Bristleback vs Backstab

I started working on the content of this blog shortly after Red Bull Battle Grounds concluded, but couldn't decide how much I wanted to do with it. Eventually, I just got tired of staring at it and so it's been sitting untouched in my Dota 2 documents folder for a while..... Yeah, it was last modified on the 1st of June, damn. Anyway, I don't recall what exactly made me remember it, but I figure I may as well throw it up here since I already put in the work. Maybe I'll be inspired to continue where I left off if people care enough (please don't XD). The red text is me differentiating between stuff I wrote today and my initial work on this. I think that's enough introduction; here's what I did.

In the first game of the second day of Red Bull Battle Grounds’s lives stage, Team Secret reflected Invictus Gaming’s challenge to HellRaisers. They played each other on day 1 with no particularly interesting picks, but HR decided to spice things up on day 2. After iG 5th-picked Bristleback, HR made the 1st Riki pick of 6.84.

A common plague of invincibility down in the darkest depths of the trench, Riki is very rarely seen in professional Dota. What was more interesting to me than the Riki pick itself, however, was the fact that HR picked it after iG’s Bristleback pick. Had HR planned on picking Riki even before iG’s last pick or did they pick it specifically with Bristleback in mind? Regardless of what HR was thinking, I shared the curiosity of the casters about how Bristleback’s namesake interacts with Riki’s Backstab. Does Riki do more damage by Backstabbing through Bristleback or just attacking from the front?

Bristleback:

Bristleback takes less damage if hit on the sides or rear. If Bristleback takes 250 damage from the rear, he releases a Quill Spray of the current level.

Side Damage Reduction: 8%/12%/16%/20%
Rear Damage Reduction: 16%/24%/32%/40%

Bristleback's side is considered to be within 110 degrees from the back.
Bristleback's rear is considered to be within 70 degrees from the back.

Backstab:

If Riki attacks from behind, bonus damage is applied based on his current agility.

Agility Damage Multiplier: 0.5/0.75/1/1.25

Illusions of Riki will play Backstab animations, but not deal any bonus damage.
Bonus damage from Backstab can't be evaded.

Before we start the analysis, we need to define each possible scenario. Rather than just two, there are, in fact, four different ways that Bristleback and Backstab can interact. Three of them are probably fairly obvious. The other is the least likely to occur and probably needs the following diagram to be properly explained.

+ Show Spoiler +

I can’t remember the last time I used MS Paint for anything more than just copying and pasting from Print Screen. Anyway, according to the tool-tip quoted above, Bristleback’s rear is the area behind him bounded by the yellow lines; his sides are the areas on his left and right between the orange and yellow lines. Backstab’s tool-tip says nothing about its angle of effect, but the Dota 2 Wiki’s Riki page says that a unit's back is considered to be within 105° from the back. Rather than just taking the wiki at its word, I made Riki attack Bristleback from a bunch of different angles in a private lobby. I can’t say whether or not 105° is the exact value, but I can confirm that there is a small window on both sides where Bristleback applies and Backstab does not. In my diagram, they are the areas between the orange and black lines.

So, as I said, we have four scenarios.

1. Riki attacks in the 140° window at Bristleback's front: neither Backstab nor Bristleback apply.
2. Riki attacks in either of the 5° windows just past Bristleback's front: Backstab doesn't apply but Bristleback's side does.
3. Riki attacks in either of the 35° windows at Bristleback's sides: both Backstab and Bristleback's side apply.
4. Riki attacks in the 140° window at Bristleback's rear: both Backstab and Bristleback's rear apply.

So, how does each scenario compare with the others? Well, there are a few conclusions that we can draw straight from the scenario definitions.

Backstab doesn’t apply in either 1 or 2, while Bristleback’s side applies only in 2. Therefore, Riki will always do more damage in 1 than 2.
Bristleback’s side applies in both 2 and 3, while Backstab applies only in 3. Therefore, Riki will always do more damage in 3 than 2.
Backstab applies in both 3 and 4, while Bristleback’s side applies in 3 but Bristleback’s rear applies in 4. Therefore, Riki will always do more damage in 3 than 4.

Of the six direct comparisons we can make between each scenario, only three (1-3, 1-4 and 2-4) will require further analysis. We can even make each remaining comparison from the same derivation. Nice la!

TD = total damage
AGI = agility
PD = plus (raw) damage
BBL = Bristleback level
BBM = Bristleback multiplier
~~BSL = Bird, Stewart and Lightfoot~~
BSL = Backstab level
BSM = Backstab multiplier
F = front (prefix for 1)
5 = 5° (prefix for 2)
S = side (prefix for 3)
R = rear (prefix for 4)

Have I mentioned that I really fucking hate being unable to use subscripts?

Bristleback reduces all damage. Backstab gives bonus damage based on Riki’s agility. Riki’s base damage is 6. Put all that together and we get this total damage formula.

(TD) = (BBM){6 + (AGI) + (PD) + (BSM)(AGI)}
(TD) = (BBM){6 + [1 + (BSM)](AGI) + (PD)}

Riki’s total damage is a function of four variables: Bristleback level, Backstab level, Riki’s agility and Riki’s plus damage. We could do some fancy linear algebra, but for the sake of concision and clarity, we’ll just look at a few specific sets of Bristleback and Backstab levels. This lowers our variable count to two (agility and plus damage), allowing us to illustrate our conclusions with a simple graph.

Some abilities, like Bounty Hunter’s pre-6.84 Shuriken Toss damage, scale differently between certain levels. Thankfully, both Bristleback and Backstab scale the same amount per level, so we can represent their values with simple expressions. This isn’t really useful for our analysis outside of making my spreadsheet easier to use, but whatever. Here it is anyway.

(FBBM) = 1
(5BBM) = 1 – {(0.04)[(BBL) – 1] + 0.08}
(SBBM) = 1 – {(0.04)[(BBL) – 1] + 0.08}
(RBBM) = 1 – {(0.08)[(BBL) – 1] + 0.16}

(FBSM) = 0
(5BSM) = 0
(SBSM) = (0.25)[(BSL) – 1] + 0.5
(RBSM) = (0.25)[(BSL) – 1] + 0.5

To make our remaining three comparisons, we equate each scenario’s total damage and simplify. For our single, general derivation, we’ll use X and Y as generic prefixes. Double spacing for reduced eye cancer.

(XTD) = (YTD)

(XBBM){6 + [1 + (XBSM)](AGI) + (PD)} = (YBBM){6 + [1 + (YBSM)](AGI) + (PD)}

[(XBBM)/(YBBM)](6) + [(XBBM)/(YBBM)][1 + (XBSM)](AGI) + [(XBBM)/(YBBM)](PD) = 6 + [1 + (YBSM)](AGI) + (PD)

[(XBBM)/(YBBM)][1 + (XBSM)](AGI) – [1 + (XBSM)](AGI) = 6 – (6)[(XBBM)/(YBBM)] + (PD) – [(XBBM)/(YBBM)](PD)

(AGI){[(XBBM)/(YBBM)][1 + (XBSM)] – 1 – (YBSM)} = (6){1 – [(XBBM)/(YBBM)]} + (PD){1 – [(XBBM)/(YBBM)]}

(AGI) = ([6 + (PD)]{1 – [(XBBM)/(YBBM)]}) / {[(XBBM)/(YBBM)][1 + (XBSM)] – 1 – (YBSM)}

Specifying the scenarios for comparison and particular Bristleback and Backstab levels gives us BBM and BSM values to plug into our equation, which then becomes a nice, simple line relating AGI to PD for equal total damage between scenarios. We’ll look at four sets of Bristleback and Backstab levels for each comparison.

Level 1 Bristleback & Level 1 Backstab

The Bristleback and Backstab multipliers:

(FBBM) = 1
(5BBM) = 1 – [(0.04)(1 – 1) + 0.08] = 0.92
(SBBM) = 1 – [(0.04)(1 – 1) + 0.08] = 0.92
(RBBM) = 1 – [(0.08)(1 – 1) + 0.16] = 0.84

(FBSM) = 0
(5BSM) = 0
(SBSM) = (0.25)(1 – 1) + 0.5 = 0.5
(RBSM) = (0.25)(1 – 1) + 0.5 = 0.5

And the AGI(PD) equations:

1-3 (front = side):

Show nested quote +

1-4 (front = rear):

Show nested quote +

2-4 (5° = rear):

Show nested quote +

Open the spoiler to see the graph or click here for the slightly larger original.

+ Show Spoiler +

Plugging some PD values into each equation shows that the right side of each legend equation is greater than the left side at all points above each line. If you’re wondering what the fuck that means, sorry, I couldn’t think of a more concise way to explain that, but hopefully a few examples will make things clear for you.

At level 2 with Poor Man’s Shield and Ring of Aquila, Riki has 52 agility and 9 plus damage. This point is above every line.

Front < Side
Front < Rear
5° < Rear

At level 3 with Phase Boots and Ring of Aquila and under the effects of level 2 Lunar Blessing and 14 Essence Shift stacks (yeah, I’m reaching, but look at the god damn graph), Riki has 35 agility and 55 plus damage. This point is above every line except for Front = Rear.

Front < Side
Front > Rear
5° < Rear

At level 3 with Phase Boots, Ring of Basilius and Poor Man’s Shield and under the effects of level 2 Lunar Blessing, Whirling Death and 30 Essence Shift stacks (I don’t know dude, Dazzle and Oracle and Abaddon or some shit), Riki has 10 agility and 52 plus damage. This point is below every line.

Front > Side
Front > Rear
5° > Rear

The right side is greater than the left side if the point is above the corresponding line, while the left side is greater than the right side if the point is below the line.

The ridiculous lengths to which we had to go for the second and third examples are very indicative of how unlikely they are to occur. The plus damage and agility values required to make attacking without Backstab the more damaging option are just too high and low, respectively.

Level 1 Bristleback & Level 4 Backstab

The Bristleback and Backstab multipliers:

(FBBM) = 1
(5BBM) = 1 – [(0.04)(1 – 1) + 0.08] = 0.92
(SBBM) = 1 – [(0.04)(1 – 1) + 0.08] = 0.92
(RBBM) = 1 – [(0.08)(1 – 1) + 0.16] = 0.84

(FBSM) = 0
(5BSM) = 0
(SBSM) = (0.25)(4 – 1) + 0.5 = 1.25
(RBSM) = (0.25)(4 – 1) + 0.5 = 1.25

And the AGI(PD) equations:

1-3 (front = side):

Show nested quote +

1-4 (front = rear):

Show nested quote +

2-4 (5° = rear):

Show nested quote +

Open the spoiler to see the graph or click here for the slightly larger original.

+ Show Spoiler +

If you’re wondering why this graph’s AGI axis goes to 120, it’s because I wanted each of these graphs to have the same scale for easy comparability. As far as we had to reach to make the second and third examples for level 1 Bristleback and level 1 Backstab, we would have to reach even farther here. The required plus damage and agility values are even higher and lower, respectively.

Level 4 Bristleback & Level 1 Backstab

The Bristleback and Backstab multipliers:

(FBBM) = 1
(5BBM) = 1 – [(0.04)(4 – 1) + 0.08] = 0.8
(SBBM) = 1 – [(0.04)(4 – 1) + 0.08] = 0.8
(RBBM) = 1 – [(0.08)(4 – 1) + 0.16] = 0.6

(FBSM) = 0
(5BSM) = 0
(SBSM) = (0.25)(1 – 1) + 0.5 = 0.5
(RBSM) = (0.25)(1 – 1) + 0.5 = 0.5

And the AGI(PD) equations:

1-3 (front = side):

Show nested quote +

1-4 (front = rear):

Show nested quote +

2-4 (5° = rear):

Show nested quote +

Open the spoiler to see the graph or click here for the slightly larger original.

+ Show Spoiler +

This graph looks a lot different than the previous two. Not only did Front = Side and 5° = Rear both jump way up the AGI axis, but Front = Rear changed so much that it’s not even in the same quadrant. Let’s zoom out.

+ Show Spoiler +

It’s impossible for a hero to have less than one point in any attribute, which is why the previous graphs only showed non-negative AGI values. This case’s Front = Rear line doesn’t change that; it just means that Front > Rear for every possible combination of AGI and PD values. That’s right; Front > Rear, not Front < Rear. Our previously established rule has been flipped along with the slope of the line (positive to negative). Riki deals more damage from the rear than from the front at all points below the line, which is impossible because a hero can’t have less than one point in any attribute. Therefore, attacking from the front is always the better option for Riki when Bristleback is level 4 and Backstab is level 1.

But Acetone, what about negative PD values? Static Link, dude!

Yes, there are a few heroes (e.g. Razor, Rubick and Vengeful Spirit) that can give enemy heroes a negative PD value. However, this analysis is long enough already (I have no more fucks to give), so we’re not going to worry about that. Our focus is on Bristleback and Riki.

Now that we’ve encountered one situation (among the three comparisons requiring extra analysis) in which it’s always better to attack from the front, the obvious follow-up question is whether or not there are others. The answer is no; this is the only one.

But Acetone, when will Bristleback ever be maxed before Backstab? It seems really unlikely.

Indeed it does, and Dotabuff agrees. A roaming/support/offlane/new meta safe lane 4/4/1 ~~kitten~~ Riki, I guess?

Level 4 Bristleback & Level 4 Backstab

The Bristleback and Backstab multipliers:

(FBBM) = 1
(5BBM) = 1 – [(0.04)(4 – 1) + 0.08] = 0.8
(SBBM) = 1 – [(0.04)(4 – 1) + 0.08] = 0.8
(RBBM) = 1 – [(0.08)(4 – 1) + 0.16] = 0.6

(FBSM) = 0
(5BSM) = 0
(SBSM) = (0.25)(4 – 1) + 0.5 = 1.25
(RBSM) = (0.25)(4 – 1) + 0.5 = 1.25

And the AGI(PD) equations:

1-3 (front = side):

Show nested quote +

1-4 (front = rear):

Show nested quote +

2-4 (5° = rear):

Show nested quote +

Open the spoiler to see the graph or click here for the slightly larger original.

+ Show Spoiler +

This graph is very similar to the first (level 1 Bristleback and level 1 Backstab). The Front = Rear line has jumped up the AGI axis a bit, but the other two are in about the same position. This doesn’t mean that the same conclusions apply, though. According to Dotabuff, Bristleback and Backstab are most commonly maxed at levels 10 and 7, respectively. Keep in mind that the difference between AGI and PD values when Bristleback and Backstab are both level 4 and when they’re both level 1 will, depending on various factors, probably be significantly large.

Whew. We finally made it through four sets of Bristleback and Backstab levels. Maybe that was too many. OH WELL. I’ve been sitting on this shit for long enough that I’m at the point where I just want to put it out there before I say ‘fuck it’ and throw it away. Insert regular EE passion lost tweet here.

Anyway, we looked at just 4 of the 16 possible combinations of Bristleback and Backstab levels. You’ll likely never see some of them in an actual Dota game, but I did the work for them anyway because completion or some shit. Open the spoiler if you’re really, really curious (just skip it).

+ Show Spoiler +

Level 1 Bristleback and Level 1 Backstab
Front = Side

(AGI) = (4/19)(PD) + (24/19)

Front = Rear

(AGI) = (8/13)(PD) + (48/13)

5° = Rear

(AGI) = (4/17)(PD) + (24/17)

Bristleback vs Backstab

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