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I was watching Arteezy's stream a few days ago when he commented on how he didn't feel like he was farming any faster with the new Battle Fury (I think he was losing a Phantom Assassin game). It sparked my curiosity, so I did some math. I wasn't sure where to post this, but since every non-shitpost I've ever submitted to Reddit went nowhere, I figured I'd just put it here.
A few definitions first.
BD = base damage
PD = plus (raw) damage
QM = Quelling Blade/Battle Fury damage multiplier
TD = total damage
And the relevant patch notes second.
Quelling Blade - Changed from full damage bonus to base damage bonus
- Quell damage bonus from 32/12% to 40/15%
Battle Fury - Battle Fury now requires and provides the bonuses of Quelling Blade
- Quell bonus is now 60% for melee heroes and 25% for ranged heroes
- Attack damage reduced from 65 to 55
- Cleave radius from 250 to 280
These aren't all the Quelling Blade changes, but they're the ones relevant to farming speed.
Quelling Blade
In 6.83, it multiplied each attack's full damage.
(TD) = (QM)[(BD) + (PD)]
Melee: (TD) = (1.32)[(BD) + (PD)]
Ranged: (TD) = (1.12)[(BD) + (PD)]
In 6.84, its multiplier values increased, but it no longer multiplies plus damage.
(TD) = (QM)(BD) + (PD)
Melee: (TD) = (1.4)(BD) + (PD)
Ranged: (TD) = (1.15)(BD) + (PD)
To find the dividing line, we set the corresponding expressions to be equal and simplify.
Melee:
(1.32)[(BD) + (PD)] = (1.4)(BD) + (PD)
(1.32)(BD) + (1.32)(PD) = (1.4)(BD) + (PD)
-(0.08)(BD) = -(0.32)(PD)
(BD) = (4)(PD)
(BD)/(PD) = 4
Ranged:
(1.12)[(BD) + (PD)] = (1.15)(BD) + (PD)
(1.12)(BD) + (1.12)(PD) = (1.15)(BD) + (PD)
-(0.03)(BD) = -(0.12)(PD)
(BD) = (4)(PD)
(BD)/(PD) = 4
Conveniently, the relationship is identical for melee and ranged heroes. When a hero's base damage is 4 times its plus damage, it deals exactly as much damage to creeps with a Quelling Blade in 6.84 as it did in 6.83. I wanted to show this relationship graphically (people like pictures and hate math, right?), but to make it more straightforward, I decided to instead graph the ratio of total damage in 6.84 and 6.83 against the ratio of base damage and plus damage (maybe that doesn't sound more straightforward, but whatever, fuck you, it's elegant).
BPR = base/plus damage ratio = (BD) / (PD)
Melee:
(TD6.84) / (TD6.83) = [(1.4)(BD) + (PD)] / {(1.32)[(BD) + (PD)]}
(TD6.84) / (TD6.83) = [(1.4)(BPR) + 1] / {(1.32)[(BPR) + 1]}
Ranged:
(TD6.84) / (TD6.83) = [(1.15)(BD) + (PD)] / {(1.12)[(BD) + (PD)]}
(TD6.84) / (TD6.83) = [(1.15)(BPR) + 1] / {(1.12)[(BPR) + 1]}
You can either open the spoiler below to see the graph or click here for the slightly larger original.
+ Show Spoiler +
You can see that with a BPR (BD/PD) of 4, the ratio of TD in 6.84 and 6.83 is 1, as we'd already established; exactly as much damage is dealt to creeps with a Quelling Blade in 6.84 as in 6.83. Furthermore, this graph also shows that when a hero's base damage is greater than 4 times its plus damage (BPR > 4), it deals more damage to creeps in 6.84 than in 6.83, while the opposite is true when its base damage is less than 4 times its plus damage.
Battle Fury + Quelling Blade
Because 6.84 made Quelling Blade build into Battle Fury, I compared the 6.84 Battle Fury with the 6.83 combination of Battle Fury and Quelling Blade.
PD = plus damage (excluding Battle Fury's)
In 6.83, Battle Fury cleaved in a 250 radius and gave 65 plus damage, while Quelling Blade multiplied each attack's full damage.
(TD) = (QM)[(BD) + (PD) + 65]
Melee: (TD) = (1.32)[(BD) + (PD) + 65]
Ranged: (TD) = (1.12)[(BD) + (PD) + 65]
In 6.84, Battle Fury's cleave radius increased and it gained improved multipliers from Quelling Blade, but it no longer multiplies plus damage and gives less plus damage.
(TD) = (QM)(BD) + (PD) + 55
Melee: (TD) = (1.6)(BD) + (PD) + 55
Ranged: (TD) = (1.25)(BD) + (PD) + 55
Once again, we set the corresponding expressions to be equal and simplify to find the dividing line.
Melee:
(1.32)[(BD) + (PD) + 65] = (1.6)(BD) + (PD) + 55
(1.32)(BD) + (1.32)(PD) + 85.8 = (1.6)(BD) + (PD) + 55
-(0.28)(BD) = -(0.32)(PD) - 30.8
(BD) = (8/7)(PD) + 110
Ranged:
(1.12)[(BD) + (PD) + 65] = (1.25)(BD) + (PD) + 55
(1.12)(BD) + (1.12)(PD) + 72.8 = (1.25)(BD) + (PD) + 55
-(0.13)(BD) = -(0.12)(PD) - 17.8
(BD) = (12/13)(PD) + 136.923076
(BD) = (12/13)(PD) + (1780/13)
Unfortunately, the change to Battle Fury's plus damage bonus prevents us from elegantly expressing these relationships with ratios. Shit happens; math can be messy. Let's just get to the graph. Again, you can either open the spoiler below or click here for the slightly larger original.
+ Show Spoiler +
Plug some BD and PD values into the TD equations and you'll find that the damage dealt to creeps at all points above and left of each line is greater in 6.84 than in 6.83, while the opposite is true at all points below and right of each line. In math terms:
Melee - If (BD) > (8/7)(PD) + 110 then TD6.84 > TD6.83
- If (BD) < (8/7)(PD) + 110 then TD6.84 < TD6.83
Ranged - If (BD) > (12/13)(PD) + (1780/13) then TD6.84 > TD6.83
- If (BD) < (12/13)(PD) + (1780/13) then TD6.84 < TD6.83
So, was Arteezy's feeling correct? Well, I don't remember exactly what level he was or what his items were, so let's just define a plausible situation and analyze it.
Arteezy is HEHEHing away more MMR. The game has hit the 25 minute mark; his team is behind multiple towers, teammates are BabyRaging and he only just got his Battle Fury on his level 17 Phantom Assassin, rounding out his inventory with Bottle, Wraith Band, Poor Man's Shield and Power Treads (set to Strength because he's been feeding). He's Arteezy, so he just wants to hit creeps. The enemies have invaded his jungle, so he heads into the enemy jungle, praying to EE-sama for a few minutes of uninterrupted farm. Soon after starting his venture into enemy territory, he recalls the changes to Quelling Blade and Battle Fury and starts to wonder if he's actually farming any faster than he was in the last patch. At level 17 with Strength Treads, Wraith Band, Poor Man's Shield and Battle Fury, Phantom Assassin's attacks hit for 113+58 (+3 from Wraith Band, the balance from Battle Fury). 6.84: 6.83
The difference is negligibly small, but his feeling was correct! He was, in fact, doing slightly less damage to creeps than he would have been doing in the previous patch.
BabyRage
Christ, this took forever. Whatever. Maybe I'll work on leveling my PA arcana now (haha, yeah right, let's see who's streaming).
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hey now twitch meme not allowed
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Bold strategy to include twitch emotes in such an elegant post.
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On May 04 2015 20:04 mnck wrote:
Bold strategy to include twitch emotes in such an elegant post.
Lets see if it pays off, nice post love the graphs
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Northern Ireland22213 Posts
On May 04 2015 21:36 YourGoodFriend wrote:Show nested quote +On May 04 2015 20:04 mnck wrote:
Bold strategy to include twitch emotes in such an elegant post. Lets see if it pays off, nice post love the graphs
literally like how katniss stood up to president snow ayy
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Your math that "proves" that you somehow need 4 times more base damage than additional damage to hit as hard in this patch is just a bunch of nonsensical numbers and you forgot your basic rules for simplifying basic linear equations.
What happens if I am dealing 50 base damage with Dragon knight in 6.84 with no items granting raw damage bonus? Uhm... nothing right? When is a quelling blade important? Uhm, in the early game when most heroes have no items granting raw damage... In 6.83 I deal 66 damage. In 6.84, I deal 70... If I add an item like a bracer which gives 3 raw damage, the difference would be that I add the 3 damage before adding the QB bonus in 6.83. This would mean that DK's damage would be 70 in 6.83 with the bracer and 73 in 6.84, if we just disregard the added strength for mathematical simplicity, as both patches would recognise the additional 3 damage, leading me to the conclusion that no, your damage really doesn't need to be higher...
In the early game your quelling blade will be, as the basic math suggests, 8% better. When your damage starts getting based on raw damage, it's effect is lessened. Why? Well it would be idiotic to have an item like battlefury incorporating that damage bonus into it for the late game because the late game push would become stupid.
Just check your math early in the post. You seem to have forgotten that raw damage makes no difference and there really is no correlation between raw damage and base damage that makes logilogical sense... They are unrelated in this patch and having higher base damage is all that matters. Your base damage does not have to be high in relation to your raw damage, it simply has to be high. That is the simplicity of it. Early game, where it is needed, it is better. Late game, when you wouldn't have a slot for it without the battlefury change anyway, it is worse. In other words, they did our carries an early game favour and prevented the creep push from being crazy from a simple 200 gold item.
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I dont think it really matters though, cause by the time you have battlefury creep waves just disappears in a second or two.
The biggest deal is you actually keep the item slot of the quelling blade. Especially with stuff like roshan, you usually had to drop it pretty early in favor of TP scrolls or aegis or whatever.
Now you'll do some decent extra dmg (and cut trees) for the rest of the game.
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✔ Interesting topic
✔ Concise and clear analysis
✔ Well-formed conclusions
✔ Graphs and dank memes
5/5 nice la
My thoughts on the topic:
I think the patch was an overall nerf to Battlefury's effectiveness and made it an overall more specialized item.
It's now less effective versus heroes (as it gives less damage now), grows relatively more inefficient as the game goes later (most BF carriers tend to favor raw damage items), and losing the bonus against Roshan sucks too (no one who got BF didn't get a QB too).
In exchange we get to save a slot and keep the QB active, which is nice but probably only an even trade if we intend only to farm with it (which, luckily, is usually the case).
Overall it seems to focus the item's niche as a mid-game farming tool, which I think is nice as it makes item choices more of a strategic decision, especially since there are more items to choose from now.
With the nerf to jungle stack camps and lane creep gold/exp though, I think the item was generally nerfed this patch (that may have been the actual reason Arteezy felt he was farming slower). Maybe this is also part of Icefrog encouraging fighting this patch? Who knows.
I already saw BF picked up pretty rarely last patch; eh, oh well.
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Thanks for the replies, folks. 
On May 05 2015 04:19 Baozi wrote:
With the nerf to jungle stack camps and lane creep gold/exp though, I think the item was generally nerfed this patch (that may have been the actual reason Arteezy felt he was farming slower). Maybe this is also part of Icefrog encouraging fighting this patch? Who knows.
I'd originally intended to comment on the increased cleave radius as well, but I just forgot about it, and I suppose there isn't much to say about it anyway. 250 to 280 isn't much, but I suppose it makes taking big stacks a bit easier. I don't remember him saying anything about the bounty changes, so I doubt he was thinking about it then, but it still could have been the reason behind his feeling of sluggish farming.
The implications of the Battle Fury changes are indeed quite interesting. As you said, it's been a pretty rarely seen item for quite some time, and these changes don't seem to widen its viable hero pool. Anti-Mage has been THE Battle Fury builder for a very long time and until Ember Spirit was added, he was pretty lonely in that regard. Phantom Assassin and Faceless Void are the only other heroes that come to my mind, but PA's build rate is definitely lower, and Void's even more so (at least since EG brought him back to the competitive scene in the offlane). I imagine someone might want to mention Alchemist making a comeback, but he was gone for so long, and even now, as much as I'd like to see him again, I doubt he'll be the competitive force he was pre-TI3. He's the kind of hero whose farming got hit hardest by these changes; he has shitty stat gain and doesn't like building stats items.
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On May 05 2015 01:07 Pinkie_Dota wrote:
Your math that "proves" that you somehow need 4 times more base damage than additional damage to hit as hard in this patch is just a bunch of nonsensical numbers and you forgot your basic rules for simplifying basic linear equations.
What happens if I am dealing 50 base damage with Dragon knight in 6.84 with no items granting raw damage bonus? Uhm... nothing right? When is a quelling blade important? Uhm, in the early game when most heroes have no items granting raw damage... In 6.83 I deal 66 damage. In 6.84, I deal 70... If I add an item like a bracer which gives 3 raw damage, the difference would be that I add the 3 damage before adding the QB bonus in 6.83. This would mean that DK's damage would be 70 in 6.83 with the bracer and 73 in 6.84, if we just disregard the added strength for mathematical simplicity, as both patches would recognise the additional 3 damage, leading me to the conclusion that no, your damage really doesn't need to be higher...
In the early game your quelling blade will be, as the basic math suggests, 8% better. When your damage starts getting based on raw damage, it's effect is lessened. Why? Well it would be idiotic to have an item like battlefury incorporating that damage bonus into it for the late game because the late game push would become stupid.
Just check your math early in the post. You seem to have forgotten that raw damage makes no difference and there really is no correlation between raw damage and base damage that makes logilogical sense... They are unrelated in this patch and having higher base damage is all that matters. Your base damage does not have to be high in relation to your raw damage, it simply has to be high. That is the simplicity of it. Early game, where it is needed, it is better. Late game, when you wouldn't have a slot for it without the battlefury change anyway, it is worse. In other words, they did our carries an early game favour and prevented the creep push from being crazy from a simple 200 gold item.
I don't understand what disagreement you have with the OP's math. He is answering a basic question, "If I have X base damage and Y raw damage, do I do more damage in 6.83 or 6.84?"
You repeatedly make the point that the QB bonus does not apply to raw damage in 6.84. This is obvious. However, in 6.83 the raw damage does matter. For the purposes of comparison, you need to include the raw damage into the calculations.
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ITT: people who don't realize that keeping QB throughout the game significantly increases farming on farm-heavy heroes(most obvious example is, of course, AM).
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BF, Treads, QB, TP, PMS, Yasha. <- 6 slots already filled. The New BF saves a slot, that is huge.
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@Pinkie_Dota: 1 post, misunderstands OP and starts flaming immediately... seems like he got linked from Reddit.
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The new BF is slightly less effective, but one could argue that the larger cleave radius deals better with creeps en masse.
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It does seem kind of unfortunate that 99% of real-game situations occur in the tiny corner in the bottom left of the last graph. I'd have thought it would be a bit clearer to plot a 6.84 melee and a 6.83 melee line on a BD vs PD graph from like 30-200 each.
Ratios are elegant, certainly, but they're less approachable to the average dude. Nobody really cares about ranged QB either.
Useful info, either way.
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tldr
User was warned for this post
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On May 06 2015 11:33 Belisarius wrote:
It does seem kind of unfortunate that 99% of real-game situations occur in the tiny corner in the bottom left of the last graph. I'd have thought it would be a bit clearer to plot a 6.84 melee and a 6.83 melee line on a BD vs PD graph from like 30-200 each.
Fair point. Here's the Battle Fury + Quelling Blade graph with a smaller scope.
+ Show Spoiler +
However, you can take my axes, but you cannot take my ratios.
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Very informational, but it isn't a huge difference at all, I would neglect it even.
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EPT![[image loading]](http://i.imgur.com/esbCgbS.png)
![[image loading]](http://i.imgur.com/8P2xBkm.png)
![[image loading]](http://i.imgur.com/tFZuIHI.png)
