Saturday, December 5, 2015

The Nonlinear Effectiveness of Armor Class

Or, why Fighting Style (Shield) and Plate is pretty great.

So you're playing ACKS and you find a ring of protection +1 (a rare and extremely valuable thing).  The wizard with AC0 wants it, because having any AC at all would be nice.  The fighter with AC10 (plate +1, shield +1, fighting style shield) also wants it, because he thinks AC11 is much better than AC10.  The wizard argues that this isn't true, the fighter has enough AC already, and he should stop hogging all the magic items.

(Then the ghouls attack while the party is arguing, but that is neither here nor there)

Who is right?

It turns out the fighter will actually gain (much) more survivability than the wizard will.  Here is why.

The more AC you have, the better each point of AC is, on average across all reasonable monsters (up to a point).  Let us consider four characters: wizard in no armor (AC0), clumsy thief in leather (AC2), barbarian in chain (AC4), and fighter in plate (AC6).  Let us also consider a couple of hypothetical monsters: an orc (THAC0 10+), an ogre (THAC0 6+), a stone giant (THAC0 3+), a woolly mammoth (THAC0 0+), and a great wyrm dragon (THAC0 -3+).  If each of these monsters attacks each of these PCs, what is the probability that they will hit?

AC 0 2 4 6

10 0.55 0.45 0.35 0.25
6 0.75 0.65 0.55 0.45
3 0.9 0.8 0.7 0.6
0 0.95 0.95 0.85 0.75
-3 0.95 0.95 0.95 0.9

Looks pretty straightforward - +2 points of AC reduces your probability of being hit by 10%, as we'd expect (up until the monsters only miss on a natural 1, which is what the 0.95s in the bottom left mean).

But what does having your probability of being hit reduced by 10% actually mean for your survival time when fighting this monster?  It can mean a lot more than a 10% increase in survival time!  Consider AC4 vs AC6 in the case of THAC0 10+.  AC4 gets hit about once every 3 attacks.  AC6 gets hit once every 4 attacks.  So assuming each PC can take the same number of hits, the guy in plate lives 33% longer on average than the guy in chain.

What happens if platefighter gets a shield and Fighting Style, bringing his AC up to 8?  Or fighting style and various levels of magic shield and plate for ACs 10, 12, and 14?

AC 8 10 12 14

10 0.15 0.05 0.05 0.05
6 0.35 0.25 0.15 0.05
3 0.5 0.4 0.3 0.2
0 0.65 0.55 0.45 0.35
-3 0.8 0.7 0.6 0.5

AC8 only gets hit by an orc once every 6-7 attacks.  He will live twice as long as the PCs in chain, and 1.5 times as long as if he were in plate.  At AC10 (shield +1, plate +1, and fighting style), he is hit by an orc only on a natural 20, once every 20 attacks, and will live five times as long as if he were in nonmagical plate with no shield (or about three times as long as if he were in nonmagical plate+shield+fighting style).

This is what I mean by the nonlinear effectiveness of Armor Class.  The more AC you have, the more each point increases your life expectancy in combat, up to the point where your opponent needs a natural 20 to hit.  Just that one point that pushes them from 19+ to 20+ to hit doubles your life expectancy.

On the flip size, the increase in AC from 0 to 6 only changes your probability of being hit by an orc from 0.55 to 0.25, slightly more than doubling your survival time (assuming the AC0 character has the same number of HP as the AC6 character - an AC0 wizard with d4 HD thus has about a quarter the combat life expectancy in melee of an AC6 fighter with d8 HD).  At higher THAC0s, the increase from AC0 to AC6 is even less effective - it means the great wyrm will hit on a 3+ instead of a 2+, which is right around a 5% increase in life expectancy.  At that point 6 stone of armor is a liability.  Go magic or don't bother.

By comparison, at high THAC0s, even small increases in AC continue to matter.  AC12 vs AC14 at THAC0 -3+ is 12 hits per 20 attacks vs 10 hits per 20 attacks is about a 20% increase in life expectancy.  Against a stone giant (THAC0 3+), AC 14 survives one and a half times as long as AC12, and twice as long as AC10.  Those last couple of points that push your AC up into the low teens are really, really good.

(I strongly suspect that there is some connection here with something like the harmonic progression or some other known sequence, but mathematical insight is failing to strike today)

Incidentally, this also explains why some of my past players felt vehemently that classes who can't use plate shouldn't be in melee, and that if you roll an 18 Dex and a 9 Str, you should use that statblock for a fighter (AC11 with shield and fighting style at 1st level and no magic?  Yes please) rather than a thief, explorer, or other dex-based class.  This is also, I suspect, why there are basically no spells that increase other peoples' AC in ACKS (except protection from evil, which has some drawbacks).  Shimmer (+2 AC for 3 turns) with range touch is probably absurdly good for keeping the monsters from breaking through your fighter-line.

To return to our AC10 fighter and AC0 wizard, the wizard gains, against opponents with THAC0 10+, only 10% survivability by increasing his AC by 1, and that falls as THAC0 increases to the point where it is irrelevant against opponents with THAC0 1+ or better.  The fighter, by comparison, doubles his survivability against opponents with THAC0 9+ by increasing his AC to 11, and even against great wyrms with THAC0 -3+, that +1 point of AC increases his expected survival time by about 8%.

Then the question becomes, "Is expected survival time in One Monster vs One PC combat a representative metric of utility in an adventuring context for item allocation purposes?"  I think it's pretty arguable - you can't do the party any good if you're unconscious in a pool of your own blood.  Likewise, if you live twice as long, you get twice as many chances to attack before dying, and your odds of successfully escaping from combat are much improved too.

This post roughly reflects the state of the dungeoneering fighter metagame in my group - shields are king.  I am curious under what circumstances other fighting styles make sense for classes who have access to plate and fighting style shield.  I think constructing an ACKS combat simulator and just running like a million trials is probably a better plan that doing math for that, though.

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