Hello again, and welcome to the second installment of Mathaga, where today we'll be talking about when defense dice are better than attack dice, and vice versa. Lots of SAGA abilities let you choose to throw additional attack or defense dice, and many of them let you choose which kind of die to throw. Note that I'm only talking about melee here, since for shooting attacks you only ever roll one or the other.
So should you roll attack dice or defense dice? Is there even a difference?
Well, yes.
Against a standard unit of warriors with no special equipment, a single attack die has a 1/3 chance of inflicting a casualty (1/2 chance to hit times 2/3 chance for a hit not to be cancelled). And it turns out that a defense die will cancel a hit 1/3 of the time as well. So, if you're attacking a unit with armor 4 in melee, your attack and defense dice are equally valuable, and you can just pick whether you want to kill more enemies or save more of your own troops for later turns.
Against armor 5, an attack die has a 2/9 chance to inflict a casualty, while a defense die still has a 1/3 (3/9) chance, so if you're just looking at inflicting vs preventing casualties, a defense die will do more for you against a heavily armored unit than an attack die. Conversely, against armor 3, an attack die has a 4/9 chance to kill, making attack dice the better choice.
It's worth noting that your own armor value doesn't impact the math at all; just the point value of your unit and strength of your defense dice. Adding defense dice will always prevent the same number of casualties: 1/3 per die added (1/2 if you're defending solid cover or closed ranks), regardless of your own armor value.
Well what about closing ranks then?
Yes, closing ranks or defending solid cover makes defense dice much more valuable. The probability of a defense die preventing a casualty increases to 1/2, which is better than the probability of an attack die inflicting a casualty against either armor 4 or armor 3. If you somehow reduce your opponent's armor to 2 (fatigue and equipment can do it), your attack dice will kill 5/9 of the time, which is better than the 1/2 chance your defense dice have of preventing deaths. Beyond that case, take defense dice if you've closed ranks.
Interestingly, your opponent closing ranks also favors you grabbing defense dice. After all, each of your attack dice has a lower chance to result in a casualty, while the power of your own defense dice remains unchanged. If the warriors in our first example had closed ranks, an attack die would have a 1/4 chance to kill, less than the 1/3 chance to prevent a casualty you get from defense dice. Your choice becomes equal again vs. an armor 3 unit which has closed closed ranks at 1 in 3.
There is an upper limit here, since defense dice only cancel hits. You can't cancel more hits than you take, , no matter how many defense dice you have. So if a high-armor unit suffers one hit, adding defense dice is really just increasing the odds of preventing one hit, while adding attack dice might result in more than one additional casualty. So when a SAGA ability lets you choose between attack and defense dice, you should take your opponent's attack pool and your own armor value into account. If your extra dice can inflict (on average) more casualties than they would be needed to prevent, throw the attack dice, even if your opponent's armor is high too.
Same thing for attack dice. If you're fairly certain of wiping your opponent out with the dice you have, put any dice you gain from SAGA abilities into defense.
Now, we need to go back to the top and add in another factor. As I implied in my last Mathaga post, not all casualties are equal. SAGA has given us a very convenient way of assigning value to casualties, however: point cost. Attacking higher cost enemies will favor attack dice, since you get more value per kill. Conversely, your own defense dice are more valuable protecting your own more expensive units.
If you're fighting a unit of armor 4 warriors, for example, each of them has a point cost of 1/8, so each attack die is going to kill 1/24 of a point (1/3 of dice will kill and each kill is worth 1/8 of a point). If your own unit is warriors, offense and defense are still on even footing: a defense die will prevent you from losing 1/24 of a point. However, if your unit is hearthguard, each defense die will rescue 1/12 of a point for you (1/3 of dice cancel hits and hearthguard are worth 1/4 of a point each), making defense twice as good as offense!
Meanwhile, if you're attacking opposing hearthguard at armor 5, adding an attack die will kill 1/18 points. If you're fielding warriors, now you favor offense again: killing 1/18 of a point per attack die vs saving 1/24 of a point per defense die. Your hearthguard were spared 1/12 points of death per defense die, so against armor 5 enemy hearthguard, they still favor defense.
"Jeezy Creezy!" I hear you exclaiming. "How many combinations are there? At this rate I'd need a spreadsheet just to keep track things!"
Not to worry, my friend, for I have got you covered.
I've outlined combinations which favor attack dice in red, and combinations which favor defense dice in blue. If the points killed per attack die is equal to the points saved by a defense die, I put "Either" in yellow. The light purple section is for when both parties are defending solid cover or have closed ranks, which is currently impossible in SAGA.
There are a couple of interesting trends. First of all, it's usually better to add defense dice than attack dice. In an even fight (hearthguard vs hearthguard, etc.), you basically always want defense dice, although if your opponent's armor is low enough, it might be better to grab attack dice. Hearthguard always want defense dice unless they're up against low-armor hearthguard. If neither side has cover, warriors want attack dice and defense dice about the same amount. Otherwise, they favor defense pretty heavily. Levy usually want attack dice, though it's about 50/50 if they have solid cover.
Anyway, there you go. Hopefully this will be useful. Your opponent may take a dim view of you reading a spreadsheet before every melee, but if they do, you should inform them that it's a time-honored tradition, and Emperor Charlemagne himself appointed a Count of Data Visualization to help him during his many campaigns.*
It's important to note that there are plenty of considerations which might alter where you want to put your dice, including both your own and your opponent's SAGA abilities and the positioning of the other units on the battlefield. All the Mathaga in the world cannot save us from having to grapple with the messy realities of the tabletop, but it can help us make the best of the situation we're in.
Until next time.
*Citation needed.
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