Seriously game statistics are no joke, by definition they are calculus, where dynamic changable values are compared and processed against other dynamic changable values.
Actually, the calculus most people think of when they think of "calculus" - differential and integral calculus - isn't often used to analyze MMOs. In a practical sense, I only really used it once, to perform an approximate analysis of the SR scaling resistances. And to be honest that analysis turned out to be somewhat rubbish which is why I followed that one up with a more discrete precise analysis.
To be frank, most of the math you need to analyze most MMOs out there is simple high school algebra, because to be frank that's all the math most MMO designers are comfortable with. For complex analysis of dynamic situations, you're looking at statistical analysis as the approximation and Markov matrices and monte carlo analyses (aka simulations) as the less-approximation.
The biggest problem isn't the math, or the calculation part of the math. Its the conceptual translation and interpretation of the math as it applies to the game. There were a lot of forum analysts with reasonable quantitative skills that nevertheless could not apply them correctly to the game. Their calculations were correct but those calculations didn't apply correctly to any part of the game. Or sometimes they got tripped up on the precision of their calculations without asking whether those calculations were useful.
Here's my favorite version of math-correct, but mathematician-wrong. Consider the infamous case of stacking resistances or stacking defense. Question, when we stack equal numeric value resistances, does the benefit improve with stacking? Here's the case that it does increase in benefit with stacking.
Suppose we have two powers granting 25% resistance each. If we go from having no resistance to 25% resistance we lower incoming damage to 75% of original. That means it takes 1/.75 = 1.33x the damage to kill us, relative to having no resistances (ignoring regeneration for simplicity sake). But when we go from 25% resistance to 50% resistance, we go from 1/.75=1.33x damage to 0.75/0.50=1.5x damage to kill us, going from 25% resistance to 50% resistance. And in fact if we were to stack one more, we'd go from 0.75/.5=1.5 to 0.5/0.25=2.0x damage to kill us - twice as much with the third stack as without. So clearly, even though each of those powers is numerically identical, stacking them on top of each other improves their net value. Each one gives more benefit than the previous one.
However, here's the case that it does *not* increase the benefit.
We start with an entity with no resistance. We shoot 1000 points of damage at it, and it takes 1000 points of damage (ignoring tohit chance, which just reduces the numbers by a percentage). We now give it a 25% resistance power. It now takes 750 points of damage. The resistance power in effect blocked 250 points of damage. Now we give it another 25% resistance power. It now takes 500 points of damage. That's 250 points less than with just one power. So that second resistance power did exactly the same thing, it blocked 250 points of damage. A third such power would do the same thing: block 250 points of damage. Ergo, when players say stacking resistances gets stronger, they are wrong. Each is just as strong and does just as much as the previous one.
Who's right? The first one. But why? Where's the math error in the second analysis? There isn't one. The problem isn't in the math, the problem is in the head of the second mathematician. Specifically, it is in the question of
what is the value of a resistance power? The second analyst looks at resistance powers literally. They block damage. So the analysis focuses on how much damage they block, and since each one blocks the same amount of damage, the second analyst concludes each stack of resistance must have the same
value to the player. If two things do the same thing, they must have the same value.
But that's not how an MMO player
values resistance. In fact there's no way for that player to directly
perceive how much damage is *blocked*. Instead, MMO players value resistance powers the way they do all damage mitigation powers: do they keep me alive? The standard of value is *not* how much they block, but in a sense how much they
admit. I called this the admittance effect on the City of Heroes forums. We don't notice the damage we avoid, only the damage we take. And we don't directly perceive the damage, we directly perceive *time* - how long we stay alive, how long before we need a heal, how long we can tank this AV. We directly perceive, in a colloquial sense, "how long does this mitigation work before I die?" And because that's how players judge mitigation, and that's what goes into their build decisions and that's how they will value a power, that's the standard we have to judge them on mathematically if we want to make a useful statement to players.
As mathematical analysts, we have to be useful to the non-mathematicians. So we have to figure out what players want, and tune our analysis to match. In this case, the mathematically correct analysis looks at increasing survivability. And that survivability varies inversely proportional to damage admittance - the inverse of damage mitigation. When we go from 1/.75 to 0.75/0.5 to 0.5/0.25 we go from 1.33 to 1.5 to 2.0. That is
escalating return on investment. The other analysis where we go from 250 to 250 to 250 is quantitatively correct, but its not talking about the same thing players talk about when they talk about "value." It takes the word "value" and perverts it into something that is mathematically correct but inconsistent with what players themselves mean, and that's deception at best and intellectual dishonesty at worst.
This is the hard part. Not the calculations. Have Wolfram Alpha do the calculations for you. The hard part is translating the world, and people's perceptions of the world, into math in the first place. You don't need calculus, you need comprehension.
