One of the things you may have noticed when I was discussing Skill and Difficulty is that I went all the way up through Grandmaster, Through Superhuman, Through Godlike, to Divine. In continuing in that vein in fact, I have set “60” as the difficulty under which anything is possible. That means when assigning difficulties to tasks I need to keep in mind the scale.
How exactly do we do this.
Finding a Scale
Typically game systems derive their metric from what tasks of various difficulty do by figuring out a maximum based on World Records, a Minimum based on an average and find a linear relationship between those things.
Sprint Speed is an easy example of this. The World Record for “sprinting” is about 28 MPH , and a person on average can do sprint about 12 MPH. So the Sprint speed of a Grand Master (score 21-23) should come to 28, and a Average Person (score 0) should be 12. So I subtract 12 from 28, getting 16, and divide by 23, giving us a little more that 2/3. That means for every point, the character will run another 2/3 a MPH.
Here’s the problem, you then extend that into the rest, and you end up with this: Legendary Running is 30 MPH, Super Human is 35 MPH, Mythical Running is 40 MPH, Godlike Running is 45 MPH, and Divine Running is 50 MPH.
That means a being with Divine running ability is too slow to keep pace on the freeway. This is fine for a game that only deals in the realm of mere mortal, but doesn’t make sense for most games including most of the ones that do this. Though, to be fair these games have generally had some sort of hack to modify this after the fact, but that’s by definition an inelegant solution.
Hyperbolic Scaling
One solution to this is to scale exponentially using three data points. In this case we still have the Average and the World Record, and additionally we define 60 as Infinity and 59 as Absurdly huge. For example, a sprinting of 59 should be a reasonable fraction of the speed of light.
Using speed again, fiddling around with probabilities based on Standard Deviation, you can come up with a complex formula that looks something like this: 1/(1-NORMSDIST(StandardDeviation(Score)/3.6))+Average.
Now, I don’t expect any of you to understand exactly what that’s doing, but what I can tell you is it results in something like this:
- Average = 12 MPH
- Skilled = 13 MPH
- Journeyman = 14 MPH
- Expert = 16 MPH
- Master = 20 MPH
- Grand Master = 27 MPH
- Legend = 40 MPH
- Super Human = 100 MPH
- Mythical = 500 MPH
- Godlike = 2500 MPH
- Divine = 30000 MPH
Wow! That seems to more closely reflect on the high end the way we’ve described the high scores.
However there are two problems with the Hyperbolic scaling. Specifically –
- It makes increases at the low end less rewarding
- Linear scaling is easy to keep track of during play, it doesn’t rely on referring to charts to figure out.
Combined Scaling
One thing of note with Hyperbolic scaling is that relative to the Legend+ Scaling, the Average to Grandmaster scaling might as well be linear. Consider the following graph:
If you look at the green line, 6 and below from the perspective of 7+ might as well be linear. So, it’s simple to refactor our scaling such that below Legend or a score of 25 the growth can involve linear scaling, and above that it can involve hyperbolic scaling, so long as the World Record pivot point is consistent between the two. That allows the scope of mortal heroes to utilize an easy to use consistent scaling that can still expand to epic scopes. This would give us something like this for sprint speed:
- Average = 12 MPH
- Skilled = 15 MPH
- Journeyman = 18 MPH
- Expert = 20 MPH
- Master = 23 MPH
- Grand Master = 27 MPH
- Legend = 40 MPH
- Super Human = 100 MPH
- Mythical = 500 MPH
- Godlike = 2500 MPH
- Divine = 30000 MPH
This means that every rank is about +3.5 MPH up to Legend and then Hyperbolic Scaling takes over.
Dealing With The Low End
For the low end we don’t really care to scale all the way to -60. We can set a low point of None at -25 where everything will equal 0. Then we apply Hyperbolic scaling rules again, but this time the NORMDIST(StandardDeviation(Score)) will be used to create a fraction. Again, the specific math here isn’t important, what is important is that we get a drop off that isn’t too punishing right below Average, but which plummets to nothing by the time a score of -25 is reached. Using a Negative Ladder for sprinting we get scores as follows:
- Mediocre – 9 MPH
- Poor – 7 MPH
- Bad – 4 MPH
- Awful – 2 MPH
- Abysmal – 1 MPH
This doesn’t have linear scaling, but like the Heroic version the need for Linear Scaling is less. When dealing in this range the use of lookups is less cumbersome.
Should RPG’s attempt Scaling? How would you handle scaling? How does one scale something like “persuasion” or intelligence?
Living Myth Design Rpg 
There’s an inherent problem in making (or more aptly, playing) characters with high non-physical ability. I can imagine and appropriately play a character with Herculean strength, but playing one with even Holmesian intelligence or Hitchens-like oration ability poses a real problem; I can’t think better than I can think. This doesn’t apply so much to ability scales as it does to game mechanics that allow those scales to be meaningful in areas like intelligence or charisma.
I think there’s something to that, however I think that /perhaps/ it can also be handled by giving players information they otherwise wouldn’t be able to have.
That being said I don’t think Intelligence scales at the same rate Speed or Strength would. Rather, there is a limit to knowing (defined in terms of computational complexity), and while knowing more faster scales, the ramifications of knowing more faster doesn’t as much.
To simulate preposterously high IQ, you could give the player access to knowledge they couldn’t possibly have.
On the other hand, Intelligence in the realm of the Divine probably exceed narrative capacitance…
What would probably work pretty well is to go through the listed ranks and describe exemplary abilities one could possess with that kind of intelligence.
Great article showing a good approach to get some scale system into rules.
The thing is: most below average abilities will be irrelevant, most players and games simply ignore the children, peasents, untrained, unabled or very small creatures like micro-fairies or ants. There is no reward in “winning against” such weak beings. Heroic games look upwards on the ladder. Generic systems that include a scale that reaches up tinto the divine (which by actually is not really scalable to human terms), let’s call it superhuman, will have problems, even if the scaling feels right. I doubt that players on a mortal level want to interact with gods that run at 300.000 MPH, single-handedly crush a moutain into pebbles or talk with a voice that makes ears explode within 10km of range. But that would actually happen if gods on such a scale level were involved AND act like mortals do – which is the case in traditional divine mythology like Greek, Roman, Asian… Gods are super-super-super-humans and humans are not meant to compete at all. I doubt there can be fun game where half-gods, gods and mortals run around together or clash, without causing Ragnarökr in a way. You could make a genre game on divine scale, where mortals represent the ants, bt ut it would not be funny to play the mortals there. If the setting is centered around mortals but gods exist and interact, I think that stats for them are irrelevant (ha! I speak for aspects it seems). Gods could probably move at 300.000 MPH, but they simply appear in front of a human they want to talk to. No human would be able to measure or experience the reality of the mind-boggling movement of that god (on what scale level would Hermes be if gods all sit at the end of the ladder?). Same goes for strength. If Odin uses his spear to strike down a mortal, you do not need a scaling ability system that allows you to represent the attack in d6 or a difficulty level, or calculate the damage done. Odin could probably wipe a town from the surface of the Earth with a strike, that’s quite enough for narrative and for stat-based gaming. In pure divine settings you would actually use the same scale and rules as for mortals, just call everything “godly” and sometimes mention the havoc created around the players if they move through the mortal world, just like to warriors duelling in the forest – who describes how many twigs they snap off or how many ant hills they tread in while fighting?
A super-hero setting is a mixture that causes many of those problems. And I do not see them solved well in comics, books or rpgs until today. They are simply ignored or talked away by hair-raising explanations which in themselves collapse in the very next situation. Flash can move at speeds that would be Divine level, Hulk is stronger than many gods are… it seems that in super-heroe games no bounds exist. It is probaly the genre that only works with genre rules, I cannot think of well done generic rules that cover super-heroes, unless you are happy if your game world is a copy if super-heroe comics and the rubber-band non-laws of nature that exist there. Which is great fun, I damit.
PS: Your blog is outstanding, so sad I didn’t find it earlier.