Erevan Sunshadow
Heroscape is a tabletop war game released in 2004 and discontinued in 2010. Wikipedia, fan site. Attack dice show a skull 50% of the time. Defense dice show a shield 33% of the time. Each excess skull rolled counts as one wound. Disclaimer: I'm not particularly aware of any existing theory/math for this game. Topics like this may have been covered in the past. This is just something I got interested in doing for myself, in my own style.
Fire Blast Special Attack Range 5. Attack Special.
Each time you attack with Erevan Sunshadow's Fire Blast Special Attack, you may choose to roll 2, 3, or 4 attack dice. If a skull is rolled on every die, you may attack again with Erevan Sunshadow's Fire Blast Special Attack. You may continue attacking with Erevan Sunshadow's Fire Blast Special Attack until you do not roll a skull on every die.
With a basic attack of 2 and 1 range, you're effectively always going to be using this special attack. It appears to be presenting a choice - many weak attacks, or few big ones. But how do the numbers work out?
Each attack die has a 50% chance to come up as a skull. With 2/3/4 attack dice, you have a 25%/12.5%/6.25% chance to repeat. For \(a\) attack dice, your expected number of attacks is:
$$ \sum\limits_{k=0}^{\infty} \frac{1}{\left(2^a\right)^k} $$In a table:
| Dice rolled | Expected attacks |
|---|---|
| 2 | 1.33 |
| 3 | 1.14 |
| 4 | 1.07 |
As you might have been able to predict, that doesn't look good for the low-dice options. Even when rolling two dice, getting even one or two bonus attacks is unlikely.
So, you should always go with 4 attack dice and just expect to get the one attack off. Sounds easy, right?
We can check a few more scenarios to be safe. Imagine you're dealing with a whole bunch of 0 or low-defense squads. It could be better to roll less dice - you have good odds for a kill either way, and have a better chance at taking out multiples. Let's draw some charts and see how close the options are then. We'll also add another strategy. There's often only so many units in range, and with 4 dice seeming so much better than 2, you could try rolling 2 dice at first, and if you get all skulls, switch to 4 dice for any remaining attacks.
Still no luck! Let's look at the zero defense case a little closer. With 4 dice, you have a 6.25% chance to whiff, an 87.5% chance to kill exactly one, and a 6.25% chance to kill one plus get another roll. We can approximately count that as 1 extra guaranteed kill - which would give us:
$$ 0.0625\cdot 0 + 0.875 \cdot 1 + 0.0625\cdot 2 = 1 \text{ avg squads killed}$$With 2 dice, you have a 25% chance to whiff, a 50% chance to kill exactly one, and a 25% chance to kill one plus get another roll. We can approximately count that as 1 extra guaranteed kill - which would give us:
$$ 0.25\cdot 0 + 0.50 \cdot 1 + 0.25\cdot 2 = 1 \text{ avg squads killed}$$In other words, the lower attack values are a near complete trap. Average performance with 4 dice is equal or better at all levels. Switching to 4 dice after the first attack of 2 helps a little, but not enough to pass the other pure options.
One final thing. Suppose there are exactly two squad units you absolutely must kill this turn. Perhaps then you want less dice? Let's plot the probability of killing two or more squads.
So the two-dice option actually has a slight use compared to the other pure strategies. When specifically aiming for two kills, the 2-then-4 strategy is clearly even better, though. This holds even for relatively decent defense stats.
For fun, I also added a 3-then-4 option - it ends up better than all-2 for 3+ defense, but remains worse than 2-then-4.
This doesn't mean you should regularly be using 2 dice against low-defense squads. The chance of killing multiples may be higher, but as we know the expected value to be equal or worse than always using 4 dice (see graph #2), the chance of killing no enemy units must also be higher. Also, all probabilities here are low enough that you really don't want to be depending on Erevan for this. He's realistically a single-target attacker with 4 dice who can rarely highroll and attack twice.