Armor and effective hitpoints in Dota2

Armor and effective hitpoints in Dota2
Whatever hero you are playing in a game of Dota2, you will be trying to be as hard
to kill as possible with the finances available to you. It is well known that armor
reduces damage caused by physical attacks, but there are many uncertainties and
urban myths concerning the exact armor mechanics.
Damage reduction
The formula for damage reduction1 is
0.06*armor
armor 0
1+0.06*armor

Rdmg(armor) :=
-(1 - 0.94-armor)
0 > armor > -20
(1)

-(1 - 0.9420)
otherwise
As you can see damage reduction (blue curve) is far from being a linear function
(besides from armor values below -20, where it becomes even a constant function).
The derivative (red curve) makes it even more visible: the change of slope decreases
rapidly towards the positive and negative direction.
From this information we learn that the closer you are to 0 armor, the
more effective armor becomes; the closer your enemy is to 0 armor, the
more effective negative armor becomes.
1http://www.playdota.com/mechanics/damagearmor
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Effective hitpoints
We yet lack a definite way to calculate whether a hero should rather invest in a
health or armor item to maximize her survivability. To do so, we will need to find
a measure that takes the hitpoints of the hero into account. The commonly used
term for this measure is effective hitpoints. Effective hitpoints are essentially the
hitpoints your hero has, plus the portion of hitpoints that will be spared by damage
reduction (assuming only physical combat for simplicity).
EHP(hitpoints, armor) := hitpoints * (1 + Rdmg(armor))
(2)
Just from studying the function graph we learn that the more hitpoints you
have, the more effective is armor; the more hitpoints your enemy has,
the more effective is negative armor.
Consider this example: a hero has 2000 hitpoints, and 5 armor. She feels that
he needs to immediately boost her endurance towards physical attacks and has 1400
gold to spend. Her current effective hitpoints are: EHP(2000, 5) 2462. She could
either go for a vitality booster (+250 hitpoints), which would raise her effective
hitpoints by EHP(2250, 5) - EHP(2000, 5) 308, or for a plate mail (+10 armor),
which boils down to EHP(2000, 15) - EHP(2000, 5) 486. She will obviously be
better off with buying the armor.
Now let us put a support hero that only has 1000 hitpoints, and 5 armor in the
same position. Her current effective hitpoints are: EHP(1000, 5) 1231. With a vi-
tality booster she would gain EHP(1250, 5)-EHP(1000, 5) 308 effective hitpoints.
With the chainmail on the other hand only EHP(1000, 15) - EHP(1000, 5) 243.
In this case, investing in raw hitpoints will be the better choice.
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