Liability, or REliability?

Someone graciously linked my Statlords of Shamanor post on Reddit the other day, in this post which discusses stat weights. I was flattered to have the attention of a major part of the community, and even better an interesting question came up. There is, and always has, been the contention that some stats for healing are “more reliable” than others. Is this true? If so, is it a meaningful statement?

Before Warlords of Draenor, the answer was simple; Critical Strike rating has a lower probability to affect any of your heals than any other stat. Now, however, there are two stats which healers care about which inherently work probabilistically; Crit- and Multi– strike! Which is more reliable, and what does that actually mean for us as healers?

First up, let’s start with a caveat – “Reliability” is a really hard thing to pin down. Theck has put significant effort into defining a rigorous reliability measure for tanks – called the TMI – which is beautiful and empirical and by its very nature not really the kind of work I like to do. Instead of parsing a log, I want to make a specific general point about the two stats under question. In doing so, in the same way as a lot of my work, I will have to make assumptions which you will need to keep in mind. The result won’t be a definitive “yes, you should take Multistrike because it is better” – more of a “Multistrike has this general property which Crit has less of”. The interpretation – the touchyfeelycraft, if you will – is down to the player to do for themselves.

It’s 100% Reliable. Probably.

I’ve talked everyone’s ears off about probability in World of Warcraft before, so if you’re a regular reader get your hymn books out and sing along. Probability and statistics is absolutely crucial to our understanding of WoW and the way it is played, especially when looking at a complicated question such as this one. The reason that it’s complicated is because “reliability” is a poorly defined question – it can mean different things to different people. As such, I’m going to start with a strong definition of what reliability means to me and work from there. What is “reliability”?

A healing spell, to me, is more reliable than another if it can be counted on to have an effect on any given heal at any time. This is in isolation of any other heal which I have made, but not in isolation of current buffs. That means that I’m going to inherently ignore the “bulk” or “average” behavior of each stat.

Figure of Merit

We need to pin down a good way to find a number which expresses the definition I just gave, and I have a way of doing it which is quite straightforward. In short, I will calculate the probability of any effect from each stat being applied to an individual heal where the healing I will gain will be of use. We’re going to ignore overhealing effects, and also ignore “cross-terms” (i.e. a Multistrike having a Critical effect), because that makes the problem very messy indeed. So, in this simple case I’ll calculate the probability of seeing a stat have an effect at all. In the case of Critical effects, that is simply equal to the Crit chance given by;

pCrit

In contrast, Multistrike has two equal chances to Multistrike for 30%. These chances are independent, so the probability of there being any Multistrike effect on any heal is equal to the following, which takes account of the chance of both effects happening;

pMulti

If you want to work out that equation for yourself, you can get it from the Binomial distribution. How do we get a meaningful assessment from this? One approach would be to assume that the two stats are equal and then plot the reliability for each stat as a function of rating. That’s pretty easy, and you come out with a graph much like this one;

LineComparisons

However, this isn’t very useful in the sense that it only tells you whether one is better than the other if your stats are equal. If they are not equal, then you have to spend some time reading off the graph carefully to see which is best. It would be much better if we could plot a line on which the two stats are equally reliable. I’ll do this by setting the two equations I’ve given above to be equal to each other, and then rearranging so that;

Rcrit

I can now plot that equation, and it should leave two distinct regions either side of the line; a region where Multistrike is more reliable, and a region where Critical Strike is more reliable. Here is just such a plot;

EqualCurveGraph_noHS

So, you can use this graph to tell you whether you are getting more reliable results from Multistrike or Critical Strike. I have one more case left to tackle, and that is the case where I am casting Healing Surge with the Tidal Waves buff active – this gives me 25% extra Critical Strike chance. I recomputed the above graph for this case;

EqualCurveGraph_HS

Notice the change in the y-axis scale? When you’re casting Healing Surge with Tidal Waves, you have to have a whole lot of Multistrike in order for it to be more “reliable” as a stat than Critical Strike.

Being Reliably Picky

That last result is actually a bit unfair, because most of the reliability value from Critical Strike doesn’t come from stacking it as a stat – it comes from the massive passive bonuses you get to Crit. Another sensible question to ask is; which stat makes you more reliable faster? To answer this, I’m going to assume that the functions I calculated in my Excel sheet are linear and apply a linear fit to them – the stat with the highest gradient makes you more reliable more quickly as you stack it. Here is a table of the results;

Values multiplied by 1000 for sanity.

Values multiplied by 1000 for sanity.

As you can clearly see; it doesn’t matter which case you have, Multistrike makes you more reliable faster than Critical Strike does.

Skepticism

Does it matter which is most reliable, or even that either is not 100% reliable?

I don’t think so.

So, there are cases where you will be casting a heal upon a low-health player and they will die unless you gain your full potential from each stat. In the case of Mastery, that’s not a problem. In the case of Haste, that’s not a problem. In the case of Crit and Multistrike, that could be a problem. However, how often are you really in that desperate need of the extra healing that there is no way that the player will survive otherwise? Well, the more Crit and Multistrike you stack, the more you will be in that situation due to the RNG nature of the stats. On the other hand, we already know that Crit and Multistrike aren’t as good as throughput stats as Mastery and Haste – so if you’re stacking a lot of them, you already know that you don’t want to be!

Equally, you are mostly going to be healing people in a situation where it is not literally the success of the raid at stake. In these situations, the “reliability” of a stat is entirely irrelevant because it’s the bulk (average) healing which matters more. Thus, regardless of context I think it’s a silly question to try to answer. The edge cases where a “reliable” stat would provide more benefit over an “unreliable” stat are irrelevant, and therefore we should collectively stop worrying about it. Remember that in Siege of Orgrimmar, Critical Strike ruled the entire instance despite its liability of RNG, and that was in a raid environment where we encountered the perfect case against it.

All in all, I think that although it’s an interesting question, the answer doesn’t really matter at all.

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About stoove

A physicist, researcher, and gamesman. Likes to think about the mathematics and mechanics behind all sorts of different things, and writing up the thoughts for you to read. A competent programmer, enjoys public speaking and mechanical keyboards. Has opinions which might even change from time to time.
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