[G] Critical Upgrade Analysis

In this matchup, the main units in the battle will be Zerglings, Roaches, Hydras, Mutas and Corrupters against Zealots, Sentries, Stalker, Immortals and Colossi.
So we analyze the different setups with normal setup, damage or armor advantage and disadvantage.
Zerglings against Zealots, Sentry, Stalker, Immortal and Colossi changes from normally taking 3, 6, 4, 2 and 2 hits and needing 35, 18, 36, 70 and 80 hits to kill. With Upgrade advantages, these numbers change to 3, 7, 4, 2, 2 taking and 29, 14, 29, 47 and 65 needing, and with Upgrade disadvantage, the numbers are 2, 5, 4, 2 and 2 hits to die, and they need 9 more hits against Zealots, 7 against Stalkers, 4 against sentries and 17 against Immortals. (on a sidenote, on +2 against +1, Colossis need 1 shot to kill Zerglings, and on +3, Armor gets irrelevant).
The same math for Roaches gives us 11, 29, 12, 3, 6 hits to take and 10, 6, 11, 24 and 22 hits needed. On Upgrade advantage, the numbers become 13, 37, 13, 4, 6 hits to take, and 9, 5, 10, 22, 21 hits needed, and on upgrade disadvantage, they become 10, 25, 11, 3, 5 hits taken and 11, 6, 11, 25 and 24.
For Hydras, the math is the following: Taking 6, 13, 9, 5 and 3 shots, and needing 14, 7, 15, 28 and 31 on original setup. With Upgrade advantage, the Hydra takes 6, 16, 9, 5 and 3 respectively and needs 12, 7, 13, 27, 29 shots, and with Upgrade disadvantage, the numbers become taking 5, 12, 8, 4, 3 shots, and needing 15, 8, 15, 30 and 33 shots to kill.

So bottom line for Ground Upgrades on Zerg is, Attack Upgrades change almost every engagement, and if going very range heavy, Carapace has huge impact aswell.
For the Protoss, against Ling Heavy armies, Attack Upgrade might be nice in case you are Zealot Sentry Heavy, against Roach heavy, Armor is really important while attack doesnt change much significantly, apart from Colossi killing Roaches faster.
Against Hydra heavy armies, armor doesn't change much, but attack upgrades change kill counts for all warpgate units.


Now a glimpse on air. We will only look at Muta vs Stalker, Sentries, and Corrupters against Colossi and from Stalkers.
So for Mutas, its needing 19 hits for a Stalker and 10 hits for a sentry, with upgrade advantage, its 17 for Stalkers and 9 for sentries, with disadvantage its 21 and 11 for a sentry. To kill the mutas, it takes 13 shots from stalkers and 21 hits from sentries, if carapace was researched, it takes 1 shots more for a stalker and 4 more for sentries, with damage upgrades for protoss, 11 shots for a stalker and 18 shots for sentries.
The Corrupter math says 19 shots for a colossi and 17 to be taken down by Stalker, on Upgrade advantage this is 17 shots to kill and 19 to be taken down, on Upgrade disadvantage it is still 19 shots to kill and 16 taking.

So Mutas and Corrupters gain strongly from attack upgrade, aswell as both upgrades for protoss against Mutalisks.

I dodged on doing Corruption math though.

For this matchup, we concentrate on the typical Bio-Ball vs Protoss-Ball scenario, on a sidenote I will investigate a bit lategame Tanks, as this is the common way this matchup is evolving at this point in time.
The Infantry Armor shall be researched first: Marines get killed by Zealots, Stalkers, Sentries, Immortals and Collosi in 3, 5, 8, 3 and 2 shots, after a Armor Upgrade, this changes to 4, 5, 9, 3 and 2, which basically means, the upgrades mainly affects Zealots and Sentries. For the Marauder, an Armor Upgrade changes their durability from 9, 10, 25, 3, 5 to 11, 11, 32, 3, 5. For the thesis for Marines can be extended to Gateway Units in General.
In the Attack Department, math gets a bit rough because of Protoss Shields and different Armors.
Marines take 26 Hits to kill a Zealot in general, 22 if he has Damage Upgrade advantage. It takes 34 hits to kill a Stalker before upgraded, 25 after. This general theme carries on for the Marine. For Marauders, 16 for a Zealot, 9 for a Stalker and 19 hits for a collosus change to 15, 8 and 17.
So the Attack Upgrades turns out quite crucial.
Last but not least, a forecast on Mech Play:
Attack Upgrade on Tanks changes the amount of hits on Stalkers from 4 to 3, on Colossi from 8 to 7, and the shots on Zealots from 5 to 4.

On the Protoss side, lets first take a view on damage upgrades.
Zealots take 1 less shot on Marauders, their hits on Marines remain, for Stalkers, the Marine hits stay same, while they need one less hit on Marauders, and aswell on tanks.
For Immortals, the Upgrades don't change anything, Collossi take down Marauders in one less hit.
So bottom line, when facing massive Marauders, upgrades can make a difference.
On a glimpse to Air, we can see how Void Rays perform against Marines and Phoenixes against Tanks: 8 Shots for a Void on a Marine before Upgrades, 7 afterwards, so a +1 might be helpful. Phoenix Shots on Tanks drop from 20 to 16.
To remain is how armor research changes the game. Marauder Damage does get affected only slightly, Stalker endurance does not change, while Zealots die in 1 less shot, Collosi dont take more shots.
Against Marines, Zealots hold of 5 more shots, Stalkers 4, Colossi 10.
Ship plating shall be viewed on aswell, 5 shots more can be taken from a Marine.

The typical matchups in this matchup that might occure from the Zerg point of view are Zerglings, Banelings, Roaches or Mutalisks against Marines, Marauders, Hellions, or Thors.

If Zerg is facing a Bio heavy play with Ling+Bling, the +1 Armor increase the amount of hits Zerglings and Banelings take to kill by a Marine by 1, so we here found our first critical upgrade. This also holds for Roaches, as the Marauder hits increase from 8 to 9, and Marine hits raise drastically from 29 to 37.
The damage upgrade depends, as Melee changes Zerglings against Marines, though banelings don't get affected. Roach hits don't change on Marines and Marauders at all.
Mutalisk hits on Marines don't change, but the Marines need 4 more hits on Mutas in case carapace is out.

For the Terran, the Damage upgrades lets Marines take less hits on Zerglings and Roaches, hits on Banelings dont change. Marauders hits don't change on Zerglings and Banelings, but the Roaches fall faster.
On Mutalisks, you need 18 compared to 21 shots, quite a drastic difference.
Armor also changes much if you encounter Zerglings, as marines remain unaffected when they counter banelings or roaches, though Zerglings take slightly more hits (9 to 11) to kill them. Marauder math doesnt change against Roaches or Banelings, Zerglings need 42 instead of 32 hits.

The whole math of course changes, if the Terran goes Mech.
For Zerg, this means Armor Upgrade is crucial if you go Zerglings, as then the Zerglins survive a (sieged) Tank shot. Banelings though die anyways, but Roaches survive one (sieged) Tank shot more aswell. Aswell to note, against hellions, the upgrade doesn't change anything., regardless of blue flame.
Zerglings of course profit from damage upgrades because of their attack speed as the Hellion hits decrease, and tanks take less hits, but even more Roaches as hits decrease from 11 to 10 on tanks.
The Mutalisk damage might be the most important, as Zerg typically ends up high on Mutalisks, and the upgrade changes hits on tanks, and, more important, on Thors from 50 to 45. On as sidenote, as common sense might suggest, rushing for Flyer Armor doesnt make sense as it does not even change the hits on the mutas for Thors, but very well does +2 carapace lower the hits taken if Terran has no additional Upgrades.

For a meching Terran, upgrades always make sense because of the superlinear damage increase compared to armor upgrades.
On 'critical' side, a damage upgrade doesnt have critical effects on ground or air.
But the Armor upgrade changes things quite alot. Thors take 5 more hits from mutalisks, 2 more hits from roaches, stunning 34 more from Zerglings.
Tanks sustain one more shot from roaches, 13 more against Zerglings, 3 more against Mutalisks. Even Hellions take longer to kill for zerglings.

A deep Analysis on Upgrades in PvP at the current state of the game doesn't seem to make much of a sense, as PvP is the shortest matchup in average game time and most of the time ends on 1 base.
Warpgate builds at the moment are designed to kill the opponent before the game gets deep, and early Forges are build for cannon rushes.
So the only cases currently that might end up in the opportunity to get upgrades seem to be Robo vs Robo builds and getting air, so I will investigate those two cases.

The first case is most likely Zealot Stalker Immortal Colossi vice versa, so we will look into that, but not looking to every matching specifically. The most interesting part might be, if one player can let his Zealots live longer, Stalkers kill faster, and both for Immortals and Colossi.
So lets look into these issues: Zealots take 12 hits from Zealots, 17 hits from Stalkers, 8 from Immortals and 6 hits from colossi. Once Armor upgrades are out, it takes 12 from Zealots, 18 from Stalkers, 8 from Immortals and 6 from Colossi, so no effect at all.
Stalkers on Upgrades kill Zealots now in 15 hits instead of 17, Immortals need 1 less shot, Collossi need 27 shots before upgrade and 25 afterwards.
'Naked' Immortal and Colossi analysis against Zealot, Stalker, Immortal and Colossi provides 8, 4, 15 and 8 respectively 6, 6, 13 and 13 shots to kill and 21, 25, 15 and 13 respectively 24, 27, 8, 13 to get killed.
With upgrades, to kill the amount of shots they need are 8, 3, 14 and 7 respectively 5, 5, 12 and 11, to get killed, the numbers change to 23, 27, 15 and 13 respectively 27, 28, 8 and 13.
So we see the Stalkers, but even more the Robo Units profit from a Attack Upgrade. On Armor Side, it slightly benefits the durability of Robo units against Gateway Units.

The second notable duel were upgrades are possible (at least for the air unit) might be VoidRay versus Stalker, so lets at last take a look at that.
Voids need 17 shots before and 16 after a +1 upgrade, which means 4 Voids can take down a stalker in 4 volleys. It takes 25 shots from a Stalker to kill the Void Ray, after Armor Upgrade this gets 27.
So after all, Air Weapons might change things in this engagement.

The three scenarios I will adress are Bio vs Bio, Bio vs Mech, and Mech vs Mech.

So let's see what a Damage Upgrade in Bio battles can change: unupgraded Marines need 8 hits for another Marine and 25 hits for a Marauder, with upgrade the hits per Marine drop to 7, and the hits per Marauder drop to 21. For Marauders, 5 hits per Marine remain untouched, and Marauder 7 hits drops to 6.
If you research armor, the hits Marines need for other Marines increase by 1, the shots for Marauders on Marine remain untouched. For Marauders, it means Marines take 32 hits to take them down, other Marauders will not be influenced.

If it happens that a Mech army battles a Bio army, things start to differ.
Siege Tanks take down Marines and Marauders with huge overkill, same goes for Hellions, so Armor Upgrades dont change much here. But to thin out the lines, Attack Upgrades are crucial for the Bio player. Marauders take 1 less hit to kill Tanks, Marine shots drop from 32 to 24. So, contrary to visual impression, the Bio army should prefer attack upgrades.
For the Mech army, the following occures: Tanks and Hellions remain untouched by damage upgrades when it comes to critical upgrades, but Thors get to kill Marauders in 2 hits instead of 3 when having a damage upgrade.
The Armor Upgrade on Mech changes mainly Marine hits. Marauders still take 9 Hits to take out a tank, Thors die in 23 instead of 22 shots, the Marines hits obviously get drastically effected as 8 more hits for Tanks, 20 more hits for Thors and 3 more hits for Hellions are required.

In Mech Battles, the most interesting Parts are Tanks vs Tanks and Thors, and Viking Battles for obvious reasons.
As Tanks kill other tanks on Siege with huge overkill (4x(50-1) - 160 = 36), Armor upgrades do not effect this setup at all. Same math applies on Thors.
But when damage upgrade is conceided, things get interested. Due to huge scaling of Siege Tank Damage, we will dig deeper into this. After one attack upgrade, Tanks kill Tanks in 3 shots instead of 4. And, as a matter of fact, once your Tanks have +2, Tanks kill Tanks in 3 shots regardless of the opponents upgrades. On a sidenote, the Thor overkill is too huge to make any reasonable conclusions.
The Viking against Vikings math presents the following:
Vikings kill Vikings on 5 Shots, regardless of upgrade advantage or disadvantage.
And, as a bonus lolly, Cattlebruiser against Viking math as upcoming Lategame opportunity: It takes a Viking 25 shots to take down a unplated Battlecruiser, 28 shots on Level 1 plating. So it highly depends on the situation if the +1 is worth is, the effect of course raises drastically and if you can afford massive upgrades on Battlecruisers, Marines and Thors do 4x2 respectively 2 damage per shot, so Vikings are the only opportunity at all left.

The last matchup to be reviewed is Zerg Mirror. In this matchup, the common confrontations are Zerglings, Banelings and Roaches against each other, and Muta Battles. I won't investigate on Hydra against Muta unless popular demand, as mostly there is Fungal around once this happens and the critical upgrades get pointless.
So first analysis on Zerglings. Against Zerglings and Roaches they need 7 and 37 hits before upgrades, and die to 7 and 3 shots, and one baneling burst.
On superior upgrades, you need 6 hits to kill Zerglings and 29 to kill Roaches, and need 9 and 3 shots to die, but survive one baneling burst.
If Roaches have armor, you need 50 (read: forever) hits.

As you shouldnt run banelings into Roaches, I won't touch this, so lets see how Roaches perform:
You need 3 shots to kill a Zergling and 10 to kill a Roach before upgrades, with upgrades its 2 shots for a Zergling and 9 for a Roach. Once armor is out, Roaches need 11 shots on Roaches.
So upgrades seem viable in every circumstance so far.

Muta against Muta rises the following:
14 shots non-upgraded, 13 when upgraded on damage and 16 when upgraded on carapace show how fragile this can turn out, but of course corrupters mess with this a bit.