Gentleman Gustaf here today to talk to you about Diminishing Returns. Judging from comments on my previous posts, this is a topic worth going over in depth.
What are Diminishing Returns?
In Economics and Psychology, we have observed an interesting phenomenon. My high school Economics teacher explained this concept in a very excellent way. Let's say I offer you a chocolate bar for free. Do you want it? Let's say you do.
Gentleman Gustaf! What a sucker you are! Who wouldn't take a chocolate bar for free?
Right you are. You take the chocolate bar, and I offer you the same deal. I make it quite clear that I intend to make this offer many times. You're probably saying yes until you can't possibly carry any more chocolate bars. If you somehow solve transportation and logistics, you'll take as many chocolate bars as I offer you. Now, let's add a catch. What catch? You have to eat it right now. You take the first chocolate bar, of course, savoring the flavor. If it is sweet and milky, bittersweet, or bitter you're after, you eat it all. And then I offer you another chocolate bar, still free. After the 5th chocolate bar, it's safe to say that you're about done with chocolate for the day, but still the offer remains. After 10 chocolate bars, I'd probably have to pay you to keep going. 20? 30? At the start, chocolate was a positive good: it was worth more than 0 dollars to you. But by the end, it was worth less than that. Don't think from my example that diminishing returns have to cause the good to be worth negative. I could just as easily have started at 1 dollar, or 50 cents.
The moral of the story is this: most things simply aren't as good in bulk. The more of them you've had (within some recent time frame), the less value they give you. The first one was delicious, the second, tasty, the third, a bit much, the fourth, not very satisfying, the fifth, sickening, the 10th, vomit-inducing, and so on.
So what does this have to do with LoL?
A lot of people see under the misconception that certain stats, when bought in bulk, have diminishing returns.
The most common example is Armor, and DiffTheEnder has done a video on the topic. However, I am going to rehash the math here for the purposes of this article.
Let's say you've got 2000 Health and 0 Armor. You have 0% damage reduction vs physical damage. You get 50 Armor. You now have ~33% damage reduction vs physical damage. You get another 50 Armor, putting you up to 100 Armor. You now have 50% damage reduction vs physical damage.
Hold on, Gentleman Gustaf! Your first 50 Armor was worth ~33% damage reduction, and your next 50 Armor was worth half that, ~ 17%! Isn't this diminishing returns? Look at this graph I made! Each 50 Armor is worth less than the last!
And this isn't just true of Resistances! Every % stat does this! Try Crit Chance! Without any Crit Damage, your first 25% increases your damage 25%, but the next 25% only increase your damage 20%
Same goes for Attack Speed; I'll let you do the math.
Not at all. In fact, it's quite natural for a flat increase to be worth less, as a percentage of your total, as you get more. Let's say I bring you 1 orange every morning (and furthermore, you have an additional orange already). As such, my orange brings you from 1 orange to 2 oranges. Most mornings, you eat the orange right away, but one morning, you decide not to. As such, the next day, when I come to your house, you already have 2 oranges. I give you 1 orange, and you say 'hey! This is fewer oranges than you brought me yesterday! You normally bring me oranges equal to 100% of my current oranges, but this time, you have only brought me 50%! The absurdity of this example should highlight the absurdity of the Armor example. The increases are simply flat increases. So the increase may be less of a percentage increase, but that's simply because you have more. In fact, this pattern is what we expect! Let's look at a % increase chart for AD, assuming you buy BT:
Look at that! The more AD we have, the less of a % increase we get from AD. Does this mean we have diminishing returns from AD? Of course not; 50 AD is 50 AD, whenever you get it.
So once again, what are Diminishing returns?
Before I go on, here is the definition of diminishing returns:
The law of diminishing returns (also law of diminishing marginal returns or law of increasing relative cost) states that in all productive processes, adding more of one factor of production, while holding all others constant ("ceteris paribus"), will at some point yield lower per-unit returns.
To qualify as 'diminishing returns', essentially, at some point, a fixed amount of a stat would have to give you LESS.
Just like BT increases your AD by a flat amount, Armor increases your Effective HP vs Physical damage by a flat rate of 1% of your Health. Assuming 2000 Health, here is the chart for Armor vs EHP:
Lo and behold, we observe a flat increase of 1000 EHP per 50 Armor. No diminishing returns after all. The same goes for Crit Chance:
You gain AD*1%*(1+CritDmg) for each point of Crit Chance.
What is interesting to me is why people seem to think there are diminishing returns for Resistances, Crit Chance, and Attack Speed, but not a stat like AD. My guess is simple: two are listed as % stats, and while resistances aren't listed as such, people tend to know they operate with percentages in some way.
Now, this doesn't mean you should just stack resistances. Imagine you're having a party, and you buy enough hot dogs for everybody. At some point, you should realize that you probably get more out of toppings for your hot dogs than you do from getting more hot dogs. This is, essentially what multiplicativity is: the toppings make all of your hot dogs better, so the more hot dogs you have, the more value the toppings get you. But the value of your hot dogs hasn't gone down; the value of condiments have simply gone up.
Similarly, in LoL, no matter how much armor you have, your next point of armor is worth the same amount. But your next point in health becomes worth more and more.
Are there Diminishing Returns in LoL?
In fact, yes. Movement Speed
These are the simplified formulas:
- If the raw speed is 415 or under, no cap is applied.
- If the raw speed is between 415 and 490, final speed is soft capped to RawMS × 0.8 + 83.
- If the raw speed is above 490, final speed is soft capped to RawMS × 0.5 + 230.
So assuming you start at 340 MS, your first 75 MS is worth 1 MS gained per MS bought. Your next 93.75 MS cost 1.25 MS per point of MS gained (you gain .8 MS per MS bought). Finally, after that, each point of MS costs 2 MS (you gain .5 MS per MS bought). This is what Diminishing Returns look like:
To avoid any confusion, this doesn't mean you should stack only resistances or only health. The best builds will typically have a balance of stats, because multiplicative effects benefit each other. But the myth that % stats like Armor, Crit, and Attack Speed suffer from diminishing returns is simply false.
For more of my work:
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