Galice/Galiks here. Long-time WoW raider (US-Azgalor, now Kil'jaeden), former GM of Rehabilitation Clinic, etc... nerded some stuff out thought I'd share if it helps anyone out.

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Introduction

The enigma shrouding the drop rate of legendary items in Diablo 3 is a center of debate and mass confusion for the most efficient methods to obtain them. The choice of whether it's worth it to increase the difficulty level for an increase in finding legendary items, and the resulting quantitative gain associated with an increase in difficulty, is still unclear. For example, is it worth the massive time increase to suffer through Torment 6 (t6) monsters for a higher drop rate? Many would say of course not (and it turns out they're right).

It seems obvious that higher difficulty levels are just not worth it as all Torment difficulties drop the same items, but supposedly at a different rate. The aim of this report is to provide a quantitative analysis of the drop rates for torment difficulties, and provide a recommendation in farming through comparing drop rates in and outside of rifts. Additionally, magic find (MF) will be factored into the consideration as an extra factor.

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2. Equation Basis http://us.battle.net/d3/en/forum/topic/12413271220?page=13#241 This is 1.15^N as confirmed in this blue post, where N = torment difficulty level.
This was edited. Before there was an equation here that I had manipulated and used this as another reference as comparison to the 1.15^N in case either were right, but alas, 1.15^N was confirmed 3 days ago after I had done any work so only that will be shown. If for whatever reason the previous equation that was here that linked known monster health percent versus difficulty (which also can be thought of as increasing time to kill a mob - a logical way to warrant a drop rate increase), and the blue is just agreeing to release a critical piece of information that has never been released before so whimsically, then that information is no longer public. Your standard 1.15^N formula is shown, as was all along before alongside a separate approach. Nonetheless, the two were similar.

Everything taken into consideration is just looking at the legendary drop rate factor increase for varying parameters. It is in no way explaining the most effective place to farm, as that is left up to the player skill level, gear, speed in clearing, and density/type of monsters/chests in your exclusive rift or outside location. Under the assumption that there is no clear choice/multiple options, these graphs can help lead you in a certain direction.

3. Rifting versus Non-Rifting and Magic Find
Using these values, one can make a comparison against the legendary drop rate factor inside a rift (1.25x multiplier) versus outside of a rift. As one can see below, certain points can be made.

Torment 1 rifting is slightly worse than drop rates from Torment 3 non-rifting.

Torment 2 rifting is slightly worse than Torment 4 non-rifting.

Torment 3 rifting is between Torment 4/Torment 5 non-rifting.

Torment 4 rifting is near Torment 6 non-rifting

More conclusions can be drawn based on self-inquiries.

The question of magic find (MF) and its effectiveness in Reaper of Souls usually ends with 'MF NOT WORTH IT!' Below shows varying MF inside of a rift (assuming MF is multiplicative and legendaries receive 10% of their MF towards legendary drop rates [i.e. 150 MF contributes to 15%] i.e. 1.15*1.25*base_torment_droprate). The plot is shown below.

Wearing no MF inside a Torment 2 rift is slightly less than Torment 4 farming outside of a rift

Wearing 50 MF inside a Torment 2 rift is equivalent to Torment 4 farming outside of a rift

Wearing 150 MF inside a Torment 2 rift is slightly less than Torment 5 farming outside of a rift

Wearing 300 MF inside a Torment 2 rift is almost equivalent to Torment 6 farming outside of a rift

Therefore, as a recommendation for actual possible MF values (Nagelring 50%, Topaz in helm 41%, Cain's set 50%), we can say that for 150 MF farming a Torment 3 rift, we are about the same as Torment 6 outside of rifts. Since Torment 3 speed runs with 150 MF seem much more plausible with the current player/state of the game, this seems to grant the biggest bang for the buck if done consistently and at a proper pace. Comparisons for outside of rift for varying MF are omitted as most interest will likely fall upon the inside-rift scenario due to the 1.25x multiplier (which inherently acts like 250 MF as legendaries receive 10% of the MF value). Nonetheless, other conclusions can be drawn, and these graphs can be used to quantify other questions for varying player skill levels and character capabilities. Choices should be made based upon your own capabilities to optimize speed, location (possible dense area of chests outside rifts), and individual skill.

Battletag is Galice#1533 for questions/inquiries. Galice/Galiks/Galrosh @US-Kil'jaeden (retired for now) Feel free to join my diablo clan <Arch Druid> for a feisty environment and anyone looking for a full-clear heroic WoW guild check out http://www.archdruidguild.com/.

Galiks is too humble to mention this publicly, but he has MS in electrical/computer engineering and is currently pursuing a PhD in the same discipline, so it seems he knows how to crunch numbers, create graphs and read schemes

I am not into crunching numbers.. hence the reason i'm in law school, but I do have a 49% nagelring and I have the 3 set cains (with the ring) and do have that 41% mf gem, so by using good ol logic, it seems as if i was on the right track baby -- GG and great post man keep it up.

All my noob friends will be salty when i'm raking in the legendaries speeding those T3 rifts 5 min at a time... Would love to read more about this i find it rather interesting! Usually i just bend a knee and do a voodoo dance to the RNG gods...

Would be cool if you would take split bounty farming into the comparison. Since with a lot of horadic caches within a small amount of time combined with doing the actual bounties end killing mobs that seems to yield quite a lot of legendaries as well.

Great graphs! but I thought I read a post some time ago that only 10% of your MF helps with yellow items and only 1% of your MF helps for legendaries? seems to be a bit of a gray area at the moment. anyone able to confirm?

The derivation will be omitted, but the end result is the following expression (where N again is a torment difficulty value):

Can you please post the derivation because I don't see where that equation comes from. It seems unreasonable. Increasing the health of the mods increases the drop rate of legendaries, what? And (health_n+1/health_n)^1/3 will always be 1.6^1/3 ~= 1.17 since health increase is 60% per T.

Also y-axis from 0 would make for a more honest comparison.

Using information released previously in http://us.battle.net/d3/en/forum/topic/12413271220?page=1\#2 when regarding calculation between Torment 1 and Torment 2 legendary drop rates in rifts, a trial-and-error approach was utilized to apply known torment difficulty statistics and solve for the drop rate increase due to the difficulty increase. Some people assume it is just a 1.15^N function (where N = the torment difficulty), but that has no grounds or proof behind it.

The derivation will be omitted, but the end result is the following expression (where N again is a torment difficulty value):

Can you please post the derivation because I don't see where that equation comes from. It seems unreasonable. Increasing the health of the mods increases the drop rate of legendaries, what? And (health_n+1/health_n)^1/3 will always be 1.6^1/3 ~= 1.17 since health increase is 60% per T.

Also y-axis from 0 would make for a more honest comparison.

He probably assumed that blizzard wanted some kind of roughly equivalent droprate based on the time it takes to kill enemies so he used health to calculate it. The math doesnt necessarily check out though, as you say. His formula has no more ground to stand on than the assumption of 1.15^N.

Really, you cant calculate the droprate bonuses when given only 2 points of data. There are an infinite number of equations that would satisfy those requirements. For now at least, the best guess might be the simplest (1.15^n)
EDIT: In light of the post above me, it really is 1.15^n

This post is built on a completely incorrect assumption. It was confirmed by Lylirra in that very same thread that it is, in fact, a 15% multiplicative per torment level buff (eg 1.15^N):http://us.battle.net/d3/en/forum/topic/12413271220?page=13#241

Taking this into account, the drop chance for 150 MF is effectively the same thing as farming a rift one torment level higher. That is to say, farming a rift on T3 with 150 MF is effectively the same thing as farming a rift on T4 with 0 MF. It is nowhere near the drop chance of T6 rifts.

Very disappointed to see such clearly incorrect and misleading information be prominently displayed on the front page of the site.

This post is built on a completely incorrect assumption. It was confirmed by Lylirra in that very same thread that it is, in fact, a 15% multiplicative per torment level buff (eg 1.15^N):http://us.battle.net/d3/en/forum/topic/12413271220?page=13#241

Taking this into account, the drop chance for 150 MF is effectively the same thing as farming a rift one torment level higher. That is to say, farming a rift on T3 with 150 MF is effectively the same thing as farming a rift on T4 with 0 MF. It is nowhere near the drop chance of T6 rifts, as shown in your standard extrapolation graph on the right.

Very disappointed to see such clearly incorrect and misleading information be prominently displayed on the front page of the site.

Does this take into account that MF towards legendaries is only 10% effective?

It should, I think, assuming your MF chance is multiplied against the separate torment/rift modifier. 10% of 150% MF is 15% legendary MF, which is the same thing as the legendary chance increase going up one torment level.

the 1.25x multiplier (which inherently acts like 125 MF as legendaries receive 10% of the MF value)

I think you made a mistake here. 125 MF is 12.5% (factor of 1.125) and not 25% (factor of 1.25). I can see how this mistake could be easily made.

But since you by mistake made MF twice as effective, you would actually need to re-run the numbers and re-draw the MF conclusions. The 0 MF conclusions are of course the same.

Really great post, glad you put time into it: You're awesome.

The other issue is whether the base drop chance of mobs inside rifts is the same as those outside. Perhaps it isn't, perhaps rifts were buffed but are still worse than adventure mode?

So, instead of this misleading information, something that would be actually frontpage worthy would be statistics on rift clear speed at different difficulties, as this is what determines which T is the most efficient. Rift mob/chest density is unaffected by difficulty, so the amount of legendaries per rift just follows rate = base*1.15^T. So stepping up a T is worth it if your clear time goes up by less than 15%.

Sadly clear time ratios will depend heavily on gear level, class, builds, legendary specials and many other factors, so it will not be possible to conclusively say that one T is the best. But by collecting clear times from players of different DPS levels it would be possible to plot legendary rate (that is, 1.15^T/time) as a function of DPS to give people a rough idea of what difficulties are best at what gear levels.

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IntroductionIt seems obvious that higher difficulty levels are just not worth it as all Torment difficulties drop the same items, but supposedly at a different rate. The aim of this report is to provide a quantitative analysis of the drop rates for torment difficulties, and provide a recommendation in farming through comparing drop rates in and outside of rifts. Additionally, magic find (MF) will be factored into the consideration as an extra factor.

-----------------------

2. Equation Basishttp://us.battle.net/d3/en/forum/topic/12413271220?page=13#241 This is 1.15^N as confirmed in this blue post, where N = torment difficulty level.

This was edited. Before there was an equation here that I had manipulated and used this as another reference as comparison to the 1.15^N in case either were right, but alas, 1.15^N was confirmed 3 days ago after I had done any work so only that will be shown. If for whatever reason the previous equation that was here that linked known monster health percent versus difficulty (which also can be thought of as increasing time to kill a mob - a logical way to warrant a drop rate increase), and the blue is just agreeing to release a critical piece of information that has never been released before so whimsically, then that information is no longer public. Your standard 1.15^N formula is shown, as was all along before alongside a separate approach. Nonetheless, the two were similar.

Everything taken into consideration is just looking at the legendary drop rate factor increase for varying parameters. It is in no way explaining the most effective place to farm, as that is left up to the player skill level, gear, speed in clearing, and density/type of monsters/chests in your exclusive rift or outside location.Under the assumption that there is no clear choice/multiple options, these graphs can help lead you in a certain direction.3. Rifting versus Non-Rifting and Magic FindUsing these values, one can make a comparison against the legendary drop rate factor inside a rift (1.25x multiplier) versus outside of a rift. As one can see below, certain points can be made.

The question of magic find (MF) and its effectiveness in Reaper of Souls usually ends with 'MF NOT WORTH IT!' Below shows varying MF inside of a rift (assuming MF is multiplicative and legendaries receive 10% of their MF towards legendary drop rates [i.e. 150 MF contributes to 15%] i.e. 1.15*1.25*base_torment_droprate). The plot is shown below.

Therefore, as a recommendation for actual possible MF values (Nagelring 50%, Topaz in helm 41%, Cain's set 50%), we can say that for 150 MF farming a Torment 3 rift, we are about the same as Torment 6 outside of rifts. Since Torment 3 speed runs with 150 MF seem much more plausible with the current player/state of the game, this seems to grant the biggest bang for the buck if done consistently and at a proper pace. Comparisons for outside of rift for varying MF are omitted as most interest will likely fall upon the inside-rift scenario due to the 1.25x multiplier (which inherently acts like 250 MF as legendaries receive 10% of the MF value). Nonetheless, other conclusions can be drawn, and these graphs can be used to quantify other questions for varying player skill levels and character capabilities. Choices should be made based upon your own capabilities to optimize speed, location (possible dense area of chests outside rifts), and individual skill.

Battletag is Galice#1533 for questions/inquiries. Galice/Galiks/Galrosh @US-Kil'jaeden (retired for now) Feel free to join my diablo clan <Arch Druid> for a feisty environment and anyone looking for a full-clear heroic WoW guild check out http://www.archdruidguild.com/.

Galiks is too humble to mention this publicly, but he has MS in electrical/computer engineering and is currently pursuing a PhD in the same discipline, so it seems he knows how to crunch numbers, create graphs and read schemes

I'm trying my best to be active on Twitter.

All my noob friends will be salty when i'm raking in the legendaries speeding those T3 rifts 5 min at a time... Would love to read more about this i find it rather interesting! Usually i just bend a knee and do a voodoo dance to the RNG gods...

~always be farming

Psimos

I thought I read MF didn't contribute to legendary drop rate? :/

Would be cool if you would take split bounty farming into the comparison. Since with a lot of horadic caches within a small amount of time combined with doing the actual bounties end killing mobs that seems to yield quite a lot of legendaries as well.

Can you please post the derivation because I don't see where that equation comes from. It seems unreasonable. Increasing the health of the mods increases the drop rate of legendaries, what? And (health_n+1/health_n)^1/3 will always be 1.6^1/3 ~= 1.17 since health increase is 60% per T.

Also y-axis from 0 would make for a more honest comparison.

And that's where you're already wrong, sorry.

1.15^N was confirmed in a blue post ( http://us.battle.net/d3/en/forum/topic/12413271220?page=13#241 )

All the rest of your post is then clearly wrong, but thanks anyway :-)

He probably assumed that blizzard wanted some kind of roughly equivalent droprate based on the time it takes to kill enemies so he used health to calculate it. The math doesnt necessarily check out though, as you say. His formula has no more ground to stand on than the assumption of 1.15^N.

Really, you cant calculate the droprate bonuses when given only 2 points of data. There are an infinite number of equations that would satisfy those requirements. For now at least, the best guess might be the simplest (1.15^n)

EDIT: In light of the post above me, it really is 1.15^n

Taking this into account, the drop chance for 150 MF is effectively the same thing as farming a rift one torment level higher. That is to say, farming a rift on T3 with 150 MF is effectively the same thing as farming a rift on T4 with 0 MF. It is nowhere near the drop chance of T6 rifts.

Very disappointed to see such clearly incorrect and misleading information be prominently displayed on the front page of the site.

Does this take into account that MF towards legendaries is only 10% effective?

I think you made a mistake here. 125 MF is 12.5% (factor of 1.125) and not 25% (factor of 1.25). I can see how this mistake could be easily made.

But since you by mistake made MF twice as effective, you would actually need to re-run the numbers and re-draw the MF conclusions. The 0 MF conclusions are of course the same.

Really great post, glad you put time into it: You're awesome.

Sadly clear time ratios will depend heavily on gear level, class, builds, legendary specials and many other factors, so it will not be possible to conclusively say that one T is the best. But by collecting clear times from players of different DPS levels it would be possible to plot legendary rate (that is, 1.15^T/time) as a function of DPS to give people a rough idea of what difficulties are best at what gear levels.