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.

D&D and IQ

I first started playing RPGs about two decades ago with Advanced Dungeons and Dragons.  Back before 4th edition finally defaulted to a point-buy system for attributes, character attributes were determined by rolling six-sided dice (d6).  The basic and most scary method was to roll 3d6 and add the results, providing a total from 3 to 18.  The "heroic" method was to roll 4d6 and sum the highest three results, which still provided a total from 3 to 18, but with a higher mean.

It had generally been said in my gaming circles that Intelligence (INT) in D&D was the equivalent of an intelligence quotient (IQ) divided by 10.  Or INT * 10 = IQ.  The average IQ is 100, and the D&D manuals stated that the average INT was 10 (really 10.5 by rolling 3d6).  Heroes, rolling 4d6 and dropping the lowest, have a mode of 13 and a mean of 12.2446.  The whole INT distribution, however, does not fit as well as its mean onto the real IQ distribution.

The old Stanford Binet calculated IQ as 100 times the ratio between a child's age and the age of children with the same performance on average on particular tasks.  So, a 6-year old who performed as well as the average 9-year old would have an IQ of 150.  This method of determining IQ scores becomes less useful as children grow up, and makes no sense to apply to adults.  The newer Wechsler tests calculate IQ by taking the scores of everyone in an age group and putting them on a normal distribution with a mean of 100 and a standard deviation (SD) of 15.  This is a much more clear way to evaluate how a person functions intellectually compared to same-age peers.

If we take the distribution of 3d6 rolls and associate the probability of each sum with the prevalence of corresponding IQs in the real world, we can get a better impression of what a D&D INT score means in terms of IQ.  If we say that 3d6 simulates a normal distribution of intelligence, then here are the translations (rounded) of each INT score into an IQ score:

INT     IQ 
 3      57
 4      66
 5      72
 6      78
 7      83
 8      88
 9      93
10      98
11     102
12     107
13     112
14     117
15     122
16     128
17     134
18     143

The IQ scores in this table are calculated from the z-scores for the middles of each range of probability that maps onto each INT score.   For example, an INT of 3 and of 18 each have a 0.463 chance of occurring (1 in 216, which is 6^3), but I cannot use the z-score for .00463 to calculate IQ for an INT of 3 and work my way up because I would be left trying to use the z-score for 1 with the INT of 18, which is infinity standard deviations.  Also, I would end up using the z-score for .5 for an INT of 10 when it truly applies to an INT of 10.5.  What works is to use the z-scores for the midpoints between probabilities.  So, an IQ of 57 (2.84 standard deviations below the mean) corresponds to the z-score for .0023 (the midpoint between 1/216 and 0, since it is impossible to roll less than 3) with some rounding.  The probability of rolling at least a 9 is .375, and the probability of rolling at least a 10 is .5, the midpoint is .4375 for an INT of 10, which has a z-score of
about -.157.  100 (the mean) minus (.157 * 15 (the standard deviation)) equals about 97.645, so the IQ for an INT of 10 is 98 after rounding.  Each IQ score was calculated this way.

As you can see, multiplying INT by 10 does not give you your character's IQ.  Now, this does not address differences in intellectual ability by age group.  A stereotypical D&D party consists of characters emerging from adolescence into adulthood, roughly in the same stage of development, so it should be okay to use this scale.  A 12-year old with a 120 IQ still lacks much of the functioning of a 20-year old with a 100 IQ, and a 30-year old with a 100 IQ functions a little differently than a 50-year old with a 100 IQ.  Age tends to result in a larger vocabulary and body of knowledge with a slower information processing speed, but I may go deeper into how RPGs simulate the effects of aging on intelligence in later posts.

For that matter, using an overall IQ score obfuscates the differences in each person's separate areas of intellectual functioning.  "Intelligence" refers to many different abilities that we have, and a dozen people with the same IQ can each function differently from each other on a variety of tasks.  Few simulation systems break down intelligence into components, and that is also a topic I will return to in future posts.