Attrition and withdrawal

Filled with confidence, I moved a Modern Armour (with CR3) adjacent to an enemy city and found that among its 16 defenders were 8 Artillery. Come the enemy turn, and the artillery one after another attacked my unit; the first three were defeated without damaging my MA, the fourth made two 12-point hits and withdrew, the fifth was killed without causing more damage, the sixth made a 13-point hit and withdrew, the seventh made four 13-point hits and withdrew, and finally number 8 killed my MA. While I reckon that that was a pretty expensive victory for the AI (and the vastly weakened artillery were easy meat for my follow-up forces), three withdrawals out of seven possibles is a whole lot better than any of my suicide attackers have managed.
Anyway, this led me to wonder how many Warriors it would take to kill a Modern Armour. So I made a WB setup of an MA + Combat 5, surrounded by a vast horde of unpromoted Warriors. each hit by a warrior causing 7HP damage. Hits were made by warriors #6, #13, #29, #45, #61, #81, #96, #107, #112, #127, #138, #140, #151 - by this time the MA had only 9 HP left - #153, and #154 killed the MA. Thank goodness for the "quick combat" option.

Lol this game is unbalanced people say, it took 154 warriors!!

I would have thought a whole lot less than that.

Unit power in Civ4 is roughly quadratic.

The ratio of hits is roughly linear in both your own power and inversely linear in theirs (ie, number of times you are hit in ratio to the number of times they hit you).

The damage ratio between your hits and theirs is roughly linear in your own power and inversely linear in theirs.

The average damage done to you is the ratio of hits times the ratio of damage per hit.

(this is ignoring stuff like first strike)

maybe you should do more trials to confirm this result

Some math:
Damage(A): 20 * (3 * A + D) / (3 * D + A)
Chance: A / (A+B)

So a 60 strength modern armor against a 2 strength warrior:

20 * (3 * 60 + 2) / (6 + 60)
= 55 damage per hit from tank.

20 * (3* 2 + 60) / (180 + 2)
= 7 damage per hit from warrior.

The hit ratio will be 30 hits by the tank for every hit by the warrior (simply the ratio of the strength) on average.

So the tank does about 7 times as much damage (after rounding) and hits 30 times more often, which means on average the tank does 210 times as much damage as the warrior.

More accurately, it takes 15 warrior hits to kill the tank and 2 tank hits to kill a warrior. The tank hits 30 times more often than the warrior, so (on average) the tank should kill 225 warriors before being overwealmed.

That places the "power" or "number of units I can kill before I die" of a unit to be roughly power^1.6.

Does this hold over different scales?

Let's do a test. Strength 2 and Strength 3 units.
Strong unit hits 1.5 times as often and does
20 * (3 * 3 + 2) / (6 + 3) = 24
vs
20 * (2 * 3 + 3) / (9 + 2) = 16
damage, or 1.5 times as much damage.

This means a strength 3 unit does about 2.25 times as much damage to a strength 2 unit as the 2 unit does to the 3, on average.

In this case, a boost of 50% gave a 125% boost -- or S^2 power increase.

A "simpler" damage equation, for our purposes, is:
Dam(A) = 20 * (3*A/B + 1) / (3 + A/B)
DR(A) = DamRatio(A) = (3*A/B + 1)^2 / (3 + A/B)^2 = Dam(A)^2/400
And
Hit(A) = A/(A+B)
HR(A) = HitRatio(A) = A/B

So DR(A) = (3*HR(A) + 1)^2 / (3 + HR(A))^2

KR(A) = KillRatio(A) = HitRatio(A) * DamRatio(A)

KR(A) = HR(A) * [9 HR(A)^2 + 6 HR(A) + 1] / [9 + 6 HR(A) + HR(A)^2]
= [9 HR(A)^3 + 6 HR(A)^2 + HR(A)] / [9 + 6 HR(A) + HR(A)^2]

Note that the top has an exponent of HR^3, and the bottom an exponent of HR^2 -- so as (A/B) gets large, the expected kill rate converges to about 9 * (A/B).

Plugging in 60 for the modern armor, and 2 for the warrior, we get:
60/2 * 9 = 270
pretty close to the "real" rate of 225 and the single-test observed rate of 150ish.

Backing up, that ratio (9) actually comes from the damage ratio of the two units. At infinity, the attacker does 60 and the defender does 6. Practically, that is 2 shots to kill (ie, effectively 50) on one hand, and as low as 7 damage defending: a real ratio of 7 times A/B instead of 9.

At smaller values, that ratio of 7 times the ratio won't hold.

The damage ratio at certain points:
25% stronger: 1.25 x ratio
50% stronger: 1.5 x ratio
75% stronger: 1.73 x ratio
2x stronger: 2.0 x ratio
3x stronger: 2.8 x ratio
5x stronger: 4.0 x ratio
8x stronger: 5.2 x ratio
10x stronger: 5.7 x ratio
15x stronger: 6.5 x ratio
20x stronger: 7.0 x ratio
30x stronger: 7.5 x ratio

Neat -- up to 3x stronger, the kill ratio of a unit is about strength ratio squared.

Beyond that, it starts to fall off. (Strength-3)/3 + 3 is a decent approximation, until you hit 7 times stronger.

So, a quick approximation:
1x to 3x stronger: strength ratio squared
3x to 12x stronger: strength * (2 + strength/3)
12x and above stronger: strength ratio * 7

...

In short, if your are attacking a unit with strength 12 with strength 6 units, you can expect to have to burn 4 attacking units per defender kill. There is of course some variance in that, and it doesn't take into account first strike effects.

If you are attacking strength 12 units damaged down to 50% by catapults with strength 6 units, you still need 2 attacking units per defending kill (I think...)

Something I forgot to mention. When the MA killed warrior #30, a Great General emerged, as did another when warrior #90 died. So although the defender died in the end, the experience gained from his successive wins was not wholly wasted. And if #154 hadn't killed him. he'd have emerged battered but with another 153 XPs to his score and a slew of promotions to come.
Now why, in my last game, did I get my first GG at 30 points whilst the second needed 75, not 60, and the third came at 120 ? I've also had 30, 60, 90, then 135 etc. All these were playing as Roosevelt, normal speed, 2 continents, monarch level: why the differences ??

Great wall?
Fighting barbarians?

No, and no.