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The formula for calculating damage reduction from armor in League of Legends is

Damage Reduction = Total Armor / (100 + Total Armor)

It seems clear to me that this formula dictates that adding more armor has diminishing returns.

For example, let's say my champion has 50 armor, which gives damage reduction of 33.33% and I buy an item that gives 25 more armor; then my champion's total armor would be 75 and damage reduction would be 42.86%. That means I saw a 28.6% increase in damage reduction (damage reduction was increased by 9.53%). Now let's say I buy another of these items with 25 armor. Now my champion's armor will be at 100 which gives damage reduction of 50%. That means I saw a 16.66% increase in damage reduction (damage reduction was increased by 7.14%).

Therefore, from the first 25 armor item I bought got me 9.53% damage reduction, while the second time I bought the item it only got me 7.14% damage reduction. It looks to me like it was 25.08% less effective the second time I bought it. That screams diminishing returns to me. Why would someone say otherwise? Would they say anything different about Magic Resistance?

Note: This question was inspired by a comment thread which started off of this question: http://gaming.stackexchange.com/questions/5601/what-is-the-formula-for-armor-magic-resist-reduction/5602#5602. I have heard similar insinuations made before, though, implying that magic resist and armor are somehow different in the department of diminishing returns (although I am not sure if that was implied in the comments on the related question).

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You might want to point out that this came from a comment discussion on this question. Because without that kind of context, it looks like a rant answer looking for a question that was never asked, to be honest. –  Grace Note Nov 4 '10 at 16:35
    
@Grace Note good point. Although, I have heard stuff like this said before, point out that Magic Resistance specifically has diminishing returns. –  Steven Oxley Nov 4 '10 at 16:42
    
By the way, this question really belongs on either GameDev or Maths. –  badp Nov 4 '10 at 17:15
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@badp How is a question about game mechanics with respect to game strategy and the effects in-game not on-topic here? Does this question also belong on Math, since it's just a matter of arithmetic? –  Grace Note Nov 4 '10 at 17:17
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@badp It couldn't be fully explained with mathematics. Mathematics alone yielded what you had in the first revision of your answer. It's the context of the game that gives meaning to the numbers and yields the "effective health" metric. Math might be behind it all, but it's these in-game constructs that shape the real essence of the question. If your concern is primarily about the lack of LaTeX support, though, I don't see the harm in making a formal request for it. –  Grace Note Nov 4 '10 at 18:00
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5 Answers

up vote 52 down vote accepted

As I mentioned in the comments, you're looking at the wrong aspect of armor. You shouldn't be looking at the percentage it reduces, but rather the amount of damage it actually prevents as a measure of how much longer that armor lets your survive.

Let's consider an example for a champion with 0 armor. Say he has 1000 health, just 'cause that's a nice round number. Now our champion gets caught in Garen's judgement, and starts taking 100 physical damage each second (at one hit per second). Clearly, our protagonist can take ten hits before dying.

But now, imagine our champion has the 50 armor you mention, meaning the incoming physical damage is reduced. Now (still with 1000 health) it takes 15 seconds for that bastard Garen to claim our champion's life. 50 armor extended our life by 5 seconds. (100 * (2/3) = 66.6 DPS; 1000 / 66.66... = 15 hits)

So our champion respawns, wises up, and buys 50 more armor. Now with 100 armor, we're sitting at a flat 50% damage reduction. Again we run into Garen, who's still dealing 100 base damage per second. It now takes him twice as long to kill us -- 20 seconds. Once again the armor has extended our life, again by 5 seconds. (100 * .5 = 50 DPS; 1000 / 50 = 20 hits)

Back in base, we buy another 50 armor. We're now at 150 armor, and 60% damage reduction. Enter Garen -- spinspinspinspinspinspin and it takes 25 seconds before our champion dies. Again, adding 50 armor extended our life by 5 seconds. (100 * .4 = 40 DPS; 1000 / 40 = 25 hits)

So even while the amount of damage reduced per point of armor diminishes (as you observed), the amount of time armor extends your life (or effective health) remains distinctly linear.

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Aside: A winner is you! Congratulations, player 2, on reaching 10k! –  Grace Note Nov 4 '10 at 17:03
    
@Grace Note -- Thanks! I still remember the two glorious days where I had about 200 more rep than Tzenes. :P –  Raven Dreamer Nov 4 '10 at 17:06
    
@Raven oh how I long for those days again... –  tzenes Nov 4 '10 at 17:07
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This is a very good explanation - looking at the amount of time you survive rather than the percent of damage reduction. Incidentally, this is why percent reduction of damage (such as Alistar's ult) is so good; being able to decrease damage by percentage is far more powerful than the way armor/magic resistance works. –  Daniel Lew Nov 4 '10 at 17:20
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@Raven Bushes that say "Demacia" should not be taken lightly. –  TheQ Nov 5 '10 at 16:57
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Okay, so this is "damage reduction" over "total armor":

damage reduction = 1 / (100 + armour)

x / (100 + x)

Let's say you took 100 raw damage. The damage you actually take is:

damage taken = (1 - damage reduction) × raw damage

100 (1 - x/(100+x))

If you have 1,000 hitpoints, the amount of hits you need to die is:

time to live = 1000 / damage taken

1000 / (100 (1 - x/(100+x))) = (x + 100)/10

Surprise! It so happens that link text = link text.

Raven's point is all about the third chart. The gains, perhaps surprisingly, are constant here.

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While technically correct, I think this answer is misleading. The effectiveness of armor does not diminish. –  Raven Dreamer Nov 4 '10 at 17:11
    
@Raven I'm only now entering this debate so I'm not fully sure of what you mean by "effectiveness." –  badp Nov 4 '10 at 17:14
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@badp Read ravens answer: He means the amount of unmitigated damage the opponent needs to deal or, given a constant rate of damage, the amount of TIME the player will survive increases at a constant rate with increasing amounts of armor. Which, at the end of the day, is the part that actually matters. –  LessPop_MoreFizz Nov 4 '10 at 17:16
    
@Raven added the chart proving your point to the answer. –  badp Nov 4 '10 at 17:36
    
Think of it this way: 400 armor vs 500 armor. At 400 and 500 armor your damage reduction is about the same. HOWEVER At 500, when someone does a 100 armor pen attack on you you take a minimal hit on damage reduction. So 0.75 vs 0.8 not much change. Basically having 2-3 hundred armor means that a 100 armor pen is not going to affect you as much as if you had only 100 armor. The only counter is % armor pen. 40% armor pen vs 500 armor is a heluvalot of armor. However at 300 armor even 40% armor pen is hard to take the guy out. Note at 500 armor 40% pen will get you down to 0.7 vs 0.8. –  Dmitriy Likhten Nov 4 '10 at 18:00
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i think i got it now. heres a simple way to answer.

max hp=1000 damage=100

3 scenarios here. first if there is no armor, you will need to attack 10 times to kill. if you have 100 armor then it reduces damage by 50% so it will take 20 shots to kill. now if you have 300 armor, you will need to land 40 shots. so every 100 armor means you need to hit 10 more shots. 300 armor gives 75% damage reduction, meaning each hits will only do 25 damage. so 1000/25 is 40. even though it looks like we didn't get much from the extra 200 armor, we actually did get the full return on our investment.

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I know i might be going off in a slightly different direction, but as stated it all depends on how you view it.

How effective is armor compared to more hp? Say you got 1000hp and buy extra hp? The first 500 extra hp you get is actually an increase of 50% hp. The next 500hp you buy is only an increase of 33%.

While you still gain the same number of seconds extra living, the percentage of extra hp is diminishing.

What you really want to know is, is there a limit to when you want to stop spending gold you increase your hp or your armor, and is there a cutoff for when one thing is better than the other? Compare that to buying extra attack damage or try adding attack speed to the equation for some fun ;)

I think the answer is not in the graphs but in the wisdom explained in Raven's Garen example.

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I'm not sure if I'm wrong, but to simplify things isn't it something like this?

Between 1-50 armor the dmg reduction is:

  • 1 armor = 0.66%
  • 50 armor = 33%

Between 51-100 armor the dmg reduction is:

  • 1 armor = 0.34%
  • 100 armor = 50%

Between 101-200 armor the dmg reduction is:

  • 1 armor = 0.17%
  • 200 armor = 67%

Between 201-400 armor the dmg reduction is:

  • 1 armor = 0.105%
  • 400 armor = 88%
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so if you buy last whisper(35%armor reduction) and the other team has 100 armor each, you will hit them as if they only got 65(38.1% dmg reduction) armor, so you will do 0.34x35=11.9% more dmg to them than without last whisper –  borreftw Nov 7 '13 at 13:47
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