## Handling Scale

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?

Zach HespeltThere’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.

February 18, 2014 at 10:36 pm

sheavalentineI 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.

February 18, 2014 at 11:26 pm