player1 fanatic said:
Only problem is that Axe beats Axe better then Archer.
-Hills and walls, Attacking axemen has city raider I
Point 1: defending archer has city garrison
Attack Axe:5
Defending Archer: 3*(100+25fortify+50archer bonus+50walls+50hills+20garrison-20city raider)=3*2,75=8,25
So it's 5 vs 8,25
Point 2: defending axemen with combat I
Attack Axe:5
Defending Axe: 3*(100+25fortify+50walls+25hills+10combatI-20city raider)=5*1,9=9,5
So it's 5 vs 9,5
If it was attacking swordsmen, it would even be more badly beaten.
EDIT:
I haven't taken in accound 1First Strike Archer has, but when you compare 9,5 to 8,25, archer is still not better then axemen, it's same at best (on hills). Plus Axmen are even more better against swordsmen.
The only balancing point is that AI is using only Archers as primary defense, so human player can get some use of Swordsmen. Otherwise if they used primarly Axemen for denfese, Swordsmen would have hardly any use.
EDIT: Zombie69 has pointed out, correctly, that the combat calculations work as described above rather than as below.
This means that Axes are pretty good on defence (against other axes and swords), but the effectiveness of axes on
defence drops faster with City Raider I/II/III than Archers and archers have defence abilities that melee units don't.
Neglect below

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I always understood that modifiers were applied to the unit itself and were additive. This actually changes the math a bit. So if I have an axeman with city raider I, its base attack strength is
5 x (1 + 0.2) = 6 (against archer)
5 x (1 + 0.5 + 0.2) = 8.5 (against melee)
A swordsman with city raider I has attack strength
6 x (1 + 0.1 + 0.2) = 7.8
So an axe is a slightly better attacker against melee and worse against archers.
If the defender is an axeman (in the situation described by you), its defense is the same whether attacked by axe or sword. It is a stiff
5 x (1 + 0.25 (fortify) + 0.5 (melee) + 0.25(hill) + 0.5(wall)+0.1(CombatI),
or a defence of 13.
But swords are pretty stiff too!
A sword on defence would be
6 x (1 + 0.25(fort) + 0.25(hill) + 0.5(wall) + 0.1 (CombatI)),
or a defence of 12.6.
If you bombard down the walls and are not on a hill, axes are better. An axe defender would be at strength 9.25, while a sword defender would be at strength 8.1.
Archers are pretty respectable, but their small base strength means that they don't get quite as tough, topping out about 9 as you note. You've underrated the archers a bit because they can get much better with garrison II and garrison III (in addition to being cheaper); melee units have comparable offensive bonuses but do not have the same defensive bonuses. A maxxed out archer (easy to get in a situation like the one you described, where waves of attackers throw themselves at a city) reaches 10.5.
Neither do do especially well against a stack of horse archers with flank (whittle them down) and shock/cover (finish them off)...but again, that tactics has a counter (spears)...and cover doesn't show up until further up the promotion tree if memory serves. Defending with archers isn't unreasonable at all, but a mixed defense will be much better than any one pure ingredient (stack of doom) until siege comes in.