Analysis on the effectiveness of increased item quantity and rarity
Alternative formatting/disccusion at http://www.reddit.com/r/pathofexile/comments/2xn0xq/analysis_on_the_effectiveness_of_increased_item/
Hillock was killed 2500 times in sets of 100 with increased item quantity values of 0/20/40/60/80 and increased item rarity values of 0/100/200/300/400. Diminishing returns for IIQ and IIR are estimated, and an estimate is given for the drop rates of non-rarity and unique items. Data summaries are included as appendices.
There is limited information available on the effectiveness of increased item quantity (IIQ) and increased item rarity (IIR). Chuca79 has a brief summary of 1000 Dominus kills http://www.pathofexile.com/forum/view-thread/725812, but details are scant. GGG staff has commented on IIQ and IIR, but without offering much detail http://www.pathofexile.com/forum/view-thread/531619/filter-account-type/staff. This post seeks to offer insight into the mechanics of IIQ and IIR, and how that might affect drop rates of items without rarity.
Two characters were used to gather data, a 73-74 crit tornado shot ranger and a 79-80 blood magic dual flame totem marauder (dedicated magic find character). The tornado shot character was used as movement speed bonuses from the skill tree were quite beneficial. Data were gathered over several weeks, and occasionally the characters were used for other purposes, hence the level changes. Character experience gained from the runs was negligible.
Hillock was killed on standard in sets of 100 with a given value of IIQ and IIR. Target IIQ was 0/20/40/60/80 and target IIR was 0/100/200/300/400, for a total of 2500 kills. Both maximum IIQ and IIR values should be achievable exclusively during a temporary league, but high values of both IIQ and IIR are only achievable on standard using legacy gear (IIQ gem and helmet and amulet with IIQ). Actual IIQ values and IIR values used to gather data vary from targeted values by +/- 2 IIQ and +/- 2 IIR with one exception (0 IIQ 409 IIR).
Hillock was chosen as a target, as opposed to Dominus, for several reasons: it is easier to count a small number of drops, strand is one of the most linear zones (saving on time searching for the correct path forward), stopping to vendor or place items in stash is convenient, and time for a complete run is short. 
The following analysis is valid for Hillock on standard. Caution should be used when applying these results to other mobs in game. Additionally, a large amount of variance is shown in the data, so conclusions are only weakly drawn.
3.1 IIQ Effectiveness
As stated by GGG, IIQ suffers from diminishing returns. A baseline of 517 drops per 100 Hillock kills is established by averaging the number of drops with no IIQ. The expected number of items dropped by increasing IIQ can be predicted by multiplying the baseline by 1.2/1.4/1.6/1.8, and this can be compared to the average number of drops for each amount of IIQ.
This shows that after the first 20 IIQ, the effective IIQ decreases non-linearly by about 5% per 20 IIQ. The results are summarized in Table 1
Table 1 (CSV)
3.2 IIR Effectiveness
The number of normal items which drop falls off sharply as IIR increases (Chart 2) while the number of magic items increases in a non-linear fashion (Chart 3).
This seems to indicate nothing has changed since what GGG poster Rhys has stated http://www.pathofexile.com/forum/view-thread/531619/filter-account-type/staff:
Due to the nature of how item rarity works (rolling a non-rarity base and/or then rolling each rarity http://www.pathofexile.com/forum/view-thread/531619/filter-account-type/staff), establishing IIR effectiveness is slightly more complex. Here, only the effectiveness on the number of rare item drops will be evaluated.
Ignoring the large amount of variance, there is a definite upward trend for the number of rare items dropped as IIR increases to 400.
To establish effectiveness, the number of rare drops is normalized per IIQ (percent of total drops is used) and averaged for each tier of IIR. As IIR is not as simple an attribute as IIQ, no baseline is given to establish effectiveness, but a summary of effectiveness is given in Table 2.
rare IIR,average % rare items
Table 2 shows that the most benefit is given by the first 200 IIR with severe penalties applied afterwards. This roughly confirms Chuca79's results.
3.3 Non-rarity droprates
With such a large amount of variance, no meaningful conclusion can be drawn from how IIQ and IIR affects drop rates of items without rarity, although it does appear IIQ increases the number of non-rarity items. However, an estimate for drop rates of items without rarity should be possible from the amount of data collected, assuming IIR and IIQ has no affect on the drop rate of items without rarity (which may be incorrect). Below is a summary of the drop rates for non-rarity items. Note: according to the wiki http://pathofexile.gamepedia.com/Drop_Rate there will be a 36.4% drop rate penalty for currency items, which may or may not apply to scrolls, skill gems and other crafting items.
Table 3 (CSV)
item,# drops,% of non-rarity,% of total
scroll of wisdom,194,41.72%,7.76%
orb of transmutation,25,5.38%,1.00%
orb of augmentation,14,3.01%,0.56%
orb of alteration,14,3.01%,0.56%
orb of chance,8,1.72%,0.32%
orb of alchemy,8,1.72%,0.32%
orb of fusing,3,0.65%,0.12%
orb of scouring,3,0.65%,0.12%
Ratio's of one item to another approximately align with previously posted values with some notable differences http://www.reddit.com/r/pathofexile/comments/2dvjol/drop_rate_of_currency_in_poetw_wiki/. This could be due to the large amount of variance, or because the previously posted values were wrong or because drop rates have been changed since originally posting.
3.4 Unique droprates
Due to the large amount of variance, no significant conclusions can be drawn to how IIQ and IIR affect unique drop rates. Summing total results and ignoring IIQ and IIR, unique drop rate is estimated to be around 0.207%.
Data seems to align with Chuca79's results, although it appears an ideal balance of IIQ and IIR to other stats is achieved at a lower value than the "C=8" value.
Currency drop rates are shown to be disproportionate to the currency upgrade path offered by vendors. Orbs of chance and alchemy are much more common as drops than other "lower" currencies.
Data summary (CSV)
IIQ,IIR,unique,rare,magic,normal,non-rarity,sum,unique %,rare %,magic %,normal %,non-rarity %,# runs,char level
Non-rarity summary (CSV)
IIQ,IIR,scroll of wisdom,portal scroll,augmentation,transmutation,alteration,jeweler's,chromatic,fuse,chance,alchemy,scouring,chaos,vaal,blessed,skill gem,whetstone,armourer's,cartographer's,bauble,sum
,% of non-rarity,41.72%,20.43%,3.01%,5.38%,3.01%,1.29%,1.51%,0.65%,1.72%,1.72%,0.65%,1.51%,0.22%,0.22%,6.88%,2.37%,6.02%,1.51%,0.22%,2500
,% of total,7.76%,3.80%,0.56%,1.00%,0.56%,0.24%,0.28%,0.12%,0.32%,0.32%,0.12%,0.28%,0.04%,0.04%,1.28%,0.44%,1.12%,0.28%,0.04%
List of skill gems dropped, in order.
chance to flee
added chaos damage
superior raise zombie
melee damage on full life
List of unique items dropped, in order.
chalice of horrors
shackles of the wretched
 Of course, one of the disadvantages of strand is the lack of masters and corrupted zones.
on Mar 2, 2015, 12:02:19 AM
No comments yet? Very informative and nice write up!
"Fixing the endgame was hard - No matter how hard we buffed red maps, people would keep spamming Gorges.
So we turned Gorge into a red map"
on Mar 2, 2015, 4:48:52 AM
Do it again, this time wear a Fairgrave's Tricorne, carry an imprint of the tricorne and a stack of rhoa feathers in your inventory.
lmk how it goes, thanks.
on Mar 2, 2015, 5:22:10 AM
Thanks for doing this in the name of PoE Science. Always love reading this stuff.
Nothing is more common than unsuccessful men with talent.
on Mar 2, 2015, 6:03:32 AM
Came from the Reddit thread to post how valuable stuff like this is for the community.
on Mar 2, 2015, 7:14:46 AM
Hi everyone, hi Tolos,
First of all, GG on the analysis, that's a nice job you did there !
I was in the process of doing something similar myself, and then it turned out to be too time-consuming (because of the shitty statistics on rarity drops), so I stopped and never did anything with it, but this seems like a good place to show however few conclusions I arrived at. So here goes.
I did a few hundred Merveil, Piety and Dominus runs, with various IIQs, and noticed a few things. First of all, the number of items that dropped was always the same plus or minus 1, and that's counting uniques, currencies, maps and gems too, from which we can conclude that at least those guys (and probably all unique monsters, I would assume) have a fixed number of items they drop, which is then influenced by IIQ (and which may not be an integer, hence the plus or minus one: if the value is 13.4, one would fint 13 items 60% of the time and 14 items 40% of the time, for instance). Once you know that, you don't need too many runs to get good results on quantity. Here are mine:
Red: Merciless Piety, base 19.5 items
Blue: Merciless Merveil, base 16.8 items
Purple: Merciless Dominus, base 21.8 items
Green: Tolos's data, Merciless Hillock, base 5.17 items
As you can see, it all falls pretty neatly on the same curve. The plotted lines are the non-diminished quantity (straight line), and a fit. For the fit, I was inspired by the formulae for Armour and Evasion effectiveness. Clearly, GGG likes fractional powers. So I tried some, and the best fit I could get with simple fractions is:
number of items = base number * (1 + 4.5*IIQ/100)^0.25
It may not be exactly that, but I'd wager it's pretty close. Only thing I don't like is that it's not close to (1+IIQ/100) for small IIQ. Better statistics may tell.
The question of rarity is much trickier. Since it is random, even for bosses, we need much better statistics to be able to say anything.
The one thing we know, from what was said in http://www.pathofexile.com/forum/view-thread/531619/filter-account-type/staff, is that when an item drops (once the number has been decided, which happens first), it needs to pass checks to decide on its rarity. First it tries for unique, and if not it tries for rare, and if not it tries for magic. Let's forget about other items for the moment (they are few enough that we can disregard them), and let's drop the uniques too, and focus on the rare, magic and normal items.
If IIR works anythig like IIQ, we expect two numbers, A and B, to determine the probability to draw a rare and a magic item, respectively. And we expect those numbers to depend on IIR in a way similar to quantity:
A = A0 * (1 + x1 *IIR/100)^y1
B = B0 * (1 + x2 *IIR/100)^y2
There are two main ways that this could work. Either A and B are the absolute numerical checks for rarity, or they are the relative probabilities of rare and magic, which is to say:
1) If RandomNumber < A then rare, if A < RandomNumber < B then magic, otherwise normal
2) If RandomNumber < A/(1+A+B) then rare, if A/(1+A+B) < RandomNumber < B/(1+A+B) then magic, otherwise normal
The main difference between those two possibilities is that in the first case, with enough IIR, you may never see normal items (if RandomNumber is between 0 and 1 and B becomes larger than 1) on some enemies. Since that seems to be the case for many bosses at least, I would expect n°1 to be correct.
To see what I could get in the way of statistics, I did a few hundred Gravicius runs (so that I would have normal items as well). Here are the plots, assuming 1):
Blue: Merciless Gravicius, base A=0.07
Green: Tolos's data, Merciless Hillock, base A=0.066 and much better statistics
Blue line: 1 + 4 IIR/100)^(1/4)
Green line: 1 + 3 IIR/100)^(1/3)
Blue: Merciless Gravicius, base B=0.39
Yellow line: 1 + 2.5 IIR/100)^(1/2.5)
There are many problems with these statistics. The first one is that even the base value needs to be determined, which doesn't help. But still, it looks pretty decent. The error bars would make you cry, though.
The things that are left to do are these:
- a table of base number of drops for all unique monsters
- better statistics on rarity
- determining how much the base values and parameters for A and B depend on the monster (they do: for Merciless Dominus, for instance, I never saw a normal item, so B0>1)
- also, statistics for negative IIRs would be nice (using one or two Aurumvoraxes... Aurumvori ? Aurumvoros ?)
- How party and map MF interacts with all that. They are supposed to be calculated separately and not have diminishing returns (if I remember right). Easy enough to check for IIQ, not so easy for IIR.
- time correlations on rarity. Some people say RNG is RNG, some believe in the RNG Cloud. This needs to be checked. But it is infinitely more difficult than the rest.
Anyway, lots of things and so little time. But I'd sure like to join forces and reverse-engineer the hell out of that b**** ! The more the merrier. Who's in ?
Save a Carrot, Eat a Rabbit!
Last edited by LazarusQ on Mar 3, 2015, 8:44:00 AM
on Mar 2, 2015, 12:26:13 PM
Does this mean the optimum mf is achieved with a full party?
(Since MF at a full party is 40iq and 200ir)
on Mar 2, 2015, 8:43:08 PM
on Mar 2, 2015, 10:54:54 PM
Great write-up. Glad to see I'm pretty optimal for solo MF (54/205).
Thanks so much!
IGN: Fiona_Falconstrike, Geldar_the_Barbarian, Natalie_Fellshadow
on Mar 3, 2015, 7:47:06 AM
Thx for your time trying to give us some info, but sadly it doesnt help me.
Because i have 360 IIR and 20 IIQ farming dominus, i can get even half items magic, sometimes only 4-5 magic, but often half items are magic.
This is in Bloodlines league, so, temp league where you need damage and surv on top of mf.
My point is that every second that i mf as solo player, i could get 5+ times more with full party with MF-er that gears up only for mf and 5 runners, so 500+ IIR and 80 IIQ, so that culler kills dom with 700 IIR and 100+ IIQ.
And with those stats, every item is rare and its really rare that 0 uniques drop, and often there are 3-4 uniques per single dom kill.
Mf-ing is balanced around party play / multi client play and this sadness me a lot, also with every new MF unique introduced in game like Greed Embrace, that sealing is raised up and disparity is even bigger :(
on Mar 3, 2015, 1:24:33 PM