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It may be that someone has already done this (and if so could someone point me to it), but I was thinking it might be useful to have some sort of classification of dice systems.  Once we have that, we could have a discussion of the relative merits and drawbacks of each general type.  Other people with a broader experience of RPGs can probably contribute more than I can to this, but here is a start.  At some point this should be a hierarchical taxonomy, but for now I'm just looking for broad families.

Uniform or "Flat" curves
Generate a target number that is a function of your skill and the difficulty and try to roll under that number on a single die (usually on a d20 or d100).

"Additive curves" (maybe there is a better name for this)
Roll dice, the number and/or sides of which are determined by your skill level, and add the result of all the die together.  If the sum is greater than a target number (determined by difficulty) then you succeed, OR just use the raw sum for determining damage, power, etc.  
EABA falls into this category, although you add together the highest three dice, not all of the dice.  

"Binomial-like curves"
In this scheme, each die is treated as an independant result that is treated as a success (or victory in Sorcerer) or failure.  The difficulty of the task can affect either the target number for each die to count as a success or the number of successes required.  Usually the number of successes also determines the degree of success.  

Burning Wheel is a straight forward application of this using d6 and usually a target number of 4, where the difficulty determines the number of successes required.  

White Wolf, of course, is the target number type, but it is a bit more complicated because a 1 counts as a negative success, which makes the distribution a bit more complicated.  I don't like this, because for high difficulties (9 or 10 on a d10) your probability of botching actually increases as your skill level increases.  

Sorcerer presents an interesting twist on this scheme in that you do two seperate rolls of this type against each other, where the target number is determined after the fact as the highest number rolled by the opposing roll.  I haven't thouht about it a lot, but this seems like it would make calculating the distribution of the degree of success against your opponent somewhat difficult analytically.  Maybe Ron has done this already?

Quote from: HenriIt may be that someone has already done this (and if so could someone point me to it), but I was thinking it might be useful to have some sort of classification of dice systems.  Once we have that, we could have a discussion of the relative merits and drawbacks of each general type.  Other people with a broader experience of RPGs can probably contribute more than I can to this, but here is a start.  At some point this should be a hierarchical taxonomy, but for now I'm just looking for broad families.

I have at least an early categorization in my series of dice articles from my System Design page at:
http://www.darkshire.net/~jhkim/rpg/systemdesign/

The issue of what is actually happens numerically in the blizzard of 'vanilla' and 'pervy' game systems on the Forge is well worth discussing.  Some sort of tutorial/resource on game statistics would seem to me to be very useful. After all look at what happened to White Wolf....  It can be real easy to design a system that doesn't work anything like what we think it does if you don't understand how the statistics work. (For an example of how necessary it can be to really understand these number look at the discussion D20 vs 3D6 in Heroquest)

This is worth at least an article series, perhaps even a set of LONG sticky posts in the game design forum......

In addition to John's page, you might want to check out my Guide to Die Rolling Methods.

Quote from: JasperIn addition to John's page, you might want to check out my Guide to Die Rolling Methods.

OK, I'm going to bite on this one.  It is a good effort, but I see a number of mistakes.  In the target number section, you say "The distribution is thoroughly bell-shaped. It always has this basic shape, though the bell becomes more sharply peaked the more dice are used. The target number has no effect on shape, only absolute height."  

That's not true at all.  Taking your example of 8d6.  8d6 with target number 6 has a very lopsided distribution where the likeliest result is 1 success, and it has a long tail going out to 8 successes.  Similarly, 8d6 with target number 2 has a lopsided distribution where many successes are likely, and a tail going the other way to 0.

Hi John.  I read your article and I liked your classification scheme.  As I said earlier, we should have a hierarchical classification, and I think fixed dice vs. dice pool is a good place to start with that.  Then with fixed dice you split it into flat and bell-shaped.  And then for dice pool you have additive, target number, and highest die.

My "additive" category lumped together your bell-curve group and your additive dice pool group, but I can see that it is useful to keep these seperate.  EABA, of course, goes into the additive dice pool group, while GURPS goes into the bell-curve group.

There is one thing that I would quibble over, which is that you list Trinity and other such games as a "binary" variant of the additive dice pool.  I feel like this category is sufficiently distinct to merit its own category.  It is a lot like the target number approach, except that the target number is fixed.

John,

Ack!  What a terrible gaff.  Thanks.  Let's take any further discussion to PM though since it doesn't really relate to the topicat hand.

This sounds like a very useful resource. Perhaps one of the die-rolling programs out there could be modified to print probability tables instead of single rolls?

Question: is it the intention to stop at a (descriptive) taxonomy, or should there also be constructive tools (I want a system with such-and-such properties, how do I make it?). I think this is distinct from searching in a taxonomy, because the current systems do not exhaust the possibilites. Example: it is perfectly possible to get bell-curve probabilities with a flat die roll (use a warping function) but nobody seems to do this.

SR
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Well, I don't think there is really a "master plan" here, but my idea was to start with a descriptive taxonomy and then discuss what the advantages and disadvantages of different schemes were.  I don't think there is one great system that is the best for any circumstance, but we could come up with guidelines that say, well, if you want a system with these general features, these are the sort of systems that would be useful.  

The thing is, I think if you want to limit yourself to probability curves that can be generated relatively easy using just common dice and simple arithmetic that you can do in your head, you are really dealing with a very limitted set of distributions.  There seem to be enough people on the forge with some knowledge of probability theory that I was hoping that we could also discuss theoretical distributions that would be hard to generate using dice.  After all, lap top computers are now very common, and if you know a little bit about programming, its pretty easy to write a program to generate samples from any distribution you like, even one that would be overly cumbersome using dice.  The advantage of this is that statisticians have described a number of distributions that arise commonly in nature and have nice statistical features and can be described with a few parameters.  

For example, take something like this.  You want to perform a task.  You take your skill level, s, and then compute r = 10*exp(-s/2).  Now roll a d10.  If the result is less than r, stop.  Otherwise, keep rolling.  When you stop, count the number of times you had to roll the die, then add one to that.  That is your result.  If the result is greated than the Obstacle of the task, you succeed.  This will create a distribution known as the Poisson.  The Poisson is nice because it is roughly bell-shaped, but with a right-skew.  So it is never negative and has a long positive tail, allowing for the occasional critical success.  Another nice feature is that the average value is equal to the variance, which inn this case is skill levl / 2.

Now, in the middle of an rpg, there is no way that I would want to do the above algorithm.  It would require taking an exponential, which requires a calculator, and would require re-rolling a d10 indefinately and counting the number of rolls.  However, my laptop can do this in considerably less time than it takes me to blink.  A lot of people are used to using a GM screen to seperate them from the players, so I see no problem with replacing the GM screen with a laptop screen.  Obviously this approach is clearly not for everyone (especially since it requires owning a laptop!).  But I figure enough of roleplayers are geeky math-science-tech types that this would not be problematic for a lot of people.

EDIT:  This was meant to be a reply to Rob, but I wandered off topic.  To answer your questions more directly...

1) I've written a couple of programs to print out probability tables.  I have them for the White Wolf system, EABA, Burning Wheel (which is just a Binomial), and a Poisson.  

2) Well, as far as what can be done simply with dice, I think the possibilities actually are pretty limitted, so a taxonomy with some guidelines on how to choose a scheme and variations would probably be not that hard.  It doesn't have to be a stictly branching hierarchy like an evolutionary tree, however.  As for other things that you don't do with dice, statisticians have already been exploring these for a couple of centuries, so we already know about a lot of simple distributions with nice properties, but there aren't that many.  As to the warping example, that is a fine way to generate normals, but I don't think people are going to want to apply a warping function in their head in the middle of a game.  Better to let your laptop do it for you (at least in my opinion).

Quote from: HenriWell, I don't think there is really a "master plan" here

OK. Just wanted to make sure I wasn't doing thread hijack.

QuoteAs to the warping example, that is a fine way to generate normals, but I don't think people are going to want to apply a warping function in their head in the middle of a game.

I sure don't! But, since dice are discreet, that function is just a table-lookup, which, with sufficiently clever rules, you can have occur at char-gen time.

I agree that stuff done easily with dice is a limited set that can be fruitfully enumerated. Using a laptop/palmtop/cell phone/programmable calculator does open up quite a bit of extra design space. Another option with the same degree of freedom (stolen shamelessly from some PBM wargames) is to use a program to create a printout with, say, 1,000 samples of your distribution on it. In game you then just use these in sequence. No computer required during play.

SR
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Ok.  Both the table and the pre-generated random number list are good ways to remove the need for a computer during play.  A lot of people seem to have a strong dislike for charts, but then there are probably people who would say the same for the computer.  If I have access to one, my preference would be for the computer because I think it is quicker than having to look up stuff on a chart.

I think one reason that you might NOT want to use a computer is color.  If you are running your game in a historical/fantasy setting and you want to create a low-tech feel, a computer is going to ruin that.  On the other hand, if you're playing a high-tech type game, the computer might actually add desirable color.  Having the source of Fortune match Color is a nice detail (like cards in Dust Devils), but I don't think its that important, and its kind of off-topic anyway.

So to get back to the topic of dice curves, I think it might be useful to have some general discussion about their features and when to use what.  I'll be using John Kim's taxonomy here.  

So for broad categories have:
Fixed Dice - Flat
Fixed Dice - Bell Curve (Additive)
Dice Pool - Additive
Dice Pool - Fixed Target Number
Dice Pool - Floating Target Number
Dice Pool - Highest Dice
Step Die (different dN)

Things to consider are variance, symmetry, tails, and modes.

VARIANCE
I think that if you want a very realistic game, you want a fairly low variance.  With a high variance, you get a lot of things happening that, realistically, are very improbable, which may break the suspension of disbelief.  It seems to me that you reduce the variance when you add dice together, so I'd go with either a Fixed Dice - Bell Curve or Dice Pool - Additive system.  

On the other hand, low variance can be less interesting, since there are fewer upsets.  I think in a Nar or Gamist game in particular, you may desire a higher variance.  This makes it much easier for a weaker opponent to overcome a stronger opponent, which makes the game less predictable.  If you have a low variance system and really min-maxed characters, they never fail at the things they are good at, and are helpless at things they are bad at.  A high variance system helpls to level the playing field and rewards min-maxing and power gaming less, which is why I think they are good for gamist games.  For a high variance system, I'd go with Fixed Dice - Flat or Dice Pool - Highest Dice.  Although with Highest Dice, as Dice Pools get high, the variance goes down pretty fast, so this is only good for fairly small dice pools.

TAILS
Simple Fixed Dice and Dice Pool systems are going to be bound somewhere.  If n is the number of dice and s is the number of sides, then a fixed dice system has a lower bound of n and an upper bound of s*n.  A dice pool system has a lower bound of 0 and an upper bound at n.  This means that certain tasks may be impossible, which you may not want.  However, there are (at least) two ways around this.

1) Automatic Success - In fixed dice systmes, make s*n an automatic success (and maybe make n an automatic failure).

2) Open-endded dice rolls - In Dice Pool Systems, if you roll an s, reroll that die and add its value as well.  I guess you could do this in a fixed dice system, but then the number of dice is no longer fixed.  

The reason that I like this is that I find it frustrating that if you have low skill or if your effective skill has been reduced by wound penalties, even moderately difficult tasks are completely impossible!  On the other hand, this makes even really hard tasks possible, which can be unrealistic.  I could throw a pencil at the charging Vampire Lord, and if I roll a 20, I kill him!  In real life, the chance of throwing a pencil through someone's heart is probably a LOT less than 1 / 20 (unless you are Bulls Eye).  

Symmetry
Like tails, this is all about the critical failures and critical successes.  With a tightly bound system, there are no critical successes or failures, but if you want to stretch the bounds, you have to decide if it will be in both directions or just in one.

If you want something thats very symmetric, you had better go with Fixed Dice or Dice Pool - Additive.  Target Number Dice Pools only start to get symmetric when you have really large dice pools.  With fewer dice, they tend to lean to the left because they are bound at 0.

However, you can get around this by making "1's" negative, as in White Wolf, so that Target Number Dice Pools can be left-bound at -n instead of 0.  As long as you don't have very high Target Numbers, this is fine.  But when you combine this with a Floating Target Number system, as in White Wolf, you get some weird things up at the high difficulties, where your probability of a botch goes up with your skill and can actually be higher than the probability of normal failure!

Personally, I like having a right tail, but no left tail, because it means that while a PC is less likely to get really screwed by one bad dice roll, they will occasionally get to do something really awesome with a good dice roll.  You can get this with a dice pool system that is open-ended for successes, but doesn't have botches.  The Poisson distribution also has this shape.

Modes
Not too much to say here, but you do want to make sure that your mode is something reasonable.  I think this is only a problem for the Highest Dice Pools.  In these, with a large number of dice, the mode gets really high, but the distribution is bound on the left and the right.  This means that you get most of the probability mass pushed up against the right bound with a tail going to the left bound.  This is really awkward looking, at least to me, so I think Highest Dice systems are silly, but people don't use them much, probably for that reason.  Even worse, if you open-end a highest dice system, as with ICON, you get a bimodal distribution (it has two local maxima), which is really ridiculous looking (IMO).

Anyway... questions? comments? criticism?
Is there anything here that is dead wrong?
What would you add to this?

EDIT: Looking back at John's site, I see that I did not touch on granularity, ease and speed of use, and availability, which are also important stuff.

Real quick, granularity is higher for Additive systems (either fixed dice or dice pool).  Fixed dice systems have better availability, since you need a fixed number of dice, like 3d6 for GURPS.  Step Die systems have really crappy availability, since you need at least one of each different type of die.  Speed depends on a whole lot of things, and is basically the main cost of more refined systems.  In actual play, I think speed of use is really important, but there is a trade-off.

Quote from: HenriIf you want something thats very symmetric, you had better go with Fixed Dice or Dice Pool - Additive.  Target Number Dice Pools only start to get symmetric when you have really large dice pools.  With fewer dice, they tend to lean to the left because they are bound at 0.  

However, you can get around this by making "1's" negative, as in White Wolf, so that Target Number Dice Pools can be left-bound at -n instead of 0.  As long as you don't have very high Target Numbers, this is fine.  But when you combine this with a Floating Target Number system, as in White Wolf, you get some weird things up at the high difficulties, where your probability of a botch goes up with your skill and can actually be higher than the probability of normal failure!  

This part is fuzzy.  Being bound at zero has nothing to do with being symmetric.  After all, fixed die rolls like 3d6 or 1d100 are always bound at higher than zero, and they are symmetric.  Target number dice pools are symmetric if the target number is exactly half.  (i.e. d10's with a target number of 6 are symmetric).  

Conversely, in Storyteller where ones are negative, then the only symetric roll is target number 10.  This is the ultimate weird result, though.  This roll always has an average of zero, and the number of dice rolled only increases the variance: i.e. both the chance of botch and the chance of success go up with the number of dice rolled.  

Quote from: HenriPersonally, I like having a right tail, but no left tail, because it means that while a PC is less likely to get really screwed by one bad dice roll, they will occasionally get to do something really awesome with a good dice roll.  You can get this with a dice pool system that is open-ended for successes, but doesn't have botches.  The Poisson distribution also has this shape.  

I'd just note that I prefer this shape as well.  Having critical successes is OK for spice, but having a long left tail (i.e. critical failures) means that you can never be sure of yourself -- so it is never valid to skip the roll.  

Quote from: HenriNot too much to say here, but you do want to make sure that your mode is something reasonable.  I think this is only a problem for the Highest Dice Pools.  In these, with a large number of dice, the mode gets really high, but the distribution is bound on the left and the right.  This means that you get most of the probability mass pushed up against the right bound with a tail going to the left bound.  This is really awkward looking, at least to me, so I think Highest Dice systems are silly, but people don't use them much, probably for that reason.  

I tend to agree with you, but something left out of my analysis is Sorcerer dice-rolling which is always an opposed roll between two highest-die dice pools.  I should add in an analysis of Sorcerer at some point.

Quote from: John Kim
This part is fuzzy.  Being bound at zero has nothing to do with being symmetric.  After all, fixed die rolls like 3d6 or 1d100 are always bound at higher than zero, and they are symmetric.  Target number dice pools are symmetric if the target number is exactly half.  (i.e. d10's with a target number of 6 are symmetric).  

You are absolutely right.  I was thinking too specifically.  What it really has to do with is the relationship of the mode to the two bounds.  Low-dice dice pool systems are asymmetric because the mode is close to the lower bound, which happens to be 0.  Likewise, Highest Die systems are asymetric in the other way, because the mode is close to the upper bound.

I guess a lot of what I was saying before about Symmetry, Tails, and modes could be said a lot more succinctly.  The point is to identify your upper and lower bounds (if any) and your mode(s) and to look at the relationship between them.  I think its desirable not to let the mode get too close to either of the bounds.  This is an advantage of fixed dice systems.  Having the mode half way between the upper and lower bounds is a necessary but not sufficient condition for symmetry.  Practically speaking, however, when the mode is half way between the two bounds, it is usually also the median and the mean, which means that it is symmetric.  

Quote from: John Kim
I tend to agree with you, but something left out of my analysis is Sorcerer dice-rolling which is always an opposed roll between two highest-die dice pools.  I should add in an analysis of Sorcerer at some point.

Yeah, Sorcerer is totally weird.  I think of it as the bastard child of Target Number and Highest Die, where the TN is determined as the Highest Die of the opponent.  This is because of the way in which you determine the number of victories as the number of your die that beat the target number of your opponent's highest die.

There are a lot of other odd mechanics you can find in use out there in terms of die rolling conventions. For example, there's the median die example. Or the Godlike matching convention. Do you want to try to categorize these all, or just stick them under "unusual"?

Also, some of these types conflate die curve with output curve. That is, the outcome of a GURPS 3d6 dice roll is a bell curve. But the outputs of the system are binary pass/fail. You're only looking at the dice curve in this case when you mention the additive pool. But in the case of the die pools you then add the condition of target numbers. IOW, you should have "fixed dice - flat - Fixed Target Number" (Shadows, maybe) as different from "Fixed dice - flat - variable target number". You're really considering multiple vectors here, and the cross-product could be very large when you look at in in total. So you might want to separate them some.

Mike