DotA-allstars.com :- SsxDoTS' Armour and EHP guide

This is taken directly from http://www.dota-allstars.com

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Armour and EHP Guide, updated 09/10/06.

Foreword
Prerequisite
Armour Type VS Armour Value
Proving
Item Analysis
Miscs
Conclusion
Q&A

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Foreword

Ok, basically an explanation of armour and its relation to EHP.

Oh yea, look through the Q&A below before asking any question, so they won't be repeated.

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Prerequisite

Understanding of mathscraft.

Definitions

Effective hitpoint, or EHP is defined as the total amount of damage you can receive after factoring damage reduction before it hits zero. If anybody has a better definition, please tell me.

Maximum hitpoint, or MHP, is the amount of hitpoint displayed on the unit.

Diminishing marginal returns, or diminishing returns in short, is an economics term and is defined as each additional unit of input yielding less and less additional output. For example, a second basher gives you 12.75% chance of bashing as compared to 15% to bash on the first, but that is on probability though.

For a detailed explanation of diminishing returns application in DotA, read the third page of this thread, starting from Malle's post.

Facts

Armour does not stack with diminishing returns. It is damage reduction that stacks with diminishing returns. This will be proven later.

Damage reduction from armour is different from armour type reduction. They are independent of each other.

Critical strike does not ignore armour.

Melee bashes are reduced by armour type spell resistance.
Ranged bashes are counted just as normal damage.

Spell damage ignores armour but is reduced by spell damage resistance.

Formula for damage reduction is :
Damage Reduction = ([armor]*0.06)/(1+0.06*[armor])

The formula above only applies for positive values. For negative values, it is calculated differently.

Formula for damage increase (negative armour):
Damage Increase = 2 - 0.94^(-[armor]) or Damage Reduction = 0.94^(-[armour]) - 1
Both formula will give you the same answer, with the first one as positive value and second one as negative.

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Armour Type VS Armour Value

There is 2 types of reduction in Warcraft, one is armour type, and another is armour value. Armour value is what that is covered in this guide, while Armour type will be mentioned briefly regarding how it works in this section.

First thing first, armour type reduction and armour value damage reduction are independent of each other. They do not stack with each other diminishingly or linearly or anything.

So how does armour type reduction works? I'll give an example here, which will illustrate clearly how armour type reduction and how armour value damage reduction interacts with each other.

Let's say, target has hero armour 0 and receives 100 normal damage. Normal damage, does 0.75 times the damage against hero armour, so this is the calculation:

100*0.75 = 75 damage.

The target will receive 75 damage. What if the target has armour value? If the target has 1 armour (5.66% damage reduction), and assuming the damage source does not change, the target will receive:

100*0.75* (1-0.0566) = 100*0.75*0.9434 = 70.755 damage.

See how it works now? Not as complicated eh? Below is a full list of all armour type vs attack type difference.

CODE
light medium heavy fort norm hero unarmoured
Chaos= 1.00 1.00 1.00 0.40 1.00 1.00 1.00
Normal= 1.00 1.50 1.25 0.70 1.00 0.75 1.00
Pierce= 2.00 0.75 1.00 0.35 1.00 0.50 1.50
Siege= 1.00 0.50 1.25 1.50 1.00 0.63 1.30
Hero= 1.00 1.00 1.00 0.50 1.00 1.00 1.00
Spells= 1.00 1.00 1.00 1.00 1.00 0.75 1.00​

A little extra information here, for those who knows a little more about such stuffs. You should have seen that under spells, it does 0.75 times the damage against hero armour. Link it to the triggered spell guide. (Attack Type, Damage Type) Familiar? Yes, attack type is affected by armour type and damage type is affected by the armour value! Hope this clears many misconceptions that people have about this issue. In the case when Damage Type is Magic, it simply means that either the hero is hit by that attack (and take 1.0x the damage), or totally immune to the attack (as with spell immunity).

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Proving

DReceived = Damage Received (after damage reduction)
DReduction = Damage Reduction
DDealt = Damage Dealt (before damage reduction)

To find out how much EHP per point of armour adds,
DReceived = (1-[DReduction]) * DDealt
DReceived = (1- ([armor]*0.06)/(1+0.06*[armor])) * DDealt <- DReduction formula added in

Assuming that the damage dealt brings the MHP directly down to zero,
DReceived = MHP
DDealt = EHP

MHP = (1- ([armor]*0.06)/(1+0.06*[armor])) * EHP
EHP = MHP/(1- ([armor]*0.06)/(1+0.06*[armor])) <- Divide both side by (1- ([armor]*0.06)/(1+0.06*[armor]))

Since we are finding the EHP per point of armour, armour = 1.
EHP = MHP/(1- 0.06/1.06)

Now we have to simplify the denominator, (1- 0.06/1.06).
EHP = MHP/(1- 0.06/1.06)
EHP = MHP/[(1.06-0.06)/1.06]
EHP = MHP/[1.00/1.06]
EHP = MHP*1.06

Therefore, each point of armour adds 6% (0.06) of your MHP to your EHP. To confirm, use EHP = MHP/(1- ([armor]*0.06)/(1+0.06*[armor])) to find the EHP of a additional second armour.

The formula can be simplified as EHP = MHP + 0.06*[armour]*MHP or EHP = MHP(1 + 0.06*[armour]).

Simplified Explanation

In laymay terms, each point of armour adds 6% of your MHP to your EHP.

We will try this out on a hero with 0 armour and 100 hitpoints. Ok, obviously 100 damage to kill.

With the damage reduction formula, a hero with 1 armour has 5.66% damage reduction. With 100 hitpoints, the amount of damage needed to kill him is (assuming x is the amount of damage needed to kill him):

x * (100% - 5.66%) = 100
x = 100/94.34%
and we get the magical value:
x = 106 damage to kill

A hero with 2 armour and 100 health will need

x * (100% - 10.71%) = 100
x = 100/89.29%
and the magical value:
x = 111.99999999999999999999999999989 = 112.

And to see whether it really is linear, we do it at 50 armour, which is supposed to give 300% EHP, so 300 extra for 100 hitpoints.

50 armour gives 75.0% damage reduction.
x * (100%-75.0%) = 100
x = 100/25%
x = 400
OMFG ARMOUR'S SUPPOSED TO HAVE DIMINISHING RETURN. <- Proven wrong. Armour stacks with diminishing returns linearly, provided that hitpoints is a constant. And if you have noticed, it increases by percentage so in fact, your EHP increases exponentially with increase in actual hitpoint or max hitpoint.

Don't believe what I said? We use 200 hitpoints 5 armour as an example.

From the equation, 5 armour will give 23.08% damage reduction.

And 5 armour = 30% EHP. On a 200 hitpoints unit, it will be 260 EHP.
x*(100%-23.08%) = 200
x = 200/76.92%
x = 259.99999999999999999999999999992 = 260.
OMG MAGIC.

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Item Analysis

Comparing gold needed for each percent of EHP.

The list of items that give armour (excluding activable spells and agility items):

• Ring of Protection - 175 Gold. 2 Armour(12% EHP). 14.58 Gold/EHP
• Chainmail - 620 Gold. 5 Armour(30% EHP). 20.67 Gold/EHP
• Platemail - 1400 Gold. 10 Armour(60% EHP). 23.33 Gold/EHP
• Netherezim Buckler - 877 Gold. 5.29 Armour, factoring in stats (31.74% EHP). 27.63 Gold/EHP
• Ring of Basilius - 500 Gold. 3 Armour (18% EHP). 27.78 Gold/EHP
• Helm of Iron Will - 950 Gold. 5 Armour(30% EHP). 31.67 Gold/EHP
• Blademail - 1770 Gold. 5 Armour (30% EHP). 59 Gold/EHP
• Helm of the Dominator - 1850 Gold. 5 Armour (30% EHP). 61.67 Gold/EHP.
• Aegis of the Immortal - 4350 Gold. 10 Armour(60% EHP). 72.5 Gold/EHP
• Mekansm - 2434 Gold. 5 Armour (30% EHP). 81.13 Gold/EHP

Quite pointless really. For more EHP against physical attack, platemail is the best choice. Though not as cost effective as Chainmail or Ring of Protection, it gives 10 Armour in one slot, the highest in game and is the second (now third, thanks to Sphex xD) most cost effective item you can get for more % increase in EHP.

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Miscs

MHP VS EHP at specific armour value
Full size (credit goes to Tunger)


Non-diminishing effect graph VS diminishing effect graph
Graph 1 Graph 2 (credits goes to LoneWanderer)


A graph with no diminishing effect. EHP vs Armour Point


A graph with diminishing effect. No. of basher and the relation to chance of bashing.

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Conclusion

Each point of armour adds 6% of your MHP to your EHP and it stacks linearly. Since it stacks by percentage, having a good mix of armour and hitpoints will help against physical attackers. They complement each other in a way that more armour can boost your overall tanking ability while pure hitpoints give you your raw strength.

Hope this helps. I had a lot of fun making it. . Will add more to this if there is a request for any part.

Thanks to Malle and Virot2 for pointing out some error which may lead to misconception and everyone who has helped.

I will dedicate the next part to questions asked.

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Q&A

Look through before asking questions.

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Q: I think when people say diminishing returns, they mean that a hero with 5 armor will benefit more from a platemail than a hero with 50 armor.

A: If you consider the % increase in EHP then yes there is diminishing returns but if you are talking about the amount of EHP increase in actual numbers, there is an equal increase irregardless of the amount of armor the hero currently has. (by wakki)

Platemail works equally well on a hero with 5 armour or 50 armour. Platemail adds 60% EHP regardless of how many armour you already have.

A hero with 100 hitpoints (percentage stuffs are easier to do with 100) and 5 armour needs 130 damage to kill, with platemail, it will be 15 armour and thus 190 damage to kill.
A hero with 100 hitpoints and 50 armour needs 400 damage to kill, with platemail, it will be 60 armour and thus 460 damage to kill.

You get this value from EHP = MHP/(1- ([armor]*0.06)/(1+0.06*[armor])) so every point of armour adds 6% EHP as proven. It adds additively from the MHP so what wakki said is correct.

The EHP increase is linear, but if you compare the % of EHP increased from 5 armour to 15 armour and the % of EHP increased from 50 to 60 armour, which is 46.15% and 12% respectively, 5 armour does seem to benefit more from platemail but in reality they benefit equally. (by LoneWanderer)

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Q: So. . . based off this information, is the following correct?

"Ignoring the +dmg and the cost of both items, a Desolator is more effective than a Buriza."

The reasoning would be something like this: 6 armor would translate to 36% EHP that they lose when hit with a Desolator. A Buriza critical adds 24% damage (from 0.8*1+0.2*2.2=1.24), statistically making you need to do 24% less damage, or the same as removing four of their points of armor.

A: Not quite. If your damage is amplified by 24% (as by a Buriza), then you need to do around 19% less damage for it to come out to the same amount. This is because you are now dealing 1.24 times your damage, so you only need to deal 1/1.24, or around 81% of the damage you were initially dealing.

To understand why this is the case, suppose Burize increased your damage by 100%. That doesn't mean you need to do 100% less damage to deal the same amount- that would be dealing no damage at all. It means you need to deal 50% as much, since 50%*200% = 100%

Desolator does not reduce EHP by 36%. Consider a unit with 10 armor. This unit has EHP equal to 160% times its base HP. After being hit with Desolator, the unit has 4 armor, and thus EHP equal to 124% times its base HP. The decrease from 160 to 124 is a 36/160 = 22.5% decrease in EHP. In other words, the damage amplification effect on Desolator is dependent on the armor of the target. The higher the target's armor, the less Desolator amplifies damage.

Buriza does not reduce EHP by 24%. As explained above, it reduces EHP by close to 19%.

The damage amplification effect on Desolator equals that of Burize when the unit being attacked has 14.33 armor. If the target has less than 14.33 armor, then Stygian's amplification is greater. If the target has greater than 14.33 armor, then Buriza's amplification is greater. If you want to know how I got this answer, ask. (all by Virot2)

First off, -6 armour does not mean their EHP gets reduced to 64%. It means their EHP gets reduced by 36% of their MHP.

Secondly, an increase of 24% in damage output would not make you deal the same average damage as before if you lowered your output before the increaase by 24%. A damage decrease of about 19.35% would do it.
(1 - 0.1935) * (1.24) = 1.00006
(1 - 0.24) * (1.24) = 0.9424

A Buriza will, regardless of the target's armour, increase your average damage output by 24%, presuming you do not have a critical strike already and that you have no abilities that add damage that isn't affected by critical strikes or in any way prevents or gets prevented by it.

A Desolator will, if the target has 6 or more armour, always reduce the EHP of the target by 36% of their HP. To calculate the damage increase, you divide the EHP pre-reduction with the EHP post-reduction.


So, disregarding damage and cost, we can find the amount of armour where the percentual damage increase from both items are equal:
( 1 + 0.06a ) / ( 1 + 0.06 ( a - 6 ) ) = 1.24

( 1 + 0.06a ) / ( 1 + 0.06 ( a - 6 ) ) = 1.24
1 + 0.06a = 1.24 (1 + 0.06 ( a - 6 ) )
1 + 0.06a = 1.24 (1 + 0.06a - 0.36 )
1 + 0.06a = 1.24 (0.64 + 0.06a)
1 + 0.06a = 0.7936 + 0.0744a
1 - 0.7936 = 0.0744a - 0.06a
0.0144a = 0.2064
a = 0.2064 / 0.0144 = 14.333333333333333333333333333333

To check that this is correct:
MHP ( 1 + 0.06 * (14 + 1/3 ) ) = 1.86 MHP
MHP ( 1 + 0.06 * (8 + 1/3) = 1.5 MHP

1.86 / 1.5 = 1.24

(all by Malle)

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Q: Buriza actually needs more damage than stygian to bring a hero down to zero hitpoint, ignoring cost and damage.

A: This is totally dependant on their armour. When an EHP reduction of 36% of their MHP equals 24% damage increase they will have the same effect. If you get more armour, the percentual decrease in EHP (percentual as in comparing the value after corruption to the value before corruption) will be less, and thus Buriza will add a better average damage output. If the target has less armour, then Desolator is correct. Note that this reasoning only applies to armour values of 6 or above, as negative armour value "reduction" follow another formula. (by Malle)

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Q: say slardar had a guinsoo or someone else had hex
and they were amplified, how much more damage wud that do
-15armour
or if he had deso as well
-21 armour
or if nevermore was there
-25armour

A: That would depend on the target's armour. (by LoneWanderer, is this question even needed at all )

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Q: Doesn't hex bring their armour to 0?

A: No. (by Malle)

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Q: Can one of the smart people in this thread formulate a chart that finds the eqiulibrium of MAXIMUM EHP between HP and armor =/ ? because imo its still true that its better to go for +HP items instead of +Armor items when you have 50 armor since say if you have 2k of hp with 50 armor, then getting HoT, for a boost roughly 1k of hp would boost your EHP by 33.33% (1k of hp / 3k of hp) rather then getting an eigis for +10 armour, increasing EHP by 16.7% (10 armour / 60 armour).

A: Yeah, you would want to figure out what gives you the most possible EHP. The problem is, there's no standardized unit you can put this in. Lets say you have a hero with 1500 HP and 10 armor, well obviously 1500 HP and 11 armor is better and so is 1600 HP and 10 armor, so you need to be able to have an amount you take away from one side to add to the other (i.e. is 1400 and 11 better or worse than 1600 and 9). However, what's an equal amount of armor and HP? How do you decide how much physical HP an armor is equivalent to?

Theoretically you'd balance it out with saying you have X gold to spend on that 1500 HP, 10 armor hero. If the gold is your balance, that'll be fine, what you don't spend on armor you spend on HP, but gold is pretty complicated. Bracers, Sange, Heart, etc all give different amounts of HP for your gold and so do the armor items.

I hope that was a pretty clear expanation. The balance you're looking for is subjective to the amount of gold you have to spend, I think. (by Akuryou)

If someone is free enough to plot a graph for that I believe we can find out the best possible combination for EHP and armour. Not sure though, I lack sleep now after sleeping so late for the past few nights. . (by LoneWanderer)

Hmm, just noticed this question. This could be done if you were to set a price on 1 HP and on one point of armor. Of course, doing so will make the answer somewhat impractical, since different items give you HP and armor at different costs. If you choose a price for HP and for armor, one could use calculus to find the most efficient way to spend your gold. (by Virot2)

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Q:Anyways, so 2000 life (given 0 armour ) is the cut off point between deciding getting a heart or +17 armour's worth?

A: Why so? 2000 hitpoints?

Heart of Tarrasque gives 19*35 + 300 = 965 hitpoints.

17 armour gives 102% of your max hitpoints to your effective hitpoints.

So, heart means 2965 EHP and 17 armour means 4040 EHP. It's way different. .

For 2000 MHP and 0 armour, the cut-off point is actually:
[armour]*0.06*2000 = 965
[armour] = 965/120
[armour] = 8.042

Any amount of armour lower than 8.042 will give less EHP as compared to a heart, for 2000 hitpoints and 0 initial armour. Any amount of armour higher than 8.042 will give more EHP as compared to a heart, for 2000 hitpoints and 0 initial armour. (by LoneWanderer)

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Q: Would negative armour mean that technically, they have less health effectively?
Example,
3000 hp
-10% reduction
would that mean he has only 2700 hp effective?

A: -10% reduction means you take 1.1 times the damage.

3000 / 1.1 = 2727.2727272727272727272727272727 (by Malle)

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Q: So a tiny in the presence of nevermore, hit by slardars amplify damage and hit by a stygian, will effectively only have half his health? In fact.... high hp heroes will suffer more?

A: The damage amplified can get close to double, but it is never double. High hitpoints hero doesn't suffer more. As compared to a hero with lower hitpoints, the latter will get his hitpoints down to zero faster than the former, provided that both armour values are equal. (by LoneWanderer)

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Q: Let's see. The funny thing with armor is as you add more armor, you get less of a % increase to EHP.
Say you have 10 armor. Your EHP = 160% BHP
Add 10 more armor, Your EHP is now 220% BHP, or a 220/160 - 1 = 37.5% bonus.
I don't get the main argument in this thread ... armor does give you diminishing returns as you increase it. Are we arguing over semantics/phrasing?

A: Armor does not have diminishing returns because every point of armor increases your survivability (or EHP) by exactly the same amount as the point before it. If you are going to argue that armor gives diminishing returns because the % increase in EHP is smaller, then you must also say that HP gives diminishing returns since each HP increases your EHP by a smaller % than the one before it.

"Diminishing returns" refers to a diminishing absolute amount, not a diminishing ratio. (by Virot2)

Yes, you get a lower % increase to EHP, but a linear increase in the EHP value as each point of armour adds 6% of your MHP to your EHP.

As mentioned before, the EHP increase is linear, but if you compare the % of EHP increased from 5 armour to 15 armour and the % of EHP increased from 50 to 60 armour, which is 46.15% and 12% respectively, 5 armour does seem to benefit more from platemail but in reality they benefit equally.

Oh, armour reduces physical damage and some spells which has a damage type of normal. For a full list of the spells, visit Tennis's Triggered Spells Guide. (by LoneWanderer)

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Q: technically armor may not give diminishing returns, depending on your definition of that term, but the fact that a 0 armor hero gaining a platemail will make the other team say "wow that guy took like twice as long to kill as before" whereas a 50 armor hero gaining a platemail will hardly be noticed by the other team means that for all intents and purposes, armor DOES have diminishing returns. Same for IAS. Your first hyperstone will multiply your DPS by more than your second hyperstone. Thus, this thread is just misleading to noobs and really doesn't help.

A: Armour is actually seen as a hitpoints multiplier and I don't exactly get what you mean by that. IAS too. IAS actually stacks additively and both points shall be proven in the following answer.

By that reasoning, the following things also have diminishing returns "for all intents and purposes":

• HP
• damage
• HP regen/mana regen
• % increases to movespeed

In fact, if you look at diminishing returns that way, there's really nothing in DotA that stacks with itself without diminishing returns, which renders the term meaningless. That is why it is important to make a distinction between things that stack directly (like HP, armor, attack speed, and damage) and things that really do stack with diminishing returns, like bash chance. (by Virot2)

Assume a hero with 50 agility, BAT = 1.7.

Before items, this hero has 1.7 / (1 + .5) = 1.13 game seconds per attack, or 0.88 attacks per game second.

N = Number of hyperstones
S = attack speed
IAS = total increased attack speed

CODE

N S IAS % improvement
0 0.88 a/s (+0.00) 50
1 1.21 a/s (+0.33) 105 37.5%
2 1.53 a/s (+0.32) 160 26.5%
3 1.85 a/s (+0.32) 215 20.9%


Well, I am convinced it is not diminishing returns. You get the same additive increase in attacks / second, although the % improvement keeps decreasing. As Virot2 pointed out, adding hearts over and over will give you less and less % hp increase.

Thanks for all the clarification guys. Virot2's point cuts right to the heart of it -- it is helpful to distinguish between what is actually stacking directly and what is stacking with diminishing returns. (by URgunnaDIEclown, calculation units edited by LoneWanderer)

This is taken directly from http://www.dota-allstars.com

Direct Link

 

Hmm interesting.. never looked at armour quite like that.

 

i have a headache now :S

 

Looking at this takes the fun away from gaming and makes it alot like what you play games for to lessen the effect of stress/work/etc

is best you go learn some code and make your own game

 

It gives a good idea of what stuff is good for. eg. Deso on a hero with crit skill.

 

no the men who do it had to do the stress...

we have it real easy... neat lil charts etc... we doh hadda go working out thing for weself while game goin on..

and its jus for a basic idea of how stuff works... so u will know when armour greater > hitpoints... when you should by a AS instead of a Heart etc.... u know... things that does make u gosu

 

Thanks

These guys were really bored, and I thank the heavens for bored people every day. The % increase on armor ingame always confused me but now it makes perfect sense.