What should the odds be?

[Douglas]

I'm thinking about trying to design my own system, and I'm getting a bit into mathematical analysis of bonus numbers and dice rolling and how they interact to produce probabilities. The big question I'm looking at is what magnitude of bonus in mechanical terms represents what amount of skill in fluff terms. I can fairly easily go from desired probabilities to bonus numbers, but I'm not sure what the desired probabilities should be. Thus, I come here to solicit opinions on the matter.

With fluff skills ranging from completely untrained, through basic competence, up to worldwide legends, what should the success probability distribution look like for tasks ranging from the easiest that even requires a roll, through tricky but routine, up to mind-boggling stunts even bards doubt are real? Feel free to insert as many gradations in between as you want to comment on.

When people of different skill levels face off in opposition for something simple and variable enough to be a single opposed roll, what probability of victory should the more skilled person have for each degree of superiority?

How much of a difference, in each direction, should having an advantage such as catching someone off guard make?

[toapat]

well, it depends on exactly how you want the dicerolls to map out, and what percentage you want a +1 roll modifier to make

in D20, there is a flat distribution, and the +1 modifier increases success rate by 5%

on 3d6, the distribution is a bellcurve, but a +1 modifier has a 7.5% distortion on the chance of success

Degree by which +1 to roll changes chance to succeed

1d4: 25%
1d6: 16.7%
2d4: 14.3%
3d4: 10%
2d6: 9%
3d6: 7.5%
d20: 5%
5d6: 4%

[Douglas]

I'm not asking for help with the math, or how much a +1 should be. My question is entirely about mapping from fluff to probabilities. Period. I've got the math covered from there, and the basic nature of the system is more like Shadowrun's core mechanic than d20's anyway.

The kind of feedback I'm looking for is more of the form "someone untrained should have an X percent chance of success on Y near-trivial task that barely even requires a roll", with various levels of skill and task difficulty substituted in.

[toapat]

I'm not asking for help with the math, or how much a +1 should be. My question is entirely about mapping from fluff to probabilities. Period. I've got the math covered from there, and the basic nature of the system is more like Shadowrun's core mechanic than d20's anyway.

The kind of feedback I'm looking for is more of the form "someone untrained should have an X percent chance of success on Y near-trivial task that barely even requires a roll", with various levels of skill and task difficulty substituted in.

how many "eras" do you want the player to experience (3.5 is divided into 4 eras depending on power level, with epic content adding a 5th)
is it a level based system (whats the standard cap)
what is the base resolution mechanic

i listed out the dice magnitude changes because its highly relevant
Bellcurve vs equal distribution is also relevant to the design process.

[Douglas]

I'm not seeing how any of that is relevant to the question I'm asking. To the math and design of the system, yes very much so, but I'm asking for advice on the probabilities that the system should output. How it produces those outputs is an entirely different matter and not what I'm asking about.

Assume you have a black box that implements a completely unknown game mechanics system. It takes pure fluff descriptions of a task and a skill level as input and produces "success" or "failure", with some random element involved. You are trying to judge whether the unknown mechanics inside are a reasonable ruleset you would feel comfortable playing with in a simulationist style game. You feed it the same combination of task and skill level 100 times. How many times, for whatever combination of task and skill you picked, would it need to output "success" to get a favorable verdict from you? How does this vary for different combinations of task and skill (again, purely in fluff terms), and also for head-to-head opposed skill tasks?

[NichG]

The point about 'different eras' is important, so it really does need to be addressed. Different game systems are used to model different types of worlds. Something like Exalted or Nobilis is attempting to go for a different feel than, say, 1ed D&D or Call of Cthulhu, and all of those are trying for something different than 3.5ed D&D.

In Exalted, if even a starting character (minimum level of explicit investment in skills allowed by the system) ever failed at what would be considered an everyday task, it would break the concept that the characters are basically next-to-gods. That would be even worse in something like Nobilis. But in Call of Cthulhu, it'd make perfect sense.

Similarly, some systems really want to amp up uncertainty, so success rates might go from 20% to 80% over the span of a character's possible level of ability - again, something like Call of Cthulhu, where the system is thematically designed around it being hard to control what happens. But other systems may want to have very sharp divisions between 'you can do this' and 'you can't do this' so success rates go from 0% to 100% over a few 'points' of skill - something like, say, Nobilis, where the focus is more on what you can do and what you choose to do than whether or not you suffer a random setback (in Nobilis, for example, the question isn't 'do you succeed?' but rather 'do you pay the number of miracle points you need to in order to succeed?', and this gives a very different feel)

So basically there are no universal answers to this question - it depends what you're trying to do.

[Douglas]

Regarding "eras", consider primarily within a single era. Conceptually I'm trying to separate raw power from skill, and trying to figure out how much of a difference skill should make. Take real world human as the target benchmark.

[Eldan]

Well, for a real world human, it really isn't all that hard. I'd say trivial task 99% chance of success, difficult task 33%, impossible task 0% of the time?

There's a lot of things you simply can't do right if you don't know how. And there's things that you simply don't fail at without weird circumstances.

I can speak a gramatically correct sentence say, 99% of the time without trying. My mother does it only by accident.

[NichG]

Hm, so lets see... For a real world human, I'd use something along the lines of a 2-factor system to figure out what the rates should look like. One factor is 'will they ever succeed at this task?', the other is 'do they succeed within the next X timeframe?'. For example, I will never be able to successfully perform a successful appendectomy without the appropriate medical information/training even if I were to try a hundred times. But if I have that training, my failure rate is actually going to be relatively small - 5% or something like that. This can be explained by, say, 'med school' being a perk that gives a flat +10 to the skill or something, while individual-level training has a +1 granularity. But its something to keep in mind.

On the other hand, if I want to 'juggle three balls for one minute straight without dropping them' I might pull it off without training if I try over and over and over again, but my individual attempt success rate will be very low.

So this tells you something about the system - you want there to be a certain skill gap where the probability of success becomes 0%, and a certain skill excess where the probability of success becomes close to 100%. You also want to take into account that for tasks that can be repeated, the statistics will work differently than one-shot things, because a repeated task without consequences means there's simply a time tradeoff for success. But lets say we standardize over these factors and say that any given skill check can only be done once, and if failed it says you will forever fail at that particular usage for, say, the rest of that session (which in real life means, until an unrelated circumstance comes up or perhaps a few months pass).


Let me define things in terms of 'task level' and 'skill level'. The system isn't linear, in the sense that a gap of 5 points will mean something different at low levels of skill than high levels of skill. I think the most realistic example would be that the gap closes, so a difference of 1 point is a much bigger deal when the skill/task is already around 10, than at low levels. I would also say that the cost to advance to the next point should grow exponentially, and that 'formal training' is a way to buy your first 5 points worth in a single go at a reduced cost (so someone with 'formal medical training' would go straight to Medicine 5 and would pay less for it than someone figuring it out by themselves, but they'd pay the same cost to go to Medicine 6 as the guy who figured it out on his own). Also, I'm not ruling out half-ranks or anything like that.

People by default have '0' in a skill, and if their skill matches the task difficulty their chance of success is 100% minus the task difficulty; someone who can take their time on it and be very careful only subtracts 1/5th of the task difficulty from 100%.

Task level 0 is a trivial task - all people should automatically succeed at this task unless they have a debilitation of some form. At this level, each point of debilitation lowers the chance of success by 15%. Speaking a grammatically correct sentence is an example of this task - someone with a speech impediment might only do so 80% or 60% or 40% of the time for a 1 point, 2 point, or 3 point debilitation.

Task level 2 is a difficult task for an unskilled person - each point of deficit lowers the chance of success by 25%. An example of this would be something like a middle school mathematics test, where a good student will score in the 90s, the average student will lag behind the test difficulty by one skill rank and get around a 75%, and a completely untrained student might still be able to figure out a half or a third of the questions.

Task level 5 is a 'professional task', where someone without professional training would be expected to be incapable of doing it, but a professional would be expected to do it successfully almost all of the time. Most routine surgical operations would go here, for example. Each point of skill deficit subtracts 40%, so if you have 2 points of skill in Medicine you simply will not be able to perform a successful surgery of this sort. Someone who takes their time only screws this up 1% of the time, otherwise they have a 5% screwup rate.

At this point, the cost of skill deficit probably should stop increasing.

Task level 6 is 'expert-level'. Someone who has professional training (skill level 5) but no extensive experience has a roughly 60/40 chance of pulling this off if they take their time. Someone who has worked in a profession for ~10 years will be at this level of skill.

Task level 7 is 'master-level', corresponding to something that someone with 30 years of experience in strenuous environments or a natural prodigy at the skill can do reliably. A sensitive surgery that has a high chance of killing the patient would be at this level.

Task level 8 is 'limits of reliable human ability' - things the top humans in history every time, but mere masters would botch occasionally.

Task levels 9 and 10 would be 'record-holding events'. These are the pinnacles of achievement of the humans who were the best in their fields throughout history. Task level 10 means that a true master of the craft would be incapable of it, and a natural prodigy who has studied/trained/worked for 30 years would achieve this level of task only 10% of the time.

[Grod_The_Giant]

Off the top of my head, I'd say that for a level-appropriate task, you should have around a 75% chance of success if you're a specialist (ie, a fighter stabbing someone), a 50% chance of success if your build doesn't care much one way or the other (a fighter picking a lock), and a 33% chance of success if the task is inappropriate (a wizard stabbing someone). Maybe +20% on a particularly easy task and -20% on a particularly difficult one.

[erikun]

I think you might want to also consider at what point someone should reasonably stop having a chance at failure at something.

For example, I can't lift 200 pounds over my head. I just can't do it. I'm not strong enough. There are people, however, who can. They might not be able to do so constantly, especially if they're running around in a fight, but they could possibly do so. There is also a point where someone who is strong enough (maybe not humanly strong) to be able to lift 200 pounds without any difficulty, and has no reasonable way to fail.

As another example, a small child would not be able to lift a 50 pound box. I can reasonably do so, although it would be difficult to do it while running around. (I'm out of shape.) There are people who can pick up and run around with a 50 pound box on their shoulders on in a backpack with little problem, though, and for extended periods of time.

I'm mentioning this because a lot of these new system projects like to model themselves off of D&D3e, and a lot of the focus on D&D3e is on the upper bound of skill rolls with little attention to the lower bound. If someone with STR 6 and someone with STR 20 are trying to break down a DC 25 door, then there is a non-zero chance that the STR 6 person will succeed where the STR 20 person failed. Is this a problem? It really depends on the type and tone of the system you are making. If you want a less realistic and more amusing system, then d20+STR works just fine in this case. If you are looking for a more serious system, or if the weaker person breaking down the door just breaks immersion, then using d10+STR would be a better idea. Or change what the range of values means, so perhaps STR 6 means 30% stronger than average.

[CarpeGuitarrem]

What does "failure" mean in your game?

A game where failure means "instead of succeeding, you have something bad happen to you" is going to require a vastly different probability setup than a game where failure means "you mostly get what you want, but there's a big catch".

And that doesn't necessarily tell you what the odds are--but you have to know "okay, so they're going to be failing this % of the time, and failure means X". If failure means "you whiff, you can't do it", you probably want your odds to be around 75%. If your game rests on a notion of super-competent characters, that is. (If not, I recommend introducing something to the failure state, because having "you can't do that" be the result of 2/3 or more of your rolls is not terribly enjoyable.)

On the other hand, if failure means "something horrible happens", you might not want to give the players a lot of opportunity to fail, because they can only put up with so much bad stuff. (A great example of this is Dread: you almost never fail, because when you fail...you die.)

You might also consider how players will feel about failing frequently vs. failing rarely.

[NichG]

What does "failure" mean in your game?

A game where failure means "instead of succeeding, you have something bad happen to you" is going to require a vastly different probability setup than a game where failure means "you mostly get what you want, but there's a big catch".

And that doesn't necessarily tell you what the odds are--but you have to know "okay, so they're going to be failing this % of the time, and failure means X". If failure means "you whiff, you can't do it", you probably want your odds to be around 75%. If your game rests on a notion of super-competent characters, that is. (If not, I recommend introducing something to the failure state, because having "you can't do that" be the result of 2/3 or more of your rolls is not terribly enjoyable.)

On the other hand, if failure means "something horrible happens", you might not want to give the players a lot of opportunity to fail, because they can only put up with so much bad stuff. (A great example of this is Dread: you almost never fail, because when you fail...you die.)

You might also consider how players will feel about failing frequently vs. failing rarely.

I definitely agree that the failure mode matters a lot in figuring out the ideal percentages. Unfortunately, for a 'realistic' system as he's trying to produce, the failure mode probably has to be task-dependent, which makes it very hard to generalize.

I don't agree that 75% success rate is 'a game about super-competent characters' though, at least not in terms of an inherent rate of success/failure. He asked for something realistic for a real-life human, and real-life humans actually tend to have much lower failure rates than 25% at tasks which untrained real-life humans can't complete at all. So I don't think 75% success rate at something you're supposed to be good at is super-competent, I'd actually say its pretty incompetent. Someone who is good at math could realistically reliably score ~95% on appropriate math tests, and that's something very finnicky where its easy to make lazy mistakes.

That said, I have the feeling you were actually talking about something else (please correct me if I'm wrong) - the success or failure rate such that performing the check actually matters in a narrative sense. In this sense, I agree that around 75% success rate its 'worth rolling' for something, and you generally might as well not roll for something with a higher success rate and just assume 'they succeed'. But in the context of the OP's post, I'd say that's a statement about the difficulty of a task at which uncertainty appears - a plumber trying to fix some guy's clogged sink probably doesn't need a 25% failure rate and also probably shouldn't bother rolling, its routine so they just succeed; a plumber-burglar trying to fetch a diamond ring from the U-bend of a sink without waking up the owners upstairs should probably have to roll.

[Douglas]

Thank you, that's some very useful feedback so far. May take me a while to translate it into my dice rolling system.

All right, some details about the mechanics of the system, which may fuel some extra brainstorming:
The core mechanic is rolling a bunch of d6s and counting the number of 4s and higher (or flipping coins, or any other 50-50 generator), and comparing the total to a target number. Let's call a 4 or higher a "hit", borrowing the term from Shadowrun.

I was originally thinking in terms of skill (and levels, but that's another matter) only providing a varying number of dice to roll, but it occurred to me while thinking about the point of some things being more threshold-based with little randomness involved that skill modifying the number of hits required would be a good way of handling that. There is a lot of variation in whether luck is involved or not, though, so I also came up with the idea of assigning a "luck factor" to tasks, which would determine the size of extra batches of dice that get rolled both for and against the character. For example, a task with luck factor 4 would have 4 blue dice rolled that add to the character's result and 4 red dice rolled that subtract from it.

So, tasks have a target number and a luck factor. Characters have a number of dice and a number of automatic extra (or penalty) hits that together represent skill.

For some examples of how this might work, brain surgery would have a very high target number and low luck factor, getting a hole in one in golf would have high target number and high luck factor, walking across a normal room would have a negative target number and low or 0 luck factor (so only someone with a penalty would ever fail), and I'm having trouble thinking of anything but gambling that might have a low target number and high luck factor.

I may have to do a lot of fiddling with specific numbers and such to get straight character vs task numbers to come up right, but the math for that is pretty straightforward. What gets more complicated is character vs character, where both sides are rolling dice and trying to score higher than the other.

One interesting result I've noticed from the math I've done for opposed tests is that, assuming equal automatic hits, winning ties is worth about 2 dice. If you have 4 dice and win ties, then you will win exactly 50% of the time against someone who rolls 6 dice (yes, it really is exact like that). X dice and winning ties goes 50-50 against X+2 dice and losing ties (except when X is 1, as I recall). For differences between opponents other than 2 dice it gets more approximate, but in general exchanging "win ties" for "roll 2 more dice" does not significantly change the odds. This gave me the intriguing idea of having an "advantage" mechanic that determines who wins ties, and possibly also tying it to things like sneak attack eligibility, though the net impact on probability goes down as total number of dice increases and I'm not sure how much of a difference I want it to make anyway.

And, of course, there's the issue of whether more power (distinct from skill) should grant more automatic hits, more dice, or both, and how much of each, and whether and how this depends on the specific activity involved. Another thing I noticed is that the effect of each single die on the probabilities goes down as the number of dice increases, so I could have abilities that give bonus dice scaling with level, and if I set the scaling rate right it actually would maintain the same probability gap between specialized and not, rather than diverging into 3.5 D&D's high level dichotomy of people who can't fail and people who can't succeed. There are some things where that dichotomy is appropriate, of course, but 3.5 made it far too universal.

[NichG]

Success-counting systems tend to have an intrinsic high variance because they use a Poisson distribution rather than a Gaussian, so it may simply not be a good match for what you're trying to achieve.

For example, lets say you have something with a 'high target number, zero luck factor' like your example of brain surgery. The number of dice I need to have to have a nearly 100% chance of success at the task is much higher than the target number associated with the task - in other words, there'll be a large range of skill point values between a 50% success rate and a 95% success rate in terms of absolute scale. Basically, I need to double my dice pool at 50% to go to 95%.

If the target number of successes is 4, I need 7 dice to have a 50/50 shot at it, and 13 dice to have a 95% shot at it. If the target number is 6, I need 11 dice for a 50/50 shot and 18 dice for a 95% shot. So if you want to model high success rates and high , the number of dice needed gets large quickly.

If on the other hand you're using a fixed dice pool but modifying the number of successes needed based on the character's skill, then I think that does tend to work better (or at least in a more controlled fashion). Lets say the fixed pool size is 5 dice. Then the success rates as a function of residual number of successes needed go like:

0: 100%
1: 97%
2: 81%
3: 50%
4: 18%
5: 3%
6: 0%

So roughly speaking, the number of dice in the default pool corresponds to the range of skill values for which success is still possible. More dice smooths out the curve, fewer dice makes it decay more sharply. So that might work for what you want.

As far as 'power' goes, thats kind of an ill-defined concept I think. Should someone really smart be able to just figure out how to do brain surgery without training? That's up to you and how you want your game to feel thematically. If you want a very gritty game, 'power' sets a limit on the amount of skill you can bring to bear, so it purely determines skill caps - e.g., if you have shaky hands, it doesn't matter how medically trained you are, you probably shouldn't be doing surgery. An intermediate solution would be to say that 'power' levels are rare and are from -2 for 'extremely incapable' to +2 'demigod-level', and modify the number you need on the d6 to count something as a hit. So if you have demigod-like power then your dice are hits from 2 to 6, if you're supernaturally able its 3 to 6, if you're normal its 4 to 6, if you're somehow debilitated in that task its 5 to 6, and if you're in very bad shape its 'only 6's are hits'.

[Veklim]

Hrm, with regards to how one deals with automatic success for trivial tasks, and impossible success for those beyond the character's ability, I'd have thought a tier system for tasks would help, much akin to the one discussed already for skills themselves (all numbers are arbitrary).

Any task tiered 2 or more higher than your skill tier is impossible for you, any 2 or more lower are guaranteed successes. One tier out garners you a 20% bonus/penalty to the attempt. So for any given skill action attempted, you have both a DC (the number you need roll) and a tier (your limiting factor). On top of that, it gives you ways of making skill checks easier or harder without changing the DC of the task, by simply increasing the tier of the skill by one or maybe even 2 tiers depending upon conditions, so something you can often succeed in doing under optimal conditions (one tier above your skill level) could be made one tier lower by granting assistance from another reasonably skilled person (removing the penalty to fail from a higher tier task), or made impossible when attempted under unfavourable or outright dangerous conditions (since you cannot succeed a task 2 tiers higher than your skill level.

[NichG]

This is sort of getting off of the topic of dice systems/odds, but one system I like to use to model 'you can have a high rate of reliability on something but it should still feel difficult' is to do something like a stamina bidding system. So you have a certain stamina pool that refreshes when you get a good night's sleep, and if you fail a roll you can pay points out of the stamina pool based on how much you failed by in order to make up the difference.

A system like that creates multiple classes of task:

- Guaranteed free success
- Success can be guaranteed, but it may cost stamina
- Guaranteed success, but can only succeed if stamina is spent*
- Can spend stamina for a chance to succeed
- Guaranteed failure

*: If the stamina pool is small this becomes 'Chance of success, but can spend stamina to improve the odds'

You can do other elaborations on something like this, like limiting the maximum amount of stamina to spend on a single roll. World of Darkness' Willpower system has a bit of this character.