Damage calculation (formula + tool) Thread - Conquest of Elysium 3

Average damage calculation:

Neglecting armor, the average damage inflicted by an attack doing 1-N (including bonuses) is given by the closed formula:

N
1 + -------
2

[This is derived by summing the infinite series Sigma[i = 1 to infinity](i*(N-1)/N^i) = 1 to account for open-ended die rolls, plus the average result of a non-open ended roll 1 to (N-1).]

Thus, a 1-1(+1) attack (soulless's fist) does 1-2 damage. N=2, so average damage is 1 + 1 = 2. On the other hand, a spear doing 1-5 damage averages out to 1 + 5/2 = 3.5, and a crossbow doing 1-8 averages out to 1 + 4 = 5 points of damage.

Implications:

Damage is very nearly linear in N. 10 goblin archers do 25 points of damage per round, whereas 5 crossbowmen do 25 points of damage every 2 rounds and 5 archers do 15 points of damage per round. Goblin archers are a good buy.

The closed-form solution which includes armor is trickier to derive. In the common case where total armor A is less than damage size N, we can treat two cases: 1/N of the time the very first roll is the maximum value N, which triggers an open-ended roll (in which case we just subtract A from the result), and the rest of the time it is not (in which case the average damage is simply (N-A)*((N-A)-1)/2*(N-1)).

N 1 (N-A) * (N - A - 1) (N-1)
( 1 + ------- ) + (N - A) ) * ----- + ------------------- * ----
2 N 2 * (N-1) N

The first point of armor always knocks exactly 1 point off average damage, but after that it depends. I've posted a damage calculator for this formula on my blog here. You can use this to calculate the results of fights before you start them. For instance, you can quickly determine that 100 longdead skeletons going up against 20 Hastati in a dark citadel (DB 2) will do 1.25 points of damage on average per skeleton, killing about 3.5 Hastati per turn with a full rank of skeletons (less at the beginning because of damage spread). Meanwhile the Hastati will be doing about 20 * 3.5 = 70 points of damage back at the longdead, wiping out about 17 skellies per turn, plus or minus because of damage overlap. Thus the skellies will probably be fighting with about a quarter-rank each turn, killing 1 hastatus or less, which means the skellies are probably going to lose the fight.

Anyway, the damage calculator should make planning your attacks easier.

-Max

Doesnt look easy to me. But then my math sucks.
Im interested in how an attacker fares against a defender that is inside a defense structure.

People comment that the AI sucks because it doesnt attack with big army against their castle with far fewer defenders. I know the AI likes a clear advantage on attack. But on a rough comparison it seems that the defense bonus and the shield icons of a structure makes those much closer in strength than people think.

You can use the tool to calculate the effect of defense structures. Just plug in the defensive bonus as armor.