Friday, December 22, 2017


In the ACKS repo, I'm working on a branch to change the way downtime works.  One of my issues with ACKS is that it doesn't really support open-table play well outside of low levels.  Domain XP slows down (or stops) level convergence within the party, and characters with domains tend to monopolize out-of-adventuring time and DM attention.  My intentions with these changes include:

  • Give everyone across the level range useful things to do during downtime.
  • Reduce overhead / level of detail / granularity in tracking downtime (two-week "downtime turns")
  • Create a structure / cycle of play around downtime, such that it happens in regular, predictable chunks alternating with adventures rather than "OK, Mr. Mage needs three months to make his item and we dare not go adventuring without him, so we're all gonna sit on our butts for three months."
    • As a side effect, reduce the variance of magic research by cutting it into multiple downtime actions and allowing partial progress
      • TODO: standardize monetary outlay between spell research and magic item creation
  • Add some mechanics which accelerate the convergence of low-level characters to the party mean (mentorship bonus)
  • Throw some bones to players who miss sessions (extra downtime actions)
  • Mix up the general proficiencies metagame.  Currently, a handful of general proficiencies (Healing, Alchemy, Bargaining, and Diplomacy/Intimidation initially, then Navigation and Riding in wilderness, then Military Strategy and Leadership when mass combat starts happening) typically dominate discourse and use in play in my groups.  Unsurprisingly, these are the general proficiencies that actually do useful things.  The trick, then, is adding really utile uses to some of the job-type general proficiencies.  So far, this has mostly been by increasing effective market class for goods related to the proficiency practiced, or by adding ability to find henchmen of certain classes (and possibly outside the typical henchman recruiting level range).
    • Relatedly, reduce the pain of operating in small markets and deprecate/distribute the Venturer / Varangian's core ability.
    • TODO Art - ???
    • TODO Labor, especially Mining - bonus to detect traps underground (probably only +2, smaller than dwarf bonus or Alertness), high-risk-high-reward downtime activity, ability to recruit dwarf henchmen.
    • TODO Profession, especially Lawyer - bonus to rolls on sentencing table
  • TODO: make sufficient quantities of carousing a downtime action, make the carousing table canon (that table was a regular source of fun in the first campaign)
  • TODO: rework mercenary hiring, rather like this.  Mercenary wages, henchman wages, and cost of living need some thought in open-table mode.
  • TODO: integration with Apocalypse World-style clock system, for easy running of living worlds.  I don't need to know a whole lot of details, just that this disgruntled NPC is going to try to come after the PCs in N sessions.  I'll fill in the rest when things get closer.
  • TODO: fix / redesign hijinks to provide adventure hooks and intel on running clocks.  Spying actually makes sense when people worth spying on are doing secret things in the background.
  • TODO: simplify domains.  Starting for now by taking this and working it into the Campaigns chapter, and by removing domain XP (mass combat XP can stay - that's combat.  XP from sacking domains can stay - that's treasure earned adventuring).  I have some vague notions for some Lords of Waterdeep-style mechanics, where domains and guilds provide you with anonymous, disposable "adventurer" resources that you can send on quests to delay clocks, but I'm not sure how to make it mesh with ACKS-as-it-is, nor how I really want it to look at the end of the day, so I'm going to put that one off.

Sunday, December 17, 2017

ACKS SRD forked on github

Once upon a time, I wrote a post about all the things I'd want to change in Traveller if I were to run it again.  The other day (...  or month), I started to draft a similar post for ACKS, in part in response to the ACKS 2e proposal that was floated on the Patreon.  In general, I dislike the direction ACKS seems to be headed; as I said a year and a half ago - "ACKS' continued development seems to be away from its slick, usably-abstracted B/X roots and off into what the forums jokingly call Advanced Adventurer Conqueror King, with more detail and more rules.".  The 2e draft reinforces that perception.

I considered digging through the pile of retroclones to find something that better suits my desires, but I find my patience for reading rulebooks has fallen over time, particularly with all the duplication that you tend to see in retroclones.  I found myself wishing that retroclones were just published as changelogs, as patchnotes, or in proper version control systems.

And then I remembered caphiend's ACKS SRD on github, and went "what the hell, somebody has to start this."  So here we are - a version of the ACKS SRD with a growing selection of our group's conventions, houserules (or proposed houserules), and clarifications integrated into it.

Thursday, December 14, 2017

Viking Midnight Redux - After Ragnarok

I was reflecting on one of the better campaigns I've run in the past couple of years, the Bjornaborg "Midnight but with vikings" game.  I think if I were to do it again, I'd go further from the Midnight source.  I've been reading the Eddas lately, and there're some interesting bits in the Voluspa and Vafthruthnismol about what happens after Ragnarok - Thor's sons and Odin's brothers survive and rebuild the realm of the gods, and the dragon Nidhogg rises against them, but the poem is cut short before that matter is concluded.

But it got me thinking - there's room after Ragnarok.  Taking Norse mythology as a loose base and playing Ragnarok as even less of a victory for the Aesir, you end up in more typical OSR post-apocalypse territory.  The giants set themselves up as petty kings over men, the dark elves of Niflheim gather slaves, the scaled spawn of Jormungandr and Nidhogg exact tribute, and the dead of Hel feast on the living.  Where men gather, they are preyed upon.  The old gods are dead, but in the hills their children hide among mortals, gathering worshipers and strength, prophecies and artifacts in preparation for a second day of reckoning.  And unlike in Midnight proper, here such a thing is plausible, and perhaps foretold.

Upsides over Midnight: hope, no wizard-hunters, no orcs.
Downsides against Midnight: what do you fight at low levels if not orcs?, subtler pitch than "Tolkien but Sauron won".

Friday, December 8, 2017

The Pareto Principle and the OSR

I've been spending a lot of time on wikipedia recently.  The other day I was reading about power laws, allometry, and the Pareto Principle, and it struck me as both a useful rule of thumb for the lazy simulationist and a convenient explanatory factor for some pieces of OSR weirdness.  Phrased simply, the Pareto Principle states that in general, 20% of causes account for / lead to 80% of effects.  Typical examples include aggregation of wealth / property (where in most countries, 20% of the population controls 80% of the land / wealth) and software bugs (where 80% of bugs spring from 20% of root causes, such that if you fix the most-common 20% of bugs, you also resolve roughly the next 60% incidentally).

One OSR peculiarity made believable by this observation is the lair / nonlair distinction.  Roughly 20% of encounters (lairs) have the vast majority of the treasure (certainly treasure-from-monsters) in any given dungeon level.  From the perspective of even, predictable progression this is a dubious choice, but from a simulation perspective, with monsters aggregating treasure or paying tribute off-screen, it seems solidly justifiable.

The Pareto Principle may also have applications in constructing believable causal / narrative structures from randomness.  You roll a result on a random table, and you're not sure why it's true.  Pareto suggests that there's an 80% chance it follows from one of the Big Causes of your world.  Roll a d10, on 3+ link it back to something already known, on 1-2 add a new minor causal agent unfamiliar to your players.  You could go even further with self-similarity on that 80% and balancing the sizes of your sets of big and small causes, but for a first cut that minimizes effort, tends to conserve campaign capital, and introduces occasional surprises, this is probably a reasonable solution.

If I were feeling extremely motivated, it might be interesting to look at Pareto as it applies to ACKS' economics / XP model.  But I am not; they're probably already reasonable consistent, given Autarch's economics background.

Other plausible correlates in gaming, absolutely unsupported by data but plausible to my eye - 20% of the bloggers get 80% of the pageviews, 20% of the systems get 80% of the playtime, 20% of the campaigns get 80% of the sessions, 20% of the gamers run 80% of the games.

Sunday, July 30, 2017

Further Obvious Thoughts on Wilderness

My prep for Dungeons and Discourse continues.  Also, I had lunch with one of my ACKS stalwarts a week or two ago, and have been re-reading some old posts (particularly this one) and some thoughts came up.

Topology - Jayquaying is considered important in dungeons, but not something you hear about much in the wilderness.  This is because the wilderness is "open" by default - you don't have to take any particular measures to provide multiple paths through it.  On the other hand, this abundance of paths can also be boring.  An inverse of jayquaying for the wilderness to restrict available paths probably makes sense.  Quayjaying, if you will.  Hard quayjaying is limiting available paths by impassable obstacles, soft quayjaying is limiting it by obstacles which are expensive to traverse.

Traps - Thinking about the resource game in dungeons, it's true that zap traps and trivial encounters aren't interactive or particularly exciting, but they perform an important function - putting chip damage on the party to keep them under resource pressure.  Seems workable in wilderness too - as simple as "if you sleep in the mountains above the treeline, all party-members take 1d3 points of damage from exposure unless you can find a cave", "every time you traverse a glacier hex, there's a 1 in x chance a crevasse opens beneath a random 1d4 party members, save vs paralysis or take 3d6 falling damage", or "if you sleep in a swamp, there's a 1 in n chance that your rations spoil".  In addition to grinding down party resources and providing tension, this helps make Land Surveying actually useful.

Specials - If you look at the dungeon-stocking guidelines, 25% of rooms have a "unique" or special contents.  Like traps, this is something I've largely neglected in my wildernesses to date, despite having quite a few ideas for this sort of thing.  25% may be high for a sensible wilderness, particularly a large one - 25% means that every hex has like 1.5 "unique" neighbors, which is very dense.  14-16% would mean that on average every hex would have a "unique" neighbor, which is still high but probably more reasonable.

Treasure - One persistent complaint about wilderness adventuring is "there's not enough treasure per game-time compared to dungeoneering."  Fair enough.  One way to boost this somewhat is unguarded treasure.  30% of trapped rooms have treasure in a dungeon; a reasonable parallel would be for a "trap" hex like crevassing glacier to have treasure at the bottom of a crevasse.  Likewise, 15% of "empty" dungeon rooms have treasure; no reason empty wilderness shouldn't, either.  Maybe it's a natural resource like rare wood or exposed metal ore; maybe it's actual treasure buried in a barrow mound that isn't haunted for once.  I expect you'd have to search a hex to find it, but that's OK - creates a resource tradeoff, time and rations for a chance of treasure.  This also provides endpoints for treasure maps.

Dungeons - Sometimes I think about abandoning the quest to figure out the wilderness game and just run dungeons all the way up.  Generally I don't like using dungeons at high levels because giants need a lot of calories and it stops making any sort of sense, but supernatural monsters are a reasonable solution to this objection (and wards can also explain why they're not out terrorizing the countryside).  Matt noted that dungeons do get old, and that he thinks this is not a great solution.  Reflecting further on this, the correct solution is fairly obvious - I need to put dungeons in my wilderness.  This is, of course, exactly what the damn manual suggests, but I've been too lazy for it in the past, in large part because the wilderness prep effort has been heavy as a result of large wildernesses with very dense lairs.  But, having figured those out, prep effort declines.  Dungeons provide high-treasure targets out in the wilderness, as well as a break from worrying about rations and overland movement and such.  Looking at page 235 of ACKS core, it suggests 3 large dungeons (6-10 sessions each), 10 small dungeons (1-2 sessions each), and 17 lairs for a 30x40-hex wilderness.  For a 10x10 wilderness, that's about a 25% chance of a large dungeon, an 84% chance of a small dungeon, and 1.5 lairs; lower density than I really want.  Given that historically I've gotten 6-8 sessions out of most of my 3-level, 60ish-room dungeons, I assume that what I'm building is on the low end of "large".  I could probably get 1-2 sessions out of a single-level 20-30 room dungeon, and could comfortably put three of those in a 10x10 microsandbox in about six hours of prep effort.  Then you drop rumors and these become places that players have to go find (as Alexis says - if you tell the players precisely where something is, they will go directly there.  If you tell them vaguely where it is, they will go everywhere.  And I want them to go everywhere if at all possible, because that's efficient use of my prep effort).

Homesteads - I was skimming Renegade Crowns the other day, and its system generates quite a few homesteads out in the wilderness.  Think Beorn and Craster.  These present a great, ambiguous opportunity.  If your players meet orcs, the first instinct is to fight them.  But if they meet hillfolk, it's hard to tell whether they're normal and friendly or cannibal cultists until you agree to join them for dinner.  If it works out, they're potentially great allies against beastmen, and at the very least a "safe and sanitary" place to rest and recover resources.  Homesteads might fall under a more general category of "oasis" features where replenishment of certain resources is possible; Rivendell's another example.

Level range - An observation I mentioned to Matt is that just as 1st level, 2nd level, and post-3rd level dungeoncrawling are all very different, it's possible that 5th, 6th, and 7th+ level wilderness adventuring are very different, with lower levels being much more resource-constrained and dangerous.  Got me wondering if maybe launching a wilderness game closer to 7th wouldn't be a bad idea.

On further reflection, I think that even if I executed successfully on all of these ideas, in an unmapped borderlands microsandbox with limited lair density, some route-blocking topology, "trap" hexes, unguarded treasure, an abundance of special features, three dungeons, and a smattering of homesteads, that would still be somewhat unsatisfactory.  Better than my previous wilderness efforts, and pretty close to my first ACKS dungeon, but still merely an open world rather than a living world.  The folks I've been talking to here seem big into Dungeon World and fronts and clocks lately, but I think something more up my alley would be "build a big list of NPCs who might do something that the players might care about, along with some Renegade Crowns-style internal conflicts between NPCs.  Every session, roll on the list, think about what resources they have at their disposal and what they want, and then something happens."  Might be NPC-vs-NPC violence in the background, might be something aimed at the PCs if they're on that NPC's shit-list, might be a job offer from that NPC.  But something happens, because somebody wanted something in the world and they've finally gotten around to acting on it.

Wednesday, July 12, 2017

Player-Controlled XP Allocation

A friend of mine asked me to give a talk on DMing, so I've been writing and rewriting endless cycles of notes on that.  One theme that I know I want to hit is putting players mostly-in-control of the difficulty of the combat that they face, by giving them free rein to scout the dungeon and avoid dangerous monsters.  Another theme is that really strong play often requires submission of the individual party member to the interests of the party - the wizard has to hold on to his one spell carefully rather than using it early for personal satisfaction, the thief has to expose himself to substantial personal risk in order to find traps so that they don't kill anyone (or everyone) else, and fighters are sometimes called upon to die for the party so that everyone else can live.  Clerics, of course, are clerics, and have always operated been expected to operate "in service" to the party as a whole.

But when you take these two ideas, players collectively in control of their destiny and subordination of the individual to the party, and combine them with XP for treasure, a funny possibility arises.

Under a certain reading of the rules, you earn 1 XP per GP that you receive from the adventure.  If the party as a whole determines (by vote, say) that some player failed to contribute and to pull their weight, by (say) fleeing from combat, the party as a whole need not allocate any GP from the treasure to that player, and hence he will earn little XP - just the monster XP portion, which is around 20% of XP on most adventures.  A persistent problem player who regularly earns 20% XP will eventually end up about 2 levels behind the rest of the party.  A player allocated only half a share of gold will tend to end up one level behind.  This is, obviously, not a form of censure that should be used lightly, but it is one that should be considered.  I suspect that, as with PC death, this is not something that needs to happen often for it to modify player behavior substantially.  The knowledge that it is a possibility encourages cooperation.

The reverse also applies - if you have a new player join a group and he's playing a class that you really need, the party can allocate greater than a share to him.  If someone's fighter died in a heroic rearguard action to cover the party's retreat and now they're playing a henchman and a level behind, you can allocate them greater than a share until they come up to level parity.  The trivial, common case is that a PC is 10 XP from leveling with standard shares, and a tiny reallocation might push them across a threshold.

Unfortunately, henchmen complicate this XP allocation scheme, for one because they receive an odd share size, and for two because traditionally they receive their shares straight from the party pool, but if a player receives an odd-sized share, do his henchmen too?  In the case of a problem player, it seems reasonable to dock his henchmen as well; in the case of a hero player, it seems reasonable to award his henchmen extra as well.  But this all complicates the accounting, or raises again the specter of "the pay for your henchmen should come out of your share of the treasure", which I suspect most sensible parties will reject on the grounds that a player with many henchmen is of great value to the party as a whole.

At the end of the day, one of the interesting facets of this game not present in more modern editions is that it is in many ways an experiment in small-scale self-governance, civics, and organizational behavior.  And yes, the gaming community seems to have settled on very regular norms regarding the allocation of treasure and XP, but one of the joys of transgressive/retrogressive gaming, allowing things like PvP and uneven allocation of treasure, is an opportunity to reinvent those norms (with good stories and object lessons about why we have those norms), or to arrive at new and strange norms.  And isn't that what the OSR is all about?  Rolling back the clock and seeing the other possibilities that could have been, not merely in the rules of the game, but in the rules of the group?

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.