EPTSo i wrote him a PM explaining why air armor should be prioritised when upgrading air in TvT (especially long drawn out games where you might bring in BCs) and also analysing air upgrades in TvT in general and since i put quite a bit of effort into it in the end i thought i might just as well post it here. The reasoning beeing that to my knowledge -and that of my use of the search funktion- this hasnt really been discussed/taken notice of and a lot of Terrans live still under the impression that weapon upgrades are preferable in TvT air too as they are generally in most situations (not all mind you) across the races.
So here is a copy of my writeup, if you find yourself in a long drawn out TvT with a tank viking basis this just might help you out.
It also counts for iEchoic's air/hellion TvT build, but on that he already mentioned in his own thread that air armor is preferable, he just didn't do the long boring analysis that iam doing further down.
-End of introduction, beginning of analysis-
Firstly lets take a look at viking fights, a viking gets +1 damage for each of his two attacks per upgrade, therefore the upgrades annihilate each other perfectly. You either gain 10% damage or take 10% less by other vikings per each upgrade. Furthermore you need 2 upgrades of each to actually make a difference in the amounts of attacks it takes to kill another viking (requiring 5 shots without upgrades; 4 shots with 2 attack upgrades and 6 shots with 2 armor upgrades. This is of course discounting any upgrades on the opponent which would just have to be substracted from your upgrade count).
Conclusion on Vikings
Therefore in terms of pure viking battles if your opponent doesnt upgrade his it would actually make no difference to get the third upgrades and would actually be preferable to go 2/2 instead therefore enabling your vikings to need one shot less and to take one more.
Now onward to why armor is preferable: In TvT there can be only three possible attackers on your vikings, discounting BCs since you can micro against those to stay out of range. Those three are Vikings, Marines and Thors. For the former as explained above it really doesnt matter which ones you take, but for the latter it very much does.
Both Thors and Marines are usually gonna attack you without your vikings shooting back (the occasions where you want a land fight against them are very limited) so you would want to get the armor already right here. Furthermore though both thors and marines suffer from opposing armor upgrades more than any other unit except lings and workers because of their 6 damage attacks. A single armor upgrade will reduce their damage by 16.6% therefore greatly increasing the survival rate of your vikings and (importantly since those get shot at even more especially when dropping) medivacs.
But thats more on a sidenote, the probably most important part are BCs.
In a viking versus BC scenario we can probably discount the BCs attack, he is just too easily kited so only armor upgrades will allow him to stay alive a little while longer.
But a marine versus BC scenario is quite common and this is where the armor upgrades really shine. An attack upgrade on the BC will increase his ground damage by 12.5 %, which certainly isnt bad (most upgrades increase the damage by ~10%), but just cant compare to the whopping 33% damage decrease that you get out of having an upgrade advantage in armor of 1 point over your opponents infantry weapons.
The numbers apply exactly the same with thors as well although that scenario is a lot less common and effective for the thor users thanks to the unit's costs.
If your upgrade levels are the same the upgrades will just annihilate each other perfectly again (with his armor negating your weapon and vice versa), but as is clearly visible in all the other scenarios air armor is clearly superior.
The effect gets even stronger if your upgrade advantage gets bigger giving you 50% further multiplicative damage decrease with a second armor upgrade and another 50% with the third (getting you down to about ~83.5% damage decrease)
In a BC versus BC scenario armor and weapon upgrades negate each other once again but an armor upgrade advantage will let yours survive 28 shots more whereas it would take only 18 shots less for an opposing battlecruiser, so armor is once again the superior choice.
I hope some of you found this helpful and if you for yourself did not please don't flame me but rather ignore me instead
