Diminishing Returns Revisited

 

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:

-- Find old posts @ the RoG forums and new posts every Wednesday (3 PM) and Sunday (9 AM).

-- Feel free to find me in the "A DIFFerent View" chatroom on the NA server.
-- Contact me at [email protected]
-- Follow me on Facebook and Youtube for updates on all of my new articles, videos, and streaming.


 Song of the Day

 

92

Comments

  • #85 zenzenzen

    Tenacity does have diminishing returns.

  • #86 nornamor

    Depends on how you look at it...  its actually linear in effectiveness, hence it does not have diminishing returns.... just think about it... it would be increasing returns if you had 50% tenacety and added another 50% tenacety too get 100%.... because the second 50% is infenitly much better than the first 50%....... Instead you will get 75% by adding 50% to 50%..... where the first 50% is a 50% increased effectiveness..... and the second 25% is also a 50% increased effectiveness...... TBH,.,.,, you are walking in the same trap as those who think armor/mr have diminishing returns

  • #87 sgtcolon

    I might have missed something in some patch notes along the line, but as far as I was aware, tenacity doesn't stack at all. Pretty diminished returns if you bought multiple tenacity items in that scenario! News to me if it now stacks multiplicatively and quite interesting to boot!

    Lets theorise...I wonder what item combos like Mercury Treads + Zephyr would be like on Irelia, or Mercury Treads + Spirit of the Ancient Golem on Dr. Mundo, even Mercury Treads + Zephyr + Spirit of the Ancient Golem on Trundle! Assuming everything stacked multiplicatively. That's 58% CC duration reduction from 2 tenacity items, 73% from 3, 84% for the Trundle example, 75% for the Irelia example and 70% for the Dr. Mundo example. This would be pretty insane. I think what is more likely is that passive effects on champion abilities will multiplicatively stack with tenacity items and masteries. E.g. Mercury Treads (+35 Tenacity), Trundle's Contaminate (40% reduction to CC duration) and Tenacious mastery (15% reduction to duration of CC) gives 0.6 x 0.65 x 0.85 = 0.33 or 66% CC reduction.

    That's about the maximum reduction we're likely to be able to achieve. However, of additional interest are the introduction of methods to reduce the potency of slows. Boots of Swiftness (25% reduction to potency of slows) and Relentless (15% reduction to potency of slows) stacked that equals (0.75 * 0.85) 36% reduction to the potency. E.g. Nasus' Wither slows for 35-95% over 3 seconds. This becomes 22-61%, over 3 seconds, with Relentless and Boots of Swiftness, not bad. But coupled with tenacity and CC reduction, this becomes 22-61% slow over 1 second! Now that is something to think about if you want to be able to dive those squishies against a very CC heavy team. An example build for Trundle would be. Wriggles --> Boots of Swiftness --> Spirit of the Ancient Golem --> Spirit's Visage --> Warmog's/Frozen Mallet etc.

  • #88 exacerberus

    Tenacity from multiple items doesn't stack. CC reduction from champion abilities and/or masteries (Tenacious) does stack multiplicatively with the Tenacity provided by items (source).

    It's fine to theorycraft about stuff, but most of the times someone has already done it:

    The highest available crowd control reduction can be achieved on Irelia and Trundle. This combines the 35% from Tenacity, 15% from Tenacious and 40% from either Ionian Fervor or Contaminate, reaching a total of 1-((1-0.35)*(1-0.40)*(1-0.15))=66.85%. Or 87.7% for 3 seconds with use of Cleanse.

    Besides, the fact that Tenacity does stack with champion abilities is quite well known.

     

  • #89 sgtcolon

    A question was asked, I tried to answer it, so sue me! lol

    Relax, I wasn't trying to reinvent the wheel. I still stand by the fact though that combining CC reduction with slow potency reduction is not that widespread. Whether that is because it is rarely necessary, there are poor builds for it or it's just not well known, I don't know. So, really  I don't see why I shouldn't share a little bit of theorising about just that.

  • #90 exacerberus

    Relax, I was just implying that you're doing it bad: for instance, Wither lasts 5 seconds (3,25 with Tenacity). Also, some of your numbers are also a bit off because you seem to repeatedly forget or round decimals a bit too much. But I guess I won't sue you for that.

     

  • #91 sgtcolon

    My bad on the 5 seconds I must of read it wrong on the tooltip. I'm working on 66% reduction for this exercise, so that would be 1.7 (1.65!!!) seconds and still shows that against certain spells, tenacity, slow potency reduction and CC duration reduction in combination can actually be very useful. As for rounding decimals or providing answer to 2 dp. I think it's largely superfluous for a discussion like this as it make almost no difference to the actual outcomes, just seems to be a point for people like you to quibble with tbh :P

    I dunno, maybe you are having a bad day, but I just cant see the point of tearing into any and every post in such a way. When a simple, 'dood, Wither is over 5 seconds not 3' would have sufficed. When did RoG become so disparaging?

    Last edited by sgtcolon: 1/22/2013 7:44:14 PM
  • #92 exacerberus

    I'm sorry... then: dood, Wither is over 5 seconds not 3

    I apologize if somehow I sounded rude, but you just provided flawed math all while introducing your whole argument with

    Quote from sgtcolon»

    I might have missed something in some patch notes along the line, but as far as I was aware, tenacity doesn't stack at all.

    when everyone knows it does stack multiplicatively with champion and mastery CC-reducing bonuses since the very beginning (and that's exactly why Irelia's passive got nerfed). But oh well, I shouldn't quibble over quibbles.

  • #93 sgtcolon

    Yes. I was trying to be nice in my explanation to the guy who posted above me that tenacity doesn't and has never stacked, basically without being a penis about it.

    I never introduced any argument, I participated in the discussion. The caveat was simply just to cover my arse in case, while I was away in NZ the last week, the tenacity relationship had actually been changed (which I was 99.9% certain it hadn't, but still). I then decided to theoretically show how crazy it would be if tenacity did stack... and then mentioned how things actually worked and took it a step further by including the relatively new Relentless mastery and Boots of Swiftness. That's all.

    I appreciate the need for precise math, but I don't think whether a percentage is 66 or 66.666667 really matters for this kind of back of the envelope calculations and I don't agree that means the information it provides is inherently flawed. Overall it gives you the same conclusions and all you need to know when planning a build or playing the game.

    I'm a stickler for details too, but there is a time and place when such things are appropriate. Anyways, apology accepted, perhaps I'm being overly sensitive aswell, sorry! Just don't want to see RoG degrade into a community of nitpickers and naysayers like most of the internet, this is normally a pretty good place to come and have an intelligent discussion about a game we all enjoy. Lets keep it that way.

  • #94 MerryLane

    Just do us 200 push ups and we'll forgive u ... Maybe.

     

  • #95 BuddyBoombox

    Are these pushups successive or can he take breaks?  how long are the breaks?  Come on MerryLane, we are theorycrafters, we deal in absolutes, not abstracts here!  :D

  • #96 sgtcolon

    I broke a sweat, just, but 200 cock push ups later I have gained precisely 2.25 inches length and 1.85 inches girth. Tenacious D has nothing on me!

  • #97 MerryLane

    200 successive this way :

    http://www.youtube.com/watch?v=GpH0Ymiri8g&feature=player_detailpage#t=22s

    And ofc we want the video.

  • #98 sgtcolon

    Considering I just mentioned cock push ups, I dare not to click on that link at work :P

  • #77 Hassohappy

    You initial example and the entire thrust of the article are at odds. Your chocolate bar example shows diminishing returns as a person gradually getting less and less benefit from the exact same transaction. The chocolate bar still gives as much chocolate and costs the same, but you eventually do not view it as a good deal.

    Then your entire article talks about how almost all the stats in League don't have diminishing returns because they give the same benefit regardless of how much of it you have. In the chocolate bar example you get the same amount of chocolate for the same cost but eventually don't want it. In the league example you get the same amount of effective hp for the same cost but that doesn't count as diminishing returns? You do mention that you want to diversify your stats because of the way that they stack together but the whole article seems kind of misleading.

  • #78 GentlemanGustaf

    The returns you get from chocolate aren't chocolate itself, but the joy you get from eating chocolate. It's that that decreasing. The EHP you get from armor doesn't decrease.

     

  • #79 MerryLane

    Man I don't agree with your example.

    There are things such as "eating chocolate" or "murdering te emos" that, on the contrary, have increasing returns.

    Each time I eat chocolate, I can't stop anymore and I actually eat exponantially faster each bite of it untill I ran out of it and have to buy more to the shop.
    Same thing about te emo. The more I see, the more I enjoy killing em.

    For both examples, there is virtually no finite amount of chocolate or te emos that would start to disgust me

    Last edited by MerryLane: 1/13/2013 2:46:44 PM
  • #80 GentlemanGustaf

    I never said there weren't increasing returns :P

    Murdering Teemos definitely has increasing returns.

  • #81 MerryLane

    I just don't understand how you could imagine that "eating chocolate" had decreasing returns xD

     

  • #82 GentlemanGustaf

    Go ahead and eat 5 pounds of chocolate in an hour and get back to me :P

  • To post a comment, please or register a new account.
Posts Quoted:
Reply
Clear All Quotes