Guide to Die Rolling
The purpose of this page is to present the basic mathematical properties of some of the common die rolling methods used in role playing games. I always think it best to consider the needs of a game, in terms of the mathematics, before actually looking at mechanics. I hope that this page will be an aid to game designers who would like to know such things but who do not have the time or mathematical inclination to calculate them.
I've presented six distinct rolling methods below (though one is arguably a type of another). This is not necessarily an exhaustive treatment of all possible methods, but I think it covers all the commonly used ones, and any other mechanic could probably be reached by modifying one of these. This does not preclude additions to the page later, however, and if you know of a rolling method that I've left out, please tell me about it.
Contents
- Notes
- Basic Probabilities
- Summing
- Target Number
- Highest Die
- Die Keeping
- Lowest from Highest
- Cut-off
- About this Page
Notes
This section covers the assumptions this document makes and some generalities that apply to all the mechanics covered.
Nomenclature
Most people who read this will probably already be familiar with the common “d” notation, e.g. 3d6, where “3” represents the number of dice being rolled at once and “6” represents the number of sides on each die. In all of the methods below I've stuck to using d6 where a die size needs to be assumed. Obviously, any size would work. Reading the Basic Probabilities section should give you all you need to consider what the effects of changing the die size will be.
Although most gamers know that the singular of dice is “die,” I commonly hear the word “dice” being used as an adjective, as in “dice mechanics.” However, the adjectival form of a word derives from the singular case. You would never say “records keeping” or “shoes box,” and the same applies to dice. Thus my use of “die mechanics” and “die keeping” throughout this page.
What's Important
Whenever considering a die mechanic, there are usually two things a game designer will be interested in: mean average and distribution. The mean describes what you can typically “expect” from the roll: after rolling 10,000 times, this is what you would guess the die would roll on any given occasion. The distribution is what other results, besides the mean, you could get and how frequently you can expect them.
One aspect of the distribution is the range, which describes the maximum and minimum values possible. Related to range is “granularity.” A large total range (meaning a lot of possible values) produces a “fine” output, with which it is possible to distinguish between many possible outcomes. For instance, a die mechanic might tell me exactly how many feet I'm able to run in a second. Conversely, a “coarse” result has a narrow range: the mechanic would only tell me whether I'm fast enough or not. A coin toss is the ultimate in granular mechanics: having just two results of heads or tails with nothing in between. A helpful analogy might be the difference between digital (coarse) and analog (fine).
Also important when looking at any mechanic is the number of variables it uses. A variable (or input) is something you can change about the mechanic to alter the way it works; to alter the output. For every variable a mechanic has it can relate to another element of the game. For instance, a mechanic with three variables could relate together the physical strength of a warrior, the sharpness of his sword, and the toughness of a dragon's skin. There are two modifications that can be added to any die mechanic to give it more variables:
Arithmetic
The first universal input is any kind of simple arithmetic operation. For example, one could roll a six sided die and the subtract 3 in order to have results that range from -2 to 3. Thus arithmetic can manipulate the range and average you get with a particular die system so that it better works in the context of your game's other mechanics (I call this calibration).
A seprate but common use of arithmetic is to simplify a situation down into a smaller number of inputs which can be put into the die mechanic. For instance, in the above example we have the inputs of warrior strength, sword sharpness and dragon hide thickness. Ideally, we would like all of these to interact to produce a final result. Actually using three inputs may be considered too awkward here though, so simplifying the situation is preferable. The easiest thing to do is combine these theoretical inputs into a smaller number of actual ones, e.g. by adding together all three terms into a single “difficulty.” Of course you can use multiplication and division, or any mathematical function, but these are generally more than players want to deal with during a game and become cumbersome.
Die Size
Gamers more than anyone are aware that dice need not come with six sides only. Thanks to a relatively wide variety of die sizes that are commonly available, a game can alter the size of die used from one situation to the next as an input in the mechanics. For instance, if I'm just an apprentice plumber I may roll a d4, while if I'm a master I roll a d20. The game Alternity by TSR/WotC uses such a mechanic, and Sometimes it is called a “die step” system.
The most significant problem with modifying die size is that you have a narrow range of input options: dice only come in about six common sizes, and even then there is a gap between the twelve- and twenty- siders. This creates a corresponding gap in the probabilities, so the d20 is commonly left off, limiting you to just five options (d4,d6,d8,d10,d12). Some interesting ways to combine dice in one roll are dicussed later.
Both simple arithmetic and die step mechanics can be employed with all the systems discussed below, and thus they receive little further discussion.
Basic Probabilities
This part of the document is intended for people who need to (re-)familiarize themselves with the basic probabilities involved when rolling dice. It is not an exhaustive guide to probability in general, but you should be able to calculate some basic things yourself after reading it. If you already know a little about probabilities, feel free to skip it.
A Single Die
The most simple case of a die “mechanic” is the rolling of a single die (and reading the result from the top face). Rolling a single die doesn't provide many inputs: zero in fact, since you always just roll one die, and never do anything else. Board games like Monopoly commonly use such mechanics.
The average for one die is easy enough to calculate: simply divide the number of faces by two and add one half. Thus, a d6 has an average of 3.5 while a d10 has an average of 5.5. The distribution for a single die is hardly even worth mentioning: every number from 1 up through the size of the die is a possible value, with all of them being equally likely. Below is a graph of the distribution for a d6, with results on the horizontal axis and the likelihood along the side (I'll keep to this convention throughout the page)
Single die distribution (chance vs result)
Distribution Types
As you can see, the above graph is a flat line, because each outcome (1 through 6) is equally likely. This means that although the average is 3.5 (still on a d6) you cannot expect any given value more than any other. If the die roll represented a character's success at a task, he might just as easily do very well as very poorly. This is in contrast to many other shapes the distribution curve might take.
Some different curves
On the left we have another linear, but this time increasing, curve, where you expect to roll on the high side of things. In the middle is the familiar “bell curve” or “normal distribution” where you expect most rolls to be close to average, with a few outlying results. On the right is a negative exponential curve. It is weighted away from very low values, towards middling and upper ones.
Averages
In addition to graphs of distributions, we will also be looking a lot at graphs of averages, as one of the variables in a mechanic is modified. For every variable, a different graph is therefore needed. All of the above curves are commonly found on average value graphs as well, although they need to be interpreted a little differently. Generally role-playing games use die rolls to simulate how successful a person was at some task, and one or more of the variables in it is indicative of his skill or training. Thus, as you read a graph from left to right, you consider how a character's chance of success increases as his skill increases. A flat curve would be very strange to find on an average graph, since that would mean that skill was irrelevant! We'll consider this sort of thing more in each method, as it is discussed.
Summing
variables: number of dice
The next step up, in terms of complexity, from rolling one die is rolling several of dice and summing their values. I just call this Summing. WEG's Star Wars (and other “d6” games) used a modified form of this system. Assuming (as we will tend to) that only one type of die is ever used, there's only one variable: the number of dice rolled.
The average value increases linearly as the number of dice is increased. To calculate the mean, simply multiply the average result for a single die (of the type being used) by the number of dice rolled.
Sum average (average vs dice)
The distribution for a summing multiple dice is much more interesting than for a single die, and produces the well-known “bell curve,” with most results falling towards the average. Below is the curve for Dungeons and Dragon's familiar 3d6 roll.
Sum distribution for 3d6 (chance vs result)
Uses
A linear progression is the easiest to understand in terms of, say, character skill or situational difficulty. Many also feel that a bell curve yields the most realistic results in terms of character success: character performance is to some degree predictable but it is still possible to do very well or very badly. Even if not realistic, the bell curve is somehow intuitive. The Sum mechanic also has a fairly wide range, especially compared to the variable input, producing a fine (as oppose to coarse) result.
The mechanic's chief drawback is probably “handling time,” since each die face must be counted. Although the variable range is theoretically infinite, there is a practical limit based on how many dice you're willing to have players roll.
The Summing mechanic seems to be used rarely, except when the variable is made into a constant by fixing the number of dice rolled. D&D's classic 3d6, for instance, is always 3d6 (except with an optional rule): it and does not vary based on anything to sometimes become 4d6 or 5d6.
Options
There is really only one possible option, which can indeed be made with any of the mechanics discussed on this page. This is to add another variable by allowing the size of the die rolled to change: 4d6 one time, 4d10 the next. This could be useful, but is limited because of the small range of dice that are widely available. Within the range of d4 to d12 the change in average and distribution is continuous, but if you allow another shift up to d20, there's a large gap. If this is avoided, the new variable will have 5 possible inputs, which may or may not be sufficient. (There have been rumors about unusually-sided dice being produced, e.g. d14, which would solve this problem. Unless you are selling your game commercially and can send along such dice in the package though, it's probably too much to expect your players to have them.)
Target Number
variables: number of dice, target number
This mechanic is a widely-used one, and instances of it can be found in many popular games. One rolls a handful of dice, some times called a pool of dice, and then considers each die's value separately in comparison with a set “target number.” If a die is higher than (or perhaps equal to) the target, that die counts (as a success or some such). The total number of counted dice is the result.
As can be seen from the following graph, increasing pool size has a positive linear affect on the average.
Target number average with target=3 (average vs dice)
Finding the average for any given pool size and target number is not that difficult. Simply use the following formula:
[ (die type - target) / die type ] * pool size
Varying the target number while holding the number of dice constant also produces a linear curve, although it is negative. Thus it is easy to use target number as an external difficulty, since an increasing difficulty score will intuitively lower the expected outcome.
Target Number average with 3 dice (average vs target number)
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.
Target number distribution with 8d6 and target=3
(chance vs result)
Common Variants
As I said initially, this is a very commonly used mechanic and can be seen in all kinds of games. At first glance, only a few may spring to mind, like the White Wolf games. However, in fact the vast majority of role-playing games use target number mechanics in a simplified form, and without the same name. By keeping the number of dice rolled constant, you end up with a simple “roll over the difficulty” system, like those used in D&D, GURPS and innumerable other games.
Uses
The Target Number mechanic has several advantages. It has two variables, and thus can relate to more than one game element. Both variables modify the average outcome linearly, which is an effect that comes fairly naturally to the human mind. The distribution is normal, which is often seen as desirable. It is also usually pretty fast, since counting successes is not hard to do (easier than summing up values). The principal disadvantage is that it produces fairly granular results: the number of successes possible ranges from zero to the number of dice rolled. This is again related to the practical concern of how many dice can comfortably be rolled and kept track of. If you want a simple way to generate a normal results, this is probably a good bet.
Options
Many smaller additions have been tacked on to the basic Target Number mechanic. You could count multiple 1s as an automatic failure and multiple 6s as automatic success. The appearance of doubles or triples could provide a bonus to the result or have some other, separate, effect. The “die pool” is also easily turned into a resource, so that a player must distribute his effort or some such between multiple tasks.
A few games have held the target number as a constant, so that rolling dice over a pre-defined value always counts as a success. Additionally, you could define multiple target numbers, so that each die could come in more than two states. For example, “failure, success, big success.”
As mentioned with summing mechanics, you can of course introduce another variable by allowing the type of die to vary. You could even allow more than one type (e.g. 1d4+1d6 is a worse attribute than 2d6). As is also possible with any other mechanic, you could also add a constant onto the result.
Highest Die
variables: number of dice
In this method, a handful of dice are rolled, and the single highest die is taken as the result. This bears some relation to the Target Number mechanic but really stands on its own. It's been used by Heavy Gear, the Star Trek games by Last Unicorn Games, and Ron Edwards' Sorceror. The average is not so easy to figure out, unfortunately, and has a somewhat interesting curve. As you can see, it's a negative exponential, so that after rolling a handful of dice, adding more has little effect.
Highest die average (average vs dice)
What might be surprising is that while the average graph is curved, the distribution is in fact linear, and also positive. This should make some sense though, since it only takes one high die for the final result to be high.
Highest die distribution with 2d6 (chance vs result)
Uses
The highest die method is primarily going to be of use if you want to model diminishing returns. There is only one input, and the output is fairly granular: from 1 to the die size you're using. At the low end of dice being rolled, adding one die has a significant effect (almost +1 on average), while at the higher end, a single die means little.
Options
One point of departure would be to do something not just with the highest die but the lowest die as well. This results (I think) in a fairly different system, which has been given its own section: see Lowest from Highest. Probably the most logical extension of the mechanic would be to take not just the single highest die, but the top two or three dice, depending on another input. As it turns out, this creates a larger category of rolling methods which this one is in fact just a part of. This larger group is the topic of the next section.
Die Keeping
variables: number of dice, number kept
A group of dice is rolled, and a certain number are “kept” while the rest are discarded. Those that are kept are added together for the final result. This has been used by the games 7th Sea and Legend of the Five Rings. The statistics involved are not at all intuitive, and it is perhaps the most mathematically complex rolling method considered here. Both variables positively influence the average, with the plot taking on the negative exponential, especially in the case of the pool size.
Die Keeping average, keeping top 2 (average vs dice)
Just like with the Highest Die method, a point of diminishing returns it quickly reached. Modification of the number of dice kept has a similar but less pronounced effect, being much closer to linear.
Die Keeping average with 5d6 (average vs kept dice)
The distribution is hump-shaped, like a bell curve, but pushed heavily towards the right. The most likely results are therefore nearly as high as is possible: no outstanding results can occur. On the other hand, the distribution trails off at the low end, so that poor results still do happen. (Assuming, of course, that high numbers are preferable.)
Die Keeping distribution with 5d6, 2 kept (chance vs result)
Uses
The curves of the die keeping method are definitely different from most of the others, reflecting its somewhat more complicated nature. Most designers seem to prefer a uniform bell curve to a heavily weighted one, but this by no means rules one out. Die keeping provides a fairly wide range of results, though this hinges heavily on the number of dice that are kept. One of its downsides is that the number of dice rolled will need to be fairly high so that the number kept can vary. Also, when the number of dice being kept becomes large, this method definitely slow down. If you're looking for a method that incorporates diminishing returns and yet also has another variable, or for one with a heavily weighted curve, die keeping is probably the best choice.
Options
Many of the options discussed previously are applicable here, especially those from the highest die and summing methods.
Lowest from Highest
variables: number of dice
With this method a number of dice are rolled and the value of the lowest die is subtracted from that of the highest die. I'm not aware of published game that uses this mechanic, but it is decidedly fast. The average value depends only on the number of dice rolled of course, and the curve is a negative exponential, as it was with the previous two mechanics.
Lowest from Highest average (average vs dice)
The distribution for Lowest from Highest is very interesting however. It takes the general form of a positive hump, somewhat like a bell curve but without the trailing ends. However, as you can see in the figure below, this curve changes shape dramatically as the number of dice is increased. With only 2 dice, the curve leans to the left, favoring low results. With 3 dice, it is fairly uniform (not quite though). With 4 or 5 dice, the curve begins to lean in the opposite direction, to the right. The more dice are added, the more heavily weighted the distribution is towards the higher values, which of course are capped by the size of the die being used (die size–1).
Lowest from Highest distributions (chance vs result) for 2, 3, and 5 dice
Uses
Interestingly, the changes in distribution weighting noted above happen to model an affect that many designers are apparently after: that with skill, a character should become less and less likely to tragically fail (the so-called “whiff” factor, as in missing a ball, not taking a sniff). As the die pool is increased, the result becomes more an more certain (more reliable performance from a character perhaps) as well as higher. Aside from this, the method is pretty fast. Its main disadvantages is that the range of results is absolutely constrained by the type of die you're rolling. If you need a fine granularity, you'll have to use a larger die.
Cut-Off
variables: number of dice, cut-off point
This is a mechanic I devised for a game of my own, and I've never seen it used elsewhere. It's somewhere in between the Summing and Target methods, though is also a close cousin of Die Keeping. Roll a handful of dice. Any die that shows a value greater than the cut-off point gets counted, while the rest get discarded. All the counted dice are then added together. The average is fairly difficult to calculate. As die pool increases though, it goes up linearly.
Cut-off average with cut-off=3 (average vs dice)
In averages, this method thus has a lot in common with the Target method. If the cut-off is varied, the average is influenced negatively, though in this method, the relationship is not linear but slightly logarithmic (below). If you used cut-off as a difficulty, this would mean that a high difficulty results in unexpectedly low values.
Cut-off average with 3d6 (average vs cut-off)
Cut-off shows itself to be extremely unique when you look at the distribution. In fact, it's really quite odd.
Cut-off distribution with 5d6 and cut-off=3 (chance vs result)
The trend is roughly hump-shaped, but it is hard to predict how likely any given value will be. For instance, if you use a cut-off value of 3, there is 0 chance of getting 1, 2, 3, or 7! For other cut-off points, there are other numbers that will never come up, always including a handful of low-end ones. Interestingly, the curve also has a bit of a tail towards the higher value, but not at the low end.
Uses
In many ways, Cut-Off is much like the Target rolling method. It has two inputs which have similar effects on the average, though one is slightly logarithmic here, so that the average changes more with higher cut-off values than with low ones. The range is fairly large, compared with that of Target, since you're adding dice instead of just counting them. The forward-pointing tail of the distribution also means that the mechanic could be used to model reliable behavior with occasional instances of truly great performance.
On the other hand, Cut-Off takes a little longer than Target, since it uses both comparison with a static number and the summing of dice. Moreover, the distribution is very strange, and low values are not even possible. If you were panning on using smaller numbers of dice, where missing a whole number would be very noticeably, you might want to avoid Cut-off.
About This Article
All graphs were generated in Microsoft Excel (despite its many problems). Much of the data came from Torben Mogensen's excellent Roll program (visit his web site.
All content is copyright Jasper L. McChesney, 2004. It may not be reproduced in any format without express permission. (You can't, however, copyright a mechanic. Use whatever ideas you find here freely!)