Graphs of Success Probability by Skill Total and Difficulty

I've given you tables of success probabilities by skill total and difficulty for two systems (World of Darkness, Shadowrun 4th ed.), plus a graph for Heavy Gear.  Here I present that information again in graphs, plus two more systems, to show some of the different patterns that exist for success probabilities with increases in skill among different systems.

Linear

Here is your standard d20 system, most popular in Dungeons and Dragons.  Each character has a skill modified by an attribute and various other junk, added to a d20 result and compared to a difficulty level.  Each increase in the skill total raises the probability of success by 5% linearly.  There is always at least a 5% chance of failure (rolling a 1).  In the D&D games, skills are not bought with general character development points, but characters are alloted a few points each level to be used only for skills.  Difficulty levels typically scale with character levels, so it behooves players to specialize in a few skills that are always increased with the character level in order to maintain good probabilities of success as characters level up.  I am not getting in to "taking 10" or "taking 20".

Inconsistent

Here is the graph for Dream Pod 9's Silhouette system, used in their Heavy Gear game.  We can see that the progression is not consistent.  The lowest skill is concave, rapidly dropping the probability of success at low difficulties relative to the drop at higher difficulties where the probability of success is already very low.  A skill of 1 has a linear descent.  Higher skills progressively maintain high success rates among lower difficulties before rapidly plunging at higher difficulties, and then there is the bent tail as it becomes more possible to roll multiple 6s.  Attribute bonuses are added to skill roll results, shifting the graph to the right without changing its shape.

Normal

Isn't that pretty?  I am not sure if I am completely representing the GURPS system accurately here, but I think players just have to roll lower than the characters' skills on 3d6 to succeed at tasks (17s and 18s fail).  So, there is no real "difficulty level" for tasks other than what is forced by skill levels.  There may be modifiers that increase or decrease a skill for the purpose of a challenge, shifting the whole curve to the left or right.  If we graphed the probabilities of each individual outcome for 3d6, the line would be shaped like a bell.  I call this "normal" because as a "normal distribution" it has higher probabilities of outcomes in the middle, progressively less likely outcomes away from the middle, and is relatively symmetrical.

Inconsistent Normal

We can see here that both Shadowrun by Catalyst Game Labs and World of Darkness by White Wolf approach the normal curve as their dice pools (skill total, or skill + attribute) increase.  With few dice in these systems, it is impossible to approximate the distribution of the normal pattern, and the results more follow the Inconsistent pattern.  These systems both involve rolling multiple dice (d6 and d10, respectively), and counting die results over a threshold as "successes".  Players need a number of successes equal to a task's difficult in order to succeed.  So, the terminology can get annoying as people get a bunch of successes but still fail at a task.

I really like how the Normal distribution of probabilities of success works in simulations, but not necessarily the way that GURPS implements it in the absence of difficulty levels.  In real life, when we encounter tasks far below our skill level, we are quite likely to succeed at them and have a low variance with our high success rate.  When we encounter tasks far above our skill level, we are quite likely to fail at them and have a low variance with our high failure rate.  Tasks closer to our skill level have increasingly variant success rates.  Because of this, I am in favor of the use of normal distributions of probability of success in simulation systems.  This typically requires rolling more than one die and summing the results.

RPG Mechanics Taxonomy: Probability Scales

It's taking me longer than I wanted to finish some analyses of Dream Pod 9's Silhouette system, so part 2 will be delayed while I tell you a bit about role-playing game mechanics taxonomy.

Humans love to find patterns and name things.  Our brains do it automatically, giving us ways to predict the outcomes of novel situations by comparing features to those of situations we have been in in the past, though not always accurately, often resulting in bad stereotypes and superstitious beliefs.  When we are mindful, we can harness this wonderful ability to organize information to facilitate analyses, searches, and predictions.  We give names to groups of items that share patterns of features.

Dr. Wayne Saunders of the Museum of Man developed a taxonomy of games that identifies 20 types of game based on how they are played, and three categories of games based on victory criteria.  He classifies role-playing games as "production" games because the goal is to create something rather than to win.  Within RPGs, I further break down games into categories based on patterns in their mechanics.

 In this post, I am going to focus on scales of probabilities that characters succeed at tasks they attempt as they relate to character creation or experience point costs.  There are other mechanics that I will address in later posts.  There are two main categories of probability scale features:  the method by which the scales are determined, and the patterns of increases in probabilities of success as they depend on point costs.  I will abbreviate these to Method and RoI (Return on Investment).

There are four Methods:

• Utility: The point costs of probabilities of success at tasks are intended to reflect how useful the tasks are in the game.  Tasks that occur frequently in the game would generally cost more points for a given success probability than for less common tasks.  In a typical RPG, Karate comes up a whole lot more often than Basketball, and would cost more points to be good at even though training in those skills may take similar effort in real life.
• Realism: The point costs of probabilities of success at tasks are intended to reflect how much effort it would take in the realm world to achieve such a success probability. Being an expert historian of spoons would cost the same points as being an expert computer programmer, even though one is clearly more useful (I won't tell you which one). 
• Constancy: The point costs of probabilities of success at tasks are all the same, regardless of task utility in-game or the effort required to learn how to accomplish the tasks in real life.  
• Arbitrary: The game creators just assigned costs to success probabilities without strictly or clearly using one of the other three methods.

There are four Returns on Investment:

• Increasing: The increase in probability of success increases with additional point expenditures.
• Decreasing: The increase in probability of success decreases with additional point expenditures.
• Equal: The increase in probability of success is the same for every point expenditure. 
• Inconsistent: The increase in probability of success is sometimes higher or lower or equal for successive point expenditures.

We may be able to place RPGs on a table based on their Methods and RoIs.

How would you classify your favorite system?