Designing a d20 RNG

[Chamomile]

Rough breakdown of the big ones:

-Flat rollover. d20 rollover has a flat curve which means a +1 is always +5% odds of success and it takes about a +3 bonus before things start to be noticeably different. This makes things slightly easier to keep track of and it means you can hand out +1 bonuses like Halloween candy so long as you have some discipline about the number of bonus types that can stack with each other. Since it's rollover, it is possible to get pushed off the RNG, so good for situations where you want player characters to reach a level where they just do not care how many orcs are on the battlemap. Maybe you have a "20 is always a hit" rule so that they can still hypothetically be threatened by 200 orcs, but no amount that can pragmatically be handled by the skirmish rules can threaten them. d100 rollover fits into this category, but listen, there is pretty much never a situation where you would actually want your players to try and add +1% bonuses together into something that matters, so don't. Just divide everything by 5 and use a d20.

-Curved rollover. 3d6 rollover has a bell curve, which means a +1 counts for +12.5% to the rolls that are most down to the wire but less than +1% for the rolls that were nearly (but not quite) off the RNG to begin with. It's good for situations where you want it to be possible to level off the RNG relative to starting mooks, but you want every +1 bonus to be significant, while still maintaining roughly the same total bonus size required to push off the RNG completely. 2d10 rollover is the same deal, except that the +1 bonus ranges from +1% to +10% and is always a nice round percentage of some kind (i.e. if you need to roll a 14, a +1 bonus gives you a 6% flat increase in odds of success, as opposed to 3d6, where it's 4.63%). 2d10 also means you're more likely to need a +2 before you really notice a difference. Also d10s aren't nearly as much fun to roll as a d20 or d6s, which is probably why this is so unpopular despite its mathematical elegance as a curved d20 alternative. 5d4 or 6d4 is starting to go crazy at the edges, with a +1 counting for about 15% in the middle but less than 0.1% for rolls that are barely on the RNG. A smaller RNG will increase the value of a +1 bonus at both the end and the middle of the curve, as well as obviously requiring fewer bonuses to get off the RNG entirely. For example, a +1 bonus counts for 2.78% at the edge and 16.67% in the middle, significantly larger all around compared to 3d6.

-FATE dice. I don't know a category for this one, but in the FATE example you roll four six siders, but each one has two red faces that subtract 1, two blue faces that add 1, and two blank faces that add nothing. You can use specialty dice but you can also just use regular dice with 1-2 being -1, 3-4 being null, and 5-6 being +1. This gives a standard RNG of -4 to +4, to which bonuses are then added. Odds are the same if you go with three blue or three red faces on each die, and in practice this is really just a 4d3 rollover RNG with a built-in -8 penalty. The main purpose of the -1/+0/+1 setup (or mathematically identical -1/+1 setup) is just to make it more intuitive. Players looking at their character sheet don't see skill ranks ranging from -8 to -3, which seems disempowering even if most checks have a TN between -7 and -4, but instead intuitively understand that the routine result is 0, all positive results represent some level of expertise, and negative results represent exceptional failure beyond what you'd expect of an average layman going for it. Usual curved rollover logic applies here: A +1 bonus is worth +1.23% at the edges where the roll was nearly a foregone conclusion anyway but 23.46% in the center where it's down to the wire. Note how extreme this particular RNG is due to the small die size: A +1 bonus is worth nearly as much as a +5 on a d20 in the circumstances where it most counts (and is worth about three times as much at the edges as on the edge of a 3d6 curve, for that matter). FATE dice mean that a +1 bonus represents a massive increase in odds of success compared to what would normally be expected. This system also makes taking 10 easy as pie: Your average die roll is 0, so your average result is equal to your skill bonus.

-Rollunder RNGs. Do not use these. They are mathematically identical to rollover RNGs, but harder to work with in every circumstance except the most Platonic "make a check against the default TN with no modifiers outside your skill level" situation. Even if this is a plurality of your die rolls, it is still not a majority of them. Use rollover.

-Fixed TN dicepools. d6 dicepools can have their average result set to either 3 dice = 1 hit or 2 dice = 1 hit depending on whether the TN is 5 or 4, respectively. With a dicepool, it is always possible to fail, no matter how many dice you have, which means it is always possible for a mook to score a decisive hit against a major character. As skill increases, odds of failure dwindle ever closer to zero, but never actually hit zero. There's also a physical limit on how big your dicepools can get before you get logistics problems. While a rollover RNG suffers no particular breakdown when bonuses reach +30 or +40, a dicepool starts to become physically difficult to roll and count up when you reach 20+ dice, which means whether your TN is 4 or 5 affects how much more competent a maxed out character can be compared to a mook. This problem doesn't apply if you're rolling electronically, though, where you can jolly well have fifty dice in the pool and that is no harder to roll than five. You could also use d10s, or d8s or whatever, but d6s are easier to acquire in very large numbers and you aren't really going to want any odds besides 2 dice = 1 hit or 3 dice = 1 hit. With a d10 and a TN of 7, for example, every five dice is two hits. That's wonky and hard to keep track of. Why would you want that?

-Variable TN dicepools are a bad idea. It is unreasonable to ask a GM to do algebra on every single check to figure out how hard it is for characters to hit. The amount a TN can vary is small enough that they can and probably will just brute force it by calculating the odds separately for each TN characters might plausibly roll at, but this extra math adds nothing but obfuscation.

-Weirder RNGs are usually a bad idea. They add learning curve and very rarely significantly affect the math compared to one of the existing RNGs, except to make it worse. By far the most common criticism of FFG's Star Wars is that its dice have unintuitive probabilities that makes it hard to get a grasp of how good a character is at something without crunching the numbers (or playing so many dozens or hundreds of hours of Star Wars that you start to build up intuitions, which is even less reasonable) and require you to go out and buy special dice to play the game. FFG was able to pull it off because 1) they were FFG and 2) they were selling Star Wars, but unless you are a middlingly-famous game studio selling an extra popular IP, it will go more poorly for you. The One Roll Engine was trendy amongst indie games for a bit and still has its devotees, but for the most part people eventually realized that it took longer to resolve that one roll than it did to just roll separately for damage (and maybe soak), and the probabilities were less intuitive anyway. Since 1974 the only new dice mechanics with any real staying power have been FATE dice which, as mentioned, were just 4d3-8 rollover in disguise, and dicepools, which have been around since at least WEG Star Wars in 1987. Everything else has just been different rollover systems (and again, FATE is mathematically in that category and just used a dicepool/rollover hybrid in actual play because that made it more intuitive). Don't be weird for weirdnesses sake. Do what works.

[Iduno]

I like that you got d100 down to so few words.

A few thoughts:

2d6 seems like it has simpler math than 3d6, and you could just grab 2 dice from anywhere. That benefit is significantly less if you can buy dice anywhere or just use a phone/computer to do the rolling for you. The same reasoning could probably be applied to the difference in 3d6 -> dice pools, but that comes down more to bonus dice vs roll + bonus.

Adding or subtracting your bonus from a static RNG either runs into the bonuses being too large for the RNG (D&D 3.5), or too small for anyone to care about (any d100 game ever). Dice pools giving you more dice for your higher bonus runs into diminishing returns (not always bad), and tends to break at low end more than the high-end with static RNGs.

For weird RNGs, something like Earthdawn's step system (skill + stat = step, which corresponds to the average roll on the dice shown in the table) is a good example. It's complicated because it's based on statistics, but drives away potential customers. Looking up/memorizing what dice you roll when is an atrocity at a table, but now (~25 years later) shouldn't be a problem if you're using a dice roller, when a computer can also handle rolling dice with an impossible number of sides. They still need to justify why they made things more complicated (and a statistics calculation works for that), AND they needed technology to catch up before it became as acceptable as other options.

Dice pools and the step system both have the (probably fringe) benefits of critical failures being rarer the more skilled you are. The most and least skilled chefs in the world having the same chances of fucking up and injuring themselves is stupid, when one person doesn't even know how to hold a damn knife.

Roll-under systems are the same as roll-over, except the designer is letting you know even they don't trust themselves to design an RNG that you stay on unless they force themselves to, and also that they're going to do "creative" bullshit like rolling 2 identical dice does something special.

[pragma]

One pedantic correction: curved RNGs aren't actually bell curves unless you roll an infinite number of dice. Because we're not using infinite dice there is a difference between the normalized variance of 2d6 and 3d6; 3d6 will roll extreme values significantly less often than 2d6.

One quirk of dice pool systems is that your variance goes up as the size of your pool goes up. This means highly skilled characters always succeed more than less skilled characters, but have more random results between whether they succeed a little or succeed a lot. Curved roll-over systems don't suffer from this because skill modifiers usually shift the RNG, which doesn't change its variance.

[Hicks]

There's also the 20+ resolution mechanic, which doesn't work like anything described above. As your bonus gets larger your minimm and maximum number rolled increases, and since you use the highest d20 in a pool the output skews higher and higher the more bonuses you have. But the maximum bonus the system can resonably handle is +6ish. With a +6 bonus you select the highest of 7d20 rolled and add 6 for your output. It works well if you want bonuses low but still want skilled actors succeed in their checks much more than unskilled actors.

[Username17]

There's also the 20+ resolution mechanic, which doesn't work like anything described above. As your bonus gets larger your minimm and maximum number rolled increases, and since you use the highest d20 in a pool the output skews higher and higher the more bonuses you have. But the maximum bonus the system can resonably handle is +6ish. With a +6 bonus you select the highest of 7d20 rolled and add 6 for your output. It works well if you want bonuses low but still want skilled actors succeed in their checks much more than unskilled actors.

20+, as you've described it, is just a dicepool with very large dice and variable target numbers. So basically Vampire: The Masquerade, First Edition. You would not want to use such a random number generator, because fucking obviously.

-Username17

[pragma]

I don't think so, Frank. 20+ is an RNG where your generated result is the maximum of N d20.

The probability mass function for 20+ is flat for 1d20, linear for 2d20, a parabola for 3d20, then cubic, quartic, etc (1). This is quite different from a dice pool, which follows a binomial distribution (2).

In particular, as skill increases in 20+, your result gets sharply skewed towards the maximum value the RNG can generate and your variance drops close to zero. In a dice pool system you maximum and mean results increase at the same time, which results in your variance increasing.

That said, I've never seen 20+ used anywhere and have no idea if it would hold water in play. It seems like you need to be _very_ careful with modifiers to make it work.

(1) https://math.stackexchange.com/question ... stribution

(2) https://en.wikipedia.org/wiki/Binomial_distribution

[pragma]

Ooh, found a plot of a "maximum of N random variables" PMF for comparison:

https://martin-thoma.com/distribution-o ... s-applied/

This is how die rolls in 20+ would break down.

[Username17]

I don't think so, Frank. 20+ is an RNG where your generated result is the maximum of N d20.

Well the suggestion was to also have you add a flat bonus equal to the number of dice you're rolling, but that's not relevant. The relevant issue is that you are rolling a dicepool of dice with a shit ton of sides and looking for 1 hit against a variable target number. That is literally how Vampire: the Masquerade worked in first edition. That was not a good system, and the underlying concept has not aged well.

Dicepools where you are only looking for 1 hit are very low variance and become lower variance the more dice are added. In the presented example of rolling 6d20 and adding +6 to the highest die, you hit TN 20 over ninety percent of the time. That just isn't enough variance to bother rolling for most things. Bonuses can't get very big, because the game becomes an immense pain in the ass when you're trying to roll twelve d20s and also because the game becomes laughably deterministic. 8 dice at +8 has a less than 3.2% chance of rolling less than a 22, and 21 is literally the highest number you can roll on 1d20+1. Your RNG is basically pretty similar to something like 3d3 in terms of how big a bonus it can accept before breaking.

-Username17

[pragma]

I agree: the more I think about 20+ the crazier the results seem. The system does have the properties of being relatively easy to adjudicate and having a different shape than other RNGs, but a shape which crowds all the results to the top of the RNG ranges from dull to useless.

[Hicks]

The point of 20+ wasn't that you always suceed, but that experts *very* rarely failed.

The odds of rolling the *lowest* for experts increase at a tremendous rate.
+0, 1:20
+1, 1:400
+2, 1:8,000
+3, 1:160,000
+4, 1:3,200,000
+5, 1:64,000,000
+6, 1:1,280,000,000

For a skill system where the numbers are small, getting a +2 circumstance makes the orphan thief actually succeed in stealing from the barbarian.

But 20+ isn't for fantasy games, it's for science fiction games like (Mongoose) Traveller, where dice modifiers are seriously +2~4 for experts on a 2d6 rng.

[Grek]

-Rollunder RNGs. Do not use these. They are mathematically identical to rollover RNGs, but harder to work with in every circumstance except the most Platonic "make a check against the default TN with no modifiers outside your skill level" situation. Even if this is a plurality of your die rolls, it is still not a majority of them. Use rollover.

Not universally true. Consider an RNG that uses Xd6, roll under skill where X varies with difficulty. There's no rollover analog to it, and it has about the same granularity as a dicepool system.

[Mord]

Not universally true. Consider an RNG that uses Xd6, roll under skill where X varies with difficulty. There's no rollover analog to it, and it has about the same granularity as a dicepool system.

Xd6 roll under has the same distribution as Xd6 roll over, so what exactly is gained by using roll under?

[Foxwarrior]

Oh, I see the tricky simplification going on here Grek. But it seems like the odds change by several times when you add just one difficulty die, so there's no subtlety here at all.

So Xd6, rolling at most 6 gives you odds of 100%, 41%, 9%, with X of 1-3.

While to get the same probabilities with rollover you'd need to change the target number to 1 for 1 die, 8 for 2 dice, and 15 for 3 dice. The same distribution but less convenient to say.

Since the gradations for difficulty are so wide while the gradations for whatever sets your TN are so much finer (Skill maybe?), I don't think this system is actually better than a percentile system where the DM forgets to ever give you circumstance modifiers so you have to stick with those 34% chances of success on every action you attempt.

[...You Lost Me]

On the topic of d20 RNGs, I had some thoughts from messing around with a fantasy heartbreaker.

Critical Success & Failure: The people I play with always love the fact that nat20 = critical success & nat1 = critical failure. Whenever it isn't a rule (like with D&D 3e skill checks), they want to houserule it in. Whenever I play with a different RNG like 3d6, one of the biggest complaints is that nat20/nat1 rolls don't happen often enough. So as far as I'm concerned, keeping critical successes and failures is non-negotiable. Do other people here get similar feedback from their play groups?

Measuring Bonuses: I think the prevailing opinion on these boards is that +3 bonuses are the minimum you should grant from a character option. That's an opinion I share as well, and I only want to give out +3 bonuses (or more) instead of handing out smaller ones. But I've identified two problems with this approach that I'd like feedback from you all on:

I have noticed in my playgroup is that bonuses can feel big even if they're mathematically small. People appreciate the +1 bonus to attacks when you have the high-ground, and the +1 bonus to attacks & AC from being small. The level 1 Barbarian's rage is considered a scary offensive ability even though it only gives a net +2 to-hit and most of your attacks were one-shots anyways. Forcing your bonuses into units of +3 seems like it's wasting an opportunity to harness player behavior. Are my instincts wrong here?

The other problem I'm running into is that you can only grant so many bonuses before outputs go far off the RNG. There are 6 "units" of +3 bonuses that you can grant before the RNG is no longer meaningful. Looking at an example of attacks: I already expect players to hit their targets 65% - 80% of the time, which means I can only grant +3 to +6 worth of bonuses before their attacks automatically hit. That's not a lot of room for an RNG if I want to reward smart builds & tactics. Flanking + entangling an opponent might just take your average attacker straight off the RNG if both conditions grant the equivalent of a +3 attack bonus. That feels very wrong to me. Are my instincts wrong here as well?

Advancement Rates: If we kept the D&D 3e idea of a given creature doubling in power as it gains 2 CR (or 2 levels or whatever), how much of that power increase should be granted to boosting your RNG? If a level 1 fighter had +6 to-hit, would a level 3 fighter have +9 to-hit? +12 to-hit? Could a good d20 RNG even accomodate this kind of power-doubling without the numbers getting out of hand?

[Pedantic]

Critical Success & Failure: The people I play with always love the fact that nat20 = critical success & nat1 = critical failure. Whenever it isn't a rule (like with D&D 3e skill checks), they want to houserule it in. Whenever I play with a different RNG like 3d6, one of the biggest complaints is that nat20/nat1 rolls don't happen often enough. So as far as I'm concerned, keeping critical successes and failures is non-negotiable. Do other people here get similar feedback from their play groups?

This always seems to come down to the statistical literacy of everyone around the table. Best compromise I've see to leave them in the game, but avoid the players getting critted to death by monsters and cartoonishly failing tasks: crits/fumble opportunities are triggered by 1s and 20s, and players/MCs have to spend meta-currency to "activate" them. Everyone gets X points of meta currency a session, and as a layer you can spend points to cancel fumble activations by the GM or crits against you instead of activating your own crits, possibly at a loss.

[DenizenKane]

On a d20 RNG, a nat 20 has a 5% chance of occuring, 19-20 = 10%, 18-20= 15%. On a 2d10 RNG, a nat 20 occurs 1% of the time, but a number between 18-20 occurs 6% of the time, 17-20 = 10%, 16-20 = 15%, so pretty much the same percent chance. So, your nat 20 in a 2d10 game would be a result of 18-20. Its also possible the actual nat 20 is an auto-confirmed crit (if it would hit) that occurs 1% of the time.

On d20 RNG, you need a +3 to notice a difference, the RNG breaks at +9, and you can only add 3 of those before you're off of it. On a 2d10 RNG, you need a +2 to notice a difference, and you can add 4 of those and still be on the RNG (99% success). I think this is a benefit of the "curved" rng in this case.

And, I think you can shrink armor values so they're less RNG breaking too. Light Armor = ~2 (72%), Medium = ~4 (85%), and Heavy = ~6 (94%), Shield =~ 2. (The d20 equivalents are +4 (75%), +6 (85%), and +8 (95%)). So, heavy armor and shield is +8 and you're still on the RNG. Also, in this game, you add your level to AC unlike in 3e where armor becomes useless eventually.

EDIT: Just thought of another way to handle crits on 2d10. Since it's already a percentile roll, you can have two different colored dice like a percentile, and have attacks crit when you roll a 95% or better, 90% or better, etc. So, you'd have a curved RNG attack roll, and a percentile chance to crit with 1 roll.

[deaddmwalking]

For a 2d10 RNG, you roll doubles 10% of the time. You can crit on any double that hits. If you need an 11+, you crit on double 6s, double 7s, double 8s, double 8s and double 10s. That works out to 5% of the time.

If you wanted fumbles, you could do the same (fumble on double 1, 2, 3, 4 and 5)

[tussock]

Critical Success & Failure: The people I play with always love the fact that nat20 = critical success & nat1 = critical failure. Whenever it isn't a rule (like with D&D 3e skill checks), they want to houserule it in. Whenever I play with a different RNG like 3d6, one of the biggest complaints is that nat20/nat1 rolls don't happen often enough. So as far as I'm concerned, keeping critical successes and failures is non-negotiable. Do other people here get similar feedback from their play groups?

Yep. Anything you roll a lot, you kinda want an extra rare success, and it's best as an immediate reward. If you're not hitting often, your hits are your rare success, but as you hit more often, you want a rare crit. Peak emotion boost, great stuff.

It should be rare enough that you can go a whole fight and not get one, so perhaps first hit in a round only, and not on too many numbers, and only if you hit regularly anyway. Then it needs to do enough that it noticeably changes the scene in some way, little crits like "max damage" or "+1d6" are terrible.

Also, make them class, or "leader" features, realistically the mass of grunts you fight don't need them, that's not something makes players happy, so don't use it. But the odd boss type, or semi-boss can have rare crits that randomly swing the fight, that's memorable drama, good stuff when the players overcome it.

Measuring Bonuses: I have noticed in my playgroup is that bonuses can feel big even if they're mathematically small. [...] Flanking + entangling an opponent might just take your average attacker straight off the RNG if both conditions grant the equivalent of a +3 attack bonus. That feels very wrong to me. Are my instincts wrong here as well?

They have to overcome natural variance, mathematically the effect of the bonus is relative to the square root of the number of dice you roll with it. You might make lots of attacks, maybe up to 20 in a fight, but you make 1 spot check before it, 1 initiative check.

So +1 to hit on 20 attacks is like +4 or +5 on a single d20 check like initiative. Arguably initiative is more important than an attack because it also reduces your opponent's attacks, but still, people have to feel like it matters.

A bonus on one save DC needs to be big, at least +4. A bonus on every save DC is already big at +1, unless it doesn't stack with other fairly common bonuses, though such things tend to just make every option feel bad.

Which, on stacking issues, like, you don't need to care if bonuses stack up huge on rare rolls, just let it work if people put the effort in, but need to be quite careful with them on "every round of combat" type rolls, for things that might break the game if they nearly always worked.

Advancement Rates: If we kept the D&D 3e idea of a given creature doubling in power as it gains 2 CR (or 2 levels or whatever), how much of that power increase should be granted to boosting your RNG? If a level 1 fighter had +6 to-hit, would a level 3 fighter have +9 to-hit? +12 to-hit? Could a good d20 RNG even accomodate this kind of power-doubling without the numbers getting out of hand?

It depends. Again, for something like a fighter that rolls attacks constantly, they notice those +1 steps, and even being +1 over another character gets to be obvious over time. It's not noticeable at all if the enemy AC goes up at the same rate, of course, and you might want a bigger rare bonus attached to it so people can feel better about it early on.

But no matter what your bonus rate, the main problem for a class progression is having numbers that you can use without breaking people's capacity to do basic addition. +9 is OK, +27 is a pain, +53 is crazy talk. There's a reason the older games started at +0.

And you don't always need constant bonus tracks. You could give a class +3 to hit at less regular intervals, as long as they have something else on other levels. Or give different bonuses later on, or whatever. They don't need +1 to hit on the level they get +1 attacks per round, for instance, that already feels good, +2 to hit the next level up instead is feeling good twice, instead of once.

[pragma]

Measuring Bonuses: I have noticed in my playgroup is that bonuses can feel big even if they're mathematically small. [...] Flanking + entangling an opponent might just take your average attacker straight off the RNG if both conditions grant the equivalent of a +3 attack bonus. That feels very wrong to me. Are my instincts wrong here as well?

They have to overcome natural variance, mathematically the effect of the bonus is relative to the square root of the number of dice you roll with it. ...

This is wrong. For one trial, the expectation of success scales linearly with the probability of success, which is 5% * size of the modifier in a d20 RNG. For N trials, the expected number of successes also scales linearly with the probability of success.

The idea that one trial of a save or an initiative roll matters more than a to-hit roll holds water with me. But the amount of significance should probably be measured in rounds to success and rounds to failure, which is a more involved calculation that also doesn't involve any square roots.

[deaddmwalking]

For a 2d10 RNG, you roll doubles 10% of the time. You can crit on any double that hits. If you need an 11+, you crit on double 6s, double 7s, double 8s, double 8s and double 10s. That works out to 5% of the time.

If you wanted fumbles, you could do the same (fumble on double 1, 2, 3, 4 and 5)

Since the conversation moved on, rather than editing, I'm adding an addendum here.

One benefit of 'doubles that hit crits' and doubles that miss fumbles' is that as you become more skilled you're less likely to fumble. On a d20, you're equally likely to roll a 20 or a 1, regardless of whether your modifier is -4 or +39 - the bonus factors in on the confirmation roll, but the odds of a master swordsman getting a critical doesn't improve because he gets better at swording automatically.

If you can hit on a 3+, you can hit on everything except double 1s, so instead of 5% chance of critical, you have a 9% chance of critical; the chance of fumble drops from 5% to 1%.

[DenizenKane]

For a 2d10 RNG, you roll doubles 10% of the time. You can crit on any double that hits. If you need an 11+, you crit on double 6s, double 7s, double 8s, double 8s and double 10s. That works out to 5% of the time.

If you wanted fumbles, you could do the same (fumble on double 1, 2, 3, 4 and 5)

Since the conversation moved on, rather than editing, I'm adding an addendum here.

One benefit of 'doubles that hit crits' and doubles that miss fumbles' is that as you become more skilled you're less likely to fumble. On a d20, you're equally likely to roll a 20 or a 1, regardless of whether your modifier is -4 or +39 - the bonus factors in on the confirmation roll, but the odds of a master swordsman getting a critical doesn't improve because he gets better at swording automatically.

If you can hit on a 3+, you can hit on everything except double 1s, so instead of 5% chance of critical, you have a 9% chance of critical; the chance of fumble drops from 5% to 1%.

I like how your crit rate goes up as accuracy increases, but one of the drawbacks I can see is that it's hard to have variable crit rates, but maybe that's an unnecessary detail.

[Foxwarrior]

You can have weapons that crit when the dice are just adjacent, like a roll of 5 and 6. I like how doubles feels numerologically significant, possibly even more than rolling the absolute max or minimum on a single die does.

[Iduno]

For a 2d10 RNG, you roll doubles 10% of the time. You can crit on any double that hits. If you need an 11+, you crit on double 6s, double 7s, double 8s, double 8s and double 10s. That works out to 5% of the time.

If you wanted fumbles, you could do the same (fumble on double 1, 2, 3, 4 and 5)

Isn't that the same terrible crap that Games Workshop roleplaying games pull? It's the same as saying you have an unreasonably high 10% chance to crit or crit fail, but also means your skill-ups have higher or lower value for no reason.

Foxwarrior's idea of increasing the amount of crit could work, but making it "similar numbers" instead of 1-X and Y-100 not only triples it to an insane 30%, but also increases the amount of time to figure out the result of rolling dice.

Don't do those things unless your goal is "make the players hate me as much as I hate them, which is a lot, because I work for Games Workshop."

[tussock]

Measuring Bonuses: I have noticed in my playgroup is that bonuses can feel big even if they're mathematically small. [...] Flanking + entangling an opponent might just take your average attacker straight off the RNG if both conditions grant the equivalent of a +3 attack bonus. That feels very wrong to me. Are my instincts wrong here as well?

They have to overcome natural variance, mathematically the effect of the bonus is relative to the square root of the number of dice you roll with it. ...

This is wrong. For one trial, the expectation of success scales linearly with the probability of success, which is 5% * size of the modifier in a d20 RNG. For N trials, the expected number of successes also scales linearly with the probability of success.

The idea that one trial of a save or an initiative roll matters more than a to-hit roll holds water with me. But the amount of significance should probably be measured in rounds to success and rounds to failure, which is a more involved calculation that also doesn't involve any square roots.

What you're measuring is the odds the PCs get 5 hits before the monsters get 6 hits, or something similar turning up to spoil the mood, that's the thing people notice. The time it takes you to get 5 hits is basically a bell curve, as is the time it takes the monsters to get 6 hits. Happily, the result of comparing a bell curve to a bell curve is a bell curve, and with dice easily enough described as a single pile of both side's dice.

You just need to move the probability involving a PC going down off the rare events and toward the vanishingly rare events, or even into the crazy outliers, so people notice the steps over time. The standard deviation of your bell curve is SQRT(x(y+1)(y-1)/12) for xdy. That's the probability you want to noticeably shift, the percent of time the monsters gank someone, and how far you have to shift it to do that is proportional to the standard deviation. Because it's roughly a bell curve.

Which is to say, you're right about how proportionately far you move an average by adding the same number, but not about how far you need to move it to matter to players of the game.

--

Simply, if you roll 1d20 vs 1d20, +5 on one contest wins 3:1.
But if you roll first to 8 successes, +5 is near 50:1, more dice, smaller bonus needed, and why PCs nearly always win fights with what may seem like small margins.

[deaddmwalking]

For a 2d10 RNG, you roll doubles 10% of the time. You can crit on any double that hits. If you need an 11+, you crit on double 6s, double 7s, double 8s, double 8s and double 10s. That works out to 5% of the time.

If you wanted fumbles, you could do the same (fumble on double 1, 2, 3, 4 and 5)

Isn't that the same terrible crap that Games Workshop roleplaying games pull? It's the same as saying you have an unreasonably high 10% chance to crit or crit fail, but also means your skill-ups have higher or lower value for no reason.

I don't think so.

If you're rolling 2d10, your range is 2 (double 1s) to 20 (double 10s). Since there is only one combination of dice for each result, you have a 1% chance of the most extreme results.

In 3.x, your chance of a critical is a function both of how often you roll a 'threat' and how likely you are to confirm. You're always 5% likely to roll a 20. If you need an 11+ to confirm, your chance of a critical is 2.5% (5% chance to roll a threat and 50% chance to confirm). If you're threatening on a 17-20 and you confirm on a 2+, you have an 18% chance of rolling a critical. Clearly there is a range of what is acceptable before you get to 'unreasonably high'.

In 2d10, you have a 2% chance of rolling a 19 and a 3% chance of rolling an 18; if you crit on an 18-20, you have a 6% chance of rolling a critical. You could also require a confirmation roll (a la 3.x) so 18-20 w/ confirmation. The issue is that as you add additional values to the threat range, the incremental benefit is higher. If you threaten on a 17, that adds 4% (10% total); if you threaten on a 16, that adds another 5% (15% total). So 18-20 is like 3.x natural 20 only; 17-20 is like 3.x 19-20; and 16-20 is like 3.x 18-20. You can map over the threat ranges relatively easily and get something that works out pretty closely, but there's also a cost - you're adding up two numbers before you add your bonus. If you roll a 9/8, you need to confirm that that is in the threat range before you add the attack bonus. If I had a +9 bonus, I'd probably prefer to add 9+9+8 - for me that's just a hair faster than 9+8+9.

Adding doubles is easy for anyone that's learned their x2 multiplication table. The way the brain works, you can recognize it before doing the actual addition. We know that there is a 6% chance of rolling an 18 or better (3% chance of an 18 [10+8; 9+9; 8+10], but we then have to recognize each of those combinations as potential threat. If you use doubles instead of a range of results, each possibility adds +1%. The highest it can ever be is 10% (all possible double results) - there's nothing saying 'doubles always hits, even if it is a very low roll, unless you choose to. But assuming you like ~5% crit range, and you like people who are better at attacking critting more often, you can pretty easily make a roll that 'if you roll doubles and you hit, it's a critical'. If the rogue hits on a 16+ and the warrior hits on a 12+, the rogue has a 3% chance of a critical and the warrior has a 5% chance. You've eliminated the confirmation roll as a requirement and more skilled attackers are rolling criticals more often.

Personally, I don't think you need 'doubles fumble on a miss'. If you use that roll, the rogue has a 7% chance of a fumble; the warrior has a 5% chance. The higher the roll needed to hit, the more likely they are to suffer a fumble. If you think those odds are too high, you just make snake eyes the only fumble result (1%); the rogue just misses on a double 7.

Since you're making the dice rolling take longer (adding two dice always takes longer than taking a value from a single die), you want to look for ways to save time in other places. Removing re-rolls (like confirmations) is one way to do that.