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Aerodynamik
2010-03-02, 03:43 PM
I've started developing a gaming system, and I need a little help deciding something. So the basic mechanic is rolling against either a number or in an opposed check to see whether or not you can complete the action. But I don't want to have people just rolling a d20. I was going for something that had people rolling multiple dice so that the distribution of the numbers rolled would be skewed towards the numbers in the middle. (Here's a link that explains how rolling multiple dice increases how close to the average roll the number rolled is.) (http://forums.relicnews.com/showthread.php?t=151889)

Anyway, I was thinking I'd have people roll 3d8 and then add or subtract bonuses or penalties. But is that too much math for the game to be playable? Is adding together four to five numbers for every action too much? Should I just use 3d6 or maybe even just 2d6? Using 3d8 allows me to made a more precise control over things like crit ranges (People can have a <1%, 1%, 3%, 4%, 7% or 11% chance of crit using a 3d8 system), but is it too full of math to be really playable? It always slows things down D&D games that I run when people have to add up more than just two dice and a flat bonus. Just wondering what people thought.

Solaris
2010-03-02, 03:44 PM
I wouldn't recommend calculators being a requirement for play. Look at the 3d6 variant from UA.

Weimann
2010-03-02, 03:51 PM
Systems should be made as simple as they can but not simpler. Personally, I think adding up three dice and a modifier isn't too complicated. If those dice are d6 or d8 are totally irrelevant.

Frozen_Feet
2010-03-02, 04:08 PM
Mathematically, the only thing that makes it harder than d20 is varying effect bonuses have on probabilities. This only slows down the game if the players are anal-retentive munckins who will stop every time to calculate their chances. It does have huge impact on game desing, and a meticulous DM might have a bit of headache over it, but in actual play 3d8+modifiers isn't much different from d20+modifiers.

DragoonWraith
2010-03-02, 04:45 PM
I'd imagine that after a few dozen rolls, people will be able to recognize any combination of dice as their sum, rather than having to add the three numbers together each time. In other words, the human brain is exceptional at simplifying information it sees often - after seeing 4, 2, 5 several times and calculating it each time, the brain will just remember that it is 11, and not have to calculate it. At that point, it's no different from the d20, really.

DracoDei
2010-03-02, 05:16 PM
This is going to sound really obvious, but it is actually something it would be easy to forget:
Since you are probably mostly making this for yourself one question is "How good are my players at math?".
If you are making this with a good expectation that other people will use it too, then that changes to "How good is my perspective/target audience at math?".

Felyndiira
2010-03-02, 05:28 PM
Too much math is only when you need Importance Sampling to simulate the results of a roll.

If you want a more normal distribution, though, asking the players to roll more dice is never a wrong thing to do (though 4d6 would give a more even distribution than 3d8, with a range of 4-24 instead of 3-24). You could always ask for 24 coin flips and take the number of heads (basically, 24d2) if you really want something close to a bell curve, though.

Still, 3d8 wouldn't change your downtime by much; remember that even many D&D spells and attacks requires players to roll dice multiple times (example: fireball's 10d6 at level 10).

Knaight
2010-03-02, 05:32 PM
The amount you have is reasonable, just minimize math in play. For instance, if the method of damage is a table look up with a line calculated by some absurdly complicated equation that correlates damage and wound penalties, do that once, before the game. If you have fifty derived statistics, derive them before the game. Etc, etc. Once it goes, try to keep things simple. If it extends beyond algebra, it is too complicated. Well, algebra and trig anyways. If I'm doing integral calculus then something has gone wrong.

Yakk
2010-03-02, 05:55 PM
Honestly, rolling more than 1 die just to add them up ends up not being a good idea.

I thought it was a good idea, then I figured out what it really does is change the magnitude of bonuses to your die roll, basically.

Outside of the 5% "tail" events, you can rescale a d20+mod vs DC situation to be nearly indistinguishable in actual play from NdY+mod vs DC situation with simple linear scaling.

This is reasonably math intensive. I can see if I can find the pretty graphs...

Naw, I misplaced them.

The trick is as follows:
Step 1: Subtract the 'average' on your dice you roll from both the DC and your roll. Now you have a "zero-centered" game.

Step 2: Scale DCs down by the standard deviation of your dice. SD of (NdY) = sqrt( N * (Y)(Y-1)/12 ).

Now compare your chances to succeed on a graph, after doing the above slide-and-scale. While the d20 is a strait line and the NdY one is a curvy line, outside of the ~5% outliers at either side the two lines are very close to each other, so close that you couldn't distinguish them in actual play.

Which means that all of the work for rolling NdY and adding it up .. turns out to be a system for critical hits/misses. That is all you get from it over d20.

Now, rolling more than 1 die when you are doing more than just adding them up can be worthwhile. Ie, if you have mechanics for doubles/etc.

But NdY roll over/under ends up being no better/worse than d20, other than a linear change in the importance of each +1/-1 modifier.

Felyndiira
2010-03-02, 06:44 PM
Honestly, rolling more than 1 die just to add them up ends up not being a good idea.

I thought it was a good idea, then I figured out what it really does is change the magnitude of bonuses to your die roll, basically.

Did the graph for you. There is a significant difference between a 3d8 roll and a flat dY (in this case, since there's 22 different outcomes in a 3d8, a d22):

http://img203.imageshack.us/img203/9237/graphs.gif

I'm not sure what you did to obtain the result that they're not similar, but even assuming [infinite]D[any] rolls (which is a normal curve), the curve will be vastly different from a straight line (in particular because the line, in this case, is the probability density of a d[infinity], which is zero, but you'll get vast differences no matter how many sided dice you use due to the curvature of the normal). Putting aside the fact that you should not be dividing by the standard deviation, even rolling 2 dice gets you results significantly different than flatly rolling one d[anything].

Fiery Diamond
2010-03-02, 07:42 PM
Definitely not too much math. I liked the comment about integral calculus though.

Pronounceable
2010-03-02, 08:28 PM
This isn't really maths. Well, yes it is, but no. If you were rolling dice, adding and subtracting the various modifiers on both sides then multiplying the difference with some coefficient before comparing with a number determined in a similar way, that would've been maths.

Any system whose extent is xdy+z>t isn't really much of maths. Even if z is made up of at least 17 different itty bitty situational modifiers, it still wouldn't be maths. That'd merely be stupidity.

Aerodynamik
2010-03-02, 10:59 PM
Thanks, Felyndiira. I really like that graph. I was looking at the statistics tables available on this (http://www.ogmiosproject.org/articles/stattables.html) site, but the graph works a lot better than the tables for illustrating just how much the difference between the 3d8 rolls and the d20 rolls is.

Anyway though, something that intrigued me is this:

NdY roll over/under ends up being no better/worse than d20, other than a linear change in the importance of each +1/-1 modifier.

So I agree with that statement wholeheartedly. 3d8 (or 3d6) is no better or worse than 1d20. It just depends on what you're looking for. In real life, if you do something a thousand times and then graph the results, it's going to come out looking more like a bell curve than a straight line. If you graph the probabilities or rolling certain numbers with multiple dice, then you wind up with something like a bell curve, the results in the middle being most likely occur. A d20 roll gives you a straight line. This means that more exceptional results are more likely to occur. And that's fine. It's less realistic, but I doubt D&D and the other systems that use it are going for being realistic. They're going for it's players being able to do awesome stuff (roll higher numbers) more often. Neither one is better or worse. One is more realistic and one is less realistic. That's all.

But something else that Yakk said was neat, and made me consider something else that I hadn't considered. He said that the importance of each +1/-1 modifier is increased. Lets look at some math.

So in D&D suppose you're a thief trying to pick someone's pockets. You have a +10 bonus, and the difficulty class to the check is 15. If you roll a 4 or less on the die, you're going to fail the check. So you have a 20% chance of failure using a d20.

But suppose you are using the variant where you need to roll 3d6 to determine success or failure. You're still a thief, and you're picking the same pocket. Same +10 bonus, same DC16 check. Your chance to fail is really small. It's only 4.6%. This is because of the fact that multiple dice make smaller and large numbers more uncommon and numbers in the middle more common. Only if you get one of those really rare low numbers do you fail. This means that bonuses work differently. A +1 bonus isn't going to do that much. but a large enough bonus effectively prevents you from ever messing up.

This is interesting to think about from a design perspective. I think that G.U.R.P.S. uses 3d6 and fixes this by making it so that the first few ranks you put into a skill only cost one point, but if you want more ranks, they start to get more expensive the more ranks you purchase.

(This is officially my new longest forum post ever.)

Fiery Diamond
2010-03-03, 12:00 AM
Actually, I have a few questions, since my maths be very bads. (I stopped doing anything beyond elementary algebra after I passed my multi-variable calculus and differential equations class.)

Can someone tell me/point me to a site that tells me/point me to a sight that can calculate what the following probability distributions are (what percent for each number)?

2d10
3d8
3d6
4d6
5d4

DracoDei
2010-03-03, 12:23 AM
The way I do it is in Excel...
I line up a bunch of ones in a row, one for each side on the die.
Then I copy that down one row lower and one to the right.
Do that until the number of rows ALSO equals the number of sides on the die.
The left-most one is the probability for rolling the lowest possible number (2) on the 2dX.
Each one after that is one more than that.

To get the results for 3dX do the same thing, but instead of all ones, use the results from the 2dX. Number of rows is still X.

If your iterate this process, then can get the possibilities for any number of dice (although it might take you a very long time).

Of course, you have to divide by X^(number of sides) then multiply by 100 to get these in the form of percentages.

Note that everything should always turn out symmetrical.

Just so you can check your work:
2d10 is
2|3|4|5|6|7|8|9|10|11|12|13|14|15|16|17|18|19|20
1|2|3|4|5|6|7|8|9|10|9|8|7|6|5|4|3|2|1

In this case those are both the number off possible ways to make the given number AND the percentage chance of rolling it.

Edit:
Here are a few more...
2d4
.2.|.3.|.4.|.5.|.6.|.7.|.8.
1|2|3|4|3|2|1
6.75%|12.5%|19.25%|25%|19.25%|12.5%|6.75%

3d4 = 64 possibilities
3|4|5| 6| 7| 8| 9|10|11|12
1|3|6|10|12|12|10| 6| 3| 1

4d4 = 256 Possibilities
4| 5| 6| 7| 8| 9|10|11|12|13|14|15|16
1|4|10|20|31|40|44|40|31|20|10| 4| 1

Just ask if you have questions, but realize I am trying to teach you how to do this, rather than just giving you all the answers. I enjoy teaching math.

Felyndiira
2010-03-03, 12:56 AM
Actually, I have a few questions, since my maths be very bads. (I stopped doing anything beyond elementary algebra after I passed my multi-variable calculus and differential equations class.)

Can someone tell me/point me to a site that tells me/point me to a sight that can calculate what the following probability distributions are (what percent for each number)?

I do it in R, which is a freely-distributed statistical package. The following pseudocode works for any language, though, such as VBA for Excel (to generate graphs):

VARIABLE sides = # of sides
VARIABLE ndice = number of dice
VARIABLE ARRAY dicerolls[1 to ndice]
VARIABLE ARRAY result[1 to sides^ndice]

For i FROM 1 TO sides^ndice
__For j FROM 1 TO ndice
____dicerolls[j] = Floor( (i MOD sides^j) / (sides^(j-1)) + 1)
__END For {j}

__result[i] = Sum(dicerolls)
END For {i}

ShneekeyTheLost
2010-03-03, 01:44 AM
I'm creating a gaming system where the die rolls are (ranks + stat mod) d (size based on class and skill). Most modifiers add or subtract dice, rather than giving a flat bonus.

So a Specialist (think Rogue) trying to pick a lock, would roll a d8, since it is a Core Skill for that class. With 4 ranks in the appropriate skill and a stat mod of +2, and a Bless from the party cleric for +1die to the roll, would roll 7d8 and add the result. If he had chosen that as his one Specialty skill, which all characters choose at creation, it would be d10's instead.

However, the party Soldier trying the same thing would be rolling d4's, since picking locks has absolutely nothing to do with being a Soldier, would likely not have as many ranks in the skill, or as high a stat mod. So his roll would likely look something like 2d4.

Gralamin
2010-03-03, 02:00 AM
Step 2: Scale DCs down by the standard deviation of your dice. SD of (NdY) = sqrt( N * (Y)(Y-1)/12 ).

I hope you mean the sqrt(N^2 * (Y^2-1)/12)) :smallwink:
For those who don't know statstics: This is a case of the Discrete Uniform Distribution (http://en.wikipedia.org/wiki/Uniform_distribution_(discrete)), which has a variance (Variance is the square of standard deviation) of (y^2-1)/12. In addition, Variance has a Property (http://en.wikipedia.org/wiki/Variance#Properties) where if you have a constant multiplied by the random variable (As you would for accounting for extra dice), its the same as the constant squared times the variance without that constant.

In math:
SD(NdY) = sqrt(Var(NdY)) = sqrt(N^2 * Var(1dY))
= sqrt(N^2 * (Y^2-1)/12)
This could be simplified to
= N sqrt((Y^2-1)/12)

Fortuna
2010-03-03, 02:02 AM
Reading this thread, I am inspired to make a system in which the combat resolution mechanic is based on how quickly you can do integral calculus in your head... just for fun, of course, not for use.

Gralamin
2010-03-03, 02:07 AM
Reading this thread, I am inspired to make a system in which the combat resolution mechanic is based on how quickly you can do integral calculus in your head... just for fun, of course, not for use.

Require all of the powerful abilities to use Convolution... Such as calculating Gamma Random Variables for values such as 2/3rds. I've yet to see someone do that in their head :smalltongue:

Saintheart
2010-03-03, 02:13 AM
I propose that math is like Dakka: you can never have enuff. :smallbiggrin:

Felyndiira
2010-03-03, 04:24 AM
I hope you mean the sqrt(N^2 * (Y^2-1)/12)) :smallwink:

It's actually sqrt(N * (Y^2-1)/12) :smalltongue:. NdY is rolling N independent dice each with discrete uniform, not rolling one die and multiplying the result by N. It doesn't quite matter, though, since the prospect of dividing by the standard deviation to measure the spread of a distribution isn't very valid itself.

Yakk
2010-03-03, 12:29 PM
http://img203.imageshack.us/img203/9237/graphs.gif
Graph the CDF, not the probability of rolling a given value.

You (A) did not do the slide-and-scale, and (B) didn't graph the chance to succeed (but instead, graphed the chance to roll a number exactly). Not surprisingly, your graph did not show what I was talking about, because you didn't graph what I was talking about. :)

The CDF is the probability of rolling a number or less. I made no claims about the probability of rolling exactly a given number -- I made claims about roll over/roll under systems.

Second, d22 is the wrong thing to compare a 3d8 with.

...

... sorry, I mistyped the variance equation!

The correct variance of 3d8 is 3 * 63/12, or 15.75. That makes the SD about 4, which lines up roughly with a d14.

The average of 3d8 is 13.5. The average of d14 is 7. So you need to shift things by that much (6.5)

They have similar SD, so we don't have to rescale there.

Graph the probability that a 3d8 will roll under (or over, I don't care) the values from 2 to 25.

Graph, along side, the probability that a d14 will roll under the values 0 through 15. Shift it over by 6.5. Place the two graphs on top of each other.

Notice how close they are.

(I'm sufficiently confident that, while I haven't done this for 3d8 and 1d14 in particular, I know what will happen before checking.)

The significant differences will be in the "tails", which encompass the outermost 5% chances. In those tails, things behave differently.

Hence the observation that "adding up multiple dice for roll over/under has the effect of changing the 'scale' of the system, and adding complexity to the equivalent of rolling 1 or 20 on a d20 -- critical fail/success -- cases".

I'm off to poke at a spreadsheet to graph this.

Here it is:

http://theorem.ca/~afn/temp/graph.png

I cheated marginally, in that I worked out the chance to roll under "7.5" instead of just rolling under 7. The real graph of a d14 would have dots at different points, but the line would be identical. Sorry, I got lazy.

You can see how there is a slight difference in the lines, but the main difference is that the 3d8 has "long tails", while the d14 just runs strait into zero.

Note that a horizontal scaling by a factor of the SD is only half-assed approximation to what would be ideal. As you can see in the above graph, the 'practical' slope of the line is steeper in the middle by a tad -- using something closer to that would probably be a half-decent idea.

Then, simply have mechanics for the rare "natural 1" and "natural max" that have the properties you want -- move critical hit/fumble complexity into their own system, while making the typical case easier to do, should speed up play.

DragoonWraith
2010-03-03, 01:15 PM
Require all of the powerful abilities to use Convolution... Such as calculating Gamma Random Variables for values such as 2/3rds. I've yet to see someone do that in their head :smalltongue:
The convolution operation would be beyond mental math, I think, for just about anyone. As an electrical engineer, we learned those for signal processing, but very quickly learned improper Fourier analysis that allows us to skip that, and never looked back. And if taking the Fourier transform (actually, the FFT, which is not the Fourier transform really) is easier, you know we have a problem.

Not familiar with Gamma Random Variables, though.

Anyway, as for calculus, the power rule is trivial to do in one's head. Differentiation in general is usually possible for mental math. Integration by parts, with trig substitution, etc etc. would be the way to go if you want calculus that one can't do in one's head.

But a very large amount of the applications for calculus fall under the power rule, which is why I am rather confused by how difficult people make it out to be. I'm quite confident I could teach the basic concept of differentiation/integration, as well as the power rule, to a sufficiently interested American 8th grader (and American 8th grade math classes are notably quite far behind most, if not all, of their peers in other developed countries). It wouldn't really even take very long, honestly.

Aerodynamik
2010-03-03, 01:44 PM
Reading this thread, I am inspired to make a system in which the combat resolution mechanic is based on how quickly you can do integral calculus in your head... just for fun, of course, not for use.

I can imagine that.

Me: "I attack the orc with my sword!"

DM: "Okay, integrate http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/intbypartsdirectory/img35.gif to see if you hit."

Me: "Um... Nine?"

(I'm not so great at maths.)

Gralamin
2010-03-03, 08:16 PM
It's actually sqrt(N * (Y^2-1)/12) :smalltongue:. NdY is rolling N independent dice each with discrete uniform, not rolling one die and multiplying the result by N. It doesn't quite matter, though, since the prospect of dividing by the standard deviation to measure the spread of a distribution isn't very valid itself.

You are mistaken, because Variance is based off of Expected Values. If we were looking at distributions, you could not do so, but when looking at Expected Values, it is entirely valid.

Yakk
2010-03-04, 10:18 AM
So, any comments on my pretty graph? I think it is pretty!

Basically, it is my claim that "Sum NdY systems reduce to a similar 1dY system with a critical-hit table in actual play".

Now, there are NdY systems that don't reduce this way. For example, a 2d10 system where you read the two d10s both ways (so a 27 is also a 72) in order to emulate rolling 1d100 twice and checking both times against a target percentage, and where doubles have a particular impact. This cannot be emulated with 1dY easily.

Die pool systems, also hard to emulate with 1dY.

Exploding sum die systems, also harder to emulate with 1dY.

I had an image of a half-dozen different rolling mechanisms (3d6, 2d10, 2d10 vs 2d10, 1d20, 1d20 vs 1d20, ... and maybe one other) somewhere, but I seem to have misplaced it.

Hmm. I think I posted it on the epic RPG board. I should go rummage through it!

You are mistaken, because Variance is based off of Expected Values. If we were looking at distributions, you could not do so, but when looking at Expected Values, it is entirely valid.
The variance of NdY is N times the variance of 1dY.
The variance of N * 1dY is N^2 times the variance of 1dY.
The SD of NdY is sqrt(N) times the SD of 1dY.
The SD of N * 1dY is N times the SD of 1dY.

since the prospect of dividing by the standard deviation to measure the spread of a distribution isn't very valid itself.
Naw, it isn't very valid with low N (at high N, the Central Limit Theorem kicks in, and it becomes more and more valid). It happens to be valid enough for horseshoes, hand grenades, and impact-noticeable-at-table-without-a-computer-for-casual-play statistical impact.

Zeta Kai
2010-03-04, 01:12 PM
So, any comments on my pretty graph? I think it is pretty!

Yes, it's very nice, Yakk. You deserve kudos for the effort. :smallcool: