Yes. This is how math actually works. Order of operations does not matter for a formula involving two factors multiplied. I'm not even going to go through your math because it's obviously wrong if it disagrees with basic math principles everyone should know from primary school.
Ok, you're right that, "order of operations doesn't matter when both are multipliers," which is what you said. I misspoke when I disagreed that "it matters"; but as was pointed out later, -% damage received isn't a multiplier, like discipline or IAC. It's a reduction of the entire casualty formula, including all the multipliers like discipline, IAC, pips, +%damage dealt, etc.
No, the order of operations doesn't matter in multiplication. What matters is that you are not multiplying the same things together. If you are multiplying X * Y * Z, then yes, all multipliers are equal weight. But this is not X * Y * Z, it is X * Y * (1 - Z), which means that whatever is going on with Z is going to act differently than X or Y.
By this logic -maintenance cost is also the single most impactful modifier because it also scales higher in effectiveness the more you have. Furthermore unlike -damage received you can actually stack it pretty dang high in a normal game, around -65% is pretty easily attainable with -80% or more for specific tags.
Yes!! Exactly! That is actually on the right track. -% maintenance cost IS really good. But there are two issues, that don't exist with -% damage taken or Admin Efficiency or -%diplo annexation cost.
1) You have other limits. A) Force Limit. Even at very very high discounts of maintenance costs, exceeding the force limit quickly increases the maintenance cost back up to unsustainable levels, so you can have a much bigger army, but you can't have an army that's exponentially bigger. B) Manpower / merc companies are limited anyway, so you can't just have an arbitrarily large army.
2) So you saved some money. Great? It's a money discount, which means it gets you more money, which is a useful thing, but which is ultimately only as useful as things you can spend money on. Meanwhile, not losing troops in phase 1 of a battle means you do more damage in the following phase of the battle, which means the enemy does less damage in the following phase, which means you do more damage in the next phase.... etc.
The problem with -damage taken is that it both matters virtually zilch for battles (because it does nothing to morale and to help you win battles) and you also can't stack it high enough for the scaling to start mattering outside of very specific meme runs.
"The food is awful! And the portions are so small!" Yeah, you can't stack it as high as would be truly broken, but it can certainly be high enough to be very very impactful. But this isn't just about meme runs like Shia Spain in Age of Reformation or theocratic revolutionary Zoroastrian Germany in AoR. It directly impacts your ability to fight all parts of the game and impacts how we should be weighting things like Divine ideas or national ideas.
Saying it doesn't help win battles is just ignorance of the casualty/damage formula again. If you don't take as many casualties in Phase 1, you do more damage in Phase 2, including morale damage. This persists across rounds and snowballs, so it absolutely helps you win battles. For example, if you a unit takes 200 casualties, it does only 80% of its normal damage. But if it's only taken 100 casualties, it does 90% of its normal damage. One of the many multipliers for determining morale damage and strength damage does is (strength / 1000), so taking fewer casualties in one round directly results in more damage the next.
I mean, what's next, tactics modifiers from tech don't help you win battles? Horde +% shock damage on flat doesn't help win battles because it's to strength casualties, not morale casualties? These things all help.
And again, let me repeat: DRILLING WEARS OFF AFTER A SINGLE BATTLE OR TWO OR JUST A BIT OF SIEGEING, PLUS EVERYONE CAN DRILL, ALL YOU ARE GETTING IS A MINORLY FASTER DRILLING. At best this decision gives you -2.9% damage received for a single battle. In practice it's probably more like -0.5% on average across a war for a player who drills a lot. Is -0.5% damage received worth +15% maintenance cost? Everyone reasonable know it isn't. I'm not even taking into consideration the fact that plenty of players will say that if you are playing well then you shouldn't be drilling your armies anyway.
Honestly you need to update your sensibilities from like, 1.28 or so. Drill loss is majorly mitigated by professionalism now. It lasts way longer than what you're describing. (And it's based on casualties, so taking fewer casualties = less drill loss...). Morale is not the single-most important modifier in EU4 combat anymore. Yes, yes, the devs don't like how things are working, but this is how the game is right now. Then again, we're all about to get thrown back into a period of figuring out the new combat meta this week, so it's basically all moot anyway.
