Understanding 7.20 Armor Changes
Patch 7.20 introduced a big change to a fundamental mechanic of Dota 2, and from our experience players still haven’t fully adjusted to it. We're taking a deep look into the changes to armor calculations and what it means for the players.
Armor Changes
Armor is now more effective across the board. The formula for calculating physical damage resistance was changed from (0.05 × armor) ÷ (1 + 0.05 × |armor|) to ( 0.052 * armor ) ÷ ( 0.9 + 0.048 * |armor|).
The changes already becomes substantial at 10 armor, with an almost 4.5% less physical damage taken under the new rules of the game. In a game where heroes often reach 30+ armor it can make a huge difference: most of the damage dealt in the late game is physical and when carry heroes start hitting for 300+ damage with potential critical hits, these increments quickly start adding up.
Armor Stacking
It should be clear from the table that stacking armor provides diminishing returns. While the first point in armor raises your overall physical resistance by 5.49%, the fifth point only increases it by 3.76%. Every point of armor after the 27th will give you less than 1% of extra physical resistance. There is still truth to the idea that every point of armor gives you ~5% extra EHP against physical damage, but the effectiveness of this EHP is still diminished.
It means that there is what we will refer to as "high value armor" and "low value armor". For simplicity, we will consider anything below 20 armor "high value armor" and extra points of armor above 20 "low value armor".
Armor Reductions
Armor reduction effects is when it comes into play as a major factor for DPS calculations. Armor is a multiplicative effect, so armor reduction scales pretty efficiently into the late game and can be devastating in the midgame. 
Desolator was perhaps one of the most underrated items, and we can already see how professional players are adapting to the new patch in the Chongqing Major qualifiers. The new armor formula is also part of the reason heroes like Dazzle and Phantom Assassin are incredibly popular and why both heroes had to be nerfed shortly after the patch 7.20 changes.
By cutting into "high value armor" through armor reduction effects you can greatly increase the amount of damage you deal from auto-attacks. For a target with 25 armor there is an almost 9% multiplier difference between an attack with and without Desolator effect, and that is only cutting into two points of "high value armor". For a target with 20 armor the multiplier difference becomes 11.55%. It is 15.75% for a target with 15 armor and a whopping 22.74% for a target with 10 armor.
Assuming you are a Phantom Assassin at level 12 with an average crit of 750 damage, you would effectively deal 68, 87, 119 and 170 more actual damage. As you can see, these values are far from trivial, especially when it comes to quickly disposing of enemy supports with low armor—you are cutting into the "highest value armor". The extra physical damage you deal on targets with negative armor can also get out of hand pretty quickly. In fact, after the patch changes, getting enemy heroes into the negative armor range might once again become a strategy.
What does it mean?
This leads us to several conclusions. Armor auras are amazing for helping your supports survive and it is the reason we often see items like 
Vladmir's Offering being built on cores or position four supports. In the beginning of the game these extra points of "high value armor" provide a lot of sustain. Small armor bonuses from 
Buckler and 
Mekansm can also play a major role in the outcome of the teamfights. Finally, 
Assault Cuirass is still pretty underrated for the benefits it provides.
Towers are much easier to destroy. Our initial impression during the general changes analysis was that the effects of extra HP and reduced armor of increased effectiveness would cancel each other out. We were wrong—while in vacuum the EHP of towers is more or less the same, when we take into account the armor reduction abilities the situation drastically changes. Once again, you will be cutting the the "high value armor"—even Tier 4 towers now have only 21 armor. You can easily reduce this value by 7 or 12 points, gaining ~11% or even ~22% effective damage from your physical attacks.
There is also still little point in trying to chip-reduce armor of targets with exceptional armor ratings, such as currently popular 
Morphling and 
Terrorblade. The latter has 32 natural armor at level 25 without any items or talents. Even when critting for some crazy value of 1.5k damage, the difference Desolator would make is less than a 100 damage. Against targets like this you either have to go all in, with heroes like 
Elder Titan or a really well-played 
Dazzle, or just find some other solutions. All conventional armor reductions sources on such targets will cut into "low value armor" and are not worth the effort.
Finally, cleave, which is now reduced by armor, is also in an interesting place. To compensate for physical damage reduction, the cleave values on Great Cleave, Empower and Tidebringer were changed. The former got completely destroyed, with a minimal increase to its effectiveness, and is more or less only useful for farming low-armor creeps. Both Empower and Tidebringer, however, were buffed quite substantially. Currently, Kunkka actually deals more cleave damage from Tidebringer on targets that resist less than ~40% of physical damage, meaning that as long as your cleave targets have 11 armor or less, you can expect to deal at least the same amount of damage as previously. It also means that armor reduction on backline targets can yield even crazier results than previously.
Conclusion
Dota mechanics are always an interesting topic to discuss, since they can fundamentally change how we approach the game. With the increase of armor effectiveness, the overall survivability of heroes should theoretically increase. At least when it comes to physical damage.
At the same time, there are now slightly more sources of spell amplification—perhaps going forward we will actually see more magic damage dealing cores and magic damage lineups will become more relevant in the later stages of the game. We've seen glimpses of it happening previously, so do keep an open mind for the next couple of months—Dota is still full of surprises.
Pepega clap
Interesting
Very well written.
so tidebringer was nerfed? anyone mind telling me how hard?
^^Tide Bringer was somewhat buffed, it deals 65% (165% total) more damage than previously, meaning that a lot of armor needs to be stacked for it to actually be weaker (11+ armor, which is fair amount for non-cores). In a lineup with a lot of negative armor, this new version of Tidebringer will be absolutely devastating since it has such a high splash radius compared to the other splashes that are otherwise way worse off. This also means it's more important to stack armor in Kunkka centric enemy lineups.
It isn't as good super late game without changing your battle strategy, but it's better for a good majority of the game by a long shot (this is when Kunkka shines anyway).
Great Cleave is useless now though. Sven can deal 90% of his damage in a splash, but every unit can receive a reduction in that damage due to their high armor, meaning it's effectively heavily nerfed once the target has more than 5 armor (not a high amount at all, most heroes will have this amount early if not at level 1).
All in all, cleave is in a bad place, but good old ValveFrog will likely overbuff splash at some point to compensate for the changes. Until then, only use splash heroes if you have -armor heroes on your team, or if you want to use kunkka with Assault Cuirass.
any single they shit on sven , first movespeed , then attackspeed, now warcry and cleave , next patch 7.21 I bet his ulti will give him 20 dmg for 5 seconds
rofl
Armor stacking does not diminish, each point of armor exponentially grows your EHP against physical attacks as in any patch.
You can simply prove it by going into a demo and stacking armor, then seeing how many attacks it takes to kill the hero.
Similar to this experiment which I made using a level 1 techies with 3 rapiers and a level 25 Axe, increasing his armor by 30 (because call is 30):
https://imgur.com/a/hDWD5xA
As can be seen, the attacks needed to kill the hero grow rather than reduce each time, with every 30 armor added.
who needs armor when u can have 50% percent evasion talent plus butterfly on bh(talents stack additively)
Now, the real question : is desolator a good item on kunkka now ?
Probably not, but still worth asking though.
@Signet Herald
You are not accounting for the Axe's natural regeneration. Level 1 techies with 3 rapiers attacks like once every 1.5 seconds, meaning that 26 hits he has to make to kill Axe also include almost 600 HP he regens in this period.
There is no point in arguing or doing uncontrolled tests when you can simply calculate hits to kill. We just take HP and divide it by Attack Damage*Damage Modifier from Armor. I just made a quick spreadsheet assuming 2000 HP and 100 damage (no regen).
https://imgur.com/Rl5mRcO
Sure, every point of Armor increases the hits to kill by ~1. However +1 HtK moving from 20 to 21 HtK is very different to +1 HtK moving from 50 to 51. In the first case you survive 5% more auto-attacks, in the second — 2% more auto-attacks. Armor used to give 6%, then 5% extra EHP, now it's a little bit more complicated, but the fundamentals stay the same — the progression is linear (more or less), the actual effects are not.
So it is like Law Of Diminishing return which is like in Economics, which means the more you get of a thing the additional utility (satisfaction) from each additional unit gives less and less. Nice concept as now the more armor doesn't simply imply that more physical resistance.
@< blank > Armour has always been like this (at least as long is I can remember). The new changes simply moved the diminishing returns slightly further back. However, the most important change IMO is the effectiveness of early (5-15) armour compared to the previous model.
How about Solar Crest? pretty decent item for somethig like Visage?
MAKE RUPTURE GREAT AGAIN
Would you mind doing a graph on armor reduction effectivity comparing now and then?
We had, in the legend/ancient bracket, some fun with Bristleback Chen lanes. 2 Goo sprays + a solar crest from Chen is -15.5 armor, which is pretty significant. It would also appear that negative armor is now more potent then it used to be.
Pepega Clap
@KawaiiSocks
"In the first case you survive 5% more auto-attacks, in the second — 2% more auto-attacks"
If you honestly think the percentage of total auto-attacks is meaningful, then you should really give up on trying to write these articles.
Just because you can take a percentage of something doesn't mean you should. (Unless you want to be intentionally misleading)
Not a lot of love for KawaiiSocks on here... I see a lot of people who clearly think they can write better articles?
Fuck any damage item , go for mkb 100 magic damage with 75% chance ez
@Kawaii
I ignored hero regen in the hits to kill numbers. I assumed 0 regeneration in order to show the effect of armor only. I simply divided health left by damage per hit (the first hit which excluded any regen).
But you're right hero regen makes the armor even more valuable. It is in no way linear.
There is an error in your calculations if you're getting linear value per point of armor.
Other sources of information such as gamepedia and liquipedia also state what my testing showed.
Test it yourself in a demo and use the numbers you get from one low variance hit with different amounts of armor and you will come to the same conclusion.
The only time that stacking armor vs physical is no longer worth it, is when you can buy an item which provides a significant amount of HP, lifesteal, regen or utility which combos well with your already high armor. Even then, if the enemies are pure physical you can stack AC and shivas on your DK or Axe all you want.
@Signed Herald
First of all, we are probably far from practical Dota and fully in the theoretical space. I don't believe either of us thinks that multiple Shivas' or Assault Cuirasses on a single hero is a good idea or at least it is a good idea in a very limited selection of games, unlikely to ever happen in an actual game of 2k+ Dota. We are not debating builds, strategies or anything of the sorts and are not making meaningful suggestions for players to improve their gameplay and win more games. Basically, "DO NOT TRY THIS AT HOME" and lets have some fun with math)
Now to the debate itself: I think we might have misunderstood each other and I have also underestimated the overall impact of the armor changes.
My points you disagreed with were:
1. You get diminishing returns from stacking Armor
2. Each point of Armor increases your effective HP more or less linearly, fully linearly in the old formula
Link to google doc with the spreadsheet. Everyone feel free to copy and play with the Health and Attack values:
https://docs.google.com/spreadsheets/d/1tNCLQEQQM4rOPBjI65p_FM_Z1xt5AtrZRWW2kTp-UqU/edit?usp=sharing
Let's start with the second point. You are absolutely correct in the fact that EHP you gain from armor now increases. I've initially assumed the effect is going to be much smaller, so wrote it off as "more or less the same as old formula". Turns out I was wrong in assuming the effect is subtle. As I said previously, the armor is more effective, but turns out it also meaningfully increases in its effectiveness for each point. The increase is small, at around ~0.07% on average for the first 50 units of armor, but it does add up to something practically useful.
*Armor stacking does not diminish, each point of armor exponentially grows your EHP against physical attacks as in any patch.* — this statement, however, is false. Spreadsheet has info on how the old armor used to work and it always gave you 5% extra EHP. The absolute EHP gain didn't change, so a 2k HP pool hero would always get 100 extra EHP for each point of armor. You can look at it from the EHP perspective or the HtK perspective, it stays the same. There was no growth previously, there is growth now.
Now to the more interesting bit and the probable source of confusion. *You get diminishing returns from stacking armor* is the statement I still stand by and wholeheartedly agree with. All the consequences of this statement such as "high/low value armor" and "armor reduction" conclusions, therefore, are also some things I am yet to be convinced to reconsider. Though, maybe, the subjective cutoff point is slightly higher. 20 armor was chosen more or less arbitrarily, to provide examples around. Looking at the math, there might be a reason to try and reduce armor on 25+ armor targets.
Now why do I believe that armor has diminishing returns? It really depends on what you consider "value" of the armor is.
There is a difference between "how many extra hits does it take to kill me with this extra armor" and "how many extra hits does it take to kill me with this extra armor, compared to how many hits it took to kill me before".
That is the reason we use percentages. @Duros
Because in the first case your grandma gifts you a shiny new plate of armor and you think: "oh wow, it now takes 5 more hits to kill me, that's awesome, thanks grandma!". And you are very happy.
In the second case, you still have the same HP and armor and are gifted the same exact new shiny plate of armor by the exact same grandma, but she also slips a note saying: "You know, it already took 100 hits to kill you before. Believe in yourself! You didn't really need that armor to be great!". And now, instead of thinking "oh wow, that's awesome", you think: "Grandma, WTF?" Needless to say, while you still love your Grandma and a gift is a gift, you are not particularly happy. Though you still awkwardly pretend you are.
Note how everything stays the same: you starting HP, Armor and Grandma. The only thing that's changing is the perspective.
Armor has diminishing returns even in the new formula. Its effectiveness rises (only in the new patch), but it doesn't rise fast enough to prevent the loss of effectiveness associated with stacking it (more on that later). For every extra point of armor you do get slightly more effective HP, but as a percentage of what your effective HP is already, the effect is diminishing. Look at the J and Q columns in the table—they showcase exactly that. First point of armor makes it so that your new EHP is 105.8% of what it was before it. Second makes it so that your new EHP is 105.53% of what it was before that and so on and so forth. At every step the factor becomes a little bit smaller. One extra point of armor that moves you from 0 to 1 is more effective than one point of armor that moves you from 10 to 11.
The interesting thing is that you are actually right in the more or less strictly theoretical Dota space. You see how I admitted to being wrong and underestimating the effect of the formula changes? It results in a pretty funny thing at 125+ Armor. The growth effect of the armor effectiveness starts to actually outweigh the diminishing returns of extra armor. Once again, looking at the Q column, you can see each extra unit of armor being less and less effective to a point where one extra point of armor makes it so that your EHP is only 101.71% of what it was before and then it slowly starts growing back. I think that is very cool and thank you for starting the discussion.
There are definitely caveats. There are healing effects, there is regeneration, there are different HP pools and STR growth on heroes, there are different bonuses from attributes and it all makes the game a lot more complicated and the calculations complex. Again, this was theoretical Dota and has nothing to do with actual games. If you want actual, helpful information, just, you know, look at the blog post above. Or if you want tl;dr follow the easy principle of not stacking what you already have a ton of.
Second sheet of the google doc has a "calculator" for EHP of agility and str heroes. You can just plug in HP and Armor of your desired hero at desired level in the top yellow section and you will have EHP of those heroes with one of the bigger items in the game. Note how for both TB (crazy Agi growth, low str growth) and TA (moderate Str and Agi growth) going for Heart is always a better option when EHP is in question (not that I'm saying it is a good option). At the same time for Doom with low Agi growth it is armor items that give the most EHP. So, yeah, please don't stack too much armor, it has diminishing effectiveness.
Finally, talking about confirming stuff on Liquipedia or Gamepedia — don't do that. Liquipedia has this paragraph:
https://i.imgur.com/oVbcMpX.jpg
Constant 5% and "exponential" only work together if it would add +5% to the previous, already increased value, not to the base HP. Old armor added +5% base HP. It was linear for positive values. There is even an old graph with a straight line for EHP multiplier: http://i.imgur.com/CmCyK1G.png
The screengrab from gamepedia is a complete mess with jumbled numbers and weird formulas that don't make any sense at all. Like, at all.
https://i.imgur.com/d0CBXHO.jpg
Personally, I think someone was trying to make numbers work so that somehow they have the same end result between their two deeply flawed "methods", but it kinda doesn't work where it is easy to prove that the answer they get is wrong.
TL;DR:
1. The effect of armor scaling was underestimated in the article
2. Stacking armor gives diminishing returns despite that, at least until 125+ armor
3. Don't fact check on wiki-type websites
4. Question everything, it leads to cool discussions.
@Duros
"If you honestly think the percentage of total auto-attacks is meaningful, then you should really give up on trying to write these articles.
Just because you can take a percentage of something doesn't mean you should. (Unless you want to be intentionally misleading)"
This is the exact type of bullshit only people who don't understand anything about the game can spout. Do you think a 2% effective increase in physical resistance overall is worth the same as pray tell 10%?
Do you think it makes the same different if you cut a hero down in 2 hits or 3 compared to if you hit him down in 20 hits compared to 21?
Stacking armor becomes more and more inefficient. It gives a ~5% increase over the base value, but if you have 20 times the base value it really loses out. If 15 armor makes the difference between 0% phys resist and 50% phys resist you go for that 15 armor in spite of say 30% miss chance. If the same amount of armor is 75% to 80% resist, you take the 30% miss. In the first case, incoming damage vs physical with armor: 50 percent, with evasion: 70 percent. In the second case it's 20 percent with armor and 17,5 percent with the evasion - and that's not accounting for the possible advantages of evading an attack instance at all (be it orb effect, or breakable items on you).
It's so basic it shouldn't even need to be explained, yet some people take their own understanding of the game and present them as hard facts.
Also the same kind as people who think getting your DPS up as high as possible equals getting as many divine rapiers as you can.
This article and most commenters don't understand how armor works. It doesn't have diminishing returns just because the percent you gain is lower, not all percentage gains are created equal.
If you had 2000 HP and they were dealing 1000 DPS, you would die in 2 seconds at 0% resistance, 2.02 seconds at 1%. You gained 0.02 seconds of effective HP from gaining +1% resistance. But if you went from 98% to 99% you would go from living for 100 seconds to living for 200 seconds. You gained a whopping 100 seconds of EHP from gaining +1% resistance. In the first example you reduced the damage by one percent, and in the second you cut it in half. If we go from 99 to 100, well I'm sure you realize how valuable that would be.
Obviously this is an extreme example, but it illustrates the point. Every point of armor you gain is actually more valuable now for every point of armor you already have.
@Vivenmort
But armor does not increase your resistance, it increases your Effective Hit Points, from which your resistance gets calculated.
Let us use your example of 2000 HP and 1000 DPS, assuming each point of armor increases your EHP by the same percentage (which it does not after the patch, lower armor values now give more %EHP per point of armor than higher ones). Let us say 1 armor is equal to +1%EHP, to make it easier to demonstrate. 100 armor would give us +100%EHP, raising it from 2000 to 4000. Since we doubled our EHP it would take twice as long to kill us, so our resistance would be at 50%.
Now let's add another 100 armor, to increase our EHP by another 100%. This means we are tripling our EHP to 6000, so it would take three times as long to kill us. However, our resistance would only be ~67%! This is what people mean when they say stacking armor has diminishing returns. Going from 0 to 100 armor increases your resistance by 50%, but going from 100 to 200 armor only increases it by 17%.
@Kawaii Isn't the point of this article to inform people so that they understand armor, not just theorycraft?
Anyone can launch a demo and test how armor works and they should. Simply because it does NOT diminish.
Each point of armor does exponentially increase your tankyness against physical damage (or the number of hits you can take before dying). That's before accounting for regen, lifesteal and everything else which can further add to your survival.
This is especially important in a patch where PA is picked every game almost, usually with a magnus.
Physical survivalbility has always worked like this by the way, I have tested it in many patches. Armor is just slightly more meaningful now.
There honestly isn't much to discuss, if your mid Invoker is running around with 15-20 armor lategame against a PA who keeps one shotting him, building items which have no armor in them because a dotabuff article told him it wouldn't be high value then report that Invoker for ruining your ranked games thanks.
(Also I did not mean buying multiple ACs and Shivas, I meant buying one of each.)
Your testing method was flawed if you got exponential increases in EHP from armor in previous patches.
I've also provided calculators to showcase exactly how even items like Skadi (I'm not even talking heart) are better for EHP on already high-armor targets and how armor items are better on heroes with high HP and low armor. That is in the new patch.
Not once did I make a claim that armor is bad. But having too much of it at the expense of having other survivability items is strictly inferior. Nothing in the game comes for free, and if you already have high enough armor you should build HP, if surviving PA crits is your only concern.
You wanted to go full theoretical with your DK+Shiva's and Cuirass. There isn't a single game of normal Dota where it is a viable combination of hero and items. Cuirass? Sure, it is great for your team and decent on DK. Shiva's? Please don't do that. I've entertained these theories in the same way I've entertained your idea about Axe and his full inventory of Plate Mails. All these scenarios are theoretical.
I've made a table with all the armor values, physical resistances and several perspectives on how to look at it, either from EHP or HtK standpoint. You can see for yourself how the change in EHP, compared to the previous value, diminishes with each step until 125+ armor. In your own words "there isn't much to discuss"—calculations have been made, the data is available to everyone and whoever doubts it can test it for themselves and come to their own conclusions, if they disagree with mine.
Also, just to be clear with your Invoker scenario: going Skadi or Heart will give a lvl 25 (late-game) Invoker higher EHP against physical than Shiva's or Assault Cuirass. That is naked Invoker with ~9 Armor at level 25. It is easy to understand that if he had 15-20 armor, like in your claim, the difference in EHP between extra armor (Shiva's\Curiass) and extra HP (Skadi/Heart) is going to be even bigger in favor of HP items. Please don't confuse people more than you've done already. Let them decide for themselves from the data.
nice maths
1 armor gives about 5,9 EHP