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• posted a message on Magic Find and its efficiency: A statistical insight
Quote from Cyeron
@Dynomait

If I find the calculation of the diminishing return on the previous page to be close to the observed (when I get that data point) I would like to add your computations about diminishing returns to the original post.

- - -

@ Loroese and Dynomait

Your charts regarding the affix-distributions are interesting, but I will need more time to really understand the underlying principles.
I am not sure, however, if it is really needed to include them for now.

Tried to PM you, but seems that the forum does not allow it before I reach 10-post limit.. Oh well!

Feel free to add the computations and thought process to the first post, and reformat/summarize them the way you see fit. I'd like to emphasize is that the calculation is based on the assumption that the information posted by Blizzard in the game guide is valid.

Regarding the distribution changes in affix amounts based on the mathematical model, I agree that they should be taken with a grain of salt for now.
Posted in: Theorycrafting and Analysis
• posted a message on Magic Find and its efficiency: A statistical insight
Quote from skolem
The question I have is: Does MF play a role in the number of affixes on the guaranteed NV rares? Specifically, do they use that same table method of rolling for legendary, 6 affix and on down for the guaranteed rare? And if so what are the numbers for each level.

Hard to say, since you dont know which of the rares dropping from an elite pack is the guaranteed one, unless that pack drops only 1. With an enormous amount of data about packs that drop 1 rare with 5 stacks, it might be possible to determine how it rolls, but even then its a difficult task since it is not trivial to determine how many affixes a given rare has.
Posted in: Theorycrafting and Analysis
• posted a message on Magic Find and its efficiency: A statistical insight
Quote from Loroese
That looks to be the same as the results I was getting.

Yea, all the results you presented match with mine!

Quote from pliiny
Looking at the blizzard list of 8 tiers magic->legendary. I think that there is a single roll modified by MF which determines what an item becomes. So for example a white item can turn into a magic item due to MF%. and a +6 rare could turn legendary.

Blizzard specifies in the game guide update that rolling takes place tier-by-tier, and is directed from top to down. This means that the rolling is not analogous to a single throw of a dice (meaning 'an initial' white could roll 'upwards' by MF), but instead very much alike to a series of coin tosses. These two processes are very different in terms of outcome, as demonstrated by me and Loroese.
Posted in: Theorycrafting and Analysis
• posted a message on Magic Find and its efficiency: A statistical insight
Quote from Cyeron
Since we know the guaranteed rare drop is minimum of rare quality we can just substract the total found rare items by the number of packs killed to get to the point where the data is not influenced by the guaranteed rare drops. E.g.: for my sample at 200 MF I found 999 rares for 500 elite packs over a total of 2245 items. removing the guaranteed rare provides 499 rares over 500 elite packs over 1745 items. The two tables in section 3.5 illustrates this.
Yea, that definately works!

Btw, I did some more calculations assuming all P(Ra#) are equal. This top to down rolling scheme seems alter the fractions of rares with different number of affixes to deviate from equipartition, favoring the higher amounts of affixes more as MF increases:

I think this is quite hard to validate, but still interesting. If the diminishing return model is correct, then this effect generally follows from it. One thing to note is that items with some high ilvl probably cant spawn with only 1 or 2 affixes, but even when omitting those two categories, the 6-affix category is favoured in the roll.
Posted in: Theorycrafting and Analysis
• posted a message on Magic Find and its efficiency: A statistical insight
Quote from Cyeron

Awesome. That's really impressive. Let's see if it holds water, I have a small test for that computation of yours.

I am currently collecting a new data point at 300 MF. It will take a little while before I get it complete, however, if the trend was linnear and we assume 10% dropchance baselined, then I would expect:

Expectation = 10% * 4 = 40% rares at 300 MF.
Based on the diminishing return this value is more likely to be
Effective diminishing return at 300 MF = 0.88244
--> %R = 40.00% * 0.88244 = 35.30%.

This is something I will be looking into when I get there, however, I am currently sitting at ~150 elites at 300 MF and I can say that for now it seems to be moving towards 36-37% (and maybe lower) rather than closing in on 40%.

Hehehehehe, back to farming!

Sounds promising!

I havent read this whole thread yet so it might have been said already, but are you separating data collected with different amounts of Nephalem stacks to different categories? The 'guaranteed rare' -mechanic most definately affect the statistics at 5 stacks, but there might also be some other hidden stack modifiers.
Posted in: Theorycrafting and Analysis
• posted a message on Magic Find and its efficiency: A statistical insight
Quote from Cyeron
So if I am not mistaken, at 400 MF we see an effective ratio of 0.8465?

Aka
What you're saying is that (based on that computation and that P(Ra#) = 0.0206), then the ratio at 200 MF = 0.9199. If I am not mistaken I can take this value and see what I should hit with %R at 200 MF:

0 MF --> %R = 10.12%
200 MF (linear) = 10.12% * 3.0 = 30.36%
Effective diminishing return at 200 MF = 0.919943 (fuck the digits)
--> Effective %R(MF=200) = 30.36 % * 0.919943 = 27.93 % (rounded to 4 digits).

That is quite close to the observed 28.60%.

Assuming I am using correct math ofc...

Yea, at 400% MF the ratio of the two curves is about 0.847, and at 200% MF about 0.920.

It really does fit your observations quite well
Posted in: Theorycrafting and Analysis
• posted a message on Magic Find and its efficiency: A statistical insight
Quote from Cyeron
Given that P(item-miss) = 1 - P(item-hit) I get:

P(item is not rare) = [ 1 - P(L) ] * [ 1 - P(Ra#) ]^5 = 0.9030
Assuming 4 digits of importance (since the chance for a legendary = 1/1000).

that is not quite equal to your calculated value of 0.9002. Care to explain it to me? Might have made a mistake. I know it's not that far from yours but the numbers should be the same.

The reason your value differs from mine is that in my calculation, I actually used a value P(Ra#) = 0.0206, which is slightly more than 1/50 = 0.0200. I chose my value just to bring the probability of getting a rare with 0% MF very close to 10%, which is the empirical value attained by you guys.

Quote from Cyeron
Edit: Actually you should not include the legendary into the P(item is not rare) unless what you want to calculate is the chance that the item is not rare or above.

So P(item is not rare) = [ 1 - P(Ra#) ]^5 = 0.9039
--> P(item is rare = 1 - 0.9039 = 0.09608

Since the legendary roll is always done first, it effects the chance of getting a rare, I think I decided to leave it into the result, as it does not chance it much. I should rephrase the the thing as "Chance of getting a rare or above", as you suggested.

Quote from Cyeron
And my second question: On the graph you have displayed I assume that the linear increase by MF is the "expected trend" while the actual ratio is what you have based on our data collections? Or how do you get to those curves of yours?
Linear increase by MF is based on the assumption that 100% increase in MF increases the chance of getting a rare by 100%, as my point was to show that this rolling scheme does not work that way. The values of the "Actual ratio"-curve are computed by first calculating P(item is rare) for MF values from 0% to 400%, and then diving them by P(item is rare) for 0% MF.

Edit: Thank a lot for the insight and comments! Keep em coming people, I'll try to answer
Posted in: Theorycrafting and Analysis
• posted a message on Magic Find and its efficiency: A statistical insight
Quote from Cyeron
Quote from Dynomait
For example, for the legendary check at step 1 with 250% MF
P(roll 1 'hits') = (1/1000)*(2.5).

First of all: Great to have another one join the board (so to speak)!
Secondly I don't have time to read through the whole posts (@ work), however, if I am not mistaken, then 250% MF gives 2.5 higher chance and hence you need to multiply with 3.5 (100% gives a multiplier of 2.0). Assuming stuff is linnear ofc. Gonna check up on it when I get home.

Correct, thanks! Luckily its just a typo, actual calculations were done with correct coefficients.
Posted in: Theorycrafting and Analysis
• posted a message on Magic Find and its efficiency: A statistical insight
Having read the official game guide update on magic find, I think its now possible to calculate the amount of rares found as a function of MF.

As already quoted in the earlier posts, it is now stated the guide that
Quote from "Blizzard" »
Item quality is checked in the following order:
• Legendary
• 6-affix rare
• 5-affix rare
• 4-affix rare
• 2-affix rare
• 1-affix rare
• magic item
-- when your roll ‘misses’ a higher-level item quality, the item generator proceeds to the next lowest item quality in the chain (in this case, checking to see if you got a 6-affix rare, then checking for a 5-affix rare, and so on). Your magic find bonus applies to each roll.

The key points deduced from this are that
• Rolling an arbitary rare is a 6-step process which happens from top to down.
• Magic find is applied to the base probability at each step as a linear bonus.
This means that assuming an item fails the roll at first (Legendary) step, the result of the roll at the next (6-affix rare) does not depend on the result of the previous roll, and so on. This means that the steps are independent.

Each step has a base probability to 'hit', which together determine the fractions of different item types dropped at 0% MF. Lets now assume some arbitary values for these base probabilities, lets say
• Legendary: 1/1000
• 6-affix rare: 1/50
• 5-affix rare: 1/50
• 4-affix rare: 1/50
• 2-affix rare: 1/50
• 1-affix rare: 1/50
These imply that on average, out of 1000 dropped items 1 is a legendary and so on.

With these values, we can calculate the probability of an item NOT rolling to be a rare, by using the multiplicative law of probability. Denoting legendary check as roll 1, 6-affix check as roll 2 and so on, we get

P(Item is not a rare) = P(roll 1 'misses')*P(roll 2 'misses')*P(roll 3 'misses')*P(roll 4 'misses')*P(roll 5 'misses')*P(roll 6 'misses'), where

P(roll i 'misses') = 1 - P(roll i 'hits').

As magic find bonus is applied additively, we know that

P(roll i 'hits') = [base probability]*[MF bonus].

For example, for the legendary check at step 1 with 150% MF
P(roll 1 'hits') = (1/1000)*(2.5).

For 0% MF, the assumed base values give
P(Item is not a rare /w 0% MF) = 0.9002
implying
P(Item is rare /w 0% MF) = 1 - 0.9002 = 0.0998 = 9.98%

Note that this value is in good agreement with the data presented in section 2.2 of the first post in this thread (10% of items are rares with 0% MF).

I crunched the numbers and calculated P(Item is not a rare) for MF values from 0% to 400% and then plotted the ratio P(Item is rare /w x% MF)/P(Item is rare /w 0% MF):

The result shows that there is a diminishing return in stacking MF, though not a very strong one.

The diminishing return does NOT apply to the legendary check though, as that roll is done first. Getting 100% MF will double your chances of getting legendaries and so forth. With 6-affix rares the diminishing effect will be very minimal, since as we all know, the legendary roll does not 'hit' very often .
Posted in: Theorycrafting and Analysis