The comparison is between two resolution mechanics for table top role-playing games. In those games, most of the participants control characters in a setting, and they try to accomplish goals. For example a character named John might be trying to shoot a target with a bow and arrow. Do they hit the target? The resolution mechanic decides for you. If we left that decision up to the story-teller (game master), it has a strong chance of degenerating into arguments about fairness. Instead, you use dice to allow a fairly random number decide for you. And this blog is meant to compare and judge two of those dice mechanics.
Basic Explanation of the Two Dice Mechanics
The 1d20 mechanic refers to using a single twenty-sided die to get your random number. This comes from a large pool of games that fit into the "d20" category, and started with Dungeons & Dragons. If I refer to a d20 game specifically in the future in this post, it'll probably be Pathfinder (PF), because that's the only d20 game I play these days. Your character will have a number (often called a modifier) that you add to the result of your 1d20 roll. That number is compared to a difficulty score of some kind. An important point is that a 1 on the die can be an automatic failure and a 20 on the die can be an automatic success. I'll clarify using the example of John the Archer shooting a Target.
John is a fairly average person with a little skill at archery. For the example, let's say that when he takes a shot with his bow, he has a +4 attack modifier. When he rolls his d20 to see if he hits a target, his possible results are 5 to 24. The target will have an Armor Class (AC) which is a number that represents how hard it is to hit. In this case it's primarily based on how small the target is. Let's say for ease of example that the AC is 15. I chose that number because it means John has a 50% chance of hitting the target. If he rolls a 10 or below, he misses. If he rolls an 11 or above he hits. As something that will become important later, D&D and PF use a 6-second round that abstracts some of the action. In that 6 seconds, John shoots once, so the effort of aiming at the target and even possibly moving to a firing position are included in that time. It'll become clear later why I mention this.
The 3d6 mechanic refers to using three six-sided dice to get your random number. I know of this choice from GURPS. But really, this can represent any dice mechanic where you use multiple dice to get your number. The important aspect is the multiple dice. And lots of good TTRPGs use multiple dice in their resolution mechanics. I just chose GURPS because I LOVE GURPS (Steve Jackson Games). And I am very familiar with the math around the GURPS die mechanic. In this case, your character will have a skill instead of a modifier like in Pathfinder. Whatever that number is, is the number you have to roll less than or equal to in order to succeed. There are modifiers too, but they refer to something different in GURPS, usually around outside factors that affect the outcome (PF has those too, but uses the word modifier to refer to the "skill" number as well). I'll make it clearer with John the Archer shooting a target...
John is a fairly average person with a little skill at archery. For the example, let's say that his bow skill is 11 (ever so slightly above average). In GURPS, one round of action is one second. There's no abstraction of action. Each second you say what your character does specifically, so for purpose of making this roughly equivalent to the D&D/PF mechanic, we have to say John is aiming for 3 seconds with his bow (3 seconds of aiming gets you the highest bonus possible). This will get John a +5 to his skill for the attempt for an effective skill of 16. But GURPS does like its modifiers. Instead of the Armor Class of the target representing an abstraction of how hard it is to hit, GURPS has modifiers for how far away it is, and how small the target is. So, for matching the examples, let's say the target is far enough away and small enough that John has a -6 to hit from where he is, resulting in an effective skill of 10. John has a 50% chance of hitting his target because half the possible results on 3d6 are 10 or below (3 to 10) and the other half are above (11 to 18).
Hopefully that gave you a good foundation. I see two factors as being important to the comparison: Reliability of Skill and Improvement of Skill. Actually... here's a collection of diagrams you might find useful in understand the linear probability versus the bell-curve.
The bottom two charts show the probability of each result. You can see that on 3d6, the probability is higher in the middle, and low on the ends. On a d20, the probability is the same across all results.
The top two charts show the same orange line for individual result probability, but includes a line for rolling less than or equal to that value in yellow.
Reliability of Skill
What I mean by reliability of skill is how much your character can depend on their skill. If a character spends resources getting "good" at something like archery, what does that mean in the game? If a character in Pathfinder spends character resources like feats or improves their dexterity to get a higher bonus on ranged weapon attacks, does it make a difference? If a character in GURPS spends character resources, like character-points, to improve their dexterity or their skill with a bow, does it make a difference? Does having a higher mod or skill make the skill more reliable than having an average mod or skill?
In d20, if I use feats and attributes bonuses to get my attack modifier from +4 to +7 at first level, that's 3 points higher, which means a 15% better chance of hitting a target because three of the twenty possible results just switched from being a miss to being a hit. In our example, we go from a 50% chance to a 65% chance. That's pretty good.
In 3d6, if I use character points to improve my bow skill from 11 to 12, so that in our example my effective skill is 11 instead of 10, the probability goes up from 50% to 62.5% The jump in percentage is close enough, and 62.5% is still a lot better than 50%.
But are those percentages reliable? The Law of Independent Probability means that each die roll has an equal chance of coming up on any value, regardless of any rolls before. So, saying something like "come on... I'm due for a 20" makes no sense. Just because you haven't rolled a 20 on a d20 in a while, doesn't mean you are more likely to roll one now. It is possible for a person to roll five or under for twenty rolls in a row. Just because you rolled low doesn't mean you're due for rolling high. With that +7 modifer trying to hit the AC 15 target, you only have to roll an 8 or above to hit it... but the d20 makes up for 20 possible points of your roll. It is responsible for 20 points. Your attack mod of +7 is only responsible for 7. So, there's no guarantee at all that you're going to hit because every value has the same chance of coming up on the die, and the die is responsible for more of the needed final result than your skill is by a wide margin.
With 3d6 we have the introduction of the bell curve. On 3d6, there are 16 possible values, but the combinations that result in them are counted at 216 possible combinations of the three dice. Only one of those combinations can make the result of 3. And only one can make the result of 18. So, each of those results has a 1 in 216 chance (0.46%) of being the result of rolling 3d6. A result of 10 or 11 however each has 27 combinations that can result in that number. So, a 10 or an 11 has a 27 in 216 chance (12.50%) of being the result of rolling the dice. You might already have figured out what this means, but just in case: rolling any extreme result is far less likely. If my effective skill for shooting that target with a bow is 11 instead of 10, the probability the dice will result in a number 11 or lower is actually 62.5% because of that lovely bell curve and the normalization of results by multiple dice.
A modifier in Pathfinder using the d20 system that has similar probability to a skill in GURPS is not as reliable because 1d20 has an equal chance of coming up as any result. Working your butt off to make a character with a high attack modifier (which they don't make easy) does not mean your skill will be reliable because of the d20. Making a little effort to make a character with a high attack skill in GURPS makes a noticeable difference in reliability. You get more from your effort as the player.
I suppose I'm diverging from objective comparison at this point. My personal experience with trying really hard to make a character in Pathfinder who has a great skill at using a weapon in hopes of not really missing has been that even if you make every choice to make that character's attack mod huge, the whim of the d20 has more sway over what actually happens than all the effort I put in. I find it frustrating when I go on a bad rolling streak and end up missing four attacks in a row when I was supposed to have an 80% chance of success. Doesn't the +7 to hit mean that person is noticeably better at hitting a target than the person with a +1? But the d20 matters more than the skill in the final result, and it is perfectly possible for four rolls in a row to just suck horribly.
For player satisfaction with their character, I believe it is important for the player to feel like their choices in building the character matter and have an effect on things like their ability to use a skill they put some effort into. In that scale, I feel like 3d6 is leaps and bounds better than 1d20, because of the tendency to roll a result as the bell curve predicts. If I get my character's skill up to 14 for example, the probability of rolling that or less is 90.74%. And when you roll dice in practice, they really follow that probability. So, I feel like I accomplished something by getting the skill up to 14 instead of 10. It makes a real difference. Even if I got an attack mod up to +7 (for a 90% chance against an AC of 10), there's no tendency by the single die to be a 3 or above.
An easier way to think about the difference between 3d6 and 1d20 as it relates to reliability is this: The d20 has 20 possibly combinations, and 3d6 has 216 combinations. With a +0 attack mod in d20 against an AC of 10, you get 10 results out of 20 that are successful. With a +5, you get 15 results out of twenty. It's a linear progression. With a skill of 10 in 3d6, you have 108 out of 216 that are successful (50%). With a skill of 14 (one quarter of the way through the possible values like an increase of 5 in d20) you have 196 out of 216 (90.74%); The improvement on the bell curve is better, and therefore more reliable.
Improvement of Skill
I would say that one of the reward feelings that makes playing the game fun is the feeling that your character became stronger in some way. In Pathfinder that means when your character gains a level. A few things change and your character is stronger in a few ways than they were before. If we're talking about a character where your attack modifier is important, it usually means your attack modifier goes up by 1. So your chances to hit go up by 5%. BUT... the way Pathfinder is (and its relatives are) designed, the challenge increases at the same rate. That Armor Class number goes up on the bad guys too. So, you're not getting a 5% boost. You're breaking even. Maybe the game tricked me into being happy about the level up in that way, but really, the attack modifier changes (and saving throws, and skills, and so on) don't really matter. The game is designed to maintain the same level of probability of success for all 20 levels. And if you reach level 20, and have a +20 attack modifier (or likely +30 with magic, high stats, etc), the die still accounts for most of whether or not you'll hit a foe because their AC is probably going to be 40 or above.
Sure if your character went back to face some of the same kinds of foes they did when they were level 1, they'd crush those foes easily. But the game generally keeps you looking through the keyhole where your challenges are equal to you.
In GURPS, when you spend character points that you earn, you pick what gets better, and if your combat skill to attack with is what you choose, even going up by one can make a big difference. If you bump your skill from 10 to 11, you're getting a 12.5% boost, AND it's more reliable that you'll roll 11 or below on 3d6 than it is that you'll roll 9 or above on a 1d20. Get your skill up to 16 and you've maxed out your probability without considering negative modifiers at 98.15% chance of success. Anything above 16 is beneficial mostly to deal with negative modifiers. But when you increase a skill in GURPS, you're actually making a noticeable difference.
Part of the strength here is that attack and defense are handled differently in GURPS than in Pathfinder. You roll to see if you succeed. The defender might have the option of rolling to see if they can defend. So, your skill at attacking doesn't have to reach a ever increasing goals of difficulty. If your opponent is particularly good at defending though... you might have to get clever by maneuvering them to a position where they can't defend for some reason (there are rules for this in GURPS combat). Or you might have such a high skill that you can use fancier attacks like a feint to make it harder for them to defend against the next attack. Or you might work with a friend to flank them, making it harder for them to defend. Or you might use some other tool in your arsenal. But for progress in the difficulty of the challenge, GURPS relies on the cleverness of the players instead of keeping the probability essentially the same for everything you try.
Final Thoughts
It's very difficult to keep emotion out of writing this because I enjoy playing in a good campaign, and when the play experience is a negative one, I become frustrated that my precious small amount of free time I can give to TTRPGs just got so thoroughly abused. I prefer the bell-curve to the linear distribution because it makes me happier about the character I created, and it makes me happier about how sessions go. Sure... things can still go wrong, but it's rarer and more meaningful (in my opinion).
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