Guide How (emerald) Stealing Works - Rev. 2

Preface
The research on how Stealing works is still in progress - as such, this post is not (yet) complete. Until all details are known, I will update this post with the latest information. Older revisions will be kept until then for referencing - however, note that the information there is (likely) outdated/incorrect, so always look at the latest revision for the most accurate information.

Spoiler: Changelog

All times in CEST
July 17th, 21:12 - Original post.
July 23rd, 03:47 - Revision 1.
July 23rd, 16:39 - Revision 2.

Original Post
Introduction
This morning I saw a small discussion on the Atlas Inc Discord server regarding how the attack speed of a weapon affects Stealing chances. The general belief seems to be that the Stealing stat applies 100% to Super Slow attack speed, and then gets diminished the higher the attack speed. With the introduction of Greed, Stealing has actually become a relevant stat, so I thought it'd be worthwhile to investigate this.

Spoiler: What is Stealing?

Stealing gives you a chance to 'steal' an emerald from a mob with any hit, including spells and poison. The higher your Stealing stat, the higher the chance to steal an emerald.

Note: in case you're just curious about the implications, there's a TL;DR at the end.

Method

Spoiler: Disclaimer

I am not a data scientist - please be so kind as to point out any mistakes I may have made. I don't know how high the number of trials should have been to reach significant results, so I chose the highest number that would not drive me mad. I did the data processing and plotting in MATLAB, and I'd be happy to send the file over if anyone would like to have a look at it.

Before I began testing, I had to take note of the fact that Stealing can only activate up to five times for each mob. To ensure I could keep track of which mobs I had 'depleted' easily, I used guards since they are stationary. Specifically, I used the guards at the gates in Corkus City, since they're a nice distance apart from one another and spawn close to walls. Furthermore, between testing weapons, I switched servers to refresh all guards. Finally, I wanted to use relatively high values for Stealing, to make outliers less probable.

I used one weapon of each attack speed, and each with two different builds: one with 26% Stealing, and one with 47%.

Spoiler: Builds and Weapons

https://wynndata.tk/s/38wprg - 26%
https://wynndata.tk/s/jxnlqe - 47%

Custom Items: Ring of Rubies, Old Keeper's Ring, Morph-Ruby, Opulenity

Weapons: Cerid's Dynamo, Cascade, Empire Builder, Bob's Mythic Wand, Bonder, Monk's Battle Staff, Eidolon.

For each weapon + build combination, I hit guards until they had dropped 5 emeralds, then moved on to the next one. I repeated this until I reached 200 hits with each combination.


Results
The collected data is shown in the figure below:

Spoiler: Collected data

As expected, the number of emeralds decreases with increased attack speed. However, it seems that for quite some weapons the number of emeralds did not increase linearly relative to the increase in Stealing.

The number of emeralds collected after a number of hits follows a Binomial distribution. This can be used to see how probable it is for a certain number of emeralds to drop:

Spoiler: Fig. 1: Binomial(200,0.26) distribution plotted with the Super Slow data point.

Fig. 1: Probability Mass Function of a Binomial distribution with n = 200, p = 0.26. The red dot shows the Super Slow data point.

Spoiler: Fig. 2: Binomial(200,0.47) distribution plotted with the Super Slow data point.

Fig. 2: Probability Mass Function of a Binomial distribution with n = 200, p = 0.47. The red dot shows the Super Slow data point.

Additionally, for each attack speed the hits per emerald and emeralds per second can be plotted:

Spoiler: Fig. 3: Hits per emerald

Fig. 3: Hits needed to gain a single emerald at each attack speed for different Stealing percentages.

Spoiler: Fig. 4: Emeralds per second

Fig. 4: Emeralds gained per second at each attack speed for different Stealing percentages, assuming constant melee attacks.

Additional Tests
There were some other things I was curious about: whether the Stealing chance of spells is influenced by attack speed, and what effect bonus attack speed tiers have on Stealing chances. For these tests, I only carried out 100 hits for the extreme weapon speeds with 47% Stealing.

After casting Ice Snake 100 times, I collected 21 emeralds with a Super Fast weapon, and 49 with a Super Slow one.
I used Peaceful Rest for the bonus tiers test since it has a base attack speed of Super Fast, but also gets -30 tiers. Using melee attacks I collected 41 emeralds, and using Ice Snake I got 54.


Conclusions
Since it's hard to come to conclusive numbers regarding by what factor Stealing is diminished as attack speed goes up due to variance in emeralds collected, and I'm not looking to abuse hundreds of guards again, I'll just make some general observations here.

While the dots on in Fig. 1 and 2 are both left-leaning, they are reasonably probable - it cannot be concluded that Stealing does not apply 100% to Super Slow attack speed. The same cannot be said for any other attack speed.

As can be seen in Fig. 4, Super Fast attack speed comes out on top for melee builds. However, note how in Fig. 3, Super Slow attack speed requires the least amount of hits to gain an emerald, while Super Fast requires the most. Since the number of spells a player can cast per second does not depend on the attack speed of their weapon but is instead constant, using Super Slow attack speed yields the most amount of emeralds per second for spell-based builds.

Regarding the additional tests - it is quite clear that Stealing by spells is also affected by the attack speed of a weapon. Furthermore, using Peaceful Rest, emeralds were dropped in line with Super Slow attack speed numbers, despite having a base attack speed of Super Fast. This leads me to believe that it is not the base attack speed of weapons which affects stealing chances, but the effective attack speed after bonus tiers instead (which is sad news for tierstack).


Further Research
While I am content with the discoveries made, there's still plenty more to learn. For one, the exact factors by which Stealing is reduced at higher attack speeds may be obtainable through many more trials and/or regression of some sorts. Furthermore, the data seems to indicate that higher Stealing values yield diminishing results, much like Loot Bonus and Loot Quality. However, this should certainly be investigated further and not be taken as conclusive evidence.


TL;DR

• The nominal Stealing stat likely applies 100% to Super Slow attack speed.
• The effective attack speed (i.e. including bonus tiers) is what affects Stealing chances, not the base weapon attack speed.
• For melee builds, Super Fast effective attack speed gives the most emeralds per second.
• For spell builds, Super Slow effective attack speed gives the most emeralds per second.

Thank you for reading! :)


Revision 1
Introduction
Stealing does not work how I thought it did.

I was doing some tests at 100% Stealing with a Super Fast weapon and noticed how the first hit on a mob would nearly always result in an emerald. So, I started hitting some more guards to depletion, but this time I was making notes of at which hit an emerald was dropped. Here is a link to the results: https://imgur.com/Sdlytzb

The first thing you should note is that there is clearly a pattern in which emeralds are dropping, as opposed to each hit being approximately equally likely to drop an emerald. This implies the emeralds dropped cannot follow a Binomial distribution since the trials are not independent.

Second of all, there is quite some variance in at which hit emeralds are dropped. Note how the first few guards take fewer hits to drop emeralds, with the number of hits increasing in the middle and falling again at the last few guards. For some context: at the start of the tests, I was still getting used to the procedure and was thus hitting the guards slowly. As I continued, my speed increased, until I sort of figured out what was happening and deliberately hit the last few guards more slowly. The results led me to conclude that the game does not track the number of hits landed for deciding whether to drop an emerald, but the time since the last emerald dropped instead.

In this video I show this distinction more clearly:

While the number of hits until depletion is evidently different, the time is approximately the same (~37 seconds).

Basically, this is all to say that the data collected in the original post is not meaningful, because I was not attacking at full speed with each weapon, and emeralds do not drop independently of one another regardless. Since I could not just let incorrect information sit on the forums and do nothing about it, I did some more tests to try and figure out how attack speed affects Stealing. My hypothesis going into it: until the first emerald, the nominal Stealing stat applies 100% to each hit, regardless of attack speed. After that, there is a set time - dependent on attack speed - until the next emerald is dropped, which can be decreased by increasing Stealing.


Method
I performed two tests - one for trying to determine the time in between emerald drops, and one for investigating the relationship between Stealing and the first emerald drop. Regarding the first test, I depleted one guard with a weapon of each attack speed, at five different Stealing levels (10%, 25%, 50%, 75%, 100%), while writing down at which hits emeralds were dropped.

At first, I tried carrying out the hitting by hand, but I noticed that the results were not very consistent. Moreover, I did not feel like developing an RSI at 21, so I decided to use an auto clicker (I am aware this is not allowed, however, I used it for the sake of science only and not for giving me a gameplay advantage, so I ask you kindly not to ban me). I used the Game Bar on Windows to record each trial, so I could count hits more easily afterward.

The second test comprised hitting 40 guards until the first emerald was dropped, at Super Fast and Super Slow speed with 50% Stealing only. If my hypothesis were to be correct, the number of hits until the first emerald should follow a Geometric distribution.


Results
Table 1, given below, shows the data of test 1, i.e. at which hits emeralds were dropped for each attack speed at different Stealing levels.

Spoiler: Tab. 1: Data of test 1.

The average number of hits between emeralds can be extracted from this data, which is shown in Fig. 1.

Spoiler: Fig. 1: Average hits in between emerald drops, as a function of Stealing.

I also plotted each attack speed individually, marking the minimum and maximum number of hits on the graph. The figures can be found here: https://imgur.com/a/PFZBroV
Note how generally at low Stealing values, there is quite some variance in the number of hits.

Through the attack speed multipliers, the average time in between emeralds can be calculated as well, which is plotted in Fig. 2.

Spoiler: Fig. 2: Average time in between emerald drops, as a function of Stealing.

At high Stealing values, the time in between emeralds increases as attack speed increases. However, this relationship does not seem to hold at low Stealing values. Furthermore, there does not seem to be much significant change between 50% and 100% Stealing.

Next up are the results of the second test. The number of occurrences of hits until the first emerald are shown in Tab. 2.

Spoiler: Tab. 2: Data of test 2.

Using this data, the probability of getting an emerald on a certain hit can be calculated. This is plotted next to the probability mass function of a Geometric (p = 0.5) distribution in Fig. 3.

Spoiler: Fig.3: Hits until the first emerald, compared to a Geometric distribution.

Conclusions
Let's start with the results of the first test. I am hesitant to make any conclusions regarding this test, especially since the data at low Stealing levels is quite messy. I wish I could have performed more trials and at more Stealing levels (especially at lower ones since that seems to be the most interesting region), however, this was the best I could do with the time I had available. What I can say, however, is that after ~50% Stealing, the time between emeralds hardly changes.

The number of trials in the second test is also relatively low, increasing the sensitivity to outliers. Nevertheless, the distribution of the measured hits seems reasonably similar to a Geometric distribution, having means quite close to the theoretical mean of 2 hits. Since this holds for both attack speeds, it would imply that at the same Stealing level, Super Fast weapons reach the first emerald faster than slower weapons. At the same time, the first dataset given in this revision (https://imgur.com/Sdlytzb) shows that having 100% Stealing does not guarantee a first hit emerald, which should be the case if the hits were to follow a Geometric distribution. Either the distribution isn't Geometric, that first hit was not registered properly, or something else is at play here. This is also a good moment to mention that I did the counting of hits in all tests by eye (albeit using rewindable video), so any inaccuracies could also be caused by human error.

All in all, it is clear that more testing is to be done, but I will leave that to a later moment. For now, I hope I have been able to correct myself somewhat and provide some more insight into how Stealing works.

TL;DR
The following is what I believe to be true as of this revision:

• The game does not track the number of hits landed for deciding whether to drop an emerald, but the time since the last emerald dropped instead.
• Increasing your Stealing stat above ~50% seems to have very little effect.
• The Stealing stat possibly applies 100% to all hits regardless of attack speed until the first emerald is dropped. This would mean that Super Fast weapons can get the first emerald the fastest.

Thank you again for reading :)


Revision 2
Introduction
There is no new data for this revision, just some more data analysis, so I'll try to keep it brief.

While discussing the findings of Revision 1 on Discord, user @iTechnically suggested the following theory on how Stealing works in regard to time: when an emerald is dropped, a cooldown is activated during which no emeralds can be stolen. Once the cooldown runs out, each hit has the chance to steal an emerald given by the nominal Stealing percentage.


Method
To investigate how probable this theory is, I will try to predict the hits and time between emerald drops using the aforementioned theory, and compare those predictions to the measurement data of Revision 1.

First, the cooldown times for each attack speed must be found. It's a bit tricky though since the number of hits is discrete but time is continuous - I'll use the number of cooldown hits when continuously meleeing since that will yield the most practical results.

If the theory holds, at 100% Stealing the number of hits between emeralds is equal to the number of cooldown hits, plus one hit to steal the emerald. Note that in the test data of Rev. 1 for some higher attack speeds at 100% Stealing, the hits between emeralds are not all the same. For these instances, I took the maximum number of hits, since latency could have caused an effective attack speed lower than the maximum attack speed.

Next, the average number of hits needed to steal an emerald after the cooldown is needed for various Stealing levels. Assuming each hit after the cooldown is equally likely to steal an emerald, the number of hits needed is a Geometric random variable. The average can then easily be computed as 1/p, where p is the probability of Stealing.

In short, the average number of hits between emeralds is theorized to be [cooldown hits] + 1/[Stealing probability].


Results
From the data of test 1 in Rev. 1, the following number of cooldown hits can be found (from Super Fast to Super Slow):
32, 17, 11, 8, 4, 1, 0

Using these values, the predicted number of hits between emeralds can be plotted alongside the measurement data of Rev. 1, as is shown in Fig. 1.

Spoiler: Fig. 1: Predicted hits between emeralds, compared to measurements.

The solid lines show predicted values, while dots indicate measurement points.

In a similar fashion, the measurement data regarding time between emeralds is plotted on top of the respective predictions in Fig. 2.

Spoiler: Fig. 2: Predicted time between emeralds, compared to measurements.

The solid lines show predicted values, while dots indicate measurement points.

Conclusions
The predictions of Fig. 1 and 2 are reasonably in agreement with the trends shown by the measurement data. Note how at low Stealing values, the predictions indicate that slower attack speeds will rise above faster attack speeds and have longer times between emeralds, which is supported by the data. While more data is needed to be able to conclusively state that this is how Stealing works, the current theory is very promising in my opinion.


TL;DR
The current theory on how Stealing works is as follows:
When an emerald is dropped, a cooldown is activated during which no emeralds can be stolen. Once this cooldown runs out, each hit has a chance (given by the Stealing stat) to steal an emerald regardless of attack speed. However, the cooldowns are dependent on attack speed.
The measurement data of the previous Revision seems to support this theory.

Thanks for reading!

 

Interesting analysis. Is it possible that the same effect which non-linearly scales with increasing stealing also causes attack speed differences to deviate from their general multipliers?

 

Thank you!

I didn't know that about attack speed differences - would you mind elaborating/showing me where I can read up on it?

 

Very interesting. Some thoughts on the matter:

1. How does attack speed affect the poison stealing?

2. The diminishing due to attack speed: that's a great find and actually the part that baffled me the most when I asked in Atlas-inc, because without it the data doesn't seem logical. It does seem like super slow is accurate 100% to the stealing chance that a player has, and I fiddled with the idea of it being an exponential function, but it occured to me that this is not the right data needed to calculate that change. You won't do it as you've done more than enough, but I feel like the next step is to test the stealing of the same attack speed with multiple amounts of stealing to figure out the diminishing effect each attack speed has.

Such a great reading and thank you for putting the time and effort to figure this out.

 

First of all, thanks for the kind words!

I can't quite wrap my head around this - I thought since Stealing diminishes with higher attack speeds, but for spells the spells/s (i.e. hits/s) stays the same regardless of attack speed, Super Slow speed will always yield more emeralds/s than Super Fast speed in spell builds.

Oh, I missed this, thanks! Add it to the list :)

 

Sorry! I completely misread a section of your post. My bad.

 

That's alright, I could've worded it better - no need to apologize!

 

The usual assumption we made was that the Stealing chance would scale based on the attack speed multipliers (4.3, 3.1, 2.5, etc. they should be pinned in #class_builds). Which doesn't seem to be the case in your graph. I'm just tossing out potential alternative scalings that may be in play.

 

you are a hero.

 

Does stealing work on passive mobs, including the citizens in towns and cities?

 

Yes, I believe it works on every mob you can hit, so that excludes NPCs and merchants.

 

Bump - edited in Revision 1.

 

The tldr was a life-saver

 

I get that, sorry for the convoluted writing - once I know exactly how stealing works, I'll redo the whole post to make it easier to follow :)

 

Edited in Revision 2. I think we're actually quite close now to solving this.

 

pog fish poggers fish