Combat advantage

Mage A(#XXX)'s large army gained a +28% defensive/defensive advantage

The meaning of this, as clarified by Laanders (Nov 2013):

OK so, the basis of the absolute power system is of course that once enough damage is done by the attacker, the winner of the battle is decided by whoever kills the most net power worth of units. Ergo, if the attacker kills 800k units and the defender kills 700k, the attacker wins.

Easy enough, and obvious to see in a battle. Kill most, and you win. However, with a strict absolute determination, you can in some cases run into some weird situations. Say an RD top mage rebounds a much smaller mono AA who uses HoM on defense, and thereby succeeds in killing 10 RD, or about 820k NP. Meanwhile the attacker wipes out the defending stack, a total of 600k NP. The attacker may have lost perhaps less than a tenth of his army, and wiped out the defender's stack. It seems unintuitive that the attack would be unsuccessful.

The current solution is to use a modified version of the absolute power system, when either side's army in a battle is more than twice as big as the other. When this happens, the side with the much bigger army gets a certain advantage, either offensive or defensive, which increases with the magnitude of the army size disparity. What this percentage means is that the mage that benefits from the advantage can afford to lose up to that much more of its army than the opponent, and still win the battle.

In the simplest possible example, this means that if an attacker has 50% offensive advantage, and kills 1 mil NP of the defender's army, s/he can afford to lose up to 1.5 mil NP and still win the attack. This is probably (more than?) enough for most people to know to understand battle reports.

However, for those who like to chomp numbers, it might be interesting to know where the numbers come from. So here goes:

The effective parameter is called the power ratio bonus, so let's shorten it to PRB. All NP referred to below are army size involved in the battle. Kingdom land size, NP or troops not fighting don't matter.

If the attacker is more than twice the size of the defender, then PRB = (defender's NP x 2) / attacker's NP. If the defender is more than twice the size of the attacker, then PRB = defender's NP / (attacker's NP x 2).

The reader will note that the PRB is a quota between the NP of the 1) the NP of the bigger army, and 2) twice the NP of the smaller army. The advantage percent given in the battle report is how much bigger than double the opposition that the larger army is.

In the shape of formulae, when the attacker is bigger, then the offensive advantage = (1 / (PBR - 1)) x 100, and when the defender is bigger, the defensive advantage = (PBR - 1) * 100.

To determine the battle, the PRB is simply multiplied by the attacker's casualties before it is compared to the defender's casualties. So with offensive advantage (PRB < 1), the attacker is allowed extra losses. And conversely, with defensive advantage (PRB > 1), the defender is allowed to lose more.

More detailed examples? You got it. Casualty percentages provided for reference only.

10 mil attacker, 4 mil defender, PRB = (2*4)/10 = 0.8, +25% offensive advantage. assume def loses 1 mil, that's 25% if att loses 1 mil, that's 10%; off adv makes it 800k (8%) = WIN if att loses 1.1 mil, that's 11%; off adv makes it 880k (8.8%) = WIN if att loses 1.2 mil, that's 12%; off adv makes it 960k (9.6%) = WIN if att loses 1.25 mil, that's 12.5%; off adv makes it 1m (10%) = BREAKPOINT if att loses 1.3 mil, that's 13%; off adv makes it 1.04m (10.4%) = BLOCK

So with absolute rules, the breakpoint at which the attacker gets outdamaged is 1.25 mil. With the old relative power rule, the breakpoint would be where attacker loses 2.5 mil (25%).

One might note that the old and new systems in some degree relate to one another. The absolute rule with offensive advantage triggered in fact corresponds to using relative rules, between half the size of the bigger army and all of the smaller one, ie in this case a 5 mil attacker vs a 4 mil defender.

Let's do a defensive advantage example too.

4 mil attacker, 10 mil defender, PRB = (0.5*10)/4 = 1.25, 25% defensive advantage. assume def loses 1 mil, that's 10% if att loses 1 mil, that's 25%; def adv makes it 1.25 mil (31%) = BLOCK if att loses 900k, that's 23%; def adv makes it 1.1 mil (28%) = BLOCK if att loses 800k, that's 20%; def adv makes it 1 mil (25%) = BREAKPOINT if att loses 700k, that's 18%; def adv makes it 875k (22%) = WIN

With the relative power rule, the breakpoint would be where attacker loses 400k (10%).

So again the defensive advantage showing up means the battle is effectively determined by relative power rules between half the bigger army (5 mil defender) and the full smaller army (4 mil attacker), meaning at the breakpoint where both sides lose the same percent (20% here), happens here when the attacker loses 800k.