Rolemaster Open Ended Rolls

Copyright Per-Anders Staav © 2009

Edited by Peter Mork for The Guild Companion

"Together these design choices form a rather attractive mix"

The topic of this article is to explore the extent to which the dice mechanics in Rolemaster are realistic. For some players, this discussion is irrelevant since they are not looking for realism. However, other players want a realistic game and they may worry that the open-ended roll (d100-OE) is unrealistic. This article discusses the issue in depth and concludes that the d100-OE mechanic is a reasonable compromise.

What is realism?

Claiming realism in a game is tricky business. Often detail-oriented mechanics are perceived as realistic. Unfortunately, some things that do happen in the real world may become highly improbable or outright impossible based on the mechanics. Others argue that more abstract rules make for a more realistic game. In some sense, this observation is necessarily true; details cannot be incorrect if there are no such details. Everything could be reduced to "the chance for success is 34%." Such a game is easy to understand mathematically, but likely results in a dull game.

Most likely there is no universally true answer about what level of detail is needed to get a feeling of realism. However, it is possible to determine if dice mechanics add a bias to the events of the game. Sound statistical properties for the dice rolls will mean that a skilled character is proportionally better than a less skilled character. Bad statistical properties in a game risk creating sweet spots where the game mechanics make sense and other parts where a character's skill has little impact on success.

In this article, I assume that if the dice mechanics generate results that are compatible with the distribution of observed events in the real world, then the dice do not detract from the game's realism.

Events in the Real World

There are many statistical distributions that model some real world situations well and other situations poorly. In the absence of a specific scenario, the Central Limit Theorem provides a reasonable basis for discussion. This theorem states that after many independent trials, or in the presence of many independent factors, the result will resemble a normal distribution.

The normal distribution is also often called the bell curve or the Gaussian distribution after the German scientist Carl Friedrich Gauss. The distribution is not always the perfect one; for example, events are not always independent as the theorem requires, and one factor may have more influence than the others. On the other hand, in many real world cases the Gaussian distribution produces results that are reasonably accurate. The reason this distribution is called a bell curve is that most of the mass is collected in the middle with long tails towards plus and minus infinity.

Properties of the Gaussian Distribution

Two mathematical aspects determine the curve of Gaussian distribution: the mean value and the spread around the mean. To describe how the values are spread out is tricky, but intuitively if the spread is large, the curve is "flatter." Luckily the spread can be defined mathematically by a single number: the standard deviation. The standard deviation and mean precisely define any Gaussian distribution.

For the Gaussian, the result falls within one standard deviation of the mean 68% of the time. Within two standard deviation of the mean you find 95% of the outcomes. Three standard deviations away from the middle you have 99.7% of the outcomes. The tails extend out to infinity, but the chance for a particular value becomes negligible far out on the tails.

Implications of the Standard Deviation

The standard deviation is of vital importance for understanding how dice mechanics interact with skill. If the skill values a character can achieve are too small compared to the standard deviation, the dice will rule and skill will matter little. Obviously the reverse situation arises if the standard deviation is too small compared to skill bonuses---the dice roll becomes mostly pointless since it is the skill bonus that determines success. It is useful to note that there are no laws about what the correct ratio between skill bonuses and standard deviation should be. The correct ratio depends on the game's assumptions regarding average skill bonuses, average penalties, and the style of game.

Evaluating the Standard Deviation of d100-OE

The average of Rolemaster d100-OE rolls is 50.5. Unfortunately the standard deviation is trickier to calculate. The classic solution to estimate the standard deviation quickly is to evaluate when you have accounted for 68%, 95% and 99.7% of the outcomes, and to use those results to determine the standard deviation. One problem with applying this approach to the d100-OE roll is that a discontinuity arises at 90%. For example, a result of 96 is impossible to roll. This discontinuity makes the range 96-100 especially bad for estimating the standard deviation.

A very simple way to determine the approximate standard deviation is to calculate the Normal Distribution for all reasonable standard deviations and to select the best fit. The simplicity of the method does of course depend on the idea that it must be easy to compare if two curves are close or not. Statistics gives the needed tool to answer that question in the form of the Least Squares Method. If the curves are close, then the squared distance between the curves becomes small. This method is standard practice to optimize numerical problems, but unfortunately the range used for the error calculation does impact the final result. Theoretically, if the Least Squares Method is applied only to a very narrow range near the mean, we can observe a perfect fit between the curves in that range, and fail to account for large discrepancies outside the range.

The good news for our analysis is that different ranges have limited impact on the final result. For example, if the range only includes the flat region between 6 and 95, the standard deviation is 37. Expanding the range to -10 to 110 produces a standard deviation of 35. As we increase the range, the standard deviation stabilizes at 34. Considering that ten percent of the probability mass is contained in the long tails these tails are of obvious interest and 34 seems likes the best candidate for the standard deviation of d100-OE.

Visual Evaluation

d100 open ended and Gaussian curve

Plot 1 [[d100OE_plot.PNG]] shows both the d100-OE and the Gaussian curves. While this plot may be fun to study, it does not tell us anything about the impact on d100-OE on the game. Plot 2 [[d100OE_successchance.PNG]] shows us the chance of succeeding at a static actions as a function of total skill bonus. This plot assumes that a total of 111 is needed for success, based on the static action tables in RMSS.

success curve for Rolemaster skills

We can observe the following phenomena:

  • The range 15 to 85 generates very little error.
  • The chance to succeed with a net penalty (e.g., due to injuries) is greater than accounted for by a Gaussian.
  • The chance to succeed with a bonus greater than 100 is lower than would be expected.
  • Plot 3 [[d100OE_error_plot.PNG]] shows the difference between d100-OE and Gaussian in greater detail.
difference between open-ended and Gaussian

Properties of a 3d6 Curve

Many gamers argue that the curve generated by rolling 3d6 is a better fit to the Gaussian. If we only concerned with minimizing the standard deviation, this argument is plausible. However, the limitation of 3d6 becomes quite clear when you look at the individual possibilities. The smallest probability of success is 1 in 216. 3d6 is simply not granular enough to be used to simulate really hard tasks.

3d6 curve

Additionally, we have the problem that a penalty has different impacts depending on the skill value. Only in the range 6-15 is the effect reasonably linear. Beyond this range, the probability of success is very non-linear. The expected number of rolls needed to observe 15 is 21.6, for 16, you need 36 rolls, for 17, 72 rolls, and for 18, 216 rolls.

3d6 success curve

Properties of a 10d10 Curve

10d10 plot

Rolling ten dice and adding them is far too cumbersome for regular table-top play, but this concern can be ignored if a computer is available. Even so, 10d10 is not attractive for game design. This distribution has extreme values that we are unlikely ever to observe. On the other hand, the standard deviation is 9 so most rolls (95%) will fall in the range 37 to 73. Falling outside the range 28 to 82 will only happen once in a thousand rolls. If this distribution is used, bonuses and target numbers need to be scaled to allow for reasonable chances of success. For example, 70 is probably a good threshold for success, although there is nothing intuitive about this threshold. Of course, the same problems regarding non-linearity apply to 10d10 as well.

10d10 success plot

Summary

How skill bonuses and penalties were chosen when Rolemaster was constructed is not known, but it seems pretty likely they were not random choices. Three standard deviations away from the mean for d100-OE is 50 + (34 x 3) = 152, giving results that are rather well aligned with RMSS skill progression and the maximum penalty of -70.

Together these design choices form a rather attractive mix that produces a piecewise linear dependency between penalty and effect at all points except the discontinuities at 1-5 and 96-100. Moreover, the difference between d100-OE is almost always less than a half a percent.

Looking at the results, it is pretty clear the skill progression, difficulty penalties, and target number are better parameters to adjust if more realism is desired in the game. One interesting question is the motivation for target number of 111 in RMSS compared to 101 in previous editions. From a user perspective, this change is suboptimal since it makes it harder to gauge difficulties. A RM2 skill bonus of +50 has a 50% success rate while a RMSS skill bonus need to be +60 for the same success rate. An in-depth exploration of this question exceeds the scope of this article. The important thing to note is that the d100-OE mechanic is a good design decision.