In my wanderings, I have recently stumbled upon a pretty detailed analysis of the RPPM mechanic. It’s been a long time since I have taken any real math courses but I’ll try to digest for you in simple terms what this report is saying. I would love to give credit where credit is due, but there’s no author or any kind of ownership on this other than ‘rwking’ so cheers to you rwking for providing this!

I’ll explain the first part today and the second part tomorrow. My goal is to simplify the paper’s topics not going into the calculations but rather explaining why equations are set up the way they are. Many people find math intimidating (it is) but a majority of the confusion lies in the inability to understand why the equations are the way they are because from an early age we’ve been taught to memorize and not understand, and also because many calculations have hidden assumptions which are at times rationally and other times irrationally justified.

The method of approximating the RPPM system they have explained in the paper is justified because of the properties of the power law. A WoW related example of a power law distribution is progression week boss kills. Almost every guild on wowprogress has killed Jin’rokh on normal, however, immediately at Horridon, this percentage decreases to 58%, then further at Council to 41%. In other words, the harder the boss gets (hardness based on boss number which is not necessarily true) the less the number of people have killed him at greatly decreasing rates with only 1 more boss.

The power law distribution can be approximated by decaying exponentials essentially for simplicity but also that at low values, the approximation is pretty similar to the real value which requires calculus as they mention. Approximations are merely ways to mirror values with simplified equations. You would rather spend 1 minute to calculate something with a answer that’s very close to the true value than 5 hours to calculate the true value.

I’ll separate this report by similar headings in the paper.

**Derivations **

Blizzard’s given chance to proc calculation:

Given the addition of a penalty term to the RPPM calculation which is defined m(t):

The proc calculation changes to a proc chance over a time frame (t) calculation.

This is simply the equivalent of the “max()” term that we already know from the equation for the penalty term. This essentially finds the maximum value within all the different values in the parenthesis. The conditions above are calculated by finding the value for t when the second term in the parenthesis becomes greater than 1. In the game this time, t is when the penalty factor goes into effect.

The last equation is for the chance of procing given the variables:

I won’t go into the math for the next step, but essentially it justifies approximating a binomial distribution to calculate the proc chance. A binomial distribution is a set of occurrences which only has 2 outcomes, in our case its proc or no proc, given a probability to proc ‘P’ as defined in the equation. The purpose of taking the limit is to generalize the form into the exponential equation we end up at and because we’re testing what happens as the time between attacks goes to infinity (no procs). I’m slightly confused at this step only because we’re calculating the chance not to proc taking the time between attacks to go to infinity where as what we really want to identify is even with a certain number of attacks, what is the chance still not to proc within a period t which was defined in an intermediate step as the following:

I may have some fundamental misunderstanding of the concept here though. S_{aps} is defined as the attack rate or how many attacks over a given time. It’s the same as the inverse of (one divided by) the time between attacks. Why? Because the time between attacks is time over attacks, like 5 seconds per 1 attack.

Anyways, what we end up with is the following equation for the chance over a time frame [0,t) for no procs.

* Note that variables in this equation have already been previously described.

I’ll stop here today. I hope this hasn’t been too confusing and I hope it has been helpful. The fun thing about this is you can now go and plug your own variables into the equations. PPM, and Haste valuese are all definite values. Attack rate is a value you can approximate yourself with some rational.

**Look out for part 2!**

Do you still have access to the original RPPM article? I loved your analysis but was just curious how much further in depth the math went (I’m a math major), and the link you posted is dead. Thanks!

Sent to your email.