Research

Some of you may remember my Reddit post on the impact of Magic Find (MF) buffs on drop rates from Unidentified Gear (UG). I'll admit that the research I did for that post was a prelude to my attempt at making a reliable model for profit in Istan. Some of the most useful feedback I received from that post was about the small sample size of my research. Opening only 2,000 Blue UG might help you determine whether MF affects drops in general, but it won't be good enough to reliably predict the precise impact MF has on UG drop rates. To do that, more sample would be needed.

I decided that the most effective way to predict MF impact on drop rates was to open manageable sample sizes of UG at a whole range of MF levels. While any one sample size of 1,000 Blue UG on its own couldn't tell you much, calculating a linear regression of 20,000 Blue UG across 20 different MF levels could potentially give you the exact formula used to determine drop rates.

Pursuing this option, I opened 42,000 total pieces of UG across a spectrum of MF levels ranging from 225% to 650%, measured at intervals of 25% (e.g. 225%, 250%, 275%, etc.) At each MF level, I opened at least 1,000 Blue, 500 Green, and 250 Yellow UG. There were some occasions where I got outlier results and did a second batch of testing for that gear at that MF level. Also, at the beginning I did some measurements at additional MF levels not on the 25% increment range which were also included in the model.

In total, I opened 27,000 Blue, 10,500 Green, and 4,500 Yellow UG, and I'm fairly confident that the resulting drop rate formulas can reliably predict drop rate patterns at any MF level.

Let's start with Yellow UG, because the drops you get from these are the simplest. You either get a yellow drop, or an exotic, and that's it. Yellow UG also has the highest chance of dropping exotics, which makes it easier to more accurately measure exotic drop rates. After opening 4,500 Yellow UG and plotting the results, I got the following graph.

All yellow data series correspond to the yellow gear drops I received, and all orange data series correspond to the exotic drops.

Each rarity level of gear drop has three data series associated with it:

• The first is the raw data series, denoted by the scatterplot dots on the chart. These represent the real measured drop rates from each test at various MF levels.
• The second data series is the linear regression of those observed data points, and is visualized here as a shaded zone, basically marking where we think the real drop rate formula would align.
• The third data series is the solid colored line, which is a data series I manually created to best fit the calculated linear regression while still taking into consideration the likely preference a programmer would have for easy-to-use, whole numbers. This is the same approach I took with the "Estimated" drop rates from material shipments and other items.

With research like this, we have to guess what type of regression we believe will fit the data most accurately. Regressions can be linear (as these are), exponential, polynomial, logarithmic, among other options. What seemed to make most sense in this research was linear.

Linear equations have two main components:

• The "Slope", which determines the rate of increase or decrease over a certain interval on the X-axis.
• The "Intercept", which determines the vertical position of the line when X is equal to 0, or in other words, where the X-axis intersects with the Y-axis.
• A linear formula follows the pattern: [(Slope * X) + Intercept]

Linear drop rate formulas can help us, because with data like this where we are measuring the chance out of 100% that you get one thing or the other, the Slopes and Intercepts of the various formulas all add up to predictable numbers:

• All Slopes should add up to 0, because as one line grows in a zero-sum percentage chart like this, the other line must shrink equally.
• All Intercepts should add up to 1, because in this system even at 0% MF (at the intersection of the X-axis and Y-axis), whatever results you get still add up to 100%.

These features of linear regression will be more helpful when we tackle Green and Blue UG further down.

Specifically for Yellow UG in the chart above, we can see definitive gains in exotic gear as MF increases from 225% to 650%. The linear regression zone has an r^2 value of only 0.596, which isn't great, but r^2 isn't the sole measure of how well a line fits the data. Nevertheless, the way to fix this is to get more data sample, and if you are willing to share yours, please stop by the appropriate Data Submissions page to help out.

Looking at the linear regression from the data, I estimated the true drop rate formula to be the following:

I only changed the Slopes by 0.006 each, and the Intercepts by 0.0004, and came up with the resulting formulas on the far right above.

At this point, I'd like to bring up a quote found on the Magic Find wiki page, where the mechanics of loot tables are described by Isiah Cartwright (emphasis added):

Although Isiah was simplifying the mechanics a bit, I think his description definitely helps understand why the estimated drop rate formulas above work.

For Yellow UG, exotics are essentially a 1 in 80 category (the Intercept), and each time your magic find increases by a factor of 100%, you add another 1 in 80 chance (the Slope) of getting an exotic.

Because Yellow UG is guaranteed to drop at least a piece of yellow gear, the only category we really need to calculate for here is the exotic category, because the drop rate of yellow gear will just be 100% minus the drop rate of exotic gear.

Recognizing this is important to helping us understand and predict the likely drop rates of other UG, as we start to introduce more gear drop rarities that complicate the process.

Next up is Green UG. Here, you either get a green drop, yellow drop, or an exotic. It's a little more difficult to estimate exotic drop rates here, but I attempted to account for that in my research by opening double the volume of Yellow UG. After opening 10,500 Green UG and plotting the results, I got the following graph.

Here the slopes for Green and Yellow gear are far more pronounced than Yellow and exotic were for Yellow UG. The r^2 for Green and Yellow are 0.75 and 0.72 respectively, indicating a much more likely fit to the linear formula. The r^2 for exotics is very low however, indicating the need for more sample to verify their trend.

That said, with only very small adjustments, I was able to come up with the following formulas that matched very well to the data:

Calling back to Isiah's quote, he talked about categories and gave an example of a "1 in 10" category. If you look at the intercepts for exotic or yellow gear here though, you'll see that they're not "1 in ____" categories, but rather they have a rate of gain (slope) that differs from their intercept. So when you open Green UG at 0% MF and 100% MF, your chance at an exotic drop does not double, but rather goes up by 1/6th in this case. Your chance at yellow gear increases by 2/5ths.

Although the r^2 for the exotic linear regression is very poor, the estimated formula only changed the Slope and Intercept by 0.0001 each, and resulted in a reasonable line very close to that original regression. As always, the more data the better, but for now the longitudinal analysis coupled with linear regression seems to help the predictions for drop rate formulas.

Finally we have Blue UG. Here, you are guaranteed a blue piece of gear, but have a chance to get anything up to and including exotics. It's much more difficult to estimate exotic drop rates here, and I again tried to compensate by doubling the volume of Green UG I tested to 1,000 Blue UG for each MF level. After opening 27,000 Blue UG and plotting the results, I got the following graph.

It's important to note that in order to effectively use the linear regression slopes and intercepts to help identify likely drop rate formulas, I needed to measure blue gear drops and green gear drops separately, and not salvage them all at once. If you're considering helping by submitting research of your own, please make sure to count and salvage blue and green drops separately.

Once again you can see very strong slopes for blue and green gear drops, with r^2 values over 0.97 for both. Even the yellow gear drops have a reasonable growth pattern and earn a 0.725 r^2. Exotics become very difficult to measure, and their r^2 is very low once again.

Part of the reason I addressed Blue UG last is to highlight the obvious difference in this chart compared to the others: a glaring bend in the estimated drop rate formula for green gear drops. This was a bit unexpected, but as you can see the linear regression for blue gear drops intersects with the x-axis around 860% MF. I was not able to measure Blue UG at MF levels anywhere near that, so this is a blind spot in my research.

However, I considered what some of the most likely patterns would be for opening Blue UG above 860%. The solid green line for my estimated drop rate formula in the chart above represents my best guess that after ~860%, green gear drops begin to decline at a slower rate than blue gear drops did earlier because your odds of getting exotics and yellow gear must necessarily increase with higher MF.

There could be wrinkles to this assumption. For example, if someone opens 1,000 Blue UG at 1,000% MF and get 2 or 3 blue gear drops, does that negate the entire assumption? Perhaps, but it's also possible ArenaNet programmed the drop rate to have a floor, whereby no matter how high your magic find you'll always have at minimum a 1% (or some other percentage) chance of getting a blue gear from from Blue UG.

The problem is I have no data at this range of MF and therefore can only guess what the truth might be. This is where I could use your help! If you are willing and able to open a large sample of Blue UG at 800% MF or above, I would greatly appreciate your assistance and research to help solve for this problem. Feel free to submit your data here, and let me know how I can credit you for your work.

Getting back to the estimated formulas, when I made my adjustments I came up with the following:

Exotic gear here has dropped to a 1 in 1,000 category, but actually has the same Slope as it did for Green UG (1/1,000). Yellow gear becomes a 1 in 100 category, and has a Slope to match that (1/100). Green gear becomes tricky here, and from what I can tell, its category is about 1.2 in 11, with a Slope of 1/11 as well. The blue gear formula is a bit messy to write out but as with the previous charts, you can assume it's 100% minus the drop rate of other gear.

The formulas for yellow and exotic gear here make sense to me, as you can compare their slopes and intercepts to the formulas for yellow and exotic gear in Green and Yellow UG. In each case, the drops more common, the resulting lines align with the raw data and calculated linear regression, and the Slopes and Intercepts adjust while remaining reasonable, easy-to-use numbers for a programmer designing this system in the first place.

All of this leads me to believe the formulas above, while by no means perfect and certainly can be improved with more data, are the closest to the true drop rate formulas used by programmers that we have. I hope as time goes along and more data is collected, I can either confirm that, or continue adjusting and refining the formulas as we gain a better understanding of it.

• Research: Unidentified Gear