Hi Deshiba, I noticed there's been a long discussion on diminishing returns on the item page. I wanted to prove that there are indeed no diminishing returns without resorting to effective HP, since that seems to be a stumbling block in the discussion. Unfortunately, it will require derivatives, if you are familiar with those.
First, we need to include %Damage Taken (%DT) because % Damage Reduction (%DR) only tells us half the story. What we really care about is what that damage reduction means in game--in other words, how much damage you will be taking from attacks, and that's %DT. %DT is just 1 - %DR, so 90% damage reduction means 10% damage taken, 50%DR is 50% DT, 10% DR is 90% DT. You buy armor to increase your %DR, thereby lowering your %DT.
The idea is the following: Just as you said, %DR appears to increase more slowly as you stack resists; however, as I will show, when stacking resists the %DT you are getting is becoming more and more valuable at exactly the same rate, which balances everything.
Unfortunately, I don't know a simpler way to prove this without resorting to EHP. Even if you can't do the math, you can maybe get from it the general idea. Here is the math without using EHP:
Since %DR and % Damage taken (%DT) are dependent on each other, we need to find out how %DR and %DT change together. We cannot analyze this by only looking at one or the other individually without entering the territory of EHP. One way we can express how the two change together is by the ratio of %DR to %DT.
We take the formula for %DR
x / (x + 100)
and divide it by the formula for %DT
1 - x / (x +100)
The result is : 0.01 * x
This is their ratio. Naturally, the ratio widens the higher the resist: You are approaching 100 %DR, but dividing this by %DT, which is approaching 0%. The key point here is whether the increase in %DR is going up faster or slower than the simultaneous decrease in %DT.
To find out, we take the derivative to see the rate of change of this ratio--and it is a constant 0.01.
This means that %DR is increasing at relatively the same rate as the %DT is decreasing, no matter how much resist you have. In other words, there are no diminishing or increasing returns; there are only static returns. As you stack resists, %DR is in fact increasing more slowly, but at the same time, the %DT is decreasing more quickly at exactly the same rate. So the smaller gains in %DR you observe with 100, 200 and 300 armor are all being perfectly offset by a more valuable %DT.
To provide an example of a diminishing return, let's say the formula for %DR were:
x / (2x + 100)
Then %DT will be given by:
1 - x / (2x + 100)
Dividing %DR by %DT, we end up with:
x / (x + 100)
Taking the derivative of the ratio like before, we have the final result of:
100 / (100 + x)^{2}
This is not a constant, and it indicates that the higher the resist value, the smaller and smaller our ratio continually becomes. We are getting per point of resist, a comparably smaller amount of %DT in return. This is a diminishing return.
I don't know if this helps, but I hope it did! It's really a lot easier if you make the conceptual transition to EHP. It's worth investing a little time to see how EHP works because it's a simple concept that makes all of this a lot easier to understand, and it will improve your understanding of the game as a whole.
Another argument that can be made as an analogy: Consider AD. Every point of AD is just one point you gain. There are no diminishing returns. As you increase it to 100 AD, then 200 AD, then 300 AD, you are getting the same gains per point of AD; however, you are having decreasing relative gains. The jump from 100 AD to 200 AD increases your damage by 100%. The jump from 200 AD to 300 AD only increases it by 50%.
This is in effect what you are noticing with your argument. Relatively speaking, your gain is less. These are called marginal gains, which in this case do have diminishing returns. However, your absolute gain is still the same: 100 AD is 100 AD. For the purposes of cost analysis and comparing items, this distinction is important. You aren't getting less value for your money, what's happening is a natural decrease in relative value which occurs with any unbounded linear gain. It's like buying 2 cars for one person. You only need one car, but the second car isn't any less valuable for its price.
Using the AD example above, in order to go against this pattern, you'd have to have exponential scaling, in other words, you buy 100 AD, but the next 100 AD you purchase is worth 200 AD, so that you can scale 100 AD, 200AD, 400AD, 800 AD etc for the same price at each step. What you are calling a non-diminishing return is actually the opposite of diminishing returns; it's not a linear gain, but an increasing'return.'
Anything that goes against what you observed with armor (the decrease in marginal value), is not a linear increase. The decrease you noticed is exactly the indication of a linear gain.
Diminishing returns are an economical term, correct?
The definition of diminishing returns "is the decrease in the marginal (per-unit) output of a production process as the amount of a single factor of production is increased, while the amounts of all other factors of production stay constant."
If we take the set up for acquiring armor this leads to 3 factors: Gold, Armor, Percent Damage Reduction. Per definition of diminishing returns the amount of Gold and Armor stays the same while the Percentage Damage Reduction diminishes, Correct?
Even the price increase per Percentage Damage Reduction increases at 20G/%DR, where at some point on the chart, buying health will be more efficient because the gained EHP/Point for HP will be higher then the EHP/Point for Armor, Correct?
I am well aware that the resilience increase in EHP will be linear, but that doesn't change anything for the diminishing returns on armor on its own. This is the same claim that Willbackbakal is making towards Gold Efficiency, where GE should be looked at in an isolated environment. That buying AP on Vayne will not get the items worth that GE implies is according to him Irrelevant.
I'm just using his own logic to point out how hypocritical the whole ordeal is. When looking only at armor EHP is irrelevant.
Per definition of diminishing returns the amount of Gold and Armor stays the same while the Percentage Damage Reduction diminishes, Correct?
This is incorrect. the Percentage Damage Reduction does not diminish in value, because even while the number gets smaller, the value per point of damage reduction becomes more valuable. The two changes balance each other perfectly. Half my post above was a proof of this.