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Hello everyone! I have produced a more concise and better formulated paper to highlight the differences between Emeralds and Rubies when socketed into different weapons! I analyze both Radiant Star and Marquise gem tiers as well as 4 builds: one hand + shield/quiver, one hand + source/mojo, two hand, dual wield. I separated the two one hand builds because the source and mojo add min/max damage and critical hit chance whereas the shield and quivers generally only add critical hit chance and attack speed (quivers only, but doesn't make one gem better than the other).
This time, I use a MonteCarlo simulation of randomly selected parameters and see what fraction of them benefit from each type of gem in each of the 4 builds! Here is the link to the new paper: https://docs.google.com/file/d/0B4DBj6yBWjVUVGV2blVtYWw2ZWM/edit?usp=sharing I have removed the old paper (version 1.0) as it had some fallacies regarding some of the assumptions as well as some major code bugs. Again, this new paper is by no means perfect but it does try to highlight which builds benefit more from each gem in general. Thank you for your patience and understanding. 

yeah i cant read the last pages give us a post about the quick and easy when to use what


Basically:
One hand users with shield or quiver can benefit slightly from one gem or the other depending on the gear One hand users with mojo or source benefits more from emeralds in general Two hand users can benefit slightly more from one gem or the other depending on the gear Dual wielders greatly benefit more from emeralds than from rubies. I think the rubies should be slightly buffed (maybe +150/180 for radiant star/marquise) to make them even overall with the emeralds. 

Or just nerf emeralds. 

Or not...fool. No reason to nerf when you can just make something else better. 

Umm, a few things are confusing me in your document. Firstly, you're using very low ChD maximums for your calculations, and you're using INCREADIBLY low WAPD and WAED minimums, and low WAPD maximums. Your choice in range on these two would show that you think the average person is using a weapon with between 200 and 600 average damage on a weapon. Even with a weapon that hits 1.6 times per second, that's a minimum of 320 DPS and a maximum of 960 DPS on a one handed weapon. NO one is using a weapon with 320 DPS, especially not in a sword and board or a xbow + quiver set up. MOST people are probably using weapons with higher average damage than your MAXIMUM on a weapon.
Secondly, why is your WDM value from .21.2, shouldn't it be 11.5? Furthermore, your ranges on two handers are even farther off. I would say 80% of people using two handers use skorn, which has an attack of 1 per second. These are ALL physical damage and even with the 50% WDM have an average damage of 800 or so. Your ranges would imply that 600 is the max before WDM. This is why I stated in the previous thread that it's better to just estimate a person's average damage range rather than estimate each component. If you estimate the ranges on each component reasonably, you WON'T get reasonable results, since getting the minimum WAPD and WAED will ALWAYS result in silly results, no matter how well they're estimated. I would bargain that 99% of ALL users are not using a weapon with less than 600 DPS in a sword and board set up at level 60, and most of those weapons are likely axes (since barbs are most likely) or maces, possibly swords and rather unlikely daggers (these are likely the least used weapon in the game). Furthermore, your CC lower bound are increadibly low. While there certainly ARE some players in this range, there are far more sword and board players at .5 than are at .1. Assuming a normal distribution most players would be at around .3, whereas I would assume the VAST majority of sword + board or 1hander + mojo/source/quiver would ALL be at .4+, even the moderately geared ones. Remember again that we're talking about radiant gems. You're assumptions need to be made for the richest of players. Clearly rubies fail at lower levels, with 20 for perfect square? Those will never reasonably beat a perfect square emerald. You pretty well NEED to get to radiant before they can reasonably be compared, and then you have to assume the people who's ranges your using are rich.
Edited by Guybrush#1585 on 2/7/2013 1:47 PM PST


I used ChD values I thought would be reasonable for players before socketing any gem. Next, the WAPD and WAED values I used were just an average figure I thought would be reasonable but maybe not for high end level 60 players. The format again for the variables were (mean, standard deviation). I later adjust these but I think I need to adjust them again, I thought 200 average to 550 average was ok considering it is before the Damage % modifier. But I should maybe update this since not many players use weak weapons anymore! Your choice in range on these two would show that you think the average person is using a weapon with between 200 and 600 average damage on a weapon. Even with a weapon that hits 1.6 times per second, that's a minimum of 320 DPS and a maximum of 960 DPS on a one handed weapon. NO one is using a weapon with 320 DPS, especially not in a sword and board or a xbow + quiver set up. MOST people are probably using weapons with higher average damage than your MAXIMUM on a weapon. The WDM value is computed by a normal distribution with mean 1.2 and standard deviation of 0.2. If the value falls outside of the range of [1, 1.5] then I fix it to that maximum or minimum value.
You are right, most players don't use those low end weapons anymore but some still do. I think all I need to do is reestimate the values for the ranges. As for mixing all the damage together, that won't work as I have shown in the formula, you cannot factor them all out together. The dpDPS term cannot be simplified and written with less than the 7 variables I have used. Furthermore, your CC lower bound are increadibly low. While there certainly ARE some players in this range, there are far more sword and board players at .5 than are at .1. Assuming a normal distribution most players would be at around .3, whereas I would assume the VAST majority of sword + board or 1hander + mojo/source/quiver would ALL be at .4+, even the moderately geared ones. Remember again that we're talking about radiant gems. You're assumptions need to be made for the richest of players. Clearly rubies fail at lower levels, with 20 for perfect square? Those will never reasonably beat a perfect square emerald. You pretty well NEED to get to radiant before they can reasonably be compared, and then you have to assume the people who's ranges your using are rich. Again, thanks for your input, I can adjust the ranges accordingly later once I get more feedback on what parameters I should use. I don't feel that my approach is totally invalid but rather just needs better estimates on the 7 parameters used for determining which gem is better. 

Or just dont post on the forums anymore. 
This nerf attitude doesn't belong in D3. You WoWNerf fanatics have practically assisted in ruining this game. 



get ya Nerfs at WalMart... leave D3 Alone!


Hi again Iria.
Thank you for these results. I'll try some configurations using your formula with my character data. I really appreciate your "author information" :) I am also a Phd student, completely on the physic side. I'm doing nonlinear quantum optic experiment and theoretical models. I just used for the first time Matlab (Octave actually, free on linux :) ) in order to simulate some stuff. That is fun, even though I'm really more use to analytic calculation. Love anime too ^^ I hope your dissertation is on a good way ! It seems that you are really interested in your job, looking at this publicationlike document that you give us. Keep it up ! Cheers 

Gotcha, I was reading that chart entirely incorrectly, however, you method still seems unnecessarily complex, as I have said before, you don't need 7 variables, and estimating average damage is more than good enough. It CAN be simplified.
[(WAPD*WDM+AD)*ED+WAED] This ENTIRE thing is actually just average damage. You could replace that entire section with average damage. The formula then becomes (1+ChC(ChD+gem))*(average damage + gem*ED) Where ED is WDM and ED added together. While this doesn't explain HOW you got the average damage, that doesn't really matter AT ALL for your calculations. If you were to simply estimate player's average damage with a standard deviation instead, it would vastly simplify the calculations and give you far less outliers. 

Gotcha, I was reading that chart entirely incorrectly, however, you method still seems unnecessarily complex, as I have said before, you don't need 7 variables, and estimating average damage is more than good enough. It CAN be simplified. I suppose one could do an approximation of the formula with your averaging method; however, the ruby's effects are boosted by the Damage +% affix found on some weapons which is why I needed to go that deep into the calculation. Also, the elemental damage boost (like that found on Inna's Favor) doesn't apply to elemental damage on weapons but does apply to rubies, base physical part of the weapon, and the additional average damage from equipment. I do agree however that my ranges are a little inaccurate, for example, if a weapon rolls a min/max damage affix (which greatly boosts WAPD) it cannot roll an elemental damage affix (which thus means WAED is 0). My code does not have a way to limit WAED to 0 if a min/max roll is applied or to reduce WAPD to just the base weapon if WAED is not 0. Your idea of averaging does partially resolve this problem but again has other issues when dealing with the elemental damage +% and weapon damage +% modifiers. 

Hi again Iria. Thanks for your reply, I am glad I am not alone here in the DIablo 3 forums! I hope Blizzard considers some of the methods I have used in helping them optimize gem strengths so that they can release the patch sooner! 

I suppose one could do an approximation of the formula with your averaging method; however, the ruby's effects are boosted by the Damage +% affix found on some weapons which is why I needed to go that deep into the calculation. Also, the elemental damage boost (like that found on Inna's Favor) doesn't apply to elemental damage on weapons but does apply to rubies, base physical part of the weapon, and the additional average damage from equipment. Again, as I said, the Damage +% affix is considered in my formula. The damage +% affix on a players stats BEFORE socketing don't matter if you're simply going to use the players average damage and standard deviation. You still need the damage +% for the calculation of the ruby, but don't need the WAED, AD or WAPD stats anymore as they are all rolled into your estimation for average damage. And you can then roll ED and WDM into one as well, since they both affect rubies in the same way. 

If I need to take this paper to version 3.0, I will consider your averaging method and cite you in the credits lol. But at this point, I have spent around 8 hours on these papers and don't foresee any further development unless I get some major support or requests to do so.


Oh and let me know what other Diablo 3 concepts desire my mathematical eye!


You could do one on the ideal CC to CD ratio for a monk as relates to the actual DPS from cyclone/cyclone procs and wave of light uptime due to fists of thunder quickness rune. That's a bit complicated however.
Let's assume no outside sources of spirit regeneration, and empowered wave, 40 spirit, since that's how most monks will run it. Perhaps calculating with and without guardian passive. 
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