The Bloodthirster vs Last Whisper

DiffTheEnder bringing you the latest mathematical analysis here, and I've got an analysis that may cause some of you to change up your build.
Without any delay, today's analysis:

 

The Bloodthirster vs Last Whisper as the 3rd item

As the title implies, this is oriented towards the AD carry role and their build path. To better understand the direction and aim of today's analysis, a background/crash course in the standard AD carry build is needed.

 

Baby Steps

 

Your standard ADC goes bot-lane with a support and in most scenarios builds the following items, with slight variations in order depending on how the lane is going.
 
 

Now here's where things get interesting. What do you build next?

The higher skilled players will most likely be answering with, "a Last Whisper or some defence", while the lower Elo players might answer with "The Bloodthirster". One of my friends once said, "Building Last Whisper before The Bloodthirster is a free 100 Elo" - I was intrigued. Could the difference really be that significant?

Well, needless to say, I pulled out the formula and the math is shown below. 

The video version can be seen here:

Read on for the full article! 

 

Assumptions

 
Since we're interested in a 3rd item analysis - assumptions have to be made about what set-up the AD carry has at the time of finishing the 3rd item.
These are the assumptions I used.
 
 - The champion will be at level 18
 - With the following mastery set up
 
 
and a rune set up similar to this:
 
 
and that at the time of finishing the 3rd item, they will have an Infinity Edge and a Doran's Blade as added damage items (Phantom Dancer as well, but that doesn't give attack damage).
 
 
Depending on the AD carry in question, the Attack Damage of the champion at this point may range anywhere from 210 to 225AD. In terms of the results of the calculation, the variation is not huge and therefore a flat value of 215 AD can be assumed to represent accurate values for AD carries. I'm taking a lower bound for the AD since you may be building the Last Whisper pre-18 if it's a good game.
 

The Comparison in Question

 
So what exactly do we need to be comparing? The biggest purpose of both items is to maximise your damage output. 
 
 
Last Whisper does this by giving you a decent amount of attack damage and the ability to penetrate a lot of your enemy's armor.
 
 
The Bloodthirster does this by giving you huge amounts of attack damage and a return of life for every hit. The comparison can be thought of as a sharp dagger vs. a blunt hammer (that somehow heals you).
 
There is one key factor (among a variety of others) in the comparison that decides which item is better to buy - the enemy's armor.
The more armor the enemy has, the more effective the sharp dagger of a Last Whisper is. The comparison is a question of, "At what enemy armor values does 40AD 46%/6 ArP outdo 60-100AD and 10%/6 ArP?".
 
Fear not. Results are here.
 

The Almighty Graph

 
 
Let me try to explain the best way to read this graph.
If you have an Infinity Edge, a Doran's Blade, full AD marks/Quints and full 21 in offence and you are at level 18 (giving you a total value of ~215 AD) and are considering whether to buy a LW or BT next - take a look at this graph.
 - If the enemy has more than 30 armor, a Last Whisper outdoes a fresh Bloodthirster.
 - If the enemy has more than 107 armor, a Last Whisper outdoes a fully stacked Bloodthirster.
 
Where did I get 30 and 107 from? Look at where the line intercepts the 'Damage Required' values of 60 and 100 (which are the values of an empty/full Bloodthirster).
For those of you who think the 215 AD value may be too high, do not fret, for changing it to a value like 180 AD, gives you values of 36-135 as opposed to 30-107.  The variations are not huge is the point to take away and the conclusions I will make are accurate :)
 

Information to be gained

 
From the graph, it can be seen that at any value greater than 107 armor, the Last Whisper clearly exceeds the Bloodthirster whether it be stacked or not. Now this begs the question, what armor values are the enemy likely to have at this point. Considering that you are at the latter parts of the game and will be building the Last Whisper at any time from level 15-18 depending on how fed you are, it only makes sense to apply the values for the times when the enemy is at the latter parts of the game as well.
 
This page: http://leagueoflegends.wikia.com/wiki/List_of_champions%27_armor, helps us answer that question. As can be seen, at level 18, an enemy's final armor ranges anywhere from 60-90 armor. 
 
At 60 armor, you would require ~78AD to equal the damage from a Last Whisper, which is a half stacked Bloodthirster - and this is for a level 18 Heimerdinger with no armor runes/masteries.
 
However this is an unlikely situation as the enemy will likely have at least an Aegis of the Legion if they are half-decent. This gives out an aura of 12 armor to anyone near the item owner. So our squishy Heimerdinger is now at 72 armor and you would need at least 84 damage to equal the output from Last Whisper.
 
 
Let's take a look at a more realistic situation vs. an average AD carry who would have ~72 armor at 18, then we add 12 armor from Aegis and 6 armor from masteries and 13 from runes to give a grand total of 103 armor. At this value, you would need a fully stacked Bloodthirster, or to be more accurate 98.48 AD to equal the output from Last Whisper.
 
These values get increasingly worse when you start looking at tanks who have nearly 200 armor.
 
It's safe to say that Last Whisper is a better option for basic attacks in pretty much all situations at this point due to the high armor values of enemies.
 

Critical Strikes

 
One of the biggest arguments for building a Bloodthirster at this point is that it amplifies critical strike damage much more. 2.5 times 80 is more than 2.5 times 40. I'd just like to say - false. Critical strikes double the final damage and not your AD. So it's not the 80AD that gets multiplied by 2.5 but the final damage value. This means that whether you're striking critically or not, the graph for damage required stays the same as the above.
 

Life Steal

 
A major component of The Bloodthirster is the Lifesteal. In all realistic situations, you will heal for more per hit with a Bloodthirster than having a Last Whisper+ Vamp. Scepter. The difference is quite significant. You will heal for anywhere between 20% and 146% more per hit with Bloodthirster depending on how stacked it is and how much armor the enemy has. I find this argument to be pointless however. The Last Whisper costs a full 710 gold less than the Bloodthirster. If Lifesteal is really that important for you at that portion of the game, a second Vampiric Scepter or an Executioner's Calling or even a Zeke's Herald (on a support) can be bought to temporarily have a comparable lifesteal value.
 

Conclusion

This one line summarises what I think of the Last Whisper vs. The Bloodthirster.
 
"If the enemy has more than 30 armor, a Last Whisper outdoes a fresh Bloodthirster."
 
By the time you build a Last Whisper, every teamfight could be the one which decides the game. Farming a Bloodthirster is not the ideal task to be carrying out at this crucial stage of the game. Having shown you all the calculations, I feel 100% safe in saying, always buy a Last Whisper as your third item if it's a choice between the Bloodthirster and Last Whisper.
 
tl;dr Last Whisper #1
 
Feel free to disagree - that is my opinion and I am not forcing it down your throat.
 
As always, I hope you guys enjoyed the analysis. If you are one of those people who couldn't be bothered reading the whole article and scrolled down for the conclusion - maybe the video version will be something you like better!
 
Cheers,
DiffTheEnder
 
 
 
 
 
 
If you enjoy reading the article - I'd really appreciate you clicking the up arrow - thank you in advance!
 
 
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Cheers,
DiffTheEnder

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