Math and Difficulty: Percentile and Fixed Scaling: Part 3

Note: This is the third in a series about math and game balance, specifically regarding to item balance.

5. Percentile bonuses that can accrue are dangerous. Percentile bonuses out of 100 that can accrue are doubly dangerous.
+400% to a statistics at late game dwarfs everything else. A small insignificant +1 damage suddenly becomes worth 5 times the initial worth. A +1% critical bonus at the beginning of the game might be worthwhile until you realize that you can't give away more than 1% or 2% because by end game, players will reach 100%.

The crux of this is when you realize that some bonuses have a maximum relevant amount. For example, once you have reached +100% damage reduction, you can't give any more because it becomes meaningless. Some attributes simply should not be raised that high. So how do you pace yourself? Do you start off giving the player miniscule bonuses to damage reduction that accumulate? Do you arbitrarily cap players to a maximum amount of any one statistic and let overflow do nothing?

How would you then give them a "better" version of say "critical strike" gear without actually increasing the amount of critical strike they get? Or do you keep making gear with the same percentages? What if you miscalculate and accidentally give them 95% critical strike? What if you need to add new content but they've already reached absurd levels of bonuses?

World of Warcraft takes the approach of factors that scale down dependant on your level. At level 1, one critical factor might give you 1% critical strike but at level 50, one critical factor gives you 0.05% critical strike. This makes it easy to create new equipment that's better than the old (Here! This one has +86 critical factor while the other one only has 43!) but this introduces another problem. Leveling up decreases the power of your equipment. Leveling up in their system simply makes all your existing gear worse and it introduces some confusion in the inner workings of the game (Just how much critical factor do I need to reach that critical % cap?)

6. Not all percentiles are equal: Reduction and additions are and must work differently.
If I have 90% damage reduction and I get a mere 5% more reduction additively; I don't take 5% less damage. I take 50% less damage because now I'm at 95% reduction. Alternatively if I have 100% bonus to damage and I get 50% more bonus to damage additively, I don't do 50% more damage, I only do a mere 20% more damage because now I'm at 150% bonus damage.

This phenomenon often pops its nasty head up when designers either scale poorly or don't reliaze the exponential growth of penalties or the possible exponential growth of bonuses.

For example, let us take a item that reduces incoming damage by 50%. This is a very nice item as it effectively doubles our lifespan. However, what do you expect should happen when I equip another item that reduces incoming damage by 25%. This new item is clearly inferior to our old one.

Well, if the math works additively... 50% + 25% -> 75%. This means that our inferior item has raised our damage reduction to 75%! This means our lifespan is now four times our original by adding the inferior item.

What gives?

What if we then equipped another item that was a mere 15% damage reduction? This is even more inferior to our other item. However, if the math works additively.. 75% + 15% -> 90% and we now have ten times our original value.

Some poor designers then take this as a sign that this statistic is too powerful in large quantities and arbitrarily cap the statistic. Or else they give out the bonuses in small measure and irk out 5 and 10% damage reductions randomly.

However, what they fail to realize is this...

A reduction scales exponentially fast after 50%.

To double your lifespan with damage reduction, you need 50%
To triple your lifespan with damage reduction, you need 66%.
To quadruple your lifespawn with damage reduction, you need 75%.

The last few % are the ones that matter if you do additive reduction.

However, let's take positive multipliers. These are the inverse. Positive multipliers suffer the inverse problem.

Positive multipliers grow exponentially if they accumulate multiplicatively.

For example, if I have a sword that doubles my damage and a necklace that doubles my damage...

If the math works like this: 2 * 2 = 4!

Then I suddenly do four times the damage! If I get another item that doubles my damage..

2 * 2 * 2 = 8! I suddenly do eight times my damage!

This quickly leads to untenable growth when you realize just how many factors can accumulate multiplicatively...

Therefore, it is best that reductions and multipliers use different math: Reductions best accumulate multiplicatively while multipliers best accumulate additively to prevent dangerous exponential growth in effect.