White Wolf's World of Darkness Probability Tables

White Wolf's game mechanics typically look as though they were created by coked-up baboons.  I experienced a lot of frustration creating the probability tables for their 2004 version of the World of Darkness because of the mechanic of re-rolling 10s.  The whole game is like rolling damage for the arquebus in old D&D.  Technically, it is possible to roll forever, so I set some limits.  I stop calculating when the probability of an outcome drops below 0.001, or after the third roll.  It was relatively easy to find the probabilities of up to three successes with one die, then much more complicated for two dice, then after some struggles I noticed a pattern in how many separate equations must be solved and combined for each possible outcome (that meets my criteria) for a given number of dice:

So, instead of trying to solve over two thousand equations or trying to figure out how to write a computer program that would do it for me, I brute forced the problem.  I used Excel to whip up 10,000 sets of 3 rolls, then mixed them up twenty different ways, then averaged the occurrences of each number of successes.  The values in the following tables should be correct to about +/- 0.002.  (Click on the pictures to see bigger versions. I'm annoyed with blogspot's layouts.)

And, of course, by adding from the right we can find the probabilities of rolling at least any specific number of successes.  Remember that these probabilities include up to two re-rolls of 10s.

I hope that these tables will be useful to game masters (storytellers) in determining difficulty levels of challenges for characters in their campaigns.  They may also be helpful to players in deciding how much to increase a skill or attribute.

Shadowrun 4th Edition Probability Tables

This is a straightforward post of dice tables for 4th edition Shadowrun.  A player rolls a number of d6s equal to the sum of the character's relevant attribute and skill.  Dice that come up 5 or 6 count as successes.  If at least half of the dice come up 1, a Glitch occurs (something bad) even if there are also successes.  A Glitch with no successes is a Critical Glitch (something very bad).  These tables assume that players do not spend edge.  The first table shows the probability of each outcome for up to 15 dice.

What I find particularly interesting here is that the likelihood of both types of Glitch actually increases from 1 die to 2 dice, and that pattern continues for each even number of dice.  There is a kind of mechanical penalty for being better at a task half of the time, though probabilities of successes always increases.  This bizarre feature of the Glitch system does not make any sense from a simulation or game balance perspective.

This table may help game masters determine task difficulties based on characters' dice totals and the probabilities of success the GM wants in each situation.

Dream Pod 9's Silhouette System part 2

When we last left our intrepid system, I had graphed probabilities of success at various difficulty thresholds for different skill levels, holding attributes constant, and I made the claim that the system encourages min-maxing.  Let us follow up on that.

This table shows the probabilities of succeeding at tasks with a difficulty threshold of 6, which is hard, for various combinations of skill and attribute.  It also shows the character creation point cost of each skill/attribute combination, and the point cost per percentage change of success.

We see that costs mirror each other across the center diagonal, and it is a more efficient use of points to buy a high attribute than a high skill.  Even more important to note is that Attributes are applicable to many skills, and are sometimes used to determine secondary traits (Health is the average of Fitness, Psyche, and Willpower).  So, not only would it be more efficient to buy a high attribute for just one skill, there is compound efficiency for buying a high attribute in general and with multiple related skills.

This table shows the average probabilities of success across tasks with difficulty thresholds from 1 to 7.  That should be a relatively standard distribution, since a threshold of 4 is considered average.

A notable difference here is that it is actually more efficient in some cases to buy a lower attribute than skill, but only when the average probability of success is less than 51%, which is not usually desirable in a heroic simulation.  At the useful levels of success, it is still more efficient to buy a high attribute, even for a single related skill.  We also see that an attribute of 2 and a skill of 3 is a kind of sweet spot for good success at a moderate price.

There are many available skills, but the overwhelming majority of them are based on just three attributes: Agility, Knowledge, and Creativity.  As far as skills go, this means there is a strong incentive to min-max, and just pick one of the three areas to focus on while taking negative scores in the other two.  If you don't care about having a lot of skills, and want more of a brute character, scrap all three and raise the Build, Fitness, Psyche, and Willpower attributes instead.

Since those three attributes apply to so many skills, it would be more appropriate to make a new version of each of the above tables for each number of desired skills, adding in only the average attribute cost per skill.  I am not currently inclined to make a dozen more tables.  This would make it drastically more evident that a high attribute score (a 4 is possible for a starting heroic character, but a 3 is practically as high as even a min-maxer should go) allows for the most success at multiple skills for an efficient price.

Something that I did not highlight is the fact that the same point pool is not used to buy attributes and skills.  It is impossible to use a huge number of points on attributes and then buy a bunch of low level skills as it is in GURPS.  This system guarantees a set block of points for skills.  If you don't want a lot of skills, use your attribute points to build a brute, and buy high levels of the couple skills you do want.  If you want a lot of skills, pick one of the three polyskill attributes to focus on, crank it up and buy many low level related skills.  I am glad that Dream Pod 9 split the pools this way, as it does slightly limit min-maxing and forces characters to have skills, but the system still does encourage attribute min-maxing within its point pool.

Also interesting is that it is incredibly difficult to increase attributes during play.  The experience point costs are  different than the character creation point costs, and strongly incentivize buying skills during play instead of saving up to increase an attribute.

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?

Dream Pod 9's Silhouette System part 1

I had a lot of fun playing Heavy Gear by Dream Pod 9 with friends back in undergrad.  DP9 came up with a game system it calls Silhouette.  I remember a friend telling me that it was mathematically optimized for 8-sided dice, but they just changed the dice to 6-siders because people tend to have d6s lying around.  Whether or not that is true, there is a little funkiness in the system.  Here is a brief intro to the dice system, and I will have a deeper report later.

In this system, characters have attributes and skills.  To perform a task, the player rolls a number of d6s equal to his character's skill level (if the skill is 0, 2 dice are rolled and the lower result is used), and the highest result is used, with any 6s over the first adding 1 to the result.  Then the appropriate attribute is added to the die result.  So, a player whose character has an attribute of 2 and a skill of 3 will roll 3d6, take the highest die result (+1 for each 6 over the first) and add 2.  The total is then compared to a difficulty threshold, and the character succeeds if the result is higher than the difficulty.  If the total equals the difficulty, there is a draw, which usually favors a defender, and I am counting as a failure.  If all 1s are rolled, there is a "fumble" and something bad happens regardless of modifiers.

Attribute scores effectively take away from a task's difficulty (or add to it if the attribute is negative, which is common; 0 is average for an attribute).  A difficulty of 4 is supposed to be average, 8 very difficult, and 10 or more practically unattainable.  Even the game creators describe the progression of success as "peculiar."

Just a cursory glance shows us that attempting tasks with a skill of 0 significantly risks fumbles, that there are diminishing returns as skills increase (I'll go more into that in part 2), and that you shouldn't count on rolling more than one 6.  Also, you can see how important attributes are, shifting the entire graph to the left or right.

We had a sniper in our party with a dexterity of 3, which is a very high attribute.  He was ridiculously successful at dexterity tasks for which he had little or no skill, such as piloting a Gear.  This system cries out for min-maxing, and game masters should be ready to impose limits and say the magic word: "no."

GURPS Optimizing: DX and Skills

This post is a continuation of the previous post on IQ and skills. When playing GURPS (Steve Jackson's Generic Universal Role-Playing System), what is the best DX score to buy to make the most efficient use of your character creation points?

DX has the same point costs and levels as IQ, but the related skills are priced differently.  Skills are only easy, average, or hard.  An easy skill costs 1 point to buy at the same level as the character's DX, an average skill costs 2, and a hard skill costs 4.  Lower levels can be bought for half the price as the next higher level, down to .5 points (so .5 points will buy a hard skill at the DX-3 level).  Higher levels cost twice as much as the previous level until the cost is 16 points (e.g.: average skill at the DX+3 level), then each level costs 8 more points than the previous level did.  Buying hard skills above one's DX score is very expensive indeed.

Here is a graph showing the probabilities of success with average DX skills by cost and DX score:

I also created a table that you can use to easily see what DX to buy after you decide how many average DX skills (or equivalent) you want and how often you want to be successful using them (success probabilities below 25% are not shown):

*Though an DX of 3 is technically optimal in these cells, a score up to 9 may be a better choice because the point cost difference is very small and DX will have impacts on game play beyond the cost of skills.

You can see that this table is much easier to use than the IQ table.  The multipliers to convert easy and hard skills to average skill equivalents do not change (they change for some inefficient arrangements, but not for any optimal arrangements).  For all white cells: easy skills count as .5 average skills, and hard skills count as 2 average skills.  For the grey cells, the multipliers vary, but are irrelevant.  

Similarly to the IQ table, we see that a score of 13 is great if you want your character to be pretty consistently successful at a good handful of skills.  A below average (10) DX score is only appropriate if you want your character to never be effective at more than a single DX skill.  Min-maxers may be tempted to trash a stat to get points to spend on the other, but I would be hard-pressed to think up a character (okay, besides a Professor Xavier type) that would be better off with a low DX.  In fact, it seems that most characters should have both a DX and an IQ of at least 13, and would not be significantly sacrificing uniqueness or wasting points.  

A nit-picker may point out that many skills have default values (e.g.: DX-6), so characters technically have lots of skills for free.  The default values are such that even with a DX of 15, the character may have a 50% chance of success with an average skill.  A DX of 17 only gives about a 75% chance of success.  High enough DX or IQ scores to make default skill levels reliable are so expensive that a character will have significant opportunity costs for other features that were passed up.  The term "Jack of all trades, master of none" comes to mind.  It is generally more effective and efficient to be a specialist, and spend the points on actually having the skills you think will be useful for you.

GURPS Optimizing: IQ and Skills

When playing GURPS (3rd ed.), what IQ should your character have in order to optimize your use of character creation points?

GURPS lets players buy skills for their characters at different costs depending on the levels of the characters' attributes and the difficulty of each skill.  I will focus on the IQ attribute and its related skills for now.  IQ is purchased with points, ranges from 1 to infinity (practically 20), and defaults to an average score of 10.  Increasing IQ costs 10 points for each of the first three increases above 10, then 15 points for each of the next two steps, 20 points for each of the next two steps, and 25 points for each subsequent step.  So, having an IQ of 18 costs 125 points.  A character can get points back by having a below average IQ, earning 10 points for each level below 10 (except, inexplicably, the only 5 points for level 8).

Skills are either easy, average, hard, or very hard.  There are very many mental skills.  For 1 point, you can get an easy skill at the same level as your IQ, an average skill at IQ-1, a hard skill at IQ-2, or a very hard skill at IQ-3.  Spending half a point instead of 1 reduces the skill level by one (this is the lowest you can go), or you can increase your skill level by 1 for an additional 1 point (additional increases cost more per point: 2 each for the next three increases, then 4 each above that).

To succeed at a task, you roll 3d6 (3 six-sided dice) and sum the results.  If the sum is less than or equal to your skill level, you succeed, but a sum of 17 or 18 always fails (critical successes and failures are not important for this analysis).  Given our knowledge of the distribution of results on 3d6, we can figure out the best way to spend our points on IQ and skills for our characters.

Step 1) Figure out what mental skills you want your character to have, and what percent of attempts you want your character to be successful using them.

Step 2) Consult the table:
This shows the optimal IQ to have for a given number of average difficulty level skills (or their equivalents, see notes below) at a given average success rate (I did not bother showing success rates below 25%).

*Though an IQ of 3 is technically optimal in these cells, a 9 may be a better choice because the point cost difference is very small and IQ will have impacts on game play beyond the cost of skills.

How to convert non-average difficulty skills into average skill equivalents:

• White cells:  Easy skills count as .5, Hard skills count as 2, Very Hard skills count as 3.
• Black cells:  Easy skills count as .5, Hard skills count as 1.5, Very Hard skills count as 2.
• Medium grey cells with white text: Easy count as .67; Hard count as 1.33, Very Hard count as 2.
• Light grey cells: Easy count as .75, Hard count as 1.25, Very Hard count as 2.
• Medium grey cells with black text: conversion is variable and not worth including

Examples:

Derp is a big tough guy who lets his sword do most of the thinking for him.  He doesn't need many mental skills at all.  He takes four, but he wants to be really good at them because failure makes him mad.  An IQ of 3 would let him be regularly successful with his four mental skills while freeing up the most points to spend elsewhere, like on ST, DX, and HT.  An IQ of 9 may be a small but worthwhile expense, though, depending on role-playing goals.

Brainiac McGee knows how to do everything.  He got his first doctorate at the age of 7, and went back for more.  He is never (hardly ever) wrong, so the desire success rate for everything is 98.1%.  Since he'll have more than 9 mental skills, he'll definitely be in the white cell area, so very hard skills count as 3 average skills, and hard skills count as 2.  Somewhere between 40 and 50 average skill equivalents is where an IQ of 18 is a better buy than an IQ of 17.

Bob is a suave dude.  He's got a good mix of social and professional skills.  He's not the best, but he gets by pretty well.  With a success rate target in the mid 80%s, an IQ of 13 is best for 5-14 average skills.  If he wants to be better at what he does, and has more than 7 skills, he should increase his IQ.  If he has 7 or fewer skills and wants to be better, he should just spend points on the skills.  An IQ of 13 is great for someone who wants to excel at a few core mental skills, especially if they are hard.


If you want to have a bunch of skills at different success rates, you're on your own.  This table was a beast, and I'm not making one for each of the thousands of likely combinations of skill success rates that people might want.  I recommend just planning to be really good at whatever skills you want, and working from there.  Or just have an IQ of 13 and don't think about it too much.