The minimum roll would need to be 0 for an average of 10.) That minor problem aside, a d20 has what is called a "flat distribution". That is to say, you have an equal chance of rolling any given face every time you roll it. So 10.5 being the "average" result is meaningless in regard to any given single roll, and it's what happens in a single roll that is precisely the problem. See, because the system talks about the average as being equivalent to the common, normal, and mundane, everything in the system treats 10 as base. But it's actually not.

How much is a modifier of 1 actually worth on a d20? 5%. It shifts your result range by 1. If you divide 20 into percentages, each face is 5%. So, shifting your range up by 1 is a flat increase of 5%. Barring optimized cheese builds in 3.5e, and T4 characters in 5e, +5 is pretty much your typical limit to bonuses on a check. That gets you +25% to your chances of success on a static DC. That means a barbarian with 20 strength is only 25% more likely to kick down a door than some schmuck tax attorney. That's hardly a difference, and certainly not worth hours of character-life-threatening work.

How does it play out at the table? Poorly, but not as bad since they pulled crits and fumbles out of the check system in 5e. This is where people bitch about swing. Here's an example: a door needs to be busted down. DC15. The barbarian with +5 to the roll tries and rolls a 6. He can't kick down the door. The wizard with -5 rolls a 20 and knocks the door off its hinges. What went wrong? Why is the system generating nonsense? Because the players and system think 10 is average, but it's actually equal. Modifiers are overvalued and ability scores are mostly irrelevant.

There's a few ways to mitigate the above scenario.

1. The DM should be using the check to determine the reality of the situation, not the other way around. When the fighter failed to kick down the door, that doesn't mean he failed despite the door being within his power to smash- it means the door was never within his abilities to begin with. Think of it as a sort of Schrodinger's door, it isn't strong or weak until someone gives it a try.
2. Given that the fighter's strength is so high, and him kicking down the door makes more sense than him failing, why was a check ever even called in the first place? Generally, if something makes sense, the DM should just allow it without question. We don't need to roll checks to walk across a room or speak freely, do we?
3. As soon as the wizard said he wanted to try, this should have been handled as a group check, not separate individual checks. That way the wizard was helping, and his help would have opened the door where strength alone would not have been enough. That makes sense. When the wizard said he wanted to try after the check was complete, the DM should have shut him down for not joining in to help in the first place- the door has been determined to be too sturdy for your group to open. Find another way or move on.

Despite these methods of more refined check calling, the swing still remains, and characters will still find themselves randomly doing cartoonishly well or idiotically bad. The worst of it will happen about 10% of the time, but any number around 5 or 15 or beyond will get you weird results if your modifier is in the opposite direction.

It isn't the results that are the problem though. We want unpredictability. We want the gamble. The problem is the frequency at which we go from high to low results and back again within seconds. This rapid shifting between high and low produced by the flat distribution of the die makes our results swing wildly from successful, to pathetic, to heroic, to failure again. That swing frequency is the problem we seek to eliminate.

The Solution: Same Range, Better Distribution

The commonly debated solution is 2d10 because it uses close to the same range as 1d20, changing 1-20 for 2-20 with an average result of 11. I'm not going to debate the official 3d6 system because it is completely retarded. The modifiers in D&D are based on the range. Shifting the range to 3-18 makes no sense at all in relation to the mod ranges. The exact distribution doesn't matter, we don't need a pretty bell curve, we just need middling results between major results most of the time to cut the swing.

First off, you get a lot of people who will dismiss this idea outright because it is

1. Unofficial, and
2. Not a d20.

Well, that's fine. Being unofficial is good because the guys making the game are just as dumb as the guys playing and houseruling it. D&D is full of stupid design decisions and it's still a good game. The d20 system happens to be one of those stupid designs. Also, I have no loyalty to any particular polyhedral chunk of plastic. The d20 is a toy. I have many other plastic polyhedrons to play with.

Next, you get math nerds who will overvalue petty variations between the d20 and 2d10 and exaggerate the disastrous unintended consequences. The common complaints are thus:

1. Your probability distribution becomes peaked, not curved.
2. Chances are not as easily calculated in your head any more because each +1 is worth a variable % rather than a consistent 5%.
3. The average is now 11, a whopping 0.5 difference.
4. The game's math wasn't intended to use 2d10.
5. How do crits work?
6. Because ACs still operate linearly, they interact with the nonlinear attack modifiers weirdly.

Ok, let's tear into that.

A peaked distribution is fine because you don't see a distribution chart when you roll. It doesn't matter what the distribution looks like as long as it does the job we want. (Spoiler: 2d10 does its job quite well.)

I'll be honest about people who calculate the odds of a check in their head: that shouldn't be possible. The DM shouldn't declare his DCs ever. When you go to attempt a task in reality, do you have absolute certainty regarding your mathematical probability for success? No. Never. Why should your character? Uncertainty is the whole point of the system. Obfuscating such calculation is a good thing. Besides, precalculating your odds doesn't change them. It's a waste of time and effort. And, finally, as I already pointed out, even "high" mods that take a lot of effort to get are only a mild shift in probability on a d20 anyways. The difference between a "hard" check and a "very hard" check was only 25%.

I'm going to let you in on a secret about human perception: our brains only care about doubles. We don't care about 5% because we can't actually feel it. We care only a little about 25% because it only mildly registers. Until something really pays off for us, nearing or exceeding double the norm or expectation, we don't really care about it. (Until we calculate it. Once we quantify things, we begin to care a great deal about very little changes. This isn't about the quantified reality of it though, this is about the feeling generated by the dice at the table.)

Regarding the shift in average, I'm not sure how this is even a problem. The average is irrelevant on a d20 for any single check in the first place. The fact that 2d10 has an average that matters is much more significant than what the number actually is. It's also kind of the point, isn't it? We wanted the average to also be the most common result. That's what you get on 2d10.

Now, about the intent of the D&D system. Let me tell you an anecdote. In the first printing of 5e, unarmed strikes were simple weapons, granting any character with simple weapon proficiency a proficiency bonus to their fists. Characters lacking this could either take a feat or multiclass to monk to get unarmed strike proficiency. Druids, for some reason, wound up unproficient. In the second printing, unarmed strike was removed as a weapon, granting all players unarmed strike proficiency, allowing wizards to slowly become pro wrestlers without ever throwing a punch.

The point is that the developers are idiots like the rest of us and what they intend is just as much nonsense as what they did not intend. D&D is a pile of bullshit they got paid to make up on the fly. Frankly, their words and opinions mean as much to me as their choice in underwear: not at all.

Now for the interesting part: criticals.

Let's start with crit fumbles because that comes up all the time for some reason. 5E DOES NOT USE CRITICAL HITS FOR CHECKS. Furthermore, 5E DOES NOT INCLUDE CRITICAL FUMBLES OR AUTOMATIC FAILURE AT ALL. They are nonsense on a d20 and they are nonsense on 2d10. Crit fumbles do not make any sense and turn your game into a cartoon. What 2d10 would mean for crit fumbles is nothing at all because they don't exist in the first place.

OK, now for critical hits. In the 2d10 community, 3 methods of determining a crit have been proposed:

1. Nat 20 is Nat 20.

Under this system, 2 tens at the same time is a crit. This reduces your chances to 1%. That means 1 in a hundred attacks will land a crit. In 3.5 that'd be a big deal! In 3.5e, crit threshold mattered a lot and whole character builds revolved around taking advantage of the critical hit system. Not so in 5e. A crit is just an extra damage die, not even double damage. At this point, by making the rate so low, you've basically rendered them nonexistent.

2. Nat 10 is Nat 20.

Under this system, if any 10s show up in the roll, it's a crit. There are 19 combinations that include 10s on 2d10. 19 in 100 is 19%. This actually establishes a higher frequency for critical and abstracts critical effects from overall effectiveness somewhat.

3. Crit Die

Under this system, you still roll a d20 with your 2d10. If it lands a 20 it's a crit. This is silly because it's an extra die for just one number to establish a binary state. On the other hand, it has no impact on critical rate and fully abstracts effectiveness and criticals.

So, there you go. Crits. Personally I just go with two 10s is a fluke and snake eyes is a flunk. Flukes are auto-success and flunks are auto-fail. No special damage. Then I just adhere to careful check calling so people don't fluke or flunk on things that don't make sense. No, you can not fluke to punch the moon in half, you just don't get a roll at all.

And now for the meat of it: how does the 2d10 check system interact with AC in the combat system?

From the perspective of a commoner with 10 in all stats and a 0 mod to all ability checks, a DC 10 task is a 64% success rate. That means an AC 10 creature, such as an unarmored commoner, is an easy target that you will hit more than half the time. That sounds right, right? If it's easy, then you should normally pass the task, right? AC 15, however, is not the old 25% chance. Here's the distribution for 2d10:

Die result chances

2=1%

3=2%

4=3%

5=4%

6=5%

7=6%

8=7%

9=8%

10=9%

11=10%

12=9%

13=8%

14=7%

15=6%

16=5%

17=4%

18=3%

19=2%

20=1%

So, adding up all the percentages, what are your chances of meeting a target number without any mods?

DC pass chances

2=100%

3=99%

4=97%

5=94%

6=90%

7=85%

8=79%

9=72%

10=64%

11=55%

12=45%

13=36%

14=28%

15=21%

16=15%

17=10%

18=6%

19=3%

20=1%

So your 15AC target should get punched in the nose about 21% of the time. That AC-optimized fighter tank with 19 AC out the gate should get smacked roughly 3% of the time when being assailed by a bunch of untalented mooks. Modifiers don't actually alter your percentages, they just shift your range up or down. So, for example, a mod of +5 shifts your range up 5 steps. Attacking an AC 10 creature with +5 to the throw gets you a hit rate of 94%, the same as what would be given to an attack against an AC of 5 for someone without a mod. Your mods could be applied to the target number with negative logic to produce the same effects.

All of this means that the mathematics and probability now match the description given for them in the corebooks. Look at the benchmark probabilities without mods:

Trivial = 94%

Easy = 64%

Moderate = 21%

Hard = 1%

Very Hard = 0%

Impossible = 0%

Compared to the original rates of 25%, 50%, 75%, and 5%, these numbers more accurately fit the name given to them. A trivial check really is a trivial check. An easy challenge really is most likely to be passed. A moderate challenge is something that actually gives a fair bit of resistance, and something hard is actually unlikely to be done.

Furthermore, because mods shift your range, not your probability curve, having a bonus means your average result, also your common result, is always higher than someone without it, and the chances of someone outcompeting you in something you are good at drop off rapidly as you get better at it. This also works for the penalties as well though. Someone who is specifically bad at stuff will be consistently bad at it, achieving success when out of his element only on a lucky roll. The wizard could still kick a door from its hinges immediately after a barbarian flailed uselessly against it- but that degree of swing will happen far, far less frequently, and so be more of a memory and less of something weird and glitchy that happens all the time.

And, if that doesn't get you on board, there's this little cherry on top:

d10s just look cooler than d20s. Seriously! A d20 looks like a weird lumpy ball. d10s look like little diamonds. LOOK:

They're just cool, OK?!