Elite and vicious are both strong rules, especially if it’s an army wide rule. The rule allows for more damage, especially guarding against that single bad dice roll that happens once every time. The rule makes units not only perform better, it increases their reliability. (which is a good thing)
Elves (and their evil kin) are widely loathed for their damage potential using these rules.
From time to time a discussion pops up on the forums that elite is better than vicious or vice versa. The truth is, while they are mechanicaly different, the mathematical result of either special rule is exactly the same. In this blog post, I’ll post why it doesn’t matter which upgrade a unit gains in terms of performance. (one small exception at the end of the article)
What does it do?
For those who don’t know the KoW rules by heart, here’s the summary:
Elite allows a re-roll of “ones” when rolling to hit. Vicious allows the same thing on rolling for damage.
Either rule increases the average damage of the unit. Morover, it especially helps improving bad rolls more then improving good rolls, so it also increases the units’ reliability. While the damage increase as a result of either rule is limited (see below for calculations, the fact that it severely limits the amount of “very bad” rolls is worth a lot.
The calculations for average damage:
Let’s asume a random unit has a chance of A/6 to hit a unit. It also has a chance of B/6 to do damage.
The average damage this unit will do is A/6 * B/6 = AB/36 per attack.
Note: a roll of 4+ is a success on a 4,5, and 6, therefore calculated as 3/6. A roll of 2+ using the same mechanics is a success on 5/6.
Examples: A shield wall facing off against a defense 4 unit has an average damage of 3/6 * 3/6 = 9/36 or 25% per attack.
a unit of elohi facing against a defense 3 unit has an average damage of 4/6*5/6=20/36 or 56% per attack.
Due to better stats, the elohi are more than twice as effective per attack.
Let’s give a unit elite
Upgrading a given unit with elite grants a reroll to hit when the to his roll is an “one”. the math behind this is as follows:
Chance to hit: A/6 (normal to hit) + 1/6 (chance of rolling a “one”) * A/6
= A/6 + 1*A/36
= 6A/36 + A/36
The damage remains the same @ B/6, which makes the total damage of this unit:
7A/36 (chance to hit) * B/6
Applied to the shield wall above improves its damage per attack to 63/216 = 29%
The Elohi increase their damage per attack to 65%
The same with Vicious
Upgrading a unit with Vicious makes no difference to the ‘to hit’ roll; it stays A/6
The chance to do damage increase to: B/6 (normal damage) + 1/6 (chance of rolling a “one”) * B/6
= B/6 + 1*B/36
= 6B/36 + B/36
which results in the following average damage per attack
A/6 (normal to hit) * 7B/36
The calculation is exactly the same calculation as Elite!
The reason behind this is, that both rules take an 1/6 chance and improve this by the same average damage per attack.
I hope to have -mathematcially- proven that both Elite and Vicious have exactly the same result.
To all good things, there must be an exception:
Some rolls DO require a to hit roll (thereby triggering elite) and DO NOT require a damage roll. (and thus not triggering vicious). This applies -for example- in spells like Heal, Surge and Wind blast. This means in these cases Elite is better than Vicious.
That is exactly the reason why Elite is marginally better than Vicious. In all other cases it doesn’t matter.