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Roleplaying Probability Models

June 18, 2012

Roleplaying, and system design in particular, has always interested me. Roleplaying and economics have much in common. Both activities are an attempt to represent the real world as a set of rules and probability distributions. Economics is mainly concerned with creating models which can capture the essence of the effects of a policy or a firm’s behavior or something similar. Roleplaying systems strive to capture the essence of a collaborative story – the conflict in particular, because without conflict there is no need for a system. For economic models, the goal is to capture tradeoffs. Unlike economic models, however, roleplaying systems require all probabilities to be generated by dice rolls.

Flat Probability, Single Roll
The simplest method of modelling probability is a single die roll. If you get at least a particular number, you succeed, otherwise, your character fails. The problem I have with these systems is that extreme events are just as likely as the average event, as every outcome has equal probability. While this makes play exciting and unpredictable, it is unrealistic for most things people face in the real world and can be frustrating for players.

Single roll systems put a burden on the GM to figure out how difficult an action should be for a wide range of tasks and skill levels. For example, in a percentile system, a highly skilled character might have a skill level of 75%, and a low skilled character 25%. If the roll is used unmodified, the difference in success rates between a high and low skilled character is quite small relative to the real world. Someone who performs a task every day would find it easy and have a 99% success rate, even as a novice would have trouble getting it right even once. That forces the GM to apply a custom modifier to nearly every situation and skill level, which defeats the purpose of having the system to begin with.

Likewise, with the d20 system, skill rolls over-weight low skilled characters relative to high skilled ones. The range of skills for a mid level campaign is about 10, but the randomness varies by 20. A strength 8 character has a 26.25% chance of beating a strength 18 character in a opposed roll. On an unmodified 3d6 roll, an 8 represents the 26th percentile, and an 18 the 99.5th+ percentile. This system is basically saying that Bill Gates has about a 20% chance of winning the gold in the Olympic deadlift. If the die were smaller, you’d run into the reverse problem. Success would depend almost exclusively on the modifier, removing the random element and making the game boring. Feng Shui runs into this problem, I think, despite having a two dice system.

Multiple Roll, Summed
As the number of dice is increased, the probability distribution more closely approximates the normal distribution, which is a distribution which crops up all the time in the real world. It approximates the intuition that the average event happens more often than the extremes. The disadvantage is that the more dice you add to the system, the harder it is to add them up, which slows down play. I think 3d6 is a good compromise for this type of system.



Multiple Dice, Test Against a Difficulty
The third popular option is for the players to roll multiple dice, but instead of adding up the numbers, players compare each result to a target number. If the die roll is greater than or equal to the target, that die is counted as a success; if not, as a failure. Some systems have variable target numbers, such as Shadowrun or World of Darkness 3rd Edition. Some systems have a fixed target number for all rolls, and control the difficulty by requiring players to accumulate a number of successes based on the difficulty of the task, such as Exalted. This style of system does not require the players to do much math during play. With the variable difficulty systems, there might be too many variables out there for players to keep track of, and GMs might not have a good intuition of how difficulties affect various probabilities at the different skill levels. With a fixed difficulty system, you need a fair number of dice before you start to approximate the normal distribution. Still, this system seems to work pretty well for me, and it is easy to use once you get the hang of it.


At difficulty 7, on d10s. Each line represents another d10, starting from 1d10 on the left and going to 8d10 on the far right.

Update: If anyone would like me to make a new graph, just let me know and I’ll post it. I’ve got a pretty good spreadsheet for making them.

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One Comment leave one →
  1. August 9, 2012 10:46 pm

    Im incidentally building the cdf for Feng Shui atm.

    I tried for WOD (not sure which edition, 4th?) but calculating what happens when you roll a 10 is stymying me.

    thanks for your insights

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