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Side note: CA reads those forums. But when there is no discussion about balance there is no need for CA to do something (some time ago i posted that the Yi infantry line seems too redundant/useless to recruit them. CA released a patch prior in which they were buffed a bit).
Last note: trained archers were able to fire more than 2-3 arrow per minute. If we would talk about musket warfare, this answer would be correct (trained musketmen were able to fire 3 bullets per minute).
In previous total wars, when an elite unit fights a lower tier unit, the elite unit's melee attack subtracts into the latter's melee defense and vice versa and naturally this means the elite unit usually has a higher chance to hit the lower tier unit and kill them faster.
In 3k this is meaningless since everyone gets between 20-40% melee evasion and there is no stat that reduces this which means your protectors of heaven or imperial sword guards have to pray to rng that they keep hitting those peasants or else they'll take all day to kill them.
On the other hand, you know what ISN'T affected by this silly system?
Cavalry and ranged units.
Cavalry in particular outright ignore melee evasion which is why a single charge can wipe out an entire unit in a matter of seconds which otherwise melee infantry would take a several minutes to kill or rout. Heroes are the most obnoxious example of this. Lu Bu cycle charging into a blob of infantry can easily rack up a couple thousand kills. Let's see your infantry do that.
Archers and crossbowmen only have to worry about missile block chance but hey the AI is too stupid to ever make clever use of their shielded troops. At best they'll use shield wall once in a while but are still very easy to exploit when using combined arms.
The 3 shots per minute ideal for musketry came from archers. While an archer could certainly manage more than three arrows per minute in extreme circumstances sustaining that rate of fire for any length of time tends to be an issue. Three volleys per minute is the average point of compromise between effectiveness and sustainability most (Western) armies seem to have arrived at for 'normal' circumstances.
Meanwhile 3K, unlike past TWs seem to have some archers loose arrows like machine guns instead of volleys. They have toned it down a lot these days but back in release it was ridiculous.
Good point. Maybe they represent your ministers. Still, they have a lot of skills related to leading an army so I still don't think CA thought out that class too much. Like how other generals have skills related to troops in their retinue that they cannot recruit. If you were to preserve yellows for assignments and ministry stuff, their role there is kind of nerfed because most of the skills only work if they are faction leader, heir, or prime minister. so realistically it would seem like having more than three yellows is kind of pointless.
On OPs question, melee infantry is a very specific use kind of unit. In general, you are almost always better off fielding glaive units in the field. spear units are only useful in a defensive capacity, like protecting your melee infantry from cav and your artillery. TBH, the only time you should have melee troops caught in a field battle is when the enemy sallies from the seiege.
so yeah, purple infantry is meant for sieges. defending and attacking. its why the purple stat increases melee evasion and decreases construction costs. purples are your administrators. you bring them along in the field when you want a more smooth siege battle.
their formations are kind of useless tho. if you are defending walls with them, you cannot use formations. and when you are attack walls, you cannot use formations. if they get caught in a field battle then formations come into play, but a cav charge basically ignores that. green units are the only ones who benefit from formations, which means you would want to pair a blue with them. and because you have a blue, why not spring for some catapults? and because you have catapults, why not attack wall breaches? and because you are rushing wall breaches, you might as well attack with glaive infantry. pretty much every army is going to look like a green, red, and a blue. if you want some melee infantry, you might as well sacrifice a couple of cav units on your red for some militia swords. they get their commanders charge buffs, and i cannot imagine why you would want purple units in the field other than to soak up some arrow fire and take the brunt of the initial enemy charge. militia are perfect for that.
Honestly, all these just highlights the flaw in the recruitment system. See, in older Total Wars, specialist units get to shine in low numbers because any general could still field them. Here in 3k, if you want a specific purple unit, you have to bring a sentinel, and most of the time, his army will only be comprised of purple units you don't need.
Imagine if this was the case in Warhammer or even Attila or Rome 2.
Say you want to bring a single beast of war like a dread saurian or a war elephant to soak up damage and terrorize the enemy, while your army will mostly be light footed skirmishers and archers. But wait, beasts of war can only be recruited by a specific general whose specialty is heavy infantry. So now you're going compromise the strategy you're going for by hauling unit types you don't want just for a single unit.
you named your user after our products. So guess you love that rubbish eh? Dolt.
What difficulty?
Yuan Shao and Lü Bu and the Suns?
Oh please do tell me how you afford more than 2 armies that are just cav?
Lu Bu's specialty is his cavalry. He literally gets 3 unique shock cavalry (Flying Riders + Xiliang cav). The camp crushers are good yes but you cannot rely on them to defeat the enemy army on their own.
The Suns are more balanced with their mercenary troops but if you were playing as them, you wouldn't be spamming just mercenary infantry. You would be using a balanced army. Mercenary infantry alone, while decent in melee, can get easily overwhelmed without the aid of archers and cavalry. They are not exactly what I'd call battle ending units. They're more for tarpitting the enemy while your mercenary archers rain flaming arrows at them.
As for Yuan Shao, I would have believed you if you said Yuan Shu since his Rapid Tiger Infantry and Warriors of the Left are better in melee compared to his anti-cav Ye Infantry and Defenders of Hebei. But alas, both these units don't have shields and will be demolished by any ranged heavy army so no, I don't see you relying on these guys to win your battles all the time, unless you just use them to pin down the enemy while your general does all the work doing killer drive bys.