It has been said that diminishing returns on armour and magic resistance (MR) don’t exist. The answer given is that every point of armour or magic resistance increases your effective health. So, what is effective health? First, let’s give a quick view to how armour and magic resistance work and how they reduce damage.

In terms of the damage reduction that armour and magic resistance provide they do have diminishing returns. In a previous article about armour and magic resistance’s damage reduction we examined how the more you buy of these statistics the less damage reduction increases.

## How does damage reduction work?

Damage reduction works like this: the percentage of damage reduced, DR = 1 – DM*100 = (1 – (100/(100 + Def)))*100; where Def is the value of armour or MR and DM is the damage multiplier. In other words the damage that you receive, D = ED*(100/(100 + Def)); where ED is the enemy damage dealt and D is the actual damage received.

An example: let’s say we have 100 armour, this means that our DR = (1 – 0.5)*100 = 0.5*100 = 50% of damage is reduced. If we are hit by an auto-attack (AA) that deals 100 damage we instead receive D = 100*0.5 = 50 damage.

## So, what is effective health?

Effective health is considered as the amount of health (HP) that a champion has plus 1% per unit of defence as extra health due to damage reduction. Continuing our example, if our champion has 1000 HP and 100 armour, she gets a bonus of 100% health. This means that her effective health, EH = HP/DM = 1000/(100/(100 + Def)) = 2000 HP.

Another way to look at it is as the amount of damage the enemy needs to deal to kill the champion. If we are preventing 50% of the damage dealt then the enemy needs to deal 100% more damage to kill the champion. In the example, with 1000 HP and 0 armour if the enemy’s auto-attack deals 100 damage then the champion can be killed with 10 AAs.

If the champion has 100 armour she prevents 50% of the damage so the enemy deals 50 damage with each AA. Therefore, as each AA deals half damage the enemy needs to execute double the amount of AAs to deal 1000 damage: 1000 damage / 50 (damage/AA) = 20 auto-attacks. So the enemy instead of dealing 1000 damage to kill the champion has to deal 2000 damage.

## What about the diminishing returns?

The more defence you buy, armour or MR, the more damage the enemy needs to deal. If the champion has 1000 HP and 100 armour she has 2000 HP, with 200 armour she has 3000 HP, with 300 armour she has 4000 HP and so on. As you can see in the table below, the increase in effective health is completely linear. Each point of armour or MR provides exactly 1% more health.

Not only is effective health linear in regards to armour or MR, but it is also linear in terms of health. The more health the champion has the more effective health. Now the question becomes, what is better to buy: lots of health or lots of defences?

## What to buy? Health or armour and MR?

The graph below gives a clear view of how effective health raises as health and armour or MR increase. The higher the surface the more effective health the champion has. The graph is just a more visual way to look at the information already provided by the table.

Let’s consider the vertical and horizontal axes first: Effective Health and Defence. The more defence bought the more EH the champion gets. Additionally, the more health the champion has, the faster EH increases. So at 500 health we see that EH doesn’t raise beyond 5000 HP, but with 2000 HP it easily surpasses 10000 EH.

Now let’s consider the vertical and depth axes: Effective Health and Health. The more health the champion has the more EH she gets. As the amount of armour or MR increases the amount of EH gotten rapidly grows. So at 100 defence EH just reaches 10000 but with 200 defence EH easily goes over 12000.

## No breaking point

Let’s suppose we have 500 HP and 30 armour, are we better buying an item that provides 500 HP or one that provides 100 armour. The table tells that with 1000 HP and 30 armour EH = 1300 and with 500 HP and 130 armour EH = 1150. There’s isn’t much difference between the two.

There isn’t much difference between buying more defence or health because both increase linearly. In conclusion, increasing both statistics is equally effective at increasing effective health. The only concern is how much physical or magical damage the enemy deals.

Due to the linear increase with both health and defence what would be interesting to analyse is which items are most effective at increasing survivability. Taking into account the amount of health, armour and MR that they provide the calculation allows a direct comparison of EH. This is an analysis that a future article will tackle. For now, we can affirm that increasing defence is an effective way of increasing survivability.

## In conclusion

Against a physical heavy team a Thornmail or a Frozen Heart is a great purchase. Against magic heavy teams Banshee’s Veil, Odyn’s Veil or Force of Nature are the best options. Against a balanced team a Guardian Angel could be the best answer. Regardless, any item that increases defence is a good investment.

While buying more health is effective against both types of damage, it’s good to remember that getting to 100 defence is relatively cheap, armour of MR, due to champions’ natural defences. Having 100 defence is enough to double the amount of damage an enemy needs to deal. As champions are usually able to reach over 1000 health by end-game, any champion can reach 2000 EH and substantially increase survivability without compromising damage output.

I think your math is off by a little… every point of resistance doesn’t increase EHP by 1%, it increases by resistance/100+resistance… The more resistances you have the less effective it becomes. so around 150 armor, a point in armor only increases your EHP by ~.4%

Note that while resistances seem to be less effective in terms of damage reduction they linearly increase EHP. EHP is defined as HP plus 1% per point of resistance due to DR, which yields the following equation: EHP=HP/DM=HP/(100/(100 + Def)), where Def is the resistance value.

Using the equations listed, if HP = 1000 and you have 150 resistance then EHP = 2500 and if resistance is 151 then EHP = 2510.

In the case that HP remains constant then the derivative of EHP with Def as only input: EHP'(Def) = -HP*(-100/(100+D)^2)*(1/(100/(100+D))^2) = HP/100. In other words EHP as a function of Def has a linear growth equal to one hundredth of HP. That’s why every point of resistance added provides the same increase in EHP: one percent.