Last month, we looked at simple dice-roll systems where the number of dice you roll is independent of the situation and where you either add up the dice or compare each to a threshold.

This month, we will look at systems where the number of dice depends on ability. These are commonly called "dice pool systems".

Linear dice pools

Many games use a system where the ability of the character is translated into the number of dice that are rolled to determine success.

In some of these systems, the dice are added to a single value, in others each die is independently compared to a threshold and the number of dice that meet or exceed this threshold are counted. Modifiers can modify either the number of dice rolled, the required sum or success count, the threshold towards which the dice are compared or combinations of these.

Some examples:

• West End Games' d6 system adds a number of d6s equal to the character's ability and compares the sum to a difficulty level.
• White Wolf's ``World of Darkness'' system rolls a number of d10s equal to the ability and counts the number of results that are 8 or more. Rerolls on 10s increase the average somewhat (from 0.3 per dice to 0.333... per dice) and allow results in excess of the character's ability.
• Earlier White Wolf systems were similar but some had a variable threshold for the dice (determined by the complexity of the task) and there were no rerolls (though, in some variants, a 10 counts as two successes and a 1 cancels a success).

If each increase in ability adds a die to the pool, you will quickly have to roll a very large number of dice unless the range of ability is limited. White Wolf's systems limit attributes to a range of 1-5 and skills to a range of 0-5, so no more than 10 dice need to be rolled, and that only rarely. West End Games' d6 system (in some versions) has levels between adding a full die, which is another way to increase granularity without requiring a large number of dice: You go from d6 to d6+1 to d6+2 to 2d6 and so on. Dice pools are usually used in games where there is no need for very fine-grained differences in ability.

Comparing each die individually to a threshold is (for most people) faster than adding them (if the number of dice is the same), but the latter gives a wider range of results. This can allow more different difficulty levels, but whether it is useful to have many more difficulty levels than ability levels is doubtful.

Since the results of each die (which may be the straight value of the die or reduced to a smaller range of values, e.g., 0 or 1) are added, the average result increases linearly with ability, as does the variance. This has the effect that characters with higher ability have a larger spread in performance than do novice characters (although only in an absolute sense - the spread divided by the average result decreases). If the results of action rolls translate to real-world figures, this may seem counter-intuitive, but since such translations rarely exist (or, when they do, are often rather arbitrary), it is largely a matter of taste whether this is good or bad.

Nonlinear dice pools

The above-mentioned dice-pools are linear in the sense that adding more dice gives a linear increase in the average result. There are also games that use nonlinear dice-pools of various kinds.

One of the more complicated examples is from ``Godlike'' by Hobgoblynn Press. Here, you roll a number of d10 equal to attribute + skill, like in many other dice pool systems, but how you determine your result is quite different: You search for sets of at least two dice that show identical numbers and select one such set. The number showing on the dice in the dice determines how well you succeed and the number of dice in the set (called the width of the set) how quickly you accomplish the goal. There are optional ``hard'' and ``wiggle'' dice, which complicates things s bit, but we will ignore these and just look at the basic idea of looking for sets of dice and taking the set with the highest showing numbers.

We observe that the higher the result, the more likely it is. All pairs are equally likely, but since you will choose the highest-valued pair if you get more than one, the final result is skewed towards higher values. If the number of dice is low, the skew is fairly small (as the chance of getting two or more sets is small), but at eight d10, a result of 10 is nearly seven times as likely as a result of 1. The chance of getting no sets (i.e., all different values) is initially quite high, but it drops to under 50% at five dice and is less than 2% at eight dice. The average result actually increases more than linearly with the number of dice (doubling the number of dice more than doubles the average), at least up to the maximum of 10 dice used in Godlike.

There are other kinds of nonlinear dice-pool systems. One of the simplest is to roll a number of dice equal to the ability and then pick the highest result, as is done in Dream Pod 9's Silhouette. Here is definitely a case of diminishing returns: With d10s, the average result starts at 5.5 when you roll one die and gets closer and closer to 10 when the number of dice increase, but the average will never quite reach 10. The spread of the results decrease with the number of dice (as you are less and less likely to roll low), so you can say that this reflects that more able persons are more consistent. However, the effect of the diminishing returns is maybe too great: Even a rank novice with ability 1 has 10% chance of getting the best possible result and will have an average result that is more than half of what is maximally possible. Additionally, 10 (the maximum) is the most likely result already at ability 2.

To solve this, you can take the second-highest result of n dice (where n > 1). There is still diminishing returns and decreasing spread, but much slower than before. In particular, the chance of getting a result of 10 increases much slower, so it isn't until 14 dice that it becomes the most likely result. The distributions (when n>2) are bell curves skewed towards higher and higher values when n increases. Additionally, a character with skill 2 has only 1% chance of getting the best possible result, so you don't see novices achieve masterful results all the time.

Both the take-highest and take-second-highest method allow a low-skilled person a probability of achieving the best possible result. Some like this possibility, but others want to put an upper limit on the results obtainable by low-skilled persons. A nonlinear dice pool that achieves this is that you (as always) roll a number of dice equal to your ability, but then count how many different results you get. The number of different dice is bounded upwards by both the number of dice and by the size of the dice used. There is also diminishing returns, as adding more dice is less and less likely to add new different values. A disadvantage is that it takes slightly longer to count the number of different values than to find the highest or second-highest of the values (though not by much). If you use ndM, the average number of different values is M(1-(^(M-1)/_M)ⁿ). For nd10, this simplifies 10(1-0.9ⁿ).

Instead of counting different dice, you can also count identical dice. This is like the "width" of rolls in "Godlike", but not requiring at least two identical dice and not caring about the numbers shown on the dice (only the size of the largest set). This method also bounds the result by the number of dice rolled (but not by their size) and also has diminishing returns, albeit less so than if you count different dice. While counting different dice gives you a saturation curve (when plotting the average against the number of dice), counting identical dice gives you a curve that grows only a bit slower than linearly. Note that when you count different dice, larger dice will give you larger averages (as there are more different values), but when you count identical dice, larger dice give lower averages (as dice are less likely to show the same value). I would prefer using the count-different system with fairly large dice (at least d10s) to get a reasonable range of results, while I would not use the count-identical system with dice larger than d6, as larger dice give too small increases in average when you add dice. For example, if you use d10s, the average increases only by around 0.2 for each extra dice you add.

Avoiding rolls

Regardless of whether you use linear or nonlinear dice pools, dice rolls tend to take fairly long to execute: You have to pick up a largish number of dice (and be sure that the number is correct) and then do something afterwards that involves looking at all the dice. This gets even worse if there is a possibility of rerolls (as in the "World of Darkness" system). So dice pools are best suited to games where you don't roll all the time, i.e., where conflicts are resolved in a single or at most a couple of rolls or, even better, where most conflicts don't even require rolls to resolve.

You can resolve conflicts without rolls if you can see that even the best (or worst) possible roll won't make a difference to the outcome. Otherwise, players will tend to insist on rolling just for the off chance that they (or their opponent) will make an extreme roll, no matter how unlikely this is. To avoid this, you want a system where results are capped both upwards and downwards by ability, so people with low ability can not hope to beat persons of high ability (unless there are special circumstances that modify the odds). Some of the above systems do that. For example, a simple additive dice pool with ndX will have a range from n to n×X, so a person can not hope to beat an opponent that has a skill that is X times higher than his own. If X is high, this is rarely going to happen, though, so you might want a narrower range of results.

Or you could use a rule similar to the "take 10" of the d20 system: Instead of rolling, a player (or GM) can simply "take" a predetermined value. If one side in a conflict can clearly win the conflict by taking instead of rolling, you don't need to roll. Players in unstressed situations could be allowed to "take" their average result (which is pretty much what "take 10" does in d20), and in stressed situations they can be allowed to "take" a value that they would achieve in most rolls (say, 75% or more of the rolls), but thereby forgoing the possibility of a better result.

You would need an easy way to determine both "take" values from the ability. In linear dice pools, the average is fairly easy to calculate: For n dice, it is simply n times the average of one roll. Finding a threshold that you will meet in 75% of all rolls is less easy, however, as the spread does not increase linearly with the number of dice. For nonlinear dice pools, even the average may have a nontrivial relation to the number of dice rolled (see, for example, the formula for averages for the "count different" method described above). You can precalculate the distribution for all possible ability numbers and provide a table of "take" values, but that is a bit cumbersome to use in play unless it is small enough to fit on the character sheet. So it might be necessary to make fairly coarse approximations. For the method where you roll n d10 and use the second-highest value, it may be reasonable to say that you can take n (though max 8) in stressed situations and n+1 (but max 9) in unstressed situations.

Conclusions

Dice pools are more cumbersome than systems that use a single die or a small fixed number of dice, and they tend to scale badly (since you don't want to roll more than a handful of dice). But they give a direct translation of ability to roll, so you avoid having to add or compare the ability to the outcome of a roll.

You can modify difficulty by modifying the number of dice rolled in addition to modifying the number you must beat. For example, if you use the pick-second-highest method, reducing the number of dice by one will affect low-skilled persons more than high-skilled persons (due to the diminishing returns) while adding one to the target number affects all more evenly. This gives the GM a bit more flexibility when setting up challenges.

Dice pools allow all sorts of extra effects, such as treating specific numbers in special ways, such as rerolling tens or letting ones cancel successes. Or you can read several numbers out of the same pool, such as the width and height of "Godlike" rolls.

Overall, dice pools allow a wide range of design options that you can consider in your design. But by the same token, you must take care not to design a system that looks cool but is far too cumbersome in practice.

Getting odder still

Next time, I will look at systems that are not easy to categorize into classes. These are systems where you might use more than one type of dice in the same roll, or where the translation from roll to value is less direct than what we have seen so far.