If durability = hp x defense^1.35 (based on the same priciple you use in this guide)
The crossover point is:
defense = 0.675 hp
A talent node give +36 hp or +18 defense, which mean in talent tree, defense point cost twice than hp point.
To determine the crossover point/turning point, I’ll need dummy variable (total point).
Let just say we are tasked with giving a hero its defense and hp stat but we are limited in points to be allocated:
Adding 1 defense will cost 2.
Adding 1 hp will cost 1.
If all points are used, then:
2 × defense + 1 × hp = total point
Hp = total point - 2 × defense
Durability = hp x defense^1.35
= (total point - 2 × defense) x defense^1.35
To make it simpler, let just say:
a = defense
b = hp
c = durability
d = total point
c = (d - 2a) × a^1.35
c = (d x a^1.35) - (2 × a^2.35)
We have c as a function of a, to find what a result in maximum c, we use derivative:
1.35d x a^(1.350-1) - (2.35 × 2) × a^(2.35-1) = 0
1.35d × a^0.35 - 4.7 × a^1.35 = 0
1.35d × a^0.35 = 4.7 × a^1.35
(1.35d × a^0.35) / 4.7 = a^1.35
a^1.35 / a^0.35 = (1.35d x a^0.35) / (4.7 × a^0.35
a = (1.35/4.7) × d
b = d - 2a
b = d - (2 × 1.35 / 4.7) × d
b = (2/4.7) × d
a /b = (1.35/4.7) × d / (2/4.7) × d = 0.675
Maximum durability is reached when defense is 0.675x hp
If defense is lower than 0.675x hp, adding defense is better
If defense is higher than 0.675x hp, adding hp is better.
Now, lets prove and compare it with @Kerridoc durability formula, which is measured with SonyaD.
Lets say we have 3 heroes:
- Hero A with 675 defense and 1000 hp.
- Hero B with 693 defense and 964 hp (+18 defense but -36 hp compared to hero A)
- Hero C with 657 defense and 1036 hp (+36 defense but +18 hp compared to hero A)
Their durability and SonyaD will be:
As we can see, hero A with def/hp ratio of 0.675 have better durability compared to the hero who trade 18 def for 36 hp and vice versa.
As the durability formula is based on the same formula as SonyaD, their result is similiar.
I the table above, I calculate durability and SonyaD with and without +20% defense bonus, it show that the turning point is still reached at defense/hp = 0.675. I have also try using my method with revised formula (c = (1.2 × a)^1.35 x b) and its result is also a/b = 0.675. Afterall, in those formula, +20% defense become a constant with value of 1.2^1.35 = 1.279. High defense or low defense hero will both gain the same 27.9% durability caused by +20% defense. (It also work the same way for % bonus in talent tree, they are treated as constant thus shall not be included in calculating def/hp ratio)
Most of the heroes have def/hp ratio below 0.675 thus chosing defense node is more beneficial, however there are also hero such as Boldtusk who have high def/hp ratio:
There are 2 crossroad in fighter talent tree which have choices between attack-health or attack-defense. If we take the path that contain health% and def% (already included in the calculation of the table below), there are 3 possibilities of how Boldtusk stat could end up:
|5 Def + 4 HP
|6 Def + 3 HP
|7 Def + 2 HP
As giving 6 Def and 3 HP will make the def/hp ratio closer to 0.675, it result in highest durability.
Judging on how close the durability between them, I have to give SG for equating +18 def with +36 hp.
NB: the 0.675 crossover point only apply if hp node give twice as much point as defense node.