Mathematical Analysis
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Background
[edit | edit source]This section examines the quantitative foundations of The Dark Eye role-playing system, treating its rules as a structured mathematical model rather than only a narrative framework. By analyzing probability distributions of attribute or talent checks, combat rolls and hit points, as well as the scaling of attributes and skills, we can assess how fair, swingy, or predictable outcomes really are for different character builds and power levels. This mathematical perspective sheds another light on the underlying design choices of the system and outlines practical impact on balance, character progression, and perceived realism at the gaming table.
TDE 5 Details
[edit | edit source]Attribute Checks
[edit | edit source]Attribute checks are made by rolling a 1d20 against the attribute value plus potential modifiers. Rolling a 20 or a 1 is treated differently. They can lead to a critical success or failure. The player needs to confirm or avoid this, respectively, by rolling again against the same effective value as the first roll. When the second check succeeds:
- a potential critical success is confirmed.
- a potential botch was avoided and the check leads to a regular failure. Otherwise, it becomes a botch.
Furthermore:
- a double 20 is always a botch, regardless of the effective value rolled against.
- a double 1 is always a critical success, regardless of the effective value rolled against.
- an effective value of 0 or less does not allow a check.
- Hence, the effective range is between 1 and — in theory — infinity.
For practical reasons, the attribute and modifier are combined to form the effective attribute value (EAV).
The chances of achieving a critical success or botching are between 0.0025 (0.25%) and 0.0475 (4.75%). For an EAV of 19 or above, only one in 400 checks results in a botch, while 19 in 400 yield a critical success. For an EAV of 1, the probabilities are exactly reversed.
The chart shows the probability of each outcome depending on the effective value.
Please note: it is not exactly documented what happens when a character rolls a critical confirmation for an EAV of 20 or greater. Will a 20 in the confirmation roll cause a failure and reduce the check to a plain success? It is not in the rule book. Here is assumed, a) symmetry is the more elegant design choice, since a second 20 is a botch no matter what and b) it makes the game more interesting when there is always the chance that a critical cannot be confirmed.
Skill Checks
[edit | edit source]The mathematics behind skill checks is rather different. This is not only because we use three dice instead of one; let's first look at critical successes and botches. Since there is no confirmation roll, their probability is unaffected by the character's attributes. They are fixed, and both are below 1%. Approximately one in 138 checks yields one of these results.
To determine the 'regular' results we need to combine the probability distributions of 3d20. But we do not want the roll result per se. We need to find out how many skill points are left. Therefore, we have to truncate and rectify the values. In other words: when we roll a result below the effective attribute, we do not gain skill points. That is why we do not use the rolled faces in this case but the value of the effective attribute. The images show how that looks like for a check against 7, 10, and 13. In the example on the left: consider a roll against an e.g. charisma 7. When you roll a 5 (or any other value below 7) you replace that result with 7. When you roll a ten, the result stays 10.
The image show the distributions for a d20 roll against the 7, 10, and 13.
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Distribution of a 1d20 against 7 -
Distribution of a 1d20 against 10 -
Distribution of a 1d20 against 13
In the next step 2 we combine (i.e. convolve) the three distributions. The is practically the distribution of the sum of the 3 dice. In this case, however, we are not using the probability distribution of three fair dice, but of three modified dice from the previous step. We can now almost determine the remaining skill points.
However, let us not forget about critical successes and botches. The probability of these can easily be determined (see above). Therefore, we can simply subtract them from the probability distribution. There is no image to illustrate this because the changes are so small that they are not visible.
We can now work with this 'purged' distribution in the following image. To find out how many skill points we have left, we only need to add up the three dice in a roll, look up the sum on the x-axis, and count the bars up to the leftmost one. Each bar represents a skill point spent minus one.
To see how the distributions change, look at the following images. All attributes are increased by one or two. While the basic shape of the right-hand tail is preserved, the entire distribution shifts to the left, with the large bar on the far left becoming relatively larger.
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Distribution of a skill check, 7, 10, 13 against 8. -
Distribution of the skill check with a bonus of +1 -
Distribution of the skill check with a bonus of +2
On the other hand, looking at the same effective attribute values but changing skill, the basic distribution does not change at all. The only difference is that the number of bars falling within the range of a successful check increases.
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Distribution of a skill check, 7, 10, 13 against 5. -
Distribution of the skill check, 7, 10, 13 against 8
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Distribution of the skill check, 7, 10, 13 against 11
Publications
[edit | edit source]Primary Sources
[edit | edit source]- The Dark Eye: Core Rules page 18-20 (attribute checks, effective attribute value), 21-24 (skill checks)
