The aim of this treatise is to develop a method to perform a qualitative analysis of dice-mechanics for role-playing systems, by evaluating how close such mechanics come to an imagined "ideal" system. In order to accomplish this I will start out by defining such an ideal, trying to encompass all the features such an imaginary perfect roll-playing dice mechanic would contain.

As time has proven the perfect role-playing system doesn't exist, and indeed two persons would hardly even be able to agree on the theoretical attributes of such a system, but by limiting myself to examining the dice mechanics I hope to be able to extract a set of common properties that most people would agree are desirable.

A note before I begin: I've strived to use the power of mathematics whereever applicable, but this text is by no means, nor is it intended to be, an attempt to make a precise mathematical formulation of the aim of dice mechanics. I will time and again in the following make assumptions and draw conclusions based upon nothing more than intuition and feelings — after all may aim is to perform a qualitative analysis with respect to what a role-player would *feel* as the best system, rather than a quantitative analysis based upon some objective (and probably non-existing) scale.

# Definitions

Before plowing on I believe it would be wise to define a set of terms to use in the following examinations. These will be the terms that I use in this text, regardless of what their name may be in the specific rulesets — although I will try to point out what terms fits with what.

To start at the beginning I'll first define what exactly I mean by the term "dice mechanic":

Definition: Dice Mechanic

A dice mechanic is a system by which it is possible, given a set of factors, to determine the outcome of a given action in a random, unforeseeable way.

As the reader has probably noticed the definition of this term does not reference dices explicitly, and it is indeed possible to create such a system without using dices i.e, by using cards or another kind of randomizer.

The above defintion contains some terms that probably deserves a definition in their own right. So here goes:

Defition: Action

Any conscious act of will by a sentient being, with the purpose of directly or indirectly affecting itself or its surroundings.

That's about as broad and general as I can make it. I emphasize the fact that this and the following terms all refer to an imaginary role-playing world, that is, the sentient being mentioned in the above definition is a being within our imaginary world and not a being like you dear reader or I.

The following definitions are all in respect to any given action, which is implicitly assumed known.

Defintion: Actor

The sentient being who is in the process of performing said action.

Definition: Target

Any entity which is directly influenced, willingly or not, by the action.

I'm sorry about that one but I can't think of a better term. Note that a target does not have to be a sentient being, as we required for an actor. In other words, a rock for example, can be the target of an action, say when someone throws it, but it cannot itself perform actions.

Definition: Outcome

The result of an action, intended or otherwise.

Definition: Goal

The intended outcome of an action.

Definition: Success

If the outcome is near enough to the goal, we say that the action is a success.

Definition: Factors

Anything that contributes causally to the outcome of the action, circumstantial or intrinsic.

Definition: Randomness

How much the outcome of an action depends on pure chance.

That should be enough with the generics. Let's get to some terms that we can recognize:

Definition: Skill

Any factor intrinsic to the actor or any sentient target. In other words, the factors that depends solely upon who the actor or target are, and not the circumstance of the specific action.

Definition: Circumstance (factor)

Any factor that is not a skill.

Boy, I'm going to regret that one later on.

Alright. So an *action* can be performed by an *actor*, on a *target*, with an *outcome* depending on the the *skills* of the given actor and target, and the *circumstances* pertaining to the situation.

And now for a key term in my analysis:

Defintion: Degree of Success

The degree to which an action can be said to have obtained the goal. Abbrevated DoS, the higher the DoS, the better the result.

The granularity of the DoS depends upon the specific role-playing system, although almost all such systems employ DoS, implicitly or explicitly, to some degree. For example in Dungeouns & Dragons the outcome of an action is determined by rolling a twenty sided dice and seeing if the result is greater than a certain number. If it is the action succeeds. However, most players of D&D will probably agree that the exact result is usually interpreted depending upon how far beyond the target number the roll got — making the roll with a margin of ten is cosidered a "better" success than beating it with a margin of one.

Please pay extra attention to the following one.

Definition: Difficulty

The DoS required to declare the action a success.

This one is crucial — and not trivial. The term "difficulty" is used in almost every roll-playing system out there, but hardly two of them agree on what the term actually means. I use the above definition in this text, which may or may not match that of any particular system.

In an attempt to make the definition a bit clearer and more understandable, here's an example: Say Alice is at an archers contest and she's attempting to hit a certain target at thirty meters. The closer she gets to the targets center, the better she scores, but as long as she hits the target at all we'll consider her shot a success. Using any dice-mechanic we can determine if she hits the target or not, and with DoS we can probably figure out approximately how many points she scores in a given roll.

Now imagine the target being moved back thirty meters, doubling the distance. We will certainly agree that the action has become more difficult. However, I'd venture the claim that Alice will make approximately the same shots — it's just that more of them will now miss the target due to the larger distance. Hence we'll have to require a larger DoS to acknowledge a successful shot — the "difficulty", in the sense from above, has risen. This also means that our previous mapping from DoS to score has changed; a DoS that would previosly have given points might do so no more, and higher DoS's will give a lower score than before.

This leads to one more term, that is nothing but a discreetization of the DoS:

Definition: Chance of Success

Abbrivated CoS, the probability that a certain action will be a success i.e., the probability that the DoS is larger than the difficulty.

And finally for our key tool in the following analysis:

Definition: Degree of Success distribution

Given all possible DoS's, the DoS distribution for any given action is the probability distribution over the DoS's.

Note that the DoS distribution will be dependant upon all the factors of a given action but not upon the difficulty. Also note that the CoS will be the area under the part of the DoS distribution that is greater than the difficulty, and hence CoS *is* dependant upon difficulty.

The aim of any dice mechanic is to create DoS distributions consistent with our intuitive perception of how likely any given outcome of the action is. Not an easy task.

I'll round off this section with a short discussion of the important properties of the DoS distribution. Like any probability distribution it will have a mean value, that is, the weighted average of all possible outcomes, weighted according to their likelihood. Additionally it will have a variance, which indicates how far from the average the outcome is likely to fall.

If the average of the DoS distribution is high, it means that the outcome of the action is likely to be good, and vice versa. If the variance of the DoS distribution is high, it means that a lot depends on chance — the randomness is high — and it is difficult to say anything with certainty as to the outcome. If the variance is low, on the other hand, the outcome is very likely be close to the average.

# Properties of the Perfect System

In this section I'll establish a series of properties, that I believe most people will agree is desirable in any dice mechanic. I emphasize once more that I am only considering properties of the dice mechanics, and hence in the following I'll treat actions as the basis of our system. I'll enumerate the properties, and will list them all once more in the end of this section for ease of reference.

As the reader will soon see, these properties are conflicting, and one of the main purposes of any real-world dice mechanic is to weigh these properties against each other and decide which ones are more important.

## Property One — Speed and Ease-of-Use

I think we can all agree, that the faster any action can be resolved, the better. While dice rolling decidedly adds to the excitement of any situation (just try throwing dices in a class and see how exciting it gets), we all want the resolution to become clear within seconds, and preferably faster, of deciding on a given action.

In this sense, dices may not be the best way of resolving actions. Rolling a dice and interpreting the outcome is definitely slower than, say drawing a card from a pile (provided you don't have to shuffle it first).

Linked to speed, but not necessarily the same, is ease-of-use. We want the system to be as easy to use as possible, and ideally we can use it without any conscious effort.

In the interest of evaluting this property, I'll make (or rather copy) an observation pertaining to dice systems only. You can interpret the result of any given dice roll in a number of ways, but some are easier than others, and inspired by [1], here's a list sorted by ease-of-use:

Comparison > Counting > Addition > Subtraction > Multiplication > Division

## Property Two — Precision

Discreetness, continuity, call it what you will — the property that factors, difficulties and outcomes can be measured with arbitrary precision. In other words, the more DoS's, the more finegrained skills, the more precise difficulties, the merrier. Preferrably we would have continuity on all those variables, although that is practically impossible. This property is definitely conflicting with property one, in that the more values we'll have to juggle, the more time-consuming the process gets, and the more conscious effort it will require on our part to use it.

## Property Three — Correctness

Another property that I hope we can all agree on, although difficult to define explicitly. What we want is, that the dice mechanic's DoS distribution for any given action corresponds to our intuitive perception of it. While we cannot give any precise definition of our intuition, we can outline the trends in the properties of the DoS distribution, that we intuitively expect, dependent upon the factors pertaining to the action.

I will in the following assume that all factors, difficulties and outcomes are measured in a continious interval from 0 to 1. This is to ease the mathematical formulations, but the results can with relative ease be generalized to any descreet, closed interval. I will not treat open-ended systems explicitly, as most of them can be closed by putting a bound on the variables, high enough that any human achievable level is contained within.

Furthermore I'll make the assumption that the ideal DoS distribution is a normal distribution. Pause, to wait out screams of protest… Yes, I know. It's a huge and limiting assumption, but as we shall see, it is not unreasonable, and we have to make some sort of assumption about the shape of the DoS distribution if we want to be able to treat it mathematically.

So — Why a normal distribution? Let me make an observation: The DoS is a number that tells us, how well a certain action did accomplish its goals. In this sense, the DoS is actually a grade, telling us how well the actor performed the action. While the number is still arbitrary, in that we haven't yet given it any meaningful interpretation, it seems logical to take a look at how grades are normally distributed in educational environments. And yes, they follow a normal distribution (often by definition actually). Furthermore, it makes some intuitive sense too: We'd normally expect a lot more average results than extreme ones.

Given that our DoS distribution is a normal distribution, what properties/variables can we change, in order to make it fit our specific needs? Exactly two actually, and we've already discussed them briefly: Average and variance.

So what do we intuitively expect of the average and variance, given knowledge about factors? Well, as for skills, we'd definitely expect the average DoS to rise when the skill does; after all that's how we tend to measure skill — the better you do at tasks, the higher your skill. Furthermore, there's a tendency that the more skillful an actor is at a certain action, the more consistent his/her results become — a skilled archer will place his arrows within a much smaller margin than a complete novice; a trained athlete will run a certain distance in a more predectable time than an untrained one. Hence we expect variance to lower when skill rises.

The best swordsman in the world doesn't need to fear the second best swordsman in the world; no, the person for him to be afraid of is some ignorant antagonist who has never had a sword in his hand before; he doesn't do the thing he ought to do, and so the expert isn't prepared for him; he does the thing he ought not to do; and often it catches the expert out and ends him on the spot.

*- Mark Twain, "A Connecticut Yankee in King Arthur's Court"*

These trends are by no means mathematically precise, and we have said nothing about the way average and variance depends on skill — in particular it's probably a bit too far a stretch of the power of intuition to claim linearity of these dependencies. However we do expect the trends to be strictly monotonous, in that any small rise in skill will result in a rise however big in the average and a likewise fall in variance.

As for circumstances things gets a lot more difficult, mainly due to the fact that these are poorly defined. Being anything that can causally affect the outcome of the action, minus the skills of the actor and target, it is hard to say anything in general pertaining to all kinds of circumstances. Indeed it is hard to even determine what is a circumstance: In the example from earlier with the archers contest, is the distance to the target a circumstance of the action or an increase in difficulty and what is the difference?

Well, assuming that we've accepted that the DoS is normally distributed, any factor having an affect on the outcome will either affect variance, average, both, or our interpretation of the DoS. Let's look at some examples, using the same setting as before:

- Moving the target further away will, as already discussed, probably not affect Alice's shooting as such, but will affect our interpretation of success. So would making the target smaller.

- High winds on the other hand, would probably raise variance and lower the average of Alice's DoS, as opposed to shooting in calm weather.

- Making the target move would definitely complicate matters - we'd expect both higher variance and lower average in such a situation.

- Giving Alice a better bow would, provided she knew how to use it, probably both raise her average and lower her variance.

Actually, come to think of it, there seems to be a tendency that anything that raises your average (i.e., raises your chances of succeeding) lowers your variance (i.e., lowers the unprediciveness of the result). This observation is purely subjective, and I may be wrong (I rather think I am), but try as I might, I cannot think of any circumstance lowering or raising both the expected variance and average of the DoS.

There is of course no certainty, even if this observation is true, that the ratio of raise/fall is the same for all circumstances, nor that it is the same as for skills.

As for the difference between difficulty and circumstances, I put forth the following claim (more of a definition really): Factors that have no influence upon average and variance of the DoS distribution, but solely upon our interpretation of the DoS, are in fact difficulty "factors" rather than circumstances.

Finally, even given that we accept the normality of the DoS distribution, the basic distribution might vary from action to action. I.e., some actions might have a higher intrinsic uncertainty than others, for example a game of dice vs. arm-wrestling.

The perfect mechanic would be able to take all of this into account, lowering and raising the variance, average and difficulty in unique ways according to the specific circumstances. Factoring in property one of the perfect system, a good system might not achieve such correctness, but will at least lower and raise variance and average approximately according to our intuition.

For future reference, I've plotted a graph of how the DoS distribution intuitively ought to morph when varying skill on figure 1. This is using linear dependency of skill and avarage and variance — the graph is not a golden goal but more of a guideline.

*Figure 1. An approximation of the ideal DoS distribution varied with respect to skill. DoS is measured on the arbitrary scale from -9 to 9.*

## The List

Based upon the above discussions, the three properties we seek in any dice mechanic are the following:

1. Speed and ease-of-use. Any resolution should be quickly and easely achieved.

2. Precision. Any factor should be highly measurable, allowing for small variances in factors.

3. Correctness. The mechanic should come as close as possible to our intuitive understanding of the DoS distribution — specifically the DoS distributions should look at least something like that of figure 1.

It is naturally necessary for any mechanic to make compromises, as these properties are conflicting. There is a tendency to weigh them in the order of the above list, as complicated systems are practically impossible to use, taking far to much of our precious game-time to ever become popular.

This means that most succesful mechanics out there scores quite high on property one, and has a high enough degree of precision that most people are satisfied — leaving only property three to be an actual point of difference.

# A Word on Unachievable Results

Before proceeding, I should probably discuss one of the more contended properties of a dice mechanic: Should all DoS's be achievable under all circumstances and with all levels of skills? I.e., should it always be possible, even for a complete novice to achieve outstanding successes, and vice versa should even the most skilled person always have a chance to fail, however remote?

This is a point of some dissention, but here's my take one it: When we employ a system in which randomness matters — i.e., when we actually roll the dice, we do so because we can't be certain as to the result. I believe that once the dice are rolled, any DoS should be theoretically possible, although I also believe in the bell-shaped normal distribution making extreme result highly unlikely, and for high or low competence levels almost, but not quite, inconcievable.

# A Word on Reflexivity

Imagine that the target of an action is a sentient being. In that case the action often has an inverse — the actors attempt to perform the action on the target can also be regarded as the targets attempt to avoid the action of the actor. The inverse of Alice attempting to hit Bob with a sword, is Bobs attempt to avoid being hit by Alices sword. A nice property of any dice mechanic is *reflexivity*: The property that the DoS distribution for an action is the inverse of the DoS distribution for the inverse action. In other words, that it doesn't matter if we choose to roll the dices for the action or for its inverse. This is also in accordance with our intuition: The outcome of an action shouldn't depend on the mere observational property of whether we choose to look at it from one side or the other.

An easy way to obtain reflexivity is to require both parties to make dice rolls. That way, inverting the action won't affect the dices being rolled, and reflexivity is obtained by default. However this has the downside that a lot more dices has to be rolled resulting in a greater complixity and lesser speed. Furthermore the added range of DoS's needs a new mechanic for determining how to interpret results — i.e., subtracting the two DoS's will result in a range twice the normal one for the DoS, and one that will tend to be centered quite differently than is normally the case.

Reflexivity usually becomes a point of interest in what is commonly called *opposed actions* i.e., actions in which two parties are trying to achieve opposite effects. I will limit my discussions in the following to unopposed actions, and thus avoid the question of reflexivity entirely, but note that using the mechanic outlined above, almost any system can become reflexive.

# Evalutations

Using my three desirable properties I can now proceed to my qualitative analysis of a couple of different common mechanics.

## Dungeouns & Dragons

The grand old man of role-playing games, D&D employs one of the simplest dice mechanics out there:

Roll a twenty sided dice, add your skill, and compare the result to a certain difficulty number (called the DC).

This uses one addition and one comparison — it is in other words both easy and speedy to use, scoring high on property one. As for precision, any skill can theoretically be infinitely high, but usually will lie between one and twenty (twenty being the highest achievable for non-epic characters). This is actually a rather fine-grained scale compared to many other systems.

As for correctness, the average DoS (the result of the die plus your skill) rises when you skill does as desired. The variance however is constant, and what is worse, the distribution is nothing close to a normal distribution. It is in fact uniform. That is, the chance of achieving extreme results is precisely the same as for achieving standard results (5%).

A plot of the various DoS distribution for skill zero through ten can be seen on figure 2. As is hopefully obvious it looks very little like what we were rooting for — figure 1.

*Figure 2. The DoS distribution of Dungeouns & Dragons, for skills zero through ten.*

## World of Darkness

White Wolf's storyteller mechanism has undergone a number of revisions and I will be looking at two of them here, which I'll aptly name the old one and the new one.

In the old one, a player rolls a number of ten-sided dices equal to his skill in the given action. Each die that comes up greater than or equal the target-number (called the difficulty NOT! to be confused with my definition of difficulty above) counts as a success. The total number of successes is the DoS, while any die showing one subtracts one from the DoS. A roll with no successes but at least one one (yup) counts as a catastrophic failure (a botch).

In the new one, a player still rolls a number of ten-sided dices equal to his skill (but modified for circumstances). Each die that comes up equal to or greater than eight is a success, and the number of successes is the DoS. Ones still subtract from the total, but botching is no longer possible (it is, but the mechanics are way different and it will only happen under circumstances which I will not go into here). Also any dice showing ten will allow the player to add another dice to the roll.

Both systems depends solely upon counting and comparisons and thus are definitely easy to use. They are a bit slower than D&D, especially for high skills where the dice pool can be a bit chaotic, but generally fast enough that they're not a source of annoyance.

Skills are measured on a scale from one to ten (possibly but very rarely higher), giving a coarser granularity than in D&D but still workable. The granularity of the DoS is another matter though… While it may seem that a possible DoS of zero through ten is a fine granularity, the reality of the system is a different one. Low-skilled characters will only have one die, and hence can at best hope to achieve a DoS of one (yes, rerolls, I know, but the chances are too low to matter). That is, for low-skilled characters the DoS is either zero or one (possibly botch in the old system), and therefore the difficulty (as per my definition) cannot be anything but one in most cases — a difficulty of zero and there's no chance of failing, a difficulty greater than one and low-skilled characters have no chance at all. I believe that either the granularity of the DoS is to coarse, or the level of unachievable DoS's is to great in this system.

The DoS distribution for the two systems are plotted on figure 3 and 4. As can be seen they are closer to (albeit still far from) a normal distribution than was the case in D&D. The average still rises with the skill but unfortunately so does the variance. That is, the better a character gets at something, the less predictable his results get. Don't get me wrong — the CoS still rises, so we can predict that he succeeds, but how much so is a different matter. This is a consequence of low skill characters only being able to achieve very low DoS's, hence giving them a low variance, while highly skilled ones can achieve both low and high DoS's.

I'd definately prefer the World of Darkness system over D&D any day — it may be a bit slower, but the approximately bell-shaped DoS destributions means a great deal to me. I do however find it problematic that the variance of the DoS distribution rises with skill, as well as consider it a problem that high DoS's are all but unachievable for low-skilled characters. Even the most fumble-fingered of us can hit that target once in a while…

*Figure 3. The DoS distribution of the old World of Darkness rules for skills zero through ten vs. a difficulty of six, and additionally for a skill of ten using specialization rules. A DoS of -1 means a botch.*

*Figure 4. The DoS distribution of the new World of Darknes for skills zero through ten.*

## Fudge

A little less known than the previous two systems, Fudge has been developed with simplicity in mind. There's a lot of tweakables in Fudge, even in the core rules, but I'll be looking at the recommended standard way of doing things here.

In Fudge an actor rolls four special "Fudge dices" — six-sided dices numbered -1 to 1 — adds them, and adds his skill to the result and subtracts the difficulty.

This time a special note as to "difficulty" is probably required: the Fudge difficulty is still different from the one I employ in this text albeit subtly so. The Fudge difficulty subtracts directly from the DoS, meaning that a more difficult task will get lower DoS's. It's the resulting DoS that should be compared to the difficulty of this text, which has no effect on the DoS by definition. Confusing.

Given that the dice-pool is always the same, the Fudge mechanic is quickly resolved. It is relatively easy to use (although it may require a subtraction).

Skills are measured from -3 to 3 given a granularity of seven — lower than in the World of Darkness but still usable. The granularity of the DoS in Fudge is a bit better than was the case in World of Darkness — there's nine different achievable DoS's for each skill level, although high DoS's are still exclusively reserved for highly skilled actors and vice versa.

The DoS distribution of Fudge is plotted on figure 5. As can be seen, the average rises with skill (most systems seems to be able to grasp this), while the variance is constant. Furthermore the distribution is the most beatiful bell-shape yet — it's definitely almost a normal distribution.

I'd say that Fudge is at a close tie with World of Darkness. Fudge is simpler, and this I believe is it's greatest strength, as well as its weakness — the granularity means that it can be difficult to achieve the degree of precision desired, but on the other hand Fudge doesn't have the problems with coarse-grained results for low-skilled characters. However do note that Fudge has unachievable DoS's too.

*Figure 5. The DoS distribution of Fudge.*

# Conclusion

While it's difficult to even define the properties of a perfect dice mechanic, I often find that the ones employed in roll-playing systems have more serious flaws than I'd like. From the uniform distribution of D&D's mechanic, over the coarse-grained DoS in the storyteller system, to the overly simplified system of Fudge, I'm still searching in vain for a system that can actually achieve normal distributed DoS's with variable variance depending on skill.

Based upon the above analysis, I'd say that it's understandable that such a system haven't yet been found — the probability distribution of dice rolls seems to have a life of their own, and even the most innocent looking mechanic often turns out to have weird, almost erratic, behavior in extreme cases. I have not given up hope though; the desired properties exists out there, they are just distributed among many different systems and mechanics at the moment. One can only hope that a system can be found that contains enough of these properties at once, that it can be considered close enough to perfection.

# Appendix

For those of you who knows Torben Mogensen's excellent dice rolling probability calculating language Troll [2], here follows the Troll definitions I've used for the above calculations.

D&D:

skill := 0

D20 + skill

Old World of Darkness

diff := 6;

skill:=8;

pool :=

foreach x in skill#d10 do

if x = 1 then -1 else if x < diff then {} else 1;

if 0 < pool then

max (0 U sum pool)

else

if 0 > pool then -1 else 0

New World of Darkness

{{skill:=8;

pool :=

foreach x in skill#(accumulate x:=d10 while x = 10) do

if x = 1 then -1 else if x < 8 then {} else 1;max (0 U sum pool) }}

Fudge:

skill:=0

sum 4#(d3-2) + skill