## Wednesday, July 5, 2017

### ACKS Melee Offense Analysis

At long last, I've figured out the computational tools needed to do the analysis of ACKS' fighter offense system.  For small cases (d6 damage vs d8 orc HP) it's easy enough to do by hand / in head, but for a long time I was sort of stuck on the big cases (d10+6 two-handed sword vs 4d8+2 ogre HP).  Getting expected values is easy, but as we saw with Weapon Focus and backstab, variance is actually super-important to get right.  The sensible way to do this with a modern computer would've been to just enumerate all 41,000 possible outcomes.  Instead in the shower I figured out generating functions (again...  they came up twice during my undergrad), which express those sort of probability distributions as big polynomials that you multiply.  This may or may not be more efficient, but it was certainly more entertaining.

In any case, on to the main event: probability that an attack will hit and kill an enemy of a given type from full HP.  A pure "killing power" analysis.  By expected damage output * survivability, sword-and-board is superior.  I'm curious if this analysis with kill probabilities and cleaving will highlight cases where two-handers and two-weapon fighters are competitive.  Additionally, I'm interesting in looking at kill probabilities on backstab.  So we're going to consider a couple of different "builds" at 1st, 3rd, 6th, and 9th levels:

• Sword-and-board fighter
• Two-hander fighter
• Two-weapon fighter
• Two-hander assassin, backstabbing
• Thief with one-handed weapon in both hands, backstabbing
• Explorer doing archery (we often hear that explorers are "very strong", and I'm curious how much of that is raw combat power vs special abilities)

At first level, the builds look something like:
• Sword-and-board: Str 16, FS: Shield, plate and shield (AC8), THAC0 8+, damage 1d6+3
• Two-hander: Str 16, FS: Two-handed, plate (AC6), THAC0 8+, damage 1d10+4
• Two-weapon: Str 16, FS: Two-weapons, plate (AC6), THAC0 6+, damage 1d6+3
• Assassin: Str 13 Dex 13, probably Skulking, THAC0 5+, damage 1d10x2+2
• Thief: Str 9 Dex 16, Weapon Finesse, THAC0 4+, damage 1d8x2
• Explorer: Str 13 Dex 13, Precise Shooting, arbalest, THAC0 8+, damage 1d8+1
Here are the probabilities each of these PCs has of hitting and one-shot killing each of the following types of monsters (taking into account that the monster's HD are actually rolled rather than just taking the average for hit points):

 pc kobold goblin morlock orc hobgoblin gnoll lizardman bugbear ogre hill giant sword and board 0.5500 0.4583 0.4750 0.3958 0.3438 0.1359 0.0973 0.0185 0.0010 0.0000 two-hander 0.5500 0.4917 0.5550 0.4625 0.4375 0.2637 0.2250 0.0839 0.0141 0.0000 two-weapon 0.6500 0.5500 0.5542 0.4750 0.4125 0.1661 0.1189 0.0226 0.0013 0.0000 assassin 0.7000 0.6283 0.6938 0.6013 0.5769 0.4341 0.4050 0.2701 0.1224 0.0016 thief 0.7031 0.6125 0.6500 0.5688 0.5250 0.3453 0.3047 0.1425 0.0343 0.0001 explorer 0.4984 0.3958 0.4031 0.3359 0.2813 0.1055 0.0738 0.0138 0.0008 0.0000

What's interesting here is that the break-even point for two-handed weapons is 2HD monsters.  Looking back at the Nonlinear Effectiveness of AC, the AC8 sword-and-board fighter is about 1.5x as survivable as the AC6 two-hander against low THAC0s, while against 2HD gnolls, the two-hander is twice as likely to instantly-kill (and be able to cleave) as the sword-and-board fighter, which means that's about where it starts to become a good proposition - if you kill them twice as fast and take 1.5x as much damage, you're going to end up taking only 75% as much damage as a sword-and-board party.  When you factor in cleaving, as the sum of a geometric series (0.26 + 0.26^2 + 0.26^3 + ...), the two-hander fighter actually kills 0.36 gnolls per turn in expectation, while the sword-and-board fighter kills 0.16 gnolls per turn.  Meanwhile, two-weapon fighting is the most effective against low-HP foes where you're THAC0-bound, but only marginally more effective against bigger opponents than sword-and-board.  Granted: this analysis doesn't take into account the initiative penalty for using two-handed weapons.

Another conclusion here is that backstab is tremendous.  A 1st-level assassin with a claymore has a 1 in 8 chance of insta-killing an ogre with a backstab.  That's way higher than I expected.

Let's look at third level:
• Sword and board: Str 16, FS: shield and combat reflexes I guess, THAC0 7+, dmg 1d6+4
• Two-hander: Str 16, FS: two-handed and combat reflexes, THAC0 7+, dmg 1d10+5
• Two-weapon: Str 16, FS: two-weapon and combat reflexes, THAC0 5+, dmg 1d6+4
• Assassin: Str 13 Dex 13, probably Skulking, THAC0 4+, damage 1d10x2+3
• Thief: Str 9 Dex 16, Weapon Finesse, THAC0 3+, damage 1d8x2
• Explorer: Str 13 Dex 13, arbalest, probably precise shooting twice, THAC0 7+, damage 1d8+2
Here are the kill probabilities:

 pc kobold goblin morlock orc hobgoblin gnoll lizardman bugbear ogre hill giant sword and board 0.6000 0.5347 0.5688 0.4813 0.4354 0.1992 0.1510 0.0340 0.0026 0.0000 two-hander 0.6000 0.5500 0.6256 0.5294 0.5088 0.3344 0.2930 0.1226 0.0241 0.0000 two-weapon 0.7000 0.6319 0.6563 0.5688 0.5146 0.2391 0.1813 0.0408 0.0031 0.0000 assassin 0.7500 0.6883 0.7600 0.6650 0.6475 0.5007 0.4702 0.3250 0.1587 0.0026 thief 0.7500 0.6563 0.6906 0.6094 0.5625 0.3719 0.3281 0.1535 0.0369 0.0001 explorer 0.5813 0.4813 0.4977 0.4211 0.3695 0.1582 0.1172 0.0256 0.0019 0.0000

With that one extra point of fighter damage bonus at 3rd, the balance on the gnoll between the sword-and-board and the two-hander has shifted dramatically, to only a 50% advantage for the two-hander, making sword-and-board and two-handed competitive against 2HD opponents.  But now two-handed fighting has a 2:1 advantage over sword-and-board against lizardmen (makes sense; they're 2+1 HD, both fighters gained one point of damage, so now the ratio should be pretty similar to the 2HD case without the extra point of damage).  Almost everyone is THAC0-bound against kobolds, and the extra point of fighter damage bonus brought the assassin's odds of one-shotting an ogre up to 16% on the backstab, about 1 in 6.  Two-weapon fighting doesn't look competitive from these numbers, but when you start looking at cleaves, it's actually quite a bit better than the other fighting styles against weak humanoids.  A 63% kill chance against goblins means that in expectation, a two-weapon fighter kills 1.7 goblins per round in expectation, while a 53% kill chance for a sword-and-board fighter nets you 1.13 goblins per round.  That's right around where two-weapon fighting starts to pay off compared to sword-and-shield; you're two-thirds as survivable, but you're 3/2 as deadly, so it balances out.  Against morlocks and orcs, the two-handed fighter's damage is more important than the two-weapon fighter's THAC0 (for now).

At sixth level, things start to get interesting.  We're going to start adding magic weapons and armor (possibly a pessimistic assumption - it's not that unusual for 3rd level PC fighters to have swords +1), and thieves and assassins get their x3 backstab.

• Sword-and-board: Str 16, FS: Shield, Combat Reflexes, Command, sword +1 and shield +1, plate +1, THAC0 4+, damage 1d6+6
• Two-hander: Str 16, FS: two-handed, Combat Reflexes, Command, two-handed sword +1, plate +1, THAC0 4+, damage 1d10+7
• Two-weapon: Str 16, FS: two-weapon, Combat Reflexes, Command, sword +1 in each hand and plate +1, THAC0 1+, damage 1d6+6
• Assassin: Str 13 Dex 13, Skulking, Combat Reflexes, two-handed sword +1, THAC0 2+, 1d10x3+5
• Thief: Dex 16, Weapon Finesse, Skulking, sword +1 in both hands, THAC0 1+, 1d8x3+1
• Explorer: Str 13 Dex 13, Precise Shooting, Precise Shooting, Command, longbow +1, THAC0 4+, 1d6+3

 pc kobold goblin morlock orc hobgoblin gnoll lizardman bugbear ogre hill giant sword and board 0.7500 0.7000 0.7833 0.6854 0.6563 0.3910 0.3250 0.1009 0.0122 0.0000 two-hander 0.7500 0.7000 0.8000 0.7000 0.6913 0.5281 0.4845 0.2497 0.0657 0.0002 two-weapon 0.9000 0.8500 0.9302 0.8323 0.7969 0.4813 0.4000 0.1242 0.0150 0.0000 assassin 0.8500 0.8000 0.9000 0.8000 0.7900 0.6867 0.6680 0.5619 0.4250 0.0725 thief 0.9000 0.8146 0.8758 0.7836 0.7570 0.6000 0.5672 0.4166 0.2368 0.0071 explorer 0.7500 0.6417 0.6333 0.5542 0.4813 0.1964 0.1405 0.0267 0.0015 0.0000

At this point, almost everyone is THAC0-bound on goblins, the gap in effectiveness between two-hander and sword-and-board against 1HD foes has practically vanished, fighters start having a small chance to one-shot ogres, bugbears are the new 2:1 advantage zone for two-hander fighters, and assassins now have a 7% chance of one-shotting a hill giant on the backstab.  x3 backstab damage produces a sharp increase in thief and assassin lethality, most noticeable against ogres, where the assassin's odds of a kill are almost triple what they were at 3rd level.  The two-weapon fighter starts to see big increases in effectiveness against 1HD opponents, killing 4.8 orcs per turn in expectation vs the sword-and-board fighter's 2.2.  Unfortunately, the addition of magic armor and shield has raised the sword-and-board fighter's AC to 10 (and the other fighters' ACs to 7), which means he's around four times as survivable against 1HD foes as the other fighters.  Against ogres, though, he's less than twice as survivable as a two-hander fighter, and against bugbears he's twice as survivable while the two-hander is three times as deadly once cleaving is taken into account (0.33 bugbears slain per turn vs 0.11 bugbears per turn), leaving the advantage with the two-hander.  Explorer gains mostly from accuracy increases, because he swapped his arbalest for a magic longbow, which kept his expected damage flat.

9th level:
• Sword and board: Str 16, FS: shield, combat reflexes, command, ???, shield +2, sword +2, plate +2, THAC0 1+, damage 1d6+8
• Two-hander: Str 16, FS: two-handed, combat reflexes, command, ???, two-handed sword +2, plate +2, THAC0 1+, damage 1d10+9
• Two-weapon: Str 16, FS: two-weapon, combat reflexes, command, ???, two sword +2, plate +2, THAC0 -3+, damage 1d6+8
• Assassin: Str 13 Dex 13, ???, two-handed sword +2, THAC0 -2+, 1d10x4+7
• Thief: Dex 16, Weapon Finesse, ???, sword +2, THAC0 -2+, 1d8x4+2
• Explorer: Str 13 Dex 13, Precise Shooting x2, Command, ???, longbow +2, THAC0 1+, 1d6+4

 pc kobold goblin morlock orc hobgoblin gnoll lizardman bugbear ogre hill giant sword and board 0.9000 0.8500 0.9500 0.8500 0.8500 0.6271 0.5583 0.2307 0.0411 0.0000 two-hander 0.9000 0.8500 0.9500 0.8500 0.8500 0.7300 0.6950 0.4313 0.1466 0.0007 two-weapon 0.9500 0.9500 0.9500 0.9500 0.9500 0.7447 0.6630 0.2740 0.0488 0.0000 assassin 0.9500 0.9500 0.9500 0.9500 0.9500 0.9263 0.9144 0.8275 0.7005 0.3045 thief 0.9500 0.9500 0.9203 0.9203 0.9055 0.8146 0.7867 0.6531 0.4898 0.0792 explorer 0.9000 0.8264 0.8313 0.7438 0.6729 0.3188 0.2417 0.0544 0.0041 0.0000

At this point, it's pretty clear that the fighter's just not scaling like he used to.  The two-hander fighter is up to a 1 in 7 chance of one-shotting an ogre... right where an assassin backstab was at 3rd level.  It's true that with cleaving he's 4x as deadly against ogres as a sword-and-board fighter, but the sword-and-board is up to AC 12 (with the other fighters at AC8), making him more than twice as survivable against ogres.  Even high-level fighters who aren't playing the spear-charge or giant strength damage multiplication game just aren't going to cleave up ogres.  That's OK, but you have to be aware of it.  The assassin, on the other hand, can kill two and a third ogres in the surprise round (in expectation).  Lethality is now very high across the board against 1HD foes.  The two-weapon fighter is going to hit his cleave cap of 10 against hobgoblins and orcs most rounds, while the sword-and-board fighter kills 5.6 per round on average and is three times as survivable, making him the winner against those target.  The two-weapon fighter is competitive with the two-hander against 2HD foes, but remains weaker against 3-4 HD opponents.

I think that's about as far as I'm going to go with this analysis.  I suppose another thing to consider at this point is how these stats translate into mass combat.  The sword-and-board fighter gets one company-scale attack at 1+, and his AC 12 means that most massed units need a 20 to hit him, even on a charge.  The two-hander fighter also gets only one attack at company-scale at THAC0 1+, and his AC of 8 means that massed units hit him on 18+ before disorder or charging.  The two-weapon fighter gets one attack at -3+, and also has an AC of 8.  So overall, I feel like the shield guy is strongest in company-scale mass combat; the damage advantages of the two-hander are abstracted away, and the -3+ THAC0 represents only a small increase in to-hit probability when you're attacking AC4 massed noobs in chainmail.

Overall, I believe this analysis supports the conclusion that across the level range, sword-and-shield tends to be the strongest fighting style for fighters.  Two-hander fighters are competitive or superior in killing power * durability against 2HD opponents in the low- and mid-levels, while two-weapon fighters are competitive against large numbers of 1HD opponents in a brief window in the mid-levels, but overall, sword-and-shield is strong by default.  While cleaving presents another exploitable nonlinearity, where high kill probabilities lead to very large expected bodycounts per round, it requires both high THAC0 and high damage output, is capped by level, and it's hard to get the required high kill probabilities against 3+HD foes with just the straight fighter damage bonus + magic weapon damage bonus (as opposed to multipliers like polearm charges, giant strength, and backstab).

Other conclusions: I was slightly surprised as how much better assassins were at backstab than the thief.  d10 weapons make a lot of difference when you're multiplying them I guess.  If there's one other conclusion to this besides "shield fighters generally good, other fighting styles situational but not totally useless forever", it is that assassin backstab is very good.  On the other hand, explorer didn't perform all that well.  Part of that is that I of phoned it in on his proficiencies (Fighting Style: Missile would've been a better choice, for example), but it's also just an issue with ranged weapons - no str bonus to damage, low base damage, and no magic weapon bonus to damage (in the absence of magic ammunition, which is scarce and expendable).  On the other hand, ranged is qualitatively different in ways that make up for its low damage.  Finally, cleaving seems to work basically as intended in ACKS - I recall reading (probably on the Autarch blog) that cleaving was intended to mimic a rule from Chainmail or OD&D that high-level fighters could outright kill a number of 1HD opponents per round equal to their level.  9th level fighters (except for two-weapon fighters) don't quite manage that, getting around 6 instead of 9 kills per round, but they do still lay waste to weak opponents and it doesn't change their performance all that much against strong opponents.

I suspect that there are two ways to use two-hander fighters.  One is full-offense polearm bumrush, with two-handed "victory or death" as the party doctrine.  The other, more reasonable thing, is to put polearm berserkers in the second row of the phalanx.  Being in the second row masks the penalties from berserkergang, and it gives them the THAC0 of a two-weapon fighter with the damage of a two-hander fighter.  If the front-line collapses, they'll hold morale while the rest of the party retreats.  Great use for a henchman.  Between fights, if the front line is chewed up, you can give them a shield and spear and yeah, they don't have the fighting style, but they're probably OK for one or two fights as a stopgap on the way out of the dungeon.

Oh, and the third way to use two-handed weapons: Thrassians.

Limitations: obviously, as stated at the beginning, this was about killing power and cleaving.  It totally neglects teamwork and multiple fighters stacking damage on the same target.  Given that sword-and-board expected damage output per attack is around 66% of the two-hander's across the level range, this probably works against sword-and-board in this style of analysis.  Also, this analysis neglected polearm and lance charges, which I suspect are a really important part of fighter play in the mid-levels against tough opponents.  Finally, my assumptions about the availability of magic items may not be representative.  Depending on how strictly your DM reads the treasure tables, magic two-handed swords may just not exist, which increases the superiority of the sword-and-board fighter over the two-hander fighter.  Relatedly, magic shields occur on the treasure table much more often than magic plate.  60% of rolls on the Armor table yield an uncursed magic shield, while only 45% yield uncursed magic armor.  Of those 45%, 25% are plate, so about 11% of Armor table rolls yield uncursed magic plate.  Thus, you're going to find about 5.5 times as many magic shields as suits of magic plate.  This remarkable availability of magic shields also contributes to the strength of the sword-and-board fighter.

Follow-up questions:

Is there a rock-paper-scissors thing with fighting other fighters, where two-weapon beats shield, two-handed beats two-weapon, and shield beats two-handed?

As a DM, what can you do if you want to favor one fighting style over another?  Say you have a two-hander player in your party, what can you do with your encounter design to make them feel good (if that's a thing you care about)?  Probably you include more 2HD opponents that they can cleave but that your sword-and-boards can't.  If you have a two-weapon guy, include high-AC 1HD opponents, where their high to-hit lets them cleave and their lower damage doesn't hurt them.

How do these fighters look on horseback?  Two-weapon loses out.  Two-hander can use a polearm instead of a lance to keep his proficiency bonus, or go shield and lance and be inferior to the shield-fighter.  Shield can use a lance one-handed to get d10 doubled damage while keeping his AC.  I think shield wins this one too...

How important is the Fighting Style proficiency?  If, say, I'm mainly a sword-and-board fighter but we're up against gnolls and I'm considering going zweihander, how much worse am I than a dedicated, FS:Two-Handed polearm fighter (one point of damage, but how much does this matter)?  If I'm a two-hander fighter but we're up against a big mob of weak opponents where shield-fighting is strong and I decide to go shield-and-sword, how much worse off am I than a dedicated shield-fighter (one point of AC, but how much does this matter)?  Sure, you could take both, but you don't get many class proficiencies and you're probably better off taking things that still work when you're in your usual fighting style.  There was a concept in some weird corners of the 3.x sphere of the "toolbox fighter", who spent his feats on things that were useful across a wide range of weapons, carried around a whole armory, and switched to whatever weapon was good at the moment.  In general, Weapon Specialization was so strong that they didn't see much play (but they were more viable in Trailblazer).  The question: how strong is a "toolbox fighter" in ACKS?

How does everyone's favorite front-liner at 1st level, the armored war dog, compare to actual fighters?

How much do ability scores matter to fighting style choice?  It's worth noting that a 16 Str provides to-hit and damage benefits comparable to both two-weapon fighting and two-handed fighting at the same time.  Is the dominant strategy different for low-Str henchman fighters?  Is the dominant strategy different at Str 18?  What about fighters with Dex bonuses?  Is the dominant fighting style strategy different with Berserkergang's to-hit or Fighting Fury's damage bonuses (interesting because most of the time, to-hit and damage bonuses go together, like magic weapons, Str bonus, bless, and bardsong)?  What about for clerics without fighter damage bonus?

Actually working out expected kills per round instead of probability of a kill - it's straightforward, but I'm way over my weekly tolerance for pasting tables into blogger.  Also, adding in initiative - I think this should be straightforward with the method I've been using.  Roll initiative for both sides, roll attacks and damage for both sides, and then check both init and HP totals.  Multi-round analysis could be tricky, but worth pursuing.

Koewn said...

No big surprises, then. I'll have to chew on this a bit.

Some other questions, perhaps:

How do thieves and assassins look without backstab applicability (much closer to the explorer I reckon)? How is their performance changed by 'gating' the ability to backstab behind a successful Acrobatics throw, so we're not measuring just the surprise round? (that may force a limit on Assassin equipment, due to the 6 stone limit)

Thieves are listed as 1d8 damage, but described as two-weapon-fighting - that'd be 1d6, no? Typo?

John said...

The thief stats were worked with a one-handed sword used in both hands, hence d8 damage. The good THAC0 is due to a combination of the +4 from backstab and the high dex with Weapon Finesse (granted, a loose interpretation of Weapon Finesse, applying the bonus to a one-handed weapon used in both hands). Without backstab, thief is weaker than explorer in damage at low levels (no fighter damage bonus) but comparable at high levels (magic weapon damage bonus) with weaker THAC0 across the range, but assassin is a little weaker than the two-hander fighter (-1 to-hit and damage from lower Str, -1 damage from lack of Fighting Style; probably competitive with the other fighter types for killing power). I'll look into an analysis with Acrobatics; shouldn't be any trouble with gear, here I was just assuming the assassin was at 3 stone with leather and a two-handed sword. I expect that even limited by Acrobatics, the assassin will be deadlier than the fighters after 1st level. Part of my reason for structuring the analysis of backstabbers as I did was advertising to my players - look how deadly these guys can be if you would bother to use them!