I used to be able to do math and stuff, but it's been 5 years since my last math class, which was population genetics. As such, probabilities have become quite foggy for me. Yet, I've been dying to know what the damage potential of Viper Legion are ... so I used what my algebra teacher from high school would call the brute-force-Mickey-Mouse approach and just wrote it all out. I'm not ready for rerolls or anything complex, so I'm just starting with the 2x1 with no upgrades. The only variable is whether the defender has a bane or not. Here is what I found.

No banes - average damage 2.39

Banes - average damage 3

Now, I know a lot of you don't like averages, but I find them useful. Still, for those who want to see the distribution ...

What is this telling us? Well, there are 64 different combinations of dice you can roll with 2 blue, so the numbers on the left show you how many of those combinations will give the damage listed on the bottom. So a 2x1 Viper Legion shooting at a defender with no banes will do only 2 damage 25 times out of 64 shots.

What I find interesting, is that there are 9 combinations of dice when attacking a defender with banes that yield damage over what is possible when attacking a unit without banes. That means roughly 1/8 shots against a "baned" target will more damage than possible without the bane.

I'm not really sure what else to say, so ... there it is.

Unless you're good with permutations and computations, you really have to long-form this out. Because we have upgrades that change exactly how much damage the dice will do, you have to start by describing the dice damages at exactly. But rather than doing the 64 different dice combinations, you can actually simplify and use a multiplication effect to represent the additional die.

That gives you the following on your 8 sides of the blue die:

1 blank: 0 damage

1 hit: 2 damage

1 hit: 2 damage

1 surge: 1 damage bane, 0 damage without bane *caveat, on rolls where you have a surge on the other die face, you've got an extra damage

1 accuracy: 0 damage

1 hit/surge: 3 damage bane, 2 damage without bane, same caveat as the 1 surge above

hit/accuracy: 2 damage

surge/surge: 2 damage bane, 1 damage without bane

That gives you the following:

bane:

0 damage 25%

1 damage 12.5%

2 damage: 50%

3 damage: 12.5%

Average per blue die: 1.5

Yeah, it ends up in the same place as you got above, but I'm thinking if you decide to apply this to melee and need to figure out the extra flank die or dice from adding dice or hits from upgrades, it is much easier to work on the basis of a single die. And I often do find average damages easier to work with. Yes, in a short game like this, what matters is getting the damage you need exactly when you need it, and because of that, we need to understand how much variance our dice will have. Bonus damage from surges, such as the Viper Legion and Darnati get, will tend to reduce variance. So as you note above, you'll be around 3 damage when the opponent has banes, but you're going to be pretty close to that 3 damage most of the time.

For example, the possibility of doing 6 damage, you need 3 damage on both dice. That only happens 12.5% percent of the time. That only coincides 12.5 x 12.5%, and yes, you really do multiple those together, or 1.5% of the total rolls. So it is possible, but not very likely. Likewise, the 0 damage outcome is 6.25%. 1 damage is 25% x 12.5% or 3% of the time. Or alternatively, you only have 5 damage on .5x .125 of the rolls, or 6.25%. You just have far fewer die faces that do no damage than what most units have, so the possibility of extreme outcomes is way down. So that's 1.5+3+6.25+6.25: 17% of the time you'll have an extreme outcome, but most of the time, you're going to be in the 2-4 damage range.

I think the very interesting thing is how this data interacts with an upgrade like rank discipline, and how it might affect the different bane/no bane situations. There, you have to go into which individual die outcomes you might want to reroll. The single surge is automatically rerolled when you do not have a bane, and probably rerolled if you do. You probably don't reroll the double-surge though. Anything you reroll though allows you to fish for that hit/surge, which can really boost your overall damage effects. To me, I think the open question is whether the surge effect can make it so I'm a bit less inclined to take Rank Discipline, either resulting in a cheaper unit or another upgrade effect that I like better. Right now, I only see cheaper unit in the 2x1. Tempered Steel is interested on them because it makes use of leftover surges when the target is baneless, and always improves by two when it does. I think the 2x2 might be an interesting carrying platform for Champion/Music upgrades when applied with Simultaneous orders.

Without rerolls, he is right and it’s 1xThreat+1 damage against a baned target.

With rerolls is a little fuzzier

Always fishing for hits nets you... 1.5xThreat+.75 I think

- 3.75 at 2 threat,

- 5.25 at 3 threat

Fish for hits, keep double surge... 1.375xThreat+1.03 I think.

- 3.78 at threat 2

- 5.16 at threat 3.

Basically, at 2 threat, keep double surges and hits. At 3 threat, keep hits, reroll double surges. Overall difference is pretty slim either way

EDIT: This math is all for baned targets. For comparison, with Rank, Reanimate archers average 3.56 damage. If you assume a surge modifier for Viper Legion, then add 1 to all those averages.