A Question of Scale

[TooManySecrets]

Imagine that you were a normal human being (this shouldn't be too hard for most of you). Now, imagine that a super-strong deity punched you. You'd probably be dead - your body comically imitating a crash test dummy. Now, imagine that a slightly strong super-strong deity punched you. Pretty much the same thing, huh? The difference in strength between them is immaterial to you. But if the two deities got into an arm-wrestling match, it would matter.

Violent example aside, what I'm sort of groping towards is the concept of scale and how slight differences in ability don't matter all that much if one isn't operating at the same scale.

There's also something else: the closer two people are in ability/power/whatever, the more random the outcome is; contrariwise, the farther away two people are in ability, the less random the outcome is. It also seems that the change in probability is non-linear (diminishing returns) when compared to the increase in ability.

So, what does all this blabber have to do with RPGs? I think something like this (by the by, this definitely isn't for tabletop, unless you are the type of person who brings a calculator with probability and random number generator to your tabletop RPGs):

The base chance of success for an action is 50%. Increase (or decrease) the chance of success by (50%*(1-(1/(D*F+1)), where D is the absolute value of the difference in ability and F is some factor [F is basically used to make the function's curve appropriate for both a game like Shadowrun, which has stats in the 1 to 9 range, and a game like D&D, which has stats in the 1 to 30+ range].


Examples:
(F = 1, for simplicity; Also to note: you only need to determine who succeeds once, but I included both probability for completeness)

Dude A and Dude B are getting into an arm-wrestling match. They both have 6 Strength, so they both have a 50% chance of succeeding.

Dude A starts working out and increases his Strength to 8. He now has a 50%+(50%*(1-(1/2+1))[50%+16%] = 66% chance of success, while Dude B has a 34% chance of success.

Dude B won't let this stand, so he makes a deal with Dark Powers^TM which grant him a 20 Strength. Now the difference is 12. Dude B now has a 50%+(50%*(1-(1/12+1)))[50%+46%] = 96% chance of success, while Dude A only has a 4% chance of success.

Dude A calls in some favors and makes a deal with Light Powers^TM who support him in his battle against the Dark. His Strength is now 20, as well, and we're back at 0 difference in ability, which means both have a 50% chance of success.

Small conversion example: Hero A is a D&D character with +12 to Jump. The DM determines that the jump he needs to make is a DC 25. The difference is 3 (D&D normalizes at 10 so +12 is 22 for our purposes), so Hero A has a 50%-(50%*(1-(1/3+1)))[50%-37.5%] = 12% chance of success. (Normal chance would be 35%, so you can see that I need to modify that F value)


Okay, now for all three of you who were able to follow my complicated ramblings, do you cats and kitties have any thoughts about it? I know that it's complicated, but I'm not planning on using a system like this for tabletop play, but for forumplay, which gets rid of a bit of that problem (doesn't really matter if it takes you a minute or two to figure out whether you've hit if most people only post every 30 minutes or so). One of the big problems that I see right now is that the initial change in chance is extremely steep. If anybody knows a function that is asymptote to 0 (or 1) and is a bit gentler, that would be greater.

[Tyndmyr]

Where are you getting your assumptions from regarding the rate of change in randomness?

BTW, the above all seems needlessly complicated for a relatively simple idea.

[TooManySecrets]

Where are you getting your assumptions from regarding the rate of change in randomness?

Which specifically? That the rate of change approaches zero?

My reasoning goes like this: that as skill improves, the chance of failure approaches zero. It's impossible to be absolutely certain that you'll succeed at a task no matter how good you are, after all. Therefore, that implies that the rate of change has to approach zero as well.

BTW, the above all seems needlessly complicated for a relatively simple idea.

Yeah, that's pretty much how I roll.

But here's the thing: I don't know of any other way. A simple 1dX+Y roll obviously doesn't have the right curve. Dice pools come relatively close, but have the problem that they also increase the range e.g. someone with a skill of 6 can get anywhere between 0 and 6 successes.

And I don't think that it's all that complicated. You fill in one number into an equation and then do one roll.

EDIT: But then again, that's why I'm posting. If you see a way of implementing the idea in a simpler way, I'd love to hear it. One of my great failings is over-complication.

[Siosilvar]

So, what does all this blabber have to do with RPGs? I think something like this (by the by, this definitely isn't for tabletop, unless you are the type of person who brings a calculator with probability and random number generator to your tabletop RPGs):

A table and d% will suffice.

Sample table for F=1/5:
{table=head]Difference|Success%
-9|18
-8|19
-7|21
-6|23
-5|25
-4|28
-3|31
-2|36
-1|42
0|50
1|58
2|64
3|69
4|72
5|75
6|77
7|79
8|81
9|82
[/table]

[Harperfan7]

I remember reading somewhere that when arm wrestling, the character with the highest strength automatically wins, but if they have the same str score they roll 1d20, highest wins.

I would say they get their size modifiers to the 1d20 check assuming same strength, and if someone has a modifier to str checks, they should get those too.

[Tyndmyr]

Which specifically? That the rate of change approaches zero?

My reasoning goes like this: that as skill improves, the chance of failure approaches zero. It's impossible to be absolutely certain that you'll succeed at a task no matter how good you are, after all. Therefore, that implies that the rate of change has to approach zero as well.

So you want basically a bell curve. How steep of one?

You can use 2d10 for a moderately steep curve, 3d6 for a steeper curve, etc. In general, more dice result in a curve with a higher bias toward averages.

[TooManySecrets]

So you want basically a bell curve. How steep of one?

You can use 2d10 for a moderately steep curve, 3d6 for a steeper curve, etc. In general, more dice result in a curve with a higher bias toward averages.

Er, not really a bell curve. First, bell curve does nothing about range. While 3d6 has a relatively strong bias towards 10.5, it still ranges from 3 to 18. Which has also been the problem with dice pools, at least for me. Pet peeve, maybe.

Second, I really need to go write up a probability tester (I was going to make an actual second point, but I couldn't figure out whether it was valid or not). Be right back.

[Rixx]

I have it on good authority that 2d100/2-1 is a great bell curve generator, and good to use in tabletops all the time, forever.

[Tyndmyr]

Er, not really a bell curve. First, bell curve does nothing about range. While 3d6 has a relatively strong bias towards 10.5, it still ranges from 3 to 18. Which has also been the problem with dice pools, at least for me. Pet peeve, maybe.

Second, I really need to go write up a probability tester (I was going to make an actual second point, but I couldn't figure out whether it was valid or not). Be right back.

So, what range do you want, and what curve do you want?

Use the appropriate number of dice for the curve, and the appropriate size of dice to get the desired range.

[TooManySecrets]

I have it on good authority that 2d100/2-1 is a great bell curve generator, and good to use in tabletops all the time, forever.

That good authority wouldn't happen to be FATAL, would it? (They use 4d100/2-1, which was changed to 10d100/5-1 in the later edition)

So, what range do you want, and what curve do you want?

Use the appropriate number of dice for the curve, and the appropriate size of dice to get the desired range.

Okay, so I'm back from some programming.

Conclusion that I've come to is that XdY+Z would be serviceable for a game (it's possible, obviously, to get to 100% so it isn't an asymptote, but that's relatively minor) and close to what I would like, the thing is is that instead of using dice to simulate a probability curve, why shouldn't I just use a function for the probability curve? Plus, as I said, I'm not trying to run this in meatspace (though Siosilvar's table shows that it would be pretty simple to do), so I can be assured that everyone has a calculator.

Plus, tell me, what's the probability that you'll get a 20 or higher with 4d6+8? I don't know about you, but most people I know wouldn't be able to figure that out quickly (I know where to start looking - Bayesian probability and all that - but not off the top of my head). Using the actual function for the probability curve means that it's instantaneous to see the chance of success since it's right there. Now, this might not be important if the DM is hiding target numbers, but it is when you're trying to compare your character with known values ("What's the chance I'll hit an orc? How does that change if I improve my attack by +3? Now that I know that, would I rather do this other thing instead?").

[zoobob9]

dude, you are a genius! i checked it out, using my friend's dragonborn fighter and an npc goblin, he was freaked out when the goblin won.

[TooManySecrets]

Um, what? Are you sure you posted in the right thread?

[PhoenixRivers]

He describes a Bell curve. It's correct. It's shown that at the more extreme ends of probability, odds drop.

It's also been documented for over a hundred years. So it's not exactly new news.

[TooManySecrets]

Nothing new under the sun, eh?

I didn't realize that a bell curve gave diminishing returns. It's not exactly the probability curve I wanted but that's relatively minor.

I think the question is why don't more people use a bell curve i.e. multiple dice, add 'em together. Question of time and expediency?

[bosssmiley]

What Price Glory? Grappling Rules: contested 2d6+bonuses roll.

There really is nothing new...

[TooManySecrets]

Thanks for the link. The blog looks interesting. You don't see many people still playing OD&D, though I'm sure if I spent some time searching for it, I'd find it pretty easily (yay internet!).

But I wasn't implying that nobody uses XdY dice (heck, UA lists that as a variant for D&D, along with what changes would have to be made to the game and additional spells). I was just wondering more people don't use it. In my experience, most games seem to be a straight 1dY plus bonus roll.

[sonofzeal]

I have it on good authority that 2d100/2-1 is a great bell curve generator, and good to use in tabletops all the time, forever.

Not a bell curve at all. Actually, if you graph it you'll get a pyramid shape. You need three dice to get something approximating a Normal curve.

[Demented]

But I wasn't implying that nobody uses XdY dice (heck, UA lists that as a variant for D&D, along with what changes would have to be made to the game and additional spells). I was just wondering more people don't use it. In my experience, most games seem to be a straight 1dY plus bonus roll.

Ease of play takes priority over accuracy, and boardgames with dice are generally intended to be played in meatspace. Also, many things are deterministic rather than probabilistic, which makes them dull and unexciting. Randomness is often injected to give players a sense of risk/reward, rather than to achieve a genuine level of probabilistic behavior. (The majority of macroscopic events are deterministic, not probabilistic.)

A bell curve reduces unpredictability, which is often the opposite desired effect of including a die roll in the first place.
_ _ _ _ _ _ _

I believe you can roughly achieve an asymptote with dice by subsequent die rolls if you get a minimum or maximum result. (Congratulations on rolling 1, 1, 1, and then 4 with your dice. Something bad on the order of 0.3% probability occurs!)

[Tyndmyr]

But I wasn't implying that nobody uses XdY dice (heck, UA lists that as a variant for D&D, along with what changes would have to be made to the game and additional spells). I was just wondering more people don't use it. In my experience, most games seem to be a straight 1dY plus bonus roll.

Why? Not a lot of people think about the mechanical side of D&D in detail...at least not in this sort of meta sense.

I've got a friend that's designing an RPG, and he's enamored with the idea of pure D6s. Nothing wrong with that, but he's rolling five or six at a time for things like attack rolls. Obviously, crits and such are ridiculously rare, and near-average results are incredibly common. The math based reasoning just wasn't important to him.

It probably wont change until someone designs an RPG based on this sort of system that is wildly popular for other reasons.


I do advise against anything involving a calculator. This brings to mind the Hero system, which makes me want to stab my own eyes out. See, the more math-heavy the game feels, the more distracting that aspect is. Sure...the computer doesn't care if it's using a calculator for a given odds distribution or "rolling dice", but players are generally not fond of entering numbers into a formula.

[TooManySecrets]

Ease of play takes priority over accuracy,

I agree in most respects, but then you get something like ragdoll physics in a game where the dead body starts spastically flailing around and the entire sense of immersion goes zip-a-dee-doo-dah. (And it is a ease of play issue that makes them design video game physics systems the way they do. The more accurate it is, the bigger the slow down.)

Also, many things are deterministic rather than probabilistic, which makes them dull and unexciting.

Movies and books don't change any from when you watch them and yet many people find them exciting despite the fact that the outcome is already a foregone conclusion. If things are more reliant are deterministic factors, than I would guess that it would force the players to pay more attention to those factors in order to get the biggest advantage which could increase immersion (if you're concentrating so hard to get every single advantage - and I'm taking about well-integrated, story-focused advantages - than you have to believe in the verisimilitude of the world).

But, I also know the thrill of rolling lucky (or unluckily). I'm not arguing for complete determinism (though I found Amber Diceless to be pretty fun), but just more determinism.

Randomness is often injected to give players a sense of risk/reward, rather than to achieve a genuine level of probabilistic behavior. (The majority of macroscopic events are deterministic, not probabilistic.)

True, though I think it has more to do with the fact that the randomness glosses over all the hundreds of thousands of different factors. Why try to keep an accurate weather simulation program running for your game when a d20 and a table serves pretty much the same purpose.

I believe you can roughly achieve an asymptote with dice by subsequent die rolls if you get a minimum or maximum result. (Congratulations on rolling 1, 1, 1, and then 4 with your dice. Something bad on the order of 0.3% probability occurs!)

I was thinking that, but when you start doing stuff like that, why not just spend the extra second or two using a d100 and a table (like the one posted before)? It's like square-peg-in-round-hole: trying to get XdY dice to act the way I want them to, instead of just doing what I want to do with a quick equation and one or two rolls.

Why? Not a lot of people think about the mechanical side of D&D in detail...at least not in this sort of meta sense.

[...]

It probably wont change until someone designs an RPG based on this sort of system that is wildly popular for other reasons.

Yeah, but (to stereotype) the RPG fans tend to be nerds. And nerds tend to think about stuff like this. It just seems like there should be greater variance in dice systems (I know of 1dX+Y, roll-under, dice pools, and dice poker).

I agree that it won't change until someone has a super-popular RPG that uses a different system. Follow-the-leader and all that.

I do advise against anything involving a calculator. This brings to mind the Hero system, which makes me want to stab my own eyes out. See, the more math-heavy the game feels, the more distracting that aspect is. Sure...the computer doesn't care if it's using a calculator for a given odds distribution or "rolling dice", but players are generally not fond of entering numbers into a formula.

For real life games, very yes (though I often use a calculator just as a form of note taking and for quick calculations). But remember - don't need to use a calculator if you use the above table that's already been posted. The formula can be there so that people who want to know it can (I hate when people hide the numbers so much that people who want to see them can't - Mass Effect was especially painful in that regard in what was otherwise a great game), while those who could care less don't have to pay attention.

See, if we use a pre-done table for an action it's 1) one subtraction to calculate difference and 2) roll d100. No calculator, no adding up multiple die. And if it becomes important to know the outcome of corner cases (the actual chance is 72.06% and you rolled a 72) then you can just roll a d100 again (in the example, you're now rolling against 6). (The F value is something that should be constant throughout the game, if that wasn't clear).


I should also mention that the reason I'm doing it like this is so that I can plop this different method for rolling into any game system with stats, just changing the F value to make the curve right (1 or 2 is a bigger effective difference than 1 or 2 in D&D and far bigger than in Call of Cthulhu).


Jeesh, that was a lot of writing, relatively speaking.

[John Campbell]

If I'm understanding you correctly, the Shadowrun system - the original, with its variable TNs, not this fixed-TN SR4 nonsense - gives you most of what you're looking for for free - by which I mean without performing mathematical manipulations on the numbers. With task difficulty setting the TN and skill or ability setting the number of dice, and number of successes determining not just success or failure, but degree of success, 1s always being failures (but only on that die, not for the test in general) and 6s exploding, and botches being all 1s, you have a system where nothing is ever impossible, or guaranteed, but probability of success can come arbitrarily close to zero, highly skilled fighters don't drop their sword or hit themselves in the face every 30 seconds, and where raising skills or attributes has diminishing returns, but is never quite useless.

(There's another diminishing-returns curve built into the Karma costs of raising skills and attributes, but I'm referring only to the dice system itself here.)

[Tyndmyr]

Yeah, but (to stereotype) the RPG fans tend to be nerds. And nerds tend to think about stuff like this. It just seems like there should be greater variance in dice systems (I know of 1dX+Y, roll-under, dice pools, and dice poker).

Nerds vary wildly. A great deal of D&D nerds aren't math or statistics nerds. Plus, while there is a large pool of fans...there's relatively few designers.

That said, there are more systems. Roll and Keep is used in a few, roll on a table is used pretty often, and then you've got systems without dice. It all depends on exactly what you're looking for.

[sonofzeal]

GURPS, to my memory, relies almost exclusively on 3d6-under-target for most of the important rolls, the way D&D uses 1d20-plus-mod-over-target. This produces a roughly normal distribution with a fairly high degree of central tendency. The majority of your rolls will be between 8 and 13, with few in the 3-7 or 14-18 ranges.

[Sir_Ophiuchus]

The BRP (Basic Roleplaying) system has a resistance table like this.

See here: http://www.chaosium.com/forms/coc_quick_start_color.pdf

It's on page 10.

The Call of Cthulhu Keeper's Handbook included a version with a smoother bell curve.

[TooManySecrets]

The BRP (Basic Roleplaying) system has a resistance table like this.

See here: http://www.chaosium.com/forms/coc_quick_start_color.pdf

It's on page 10.

The Call of Cthulhu Keeper's Handbook included a version with a smoother bell curve.

Cool. Thanks for the head's up (though looking at the table, they really should have saved space by using the difference instead of having to line stuff up like that).