You found the function to be curved because you are trying to take this percentage increase in damage. If you subtract the one function from the other, you will find the linear function you are expecting. The line starts at 0 (there is a 0% increase in damage with 0 crit chance) and ending at 0.3 (with 100% crit chance, the set provides 30% more crit damage).
When you are trying to take this proportion increase in damage for average damage, you are talking about an increase over an increase. At 50% crit chance, you see that for any particular condition (set or non-set), you see an increase of 50% of the potential damage increase by crits as opposed to if no crits occurred. But it is the proportion of these increases that is making the curved graph. Your situation ONLY looks at crits, so sure, you see a linear increase in whatever numbers you are using. But when you go into battle, you don't score JUST crits. You score crits and non-crits. The offense set provides a linear increase for both. The cd set provides only an increase for crits, but for non-crits, will be worth less than the offense set. So when you talk about crit chance break point, you are concerned with the average crit and not crit damage. As I said before, if you only look at crits, there is no cc break point. CC might as well be 1. If you are talking about CC, you are also talking about non-crits.
"Woodroward;c-1627800" wrote:
These statements are absolute truth:
Crit damage set bonus can only increase damage on crits.
If you crit 50% of the time, you will get the increase 50% of the time
Getting the increase half the time is mathematically the same as getting half the increase.
His formula gives 67% of the maximum increase at 50% crit. It is flawed, period.
I don't know how he messed it up. I didn't want to figure out how to properly include all those factors myself ...which is why i factored them out. Either way, simple math reveals the results are flawed.
So, I realized your assertion about crit chance here, while seeming logical at first glance, is actually
false . If crit chance is 50%, you will not see the cd increase 50% of the time. The reason is because crit
chance is a random variable (or rather, critical hit rate is a random variable with parameter crit chance). That is, on any given run of any given battle, we wouldn't necessarily see a critical hit rate equal to critical chance. We would expect our critical hits to average out to critical chance over time. With a critical chance of 0.5, in any particular battle, if we score 10 hits with the character, the probability of critical hit
rate of five crits is 24.6%. The cumulative probability of 4-6 crits is 65.6%. But there is a cumulative probability of 5.5% that we'll see eight or more crits.
Furthermore, having different sets on a character are independent conditions. You cannot simultaneously put two sets on a character, go into battle and fire a shot, and expect to see two damage numbers corresponding to the different sets. So to say that a 50% cc results in whatever percentage increase 50% of the time supposes that trying a battle-real or theoretical-- with the different sets will result in the same order and number of crits, with the crits and non-crits perfectly lining up. Instead, there is an X% increase in damage with the cd set 0.5x0.5 = the 25% of the time they both crit on the same hit. 25% of the time, they will both non-crit, and there is no increase in damage. 25% of the time the set situation will crit over the non-set not, and will result in even greater damage. The last 25% of the time, the non-set will crit and the cd set will actually show a loss in damage. So you would then have to set up the whole problem with all this information. Of course, the weights will be different depending on crit chance, but they should add up to 1. Needless to say, when I computed the value of c with this method from the numbers used in our previous discussion, I got the same conclusion as my other formula.
But if you think this is an overly complicated and messy way of doing it, you'd be right. It's just a more roundabout way of finding average damage differences between the sets, which is what I did in the first place. It is much easier to just compute the average difference rather than set it up this way (and doesn't involve factoring a quadratic equation).
If you ask me, trying to take percentages of percentages leads to more conceptual confusion where you can lose track of what you're measuring. If the end equation looks conceptually confusing to you--that's OK. Math was invented to solve complex problems that aren't always easy to figure out by intuitive logic. And it's possible that the equation could be rearranged to make a little more conceptual sense while remaining mathematically equivalent. Or maybe not. But as long as the math was done correctly and the appropriate problem was set up, it's fine.
And my point is, I think you're setting up the wrong problem by looking only at crits and throwing half the equation away (or at worst, you're bringing the other half back in after improperly splitting a fraction). If someone is asking when to use an offense set over a cd set based on cc, I would think they are concerned with those average damage increases over time, including non-crits. You may have a very good method for finding the proportional increase in crit damage for crits, but it is not the same as crit chance, because you are throwing away the portion that makes crit chance relevant.
