Warmachine statistics I
So ladyarkham and I have been playing a lot of Warmachine lately. For those unfamiliar, Warmachine is a minis game set in a fantasy universe where the elves, dwarves and wizards, on discovering that their bows and spells weren't enough to drive off the dark lords, invented enormous steam-powered robots to do the job for them. Players in the game control a particular "warcaster," a wizard trained to remote-control these warjacks, and his retinue of soldiers, heroes, and giant mecha.
Warmachine's central task is resource management. Every turn, the warcaster generates a number of "Focus points," representing his pool of magic energy, and can spend them to cast spells, to heal his wounds, or (most commonly) to empower his warjacks to greater destructive feats.
So I got to wondering: what's the best way to spend your focus, anyway?
So, let me dig a little more into the mechanics of warjack focus usage. A warjack has, for my purposes here, four numbers of consequence, all typically somewhere in the range of 3-20:
1) Its attack rating. Technically, this is one of two numbers: MAT for melee attacks, RAT for ranged attacks; I'm going to write simply AT to cover both cases.
2) Its damage rating. Again, this can be derived in a couple of ways from other numbers, but I'm going to ignore those and just write POW to represent the total power of its attacks. Again, higher numbers mean more damage.
3) Its defense, or DEF. This is the measure of how good a warjack is at getting out of the way of an incoming attack.
4) Its armor, or ARM. This is the measure of how resistant a warjack is to damage, should it be hit.
In general, a warjack gets to make one attack against its target with each of either its ranged or its melee weapons, depending on distance. To do so, its controller first rolls 2D6 and adds its AT; if the total is at least the target's DEF, the target is hit. So, that's a hit if
2D6 + AT >= DEF.
On a hit, you then roll for damage - again, rolling 2D6, adding POW, and then subtracting the target's ARM. So damage is
2D6 + POW - ARM.
Each warjack can hold a limited amount of Focus, and it can (for my purposes at the moment) spend that focus in one of three ways:
1) It can buy an extra attack with one of its weapons.
2) It can "boost" an attack roll, in which case it rolls an extra D6 on the attack (and so is more likely to hit the target).
3) It can boost a damage roll, in which case it rolls an extra D6 on the damage roll (and so is more likely to damage the target).
You decide how to spend Focus mid-play - so, for instance, I can buy an attack for 1 Focus, then decide to spend 1 Focus to boost the attack roll, and (if it hits) then decide to spend another 1 Focus to boost the damage roll. Alternately, I could buy an attack, resolve it normally, and then buy and resolve two more attacks.
Good so far?
So, the riddle is: which of these usages of Focus gets you the most bang for your buck?
Let's look first at a simple case - deciding whether to boost an attack roll or buy another attack. Most of the time, all we care about is hitting our target as many times as possible, so we want to choose the option that optimizes our expected number of hits.
To decide whether we want another attack or a boost on this attack, all we really need to know is the difference between our AT and the target's DEF, AT - DEF. If this number is -2 or more, hey, we're in luck! We'll always hit - well, nearly always, since a roll of all 1s is always a miss - and so boosting our attack roll does nothing for us. We definitely should spend our Focus to buy more attacks. On the other hand, if AT - DEF < -12, we'll never hit the target without a boosted attack roll - it makes no sense to spend Focus to buy more attacks. Somewhere between these two, the right decision swings from one to the other.
It turns out that the swing point is the case where AT - DEF = -8; that is, the point where we need to roll an 8 or higher to hit our target. For an unboosted roll, that'll happen about 41.7% of the time - that is, we average .417 hits every time we take a swing. If I buy another attack, that's two attacks, each producing .417 hits on average, for a total of .417 + .417 = .834 hits. If, on the other hand, I boost my (singular) attack roll, I jump up to an 84.1% chance of hitting - that is, on average, .841 hits. So, at an 8, I am (very slightly) better off boosting to hit than buying an attack.
(For comparison, if I need a 9 to hit, boosting is a lot better - it's about .56 for buying vs. .74 for boosting. If I need a 7 to hit, buying is a lot better - about 1.16 for buying vs. .91 for boosting.)
So, first rule: If you want to hit as much as possible, and you need an 8 or above to hit, boost to attack; otherwise, buy an attack.
(I'm still simplifying, o'course: some weapons can only make one attack per turn, and in that case, if you've got the focus to burn, why not boost? But assuming you have a choice to make...)
But wait - what if I don't want to hit the target as many times as possible? What if I just want to guarantee at least one hit - say, if I know that my target is so fragile that any hit will kill him?
To answer this, I need to instead consider my odds of missing. If I need an 8, my odds of missing are 1 - .417 = .583. If I buy two attacks, my odds of hitting are 1 minus my odds of missing both times; that is, my odds of hitting at least once are
1 - (.583)^2 = .66.
On the other hand, my odds of hitting on a boost are still a straight .834 - still better. As a matter of fact, in this case it turns out that my odds are better on a boost all the way down. So, second rule: If you want to guarantee at least one hit, boost to attack.
All right, let's say you do hit, and you want to do as much damage as possible. Should you boost your damage roll, or no?
This one gets more complicated. We need to first figure out what your average damage would be without boosting. It's tempting to say that your damage is just (Average of 2D6 = 7) + POW - ARM, but that underestimates the truth somewhat. Let'as say I have POW 15 vs. ARM 20 and roll a 2. I get 2 + 15 - 20 = -3 damage, but my lousy hit isn't going to heal the target. To find the real average damage, we should treat a result like this as "0 damage" instead of "-3 damage" - and that's going to raise the average.
So let's look at this in terms of the average damage for each POW - ARM.
POW - ARM
Average Damage
<=-12
0
-11
.028
-10
.111
-9
.278
-8
.556
-7
.972
-6
1.56
-5
2.28
-4
3.11
-3
4.02
>=-2
7+POW-ARM
Damage without boosting
If we boost the damage, these averages go up:
POW - ARM
Average Damage
Increase from Unboosted
<=-12
Low
>0
-11
.963
.935
-10
1.46
1.35
-9
2.08
1.80
-8
2.82
2.26
-7
3.66
2.69
-6
4.57
3.01
-5
5.52
3.24
-4
6.46
3.35
-3
7.5
3.48
>=-2
10.5+POW-ARM
3.5
Damage with boosting
So, if we boost our damage roll, we'll get average damage from the second table; if we don't, we'll get average damage from the first... and have a focus free, with which to buy a second attack. Can we just double our results from the first table, then, and say, "That'll be our average damage on these two attacks?"
Well... no, we can't, because we've still got to roll to hit on that second attack. So our average damage on the second attack is actually [Our odds of hitting with that attack]*[Our average damage on a hit]. Our odds of hitting on an unboosted attack roll are, again:
AT-DEF
Probability
<=-13
0
-12
.028
-11
.083
-10
.167
-9
.278
-8
.417
-7
.583
-6
.722
-5
.833
-4
.917
>=-3
.972
Odds of hitting an unboosted attack roll
So what we want, then, is to know the lowest possible AT-DEF total such that [Odds of hitting]*[Avg unboosted damage] is greater than the increase we'd get from just boosting damage. That turns out to look something like this:
POW-ARM
Buy attack if AT-DEF is at least
<=-4
Always boost
-3
-4
-2
-6
-1 to 1
-7
2 to 5
-8
6 to 21
-9
So that's our summary of when to buy! If your AT-DEF is lower than these numbers, you should boost damage instead.
Of course... there is one more layer of complication. What if you're already boosting your attack rolls to hit? When should you boost damage in that case?
Unfortunately, that question has two answers: one for if you're down to 1 Focus (i.e., you can't afford to buy and boost an attack), and one if you have 2 or more remaining (i.e., you can). Either way, I think I'm going to set it aside for another post.
Warmachine's central task is resource management. Every turn, the warcaster generates a number of "Focus points," representing his pool of magic energy, and can spend them to cast spells, to heal his wounds, or (most commonly) to empower his warjacks to greater destructive feats.
So I got to wondering: what's the best way to spend your focus, anyway?
So, let me dig a little more into the mechanics of warjack focus usage. A warjack has, for my purposes here, four numbers of consequence, all typically somewhere in the range of 3-20:
1) Its attack rating. Technically, this is one of two numbers: MAT for melee attacks, RAT for ranged attacks; I'm going to write simply AT to cover both cases.
2) Its damage rating. Again, this can be derived in a couple of ways from other numbers, but I'm going to ignore those and just write POW to represent the total power of its attacks. Again, higher numbers mean more damage.
3) Its defense, or DEF. This is the measure of how good a warjack is at getting out of the way of an incoming attack.
4) Its armor, or ARM. This is the measure of how resistant a warjack is to damage, should it be hit.
In general, a warjack gets to make one attack against its target with each of either its ranged or its melee weapons, depending on distance. To do so, its controller first rolls 2D6 and adds its AT; if the total is at least the target's DEF, the target is hit. So, that's a hit if
2D6 + AT >= DEF.
On a hit, you then roll for damage - again, rolling 2D6, adding POW, and then subtracting the target's ARM. So damage is
2D6 + POW - ARM.
Each warjack can hold a limited amount of Focus, and it can (for my purposes at the moment) spend that focus in one of three ways:
1) It can buy an extra attack with one of its weapons.
2) It can "boost" an attack roll, in which case it rolls an extra D6 on the attack (and so is more likely to hit the target).
3) It can boost a damage roll, in which case it rolls an extra D6 on the damage roll (and so is more likely to damage the target).
You decide how to spend Focus mid-play - so, for instance, I can buy an attack for 1 Focus, then decide to spend 1 Focus to boost the attack roll, and (if it hits) then decide to spend another 1 Focus to boost the damage roll. Alternately, I could buy an attack, resolve it normally, and then buy and resolve two more attacks.
Good so far?
So, the riddle is: which of these usages of Focus gets you the most bang for your buck?
Let's look first at a simple case - deciding whether to boost an attack roll or buy another attack. Most of the time, all we care about is hitting our target as many times as possible, so we want to choose the option that optimizes our expected number of hits.
To decide whether we want another attack or a boost on this attack, all we really need to know is the difference between our AT and the target's DEF, AT - DEF. If this number is -2 or more, hey, we're in luck! We'll always hit - well, nearly always, since a roll of all 1s is always a miss - and so boosting our attack roll does nothing for us. We definitely should spend our Focus to buy more attacks. On the other hand, if AT - DEF < -12, we'll never hit the target without a boosted attack roll - it makes no sense to spend Focus to buy more attacks. Somewhere between these two, the right decision swings from one to the other.
It turns out that the swing point is the case where AT - DEF = -8; that is, the point where we need to roll an 8 or higher to hit our target. For an unboosted roll, that'll happen about 41.7% of the time - that is, we average .417 hits every time we take a swing. If I buy another attack, that's two attacks, each producing .417 hits on average, for a total of .417 + .417 = .834 hits. If, on the other hand, I boost my (singular) attack roll, I jump up to an 84.1% chance of hitting - that is, on average, .841 hits. So, at an 8, I am (very slightly) better off boosting to hit than buying an attack.
(For comparison, if I need a 9 to hit, boosting is a lot better - it's about .56 for buying vs. .74 for boosting. If I need a 7 to hit, buying is a lot better - about 1.16 for buying vs. .91 for boosting.)
So, first rule: If you want to hit as much as possible, and you need an 8 or above to hit, boost to attack; otherwise, buy an attack.
(I'm still simplifying, o'course: some weapons can only make one attack per turn, and in that case, if you've got the focus to burn, why not boost? But assuming you have a choice to make...)
But wait - what if I don't want to hit the target as many times as possible? What if I just want to guarantee at least one hit - say, if I know that my target is so fragile that any hit will kill him?
To answer this, I need to instead consider my odds of missing. If I need an 8, my odds of missing are 1 - .417 = .583. If I buy two attacks, my odds of hitting are 1 minus my odds of missing both times; that is, my odds of hitting at least once are
1 - (.583)^2 = .66.
On the other hand, my odds of hitting on a boost are still a straight .834 - still better. As a matter of fact, in this case it turns out that my odds are better on a boost all the way down. So, second rule: If you want to guarantee at least one hit, boost to attack.
All right, let's say you do hit, and you want to do as much damage as possible. Should you boost your damage roll, or no?
This one gets more complicated. We need to first figure out what your average damage would be without boosting. It's tempting to say that your damage is just (Average of 2D6 = 7) + POW - ARM, but that underestimates the truth somewhat. Let'as say I have POW 15 vs. ARM 20 and roll a 2. I get 2 + 15 - 20 = -3 damage, but my lousy hit isn't going to heal the target. To find the real average damage, we should treat a result like this as "0 damage" instead of "-3 damage" - and that's going to raise the average.
So let's look at this in terms of the average damage for each POW - ARM.
POW - ARM
Average Damage
<=-12
0
-11
.028
-10
.111
-9
.278
-8
.556
-7
.972
-6
1.56
-5
2.28
-4
3.11
-3
4.02
>=-2
7+POW-ARM
Damage without boosting
If we boost the damage, these averages go up:
POW - ARM
Average Damage
Increase from Unboosted
<=-12
Low
>0
-11
.963
.935
-10
1.46
1.35
-9
2.08
1.80
-8
2.82
2.26
-7
3.66
2.69
-6
4.57
3.01
-5
5.52
3.24
-4
6.46
3.35
-3
7.5
3.48
>=-2
10.5+POW-ARM
3.5
Damage with boosting
So, if we boost our damage roll, we'll get average damage from the second table; if we don't, we'll get average damage from the first... and have a focus free, with which to buy a second attack. Can we just double our results from the first table, then, and say, "That'll be our average damage on these two attacks?"
Well... no, we can't, because we've still got to roll to hit on that second attack. So our average damage on the second attack is actually [Our odds of hitting with that attack]*[Our average damage on a hit]. Our odds of hitting on an unboosted attack roll are, again:
AT-DEF
Probability
<=-13
0
-12
.028
-11
.083
-10
.167
-9
.278
-8
.417
-7
.583
-6
.722
-5
.833
-4
.917
>=-3
.972
Odds of hitting an unboosted attack roll
So what we want, then, is to know the lowest possible AT-DEF total such that [Odds of hitting]*[Avg unboosted damage] is greater than the increase we'd get from just boosting damage. That turns out to look something like this:
POW-ARM
Buy attack if AT-DEF is at least
<=-4
Always boost
-3
-4
-2
-6
-1 to 1
-7
2 to 5
-8
6 to 21
-9
So that's our summary of when to buy! If your AT-DEF is lower than these numbers, you should boost damage instead.
Of course... there is one more layer of complication. What if you're already boosting your attack rolls to hit? When should you boost damage in that case?
Unfortunately, that question has two answers: one for if you're down to 1 Focus (i.e., you can't afford to buy and boost an attack), and one if you have 2 or more remaining (i.e., you can). Either way, I think I'm going to set it aside for another post.