As with any game, there are certain actions in WARMACHINE that can be seen as strictly better than others. As a mathematician, I felt impelled to figure out what those things were. As a computer scientist, I was going to use technology to help. As this is my blog, I am going to explain it to you…

Ok, enough posing – I’ve recently been playing around with a spreadsheet to figure out what the best use of a single focus/fury on a warjack or warbeast was. Specifically, given a single focus/fury and three options: Boost the attack roll, boost the defense roll, or attack a second time with the same weapon, which is expected to provide the best damage output?

At first blush, this seemed like it should be easy to answer, but it turned out there was a lot of interesting probability hiding in the background. Here’s a quick recap of my model: Assume you’re making a single attack with a weapon. Rolling two dice (three if boosted) and adding your MAT or RAT, you hit if you roll equal to or greater than their DEF, or if you roll all sixes (all ones, however, always miss). If you hit, you roll two dice (three if boosted) and add your POW + STR and deal a number of damage equal to the minimum of zero or the difference of that value with their ARM. The total expected damage then, is the chance to hit times the expected damage given that you hit (standard independent probability).

Computing total expected damage from the three possibilities and the standard attack roll, and making a table by taking total expected damage as a function of the opponent’s DEF and ARM gives the following, color coated, chart:

This shows the regions where you should use your focus to gain an additional attack, boost the attack roll, or boost the damage. There’s not too many surprises – if their ARM is high relative to your POW + STR, you’ll need to boost the damage roll to achieve optimal damage. Conversely, if their DEF is high relative to your MAT or RAT, you’ll need to boost the attack roll. The interesting part of this graph is that damage starts retreating back to boosted damage when their relative DEF and ARM are high. The reason for this is that you still have a chance to hit even if their DEF is impossibly high (namely, by rolling all sixes). In this case, it becomes most advantageous to boost the damage roll and rely on double sixes rather than try to boost the attack! The other interesting part of this graph, to me, is that the double attack maneuver is the best in a very well-defined area – when DEF-XAT <= 7 and ARM-(POW+STR) <= 3. Because of this, it's a very easy heuristic to carry around with you and think about in game.

Branching off from this discussion, I started thinking about Combo-smite, particularly on the Argos. One might infer the same result from the Marauder’s Combo-smite ability, but I would warn against that: the Marauder gains the bonus of slamming your opponent back and knocking them down, so there there’s definite tactical advantage beyond simple damage computation. Back to the Argos, here is the table, where the top-left cell is Combo-Smite:

As you can see, none of the cells are colored pink, so you shouldn’t use Combo-smite in any circumstance! At first, this seemed a little counter-intuitive, compared to boosting, there should be an advantage to +4 STR, vs. an average of 3.5 damage per die, right? Here’s a lower-level chart depicting expected damage given that you hit:

If you look at the blue colored cells – you see that a single attack with the strength bonus is better than a single attack with boosted damage. However, once their ARM gets one higher (the yellow cells), the strength bonus looses it’s effectiveness. When you note that if you aren’t doing combo-smite, you’re getting a second attack, and at low arm, that is pretty significant damage.

So, back to the Argos – this tells a story that you should never use combo-smite with the Argos. Bummer, because it looks so good on its card…