Dice mechanics lie at the heart of many tabletop role playing games. These mechanics are often part of a resolution system that players use to determine what actually happens in the game when their characters attempt to do something risky. These mechanics provide a source of randomness that is used to model any uncertainty inherent in the proposed action and their outcome provides an objective measure of the character's degree of success.
Decibel is a generic dice mechanic that can serve as the chassis on which a complete tabletop role playing game can be built. It is a dice pool mechanic. To use Decibel as part of a resolution system, players make a check which consists of three steps:
- Assemble a pool of twenty-sided dice;
- Roll the dice and sort the resulting values from largest to smallest;
- Compute the outcome of the check.
The size of the dice pool depends on the action being attempted and any circumstances that might affect a characters performance. The outcome of the check is the difference between the largest and smallest values of the rolled dice. This outcome is then compared to some other quantity, which could be a fixed target number or the outcome of another check, to determine what happens in the game.
To make a simple check, players roll a pool of three or more twenty-sided dice (d20s). The outcome of the check is the difference between the largest and smallest values of the rolled dice. We will write NdB to denote the outcome of a simple check made with a pool of N dice.
Notice that while the set of possible outcomes (i.e. {0, 1, ..., 19}) is always the same, the size of the pool determines the distribution of the outcome of a check. In general, a larger dice pool is more likely to produce greater outcomes than a smaller dice pool.
Suppose that a player is making a simple 5dB check. The player rolls five twenty-sided dice and gets a result of {14, 10, 7, 6, 3}. The outcome of the check is 14 - 3 = 11.
This figure depicts the distribution for the outcome of simple checks for various values of N.
One shortcoming of simple checks is that the distributions of the outcome of all such checks are skewed in the same direction. Distributions that are skewed in the other direction can be generated by discarding dice with the greatest values after rolling. The outcome of the check is the difference between the largest and smallest of the remaining values. We will write (N-X)dB to denote the outcome of a standard check made with a pool of N dice and where the X dice with the greatest values are discarded before computing the outcome of the check.
Notice that a standard (N-X)dB check is not the same as a simple MdB check where M = N-X. The distributions of the outcomes of these two checks are generally quite different.
Suppose a player is making a standard (7-3)dB check. The player rolls seven twenty-sided dice and gets a result of {18, 15, 11, 10, 7, 6, 4}. The player then discards the three dice with the greatest values which yields the intermediate result of {10, 7, 6, 4}. The outcome of the check is 10-4 = 6.
This figure depicts the distribution for the outcome of standard checks for various values of N and X = N-3.
A common feature of dice mechanics used in many role playing systems is modifiers. There are two kinds of modifiers, positive and negative. A positive modifier increases the probability that a player succeeds at a static resolution roll or wins a dynamic resolution roll. A negative modifier decreases the probability that a player succeeds at a static resolution roll or wins a dynamic resolution roll. In Decibel, as in many dice pool systems, modifiers change the composition of a dice pool before a check is made. Positive modifiers can be implemented by simply adding dice to the dice pool. That is, by increasing the value of N in a standard (N-X)dB check. Negative modifiers can be implemented by both adding dice to the dice pool and increasing the number of dice that are discarded before computing the outcome of the check. That is, by increasing both the value of N and X by the same amount in a standard (N-X)dB check.
Notice that, because they both add dice to the dice pool, positive and negative modifiers do not simply cancel each other out. In general, as the number of modifiers increases the variance of the outcome of a check decreases. That said, allowing positive and negative modifiers to cancel each other out before applying the remaining modifiers leads to simpler accounting and more manageable dice pools. So, any game system that uses the Decibel dice system will need to specify how to handle opposing modifiers.
Suppose that a player is making a 5dB check with one positive modifier and two negative modifiers. The net result is that the player will make a (8-2)dB check.
This figure depicts the distribution of the outcome of a modified 3dB check for various amounts of positive and negative modifiers.
In a static resolution roll, the outcome of a Decibel check is compared to a fixed target number. This is frequently called a "skill check" in many role playing systems. If the outcome of the Decibel check is greater than or equal to the target number, then the check is considered a success. Otherwise, the check is considered a failure.
This table describes the probability of succeeding at a static resolution roll (i.e. NdB >= k) for various values of N and k.
| k = 0 | k = 1 | k = 2 | k = 3 | k = 4 | k = 5 | k = 6 | k = 7 | k = 8 | k = 9 | |
|---|---|---|---|---|---|---|---|---|---|---|
| N = 3 | 1.00 | 1.00 | 0.98 | 0.96 | 0.92 | 0.87 | 0.81 | 0.75 | 0.68 | 0.61 |
| N = 4 | 1.00 | 1.00 | 1.00 | 0.99 | 0.98 | 0.96 | 0.93 | 0.90 | 0.85 | 0.79 |
| N = 5 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.99 | 0.98 | 0.96 | 0.93 | 0.89 |
| N = 6 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.99 | 0.98 | 0.97 | 0.94 |
| N = 7 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.99 | 0.99 | 0.97 |
| N = 8 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.99 | 0.99 |
| k = 10 | k = 11 | k = 12 | k = 13 | k = 14 | k = 15 | k = 16 | k = 17 | k = 18 | k = 19 | |
|---|---|---|---|---|---|---|---|---|---|---|
| N = 3 | 0.54 | 0.46 | 0.39 | 0.32 | 0.25 | 0.18 | 0.13 | 0.08 | 0.04 | 0.01 |
| N = 4 | 0.73 | 0.65 | 0.56 | 0.48 | 0.39 | 0.30 | 0.22 | 0.14 | 0.08 | 0.03 |
| N = 5 | 0.84 | 0.78 | 0.70 | 0.62 | 0.52 | 0.42 | 0.31 | 0.21 | 0.12 | 0.04 |
| N = 6 | 0.91 | 0.87 | 0.80 | 0.72 | 0.63 | 0.52 | 0.40 | 0.28 | 0.16 | 0.06 |
| N = 7 | 0.95 | 0.92 | 0.87 | 0.80 | 0.72 | 0.61 | 0.49 | 0.35 | 0.21 | 0.08 |
| N = 8 | 0.97 | 0.95 | 0.92 | 0.86 | 0.79 | 0.69 | 0.56 | 0.42 | 0.26 | 0.10 |
In a dynamic resolution roll, the outcome of a Decibel check is compared with the outcome of another Decibel check. This is frequently called an "opposed roll" in many role playing systems. These checks are often used to determine which of two opposing forces prevails in a direct conflict between the two. As such, we say that that whichever check produces the greater outcome wins while the other loses.
If the outcomes of the two checks are equal, then neither check wins (or loses). In this case, some method of breaking the tie may be required. Simple options include re-rolling one or both checks until a winner can be declared, comparing the size of the dice pools used in the checks, or comparing the results of the dice in the dice pools that were not used to compute the outcome of the checks.
This table describes the probability of winning a dynamic resolution roll (i.e. MdB > NdB) for various values of M and N.
| N = 3 | N = 4 | N = 5 | N = 6 | N = 7 | N = 8 | |
|---|---|---|---|---|---|---|
| M = 3 | 0.48 | 0.35 | 0.26 | 0.20 | 0.16 | 0.13 |
| M = 4 | 0.60 | 0.46 | 0.37 | 0.30 | 0.24 | 0.20 |
| M = 5 | 0.68 | 0.56 | 0.46 | 0.38 | 0.32 | 0.27 |
| M = 6 | 0.75 | 0.63 | 0.54 | 0.45 | 0.39 | 0.34 |
| M = 7 | 0.79 | 0.68 | 0.59 | 0.51 | 0.45 | 0.39 |
| M = 8 | 0.82 | 0.73 | 0.64 | 0.56 | 0.50 | 0.44 |
In some cases we may be interested in how well (or badly) an attempted task is accomplished instead of simply determining whether the attempt succeeded or failed. One natural way to accomplish this for a static resolution roll is to compute the difference between the outcome of a check and the target number. Similarly, a simple way to accomplish this for a dynamic resolution roll is to compute the difference between the two checks. In either case, this difference is called the degree of success for the resolution roll. Greater greater degrees of success correspond to better performance whilst lesser degrees of success correspond to worse performance. The interpretation of what a given degree of success means is task-specific. The players should, either by adopting formal mechanics or by informal agreement, decide on how to interpret the results before the dice are rolled.
Suppose a player is making a 5dB check as part of a static resolution roll with a target number of 13. The player rolls five twenty-sided dice and gets a result of {20, 17, 12, 8, 5}. The outcome of the check is 20 - 5 = 15. The degree of success is 15 - 13 = 2.
The values of dice that are not used to compute the outcome of a check can be be used to determine whether any secondary effects occur as a part of the check. Because dice pools may have as few as three dice, two of which will be used to compute the outcome of the check, secondary effects may need to be determined by the value of a single die. One way to accomplish this is to use the largest value that was not used to determine the outcome of the check to govern any secondary effects.
Suppose that a player is making a 4dB check. The player rolls four twenty-sided dice and gets a result of {17, 15, 11, 3}. The outcome of the check is 17 - 3 = 14. Discarding the two dice used to compute the outcome of the check yields a result of {15, 11}. The value that governs any secondary effects is 15.
In a compound check a dice pool is used to resolve more than one check with a single roll. To do so, compute the outcome of the first check as normal. Then discard the two dice whose values were used to compute the first outcome and treat the remaining dice as if they were the result of a simple check.
Notice that even if the first check is a standard check (X > 0), all subsequent checks are simple checks (X = 0). That is, dice are only discarded once prior to computing the outcome of the first check. Therefore, if five or more dice remain after discarding the required number of dice, then that dice pool can be used to make a compound check.
Suppose that a player is making a (6-1)dB check. The player rolls six twenty-sided dice and gets a result of {19, 13, 8, 6, 5, 2}. They then discard the die with the greatest value which yields a result of {13, 8, 6, 5, 2}. The outcome of the first check is 13 - 2 = 11. Discarding the two dice used to compute the outcome of the first check yields a result of {8, 6, 5}. The outcome of the second check is 8 - 5 = 3.