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I did a fair bit of testing in unit_test_map before posting this, but to minimize the length, I'll ask you to take the following at my word (or verify it for yourself).
The first and second bounces of the mutalisk glaive wurm do 1/3 and 1/9 of the base attack damage respectively, before armour. Although the HP totals displayed on the damaged units are rounded, it seems that they are actually stored in some format capable of handling fractions: for instance, a mutalisk with +1 attack does 10 base damage, and a unit with 0 armour that suffers 9 "second bounce" attacks from that mutalisk takes 10 damage in total. (tip: to test this, use hellions on hold position; hellions have 0 armour and 90 hp).
The mutalisk's base damage dealt at each upgrade level then looks like this:
target 1 target 2 target 3 total
muta +0 9 3 1 13
muta +1 10 3.33 1.11 14.44
muta +2 11 3.66 1.22 15.88
muta +3 12 4 1.33 17.33
The target armour works normally in every way except in one instance; that is, each target takes 1 less damage than is shown above for each point of armour it has. The exception is that it seems armour cannot reduce an attack's damage dealt to less than 0.5; any attack which would deal less than 0.5 damage after armour reduction instead does 0.5 damage. Testing an unupgraded marine against a fully upgraded battle cruiser (6 damage, 6 armour) seems to support this.
What follows is that upgrading armour tends to add less damage reduction as your armour increases, since as bounces become subject to the 0.5 rule they stop being reduceable.
For instance: assuming 0 base armour and unupgraded mutas:
target 1 target 2 target 3 total reduction added
armour +0 9 3 1 13
armour +1 8 2 0.5 10.5 2.5
armour +2 7 1 0.5 8.5 2
armour +3 6 0.5 0.5 7 1.5
Now, what's important to remember is, almost all protoss ground units come with 1 base armour (everything except probes, high templar, and archons). Since air units don't benefit from ground armour upgrades, archons only benefit from it on 10 of their 360 s/hp, and probes probably don't bear considering, we can pretty much consider "protoss ground armour" to upgrade base armours of 1. Meanwhile, the plasma shields upgrade will always upgrade base armours of 0. So as a tendancy, the shield upgrades will be more effective than the armour upgrades on their respective point pools.
Here's the "reduction added" values for different upgrades vs variously upgraded mutalisk, always assuming the target unit has 1 base armour: (the numbers after the // are for when guardian shield is active)
shields +1 armour +1 armour +2
muta +0 2.5 // 1.5 2.0 // 1.0 1.5 // 1.0
muta +1 2.588 // 1.833 2.0 // 1.0 1.833 // 1.0
muta +2 2.722 // 2.0 2.0 // 1.166 2.0 // 1.0
muta +3 2.833 // 2.0 2.0 // 1.5 2.0 // 1.0
It's true that some units have less shield points than hit points, and it's also true that the shield upgrades are more expensive.
However, all ground units except the zealot, immortal, and colossus (note: nothing with an air attack) have at least as many shield points as hit points. So in an army intended for fighting mutalisk (which among other things, should have access to guardian shield much of the time) the shield upgrade probably provides more benefit than +1 and almost certainly more benefit than +2.
As to the cost issue, +1 shields has the same cost as +2 ground armour, in addition to a faster reseach time.
As additional points, the shield upgrade will benefit pylons and cannons under harrassment, in addition to any air units you build.
So basically, I'm suggesting that all the protoss players currently frustrated with mass mutalisk styles consider upgrading shields +1 before ground armour +2. Since the cost reduction it seems like even the possibility of zerg tech switching out of muta is a pretty tolerable downside to the idea (nevermind that if you're having problems with muta, a tech switch out of muta should be low on your list of concerns).
Thoughts, people?
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Very interesting read. I would like to add that shield upgrades are better for immortals than armor because of hardened shield, so really that leaves zealots and collosus as the only units that benefit more from armor than shield. And if you decide to throw in some phoenixes to fight muta, the shield ups extend to them too, but armor does not.
Thanks for doing all the math and testing!
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Just to add one important detail you missed is that shields regen but HP does not. What this means is that units that survive can regen their shields which makes shield upgrades more valuable the more you can save units. So while for zealots this isn't as important, blink stalkers for instance can make great use of this.
From a real game perspective though you'd probably want to get +2, and probably +3 weapons before +2 armor or +1 shields.
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On November 12 2011 16:28 Carbonthief wrote:
Very interesting read. I would like to add that shield upgrades are better for immortals than armor because of hardened shield, so really that leaves zealots and collosus as the only units that benefit more from armor than shield. And if you decide to throw in some phoenixes to fight muta, the shield ups extend to them too, but armor does not.
Thanks for doing all the math and testing!
im pretty sure armor is actually better in nearly every situation, vs roaches, maruaders and stalkers ur shields still take 10 damage. vs zerglings or marines having +2 armor is better cos you have more hp than shield. you're probably never gonna make immortals vs mutas unless you are trying to lose.
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On November 12 2011 16:35 Skyro wrote:
From a real game perspective though you'd probably want to get +2, and probably +3 weapons before +2 armor or +1 shields.
Against Muta/Ling I actually wouldn't advise this personally as armor/shield upgrades will be MUCH more valuable against that composition than weapons would. You'll even see a lot of pro players opt to go for heavy armor upgrades (and soon to be shield as well) when facing Muta/Ling. Since Zerglings are killed relatively easily, each armor or shield upgrade will drastically reduce the amount of damage it is capable of dealing to Protoss units, and in conjunction with the reduced effectiveness of Mutalisk splash, I always opt to try and buff up the armor of my units over attack upgrades when facing Muta/Ling.
Though admittedly, I tend not to have problems with Muta/Ling as I open up double stargate phoenix into Zealot/Archon... Which, errrr... kicks the crap out of muta/ling lol.
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Carbonthief: shield armour applies before hardened shield (ie, first shield armour reduces the damage, then if it's still more than 10 hardened shield kicks in). So actually immortals are likely not to benefit from shield armour at all on a given attack against its shield points.
However (and maybe this is what you meant), having shield armour reducing small attacks (including muta bounces and even direct attacks with +1 or less) increases the chance you still have have shields when your immortal *does* take a big hit, and reducing one more bigger attack to 10 may prevent more damage than reducing all the big hits by 1 would. So maybe it's a wash, I couldn't say.
Skyro: true, and mutas hit and run a lot so there's good opportunity to regen. However, whenever the damage prevented by shield armour would not have otherwise spilled over into the hp pool and the unit has time to regenerate fully regenerate shields even without shield armour, the shield armour didn't end up helping any (in both cases you're full/full after regenerating). Basically only regen you get in combat and regen on units that took hp damage really matters in this respect.
Skyro and CrAzEdMiKe: I wrote this assuming you'd be upgrading weapons AND armour anyway. Although, in the case that you're only upgrading weapons, my suggestion still works as worded; if you're not getting armour at all you never upgrade +2 armour, and thus will at least never not be upgrading shields before +2 armour :p
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Also remember that when a hit drops the shields, both shield and armor values are applied to that hit. A 9 damage mutalisk hit on a 3/3/3 stalker with 1/80 HP does 2 damage only.
Because of this, if you're going to dual forge against mutas, armor/shield is far superior to upgrading weapons/shield or weapons/armor. It's also great against lings because having armor/shield upgrades means one zergling hit on every unit is basically nullified in addition to the normal benefits of shield/armor upgrades.
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@jumai
thanks for finding this out 
i will defintely try this in my next game vs mutalisk.
Because of this, if you're going to dual forge against mutas, armor/shield is far superior to upgrading weapons/shield or weapons/armor. It's also great against lings because having armor/shield upgrades means one zergling hit on every unit is basically nullified in addition to the normal benefits of shield/armor upgrades.
i dont think it´s better to go for armor and shield instead of weapon and armor/shield. it´s not about fighting the mutalisk, it´s about killing them while they harass. furthermore u should test it in unittester. there is basically no difference or rather a very small when using 3/3/0 compared to 0/3/3 in a straight fight against mutalisk.
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Also, a small side effect is that the shield applies to buildings, so mutas kill them a tiny bit slower
Dunno how significant this is though.
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On November 12 2011 19:09 BobMcJohnson wrote:Also, a small side effect is that the shield applies to buildings, so mutas kill them a tiny bit slower Dunno how significant this is though.
Could always help for timed mutas with canons surviving 1 more shot. But I'm too lazy to know how much hits canons can get before dying... But yeah, every building gets more strong as far as the shield's upgrade comes... (Also works against aggressive pylons)
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You're missing the most important numbers, though. You need to show actual shots-to-kill for a given unit (eg. stalker in this case) with and without. As you say, most protoss units already have base armour, but in every other case that's actually one of the reasons you upgrade armour first; +2 is more than twice as good as +1. It's a good point that the final glaive shot is negligible once it hits 0.5, but you need to show me that this is enough to outweigh the other benefits of armour stacking.
It's an interesting analysis and something I've intended to do myself for a long time, don't get me wrong, but you're missing the heart of it. I don't really care how much damage the mutas do with and without. I only care which upgrade makes my stuff live longer. Those can be different things.
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On November 12 2011 19:55 Belisarius wrote:
You're missing the most important numbers, though. You need to show actual shots-to-kill for a given unit (eg. stalker in this case) with and without. As you say, most protoss units already have base armour, but in every other case that's actually one of the reasons you upgrade armour first; +2 is more than twice as good as +1. It's a good point that the final glaive shot is negligible once it hits 0.5, but you need to show me that this is enough to outweigh the other benefits of armour stacking.
It's an interesting analysis and something I've intended to do myself for a long time, don't get me wrong, but you're missing the heart of it. I don't really care how much damage the mutas do with and without. I only care which upgrade makes my stuff live longer. Those can be different things.
With glaive bounces dealing as little as 0.5 damage, a shots-to-kill analysis would be pretty much useless on the contrary.
What I would really like to know, is whether I should be upgrading defense (shield or armor) at all instead of attacks when my stalkers will most porbably never take the mutas on in a straight up fight.
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I think i've seen huk get the shield upgrade vs zerg when using heavy blink stalks.
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Comment deleted. Solved my own problem.
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On November 12 2011 16:49 mR.bONG789 wrote:Show nested quote +On November 12 2011 16:28 Carbonthief wrote:
Very interesting read. I would like to add that shield upgrades are better for immortals than armor because of hardened shield, so really that leaves zealots and collosus as the only units that benefit more from armor than shield. And if you decide to throw in some phoenixes to fight muta, the shield ups extend to them too, but armor does not.
Thanks for doing all the math and testing! im pretty sure armor is actually better in nearly every situation, vs roaches, maruaders and stalkers ur shields still take 10 damage. vs zerglings or marines having +2 armor is better cos you have more hp than shield. you're probably never gonna make immortals vs mutas unless you are trying to lose.
Did you actually read the OP?
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With the newest protoss buffs decreasing the cost of upgrades, they may start getting triple forges and not have to make this type of decision haha.
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On November 12 2011 16:49 mR.bONG789 wrote:Show nested quote +On November 12 2011 16:28 Carbonthief wrote:
Very interesting read. I would like to add that shield upgrades are better for immortals than armor because of hardened shield, so really that leaves zealots and collosus as the only units that benefit more from armor than shield. And if you decide to throw in some phoenixes to fight muta, the shield ups extend to them too, but armor does not.
Thanks for doing all the math and testing! im pretty sure armor is actually better in nearly every situation, vs roaches, maruaders and stalkers ur shields still take 10 damage. vs zerglings or marines having +2 armor is better cos you have more hp than shield. you're probably never gonna make immortals vs mutas unless you are trying to lose.
Or you want to do it like whiteRa, build 5immortals with your blinkstalker/archon army, go out, tear down the spinecrawler wall in something like half a second and march on to victory. 
Also Liquid_shet said that those shield upgrades are pretty strong on blink stalkers.
Just think about how blink stalkers work in battles. Once their shields are down, you blink them out of attackrange, shields start to regenerate and voilà: your 80/80 stalker will take something like 120-160damage on its shields before it dies, while it will still only take 80 on HP.
Also against mutalisks: everytime a stalker doesn't die in a smaller skirmish, it will have full shields again for the next (which means full shield armor protection). Everytime it gets hull damage, the hull armor protection for the next battle will be as small as your HP.
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A preemptive shield upgrade might make for some interesting zealot/archon timings as well. :D
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Weapons remain the vastly superior upgrade in PvZ. It's just the most beneficial upgrade in that matchup to start out with as most PvZs end up with roach play.
Weapons is awesome no matter what Z does because all units you'll be using benefit greatly from it. Armor and shield are both very poor against roach play because roaches do high base damage AND your shield/hp ratio will almost be 1 because you have the bulk of your army as stalkers.
Armor is also still better then shield because P units have a base value of 1. Sure the extra armor upgrade will not further reduce the third glaive bounce but this effect is very minimal. Having base armor makes the procentual damage decrease bigger and thus the relative durability of your units much higher.
I do sometimes try to go 1/1/1 before getting 2/1/1 though. The reason for that is simply that I sometimes tend to delay my twilight quite a bit and upgrade quite early when playing FFE vs Z. It's usually not worth it though I guess as upgrades in general are quite poor against Z for P, most of the times stalkers from the bulk of your army and they simply don't benefit well from upgrades. Zealot heavy styles stink against roach play so I tend to use them little.
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Your comparison is flawed.
For example: reducing damage from 3 to 2 is not the same as reducing it from to 2 to 1, both reduce damage by 1, but the second doubles the effective durability, while the first increases it by 50%, only half of the effect.
In a direct engagement armor is better than shields, since most toss units already have 1 base armor and in small engagements the bounce damage is mostly absorbed by shields, which will regenerate anyway no matter if you have shield or armor upgrades, so the effect of the bounces will be minimal.
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EPT
