Thursday, January 31, 2013

Brawn vs. grace

**WARNING: PEDANTIC ANALYSIS OF NARROW RULES CATEGORY AHEAD**

One of the peculiarities of old versions of D&D (I can't speak at all of the new ones, as I'm barely familiar with them) and similar games is the very high relevance of the Strength ability score to melee combat.  In fact, as I'll demonstrate shortly, a high Strength score is much better than an average one, and even superior to an equally high Dexterity in melee combat.

Fantasy fiction thrives on tropes like the immensely strong warrior who defeats his opponents by dealing mighty crushing blows.  But there is also the equally popular archetype of the quick and wily fighter, the swashbuckling swordsman or Robin Hood-type who, physically fit though he may be, is never depicted lifting wagons in a military press or tearing ironbound doors from their hinges with his bare hands.  Instead he overwhelms his foes with quickness, grace, nimble footwork, and skillful swordplay.  Quite often in fiction, in fact, this type of hero is portrayed as the superior combatant, though he may be painted initially as the underdog for dramatic purposes.

In D&D, it's the musclebound basher who is the odds-on favorite.  This stems from a simple quirk of the rules:  Strength adjustments apply both to the attack roll and the damage caused.  Dexterity applies only to AC.  It affects the opponent's attack rolls negatively, but does nothing else, either offensive or defensive.

First, let's take the case of a strong fighter vs. one of equal experience but only ordinary brawn.  We'll give them both AC5 mail and standard 1d8 swords.  The first one has an outstanding, but not superhuman, Strength of 16, for a +2 bonus to hit and damage, while his opponent has a perfectly respectable score of 12 which grants him no bonuses.

The base number needed to hit for either man is 14, but Brawny Bob's great Strength drops that to a 12.  He hits Average Joe 9 times in 20, or 45%.  Joe hits him 7 times in 20, or 35%.  When Bob hits Joe, he does an average of 6.5 points of damage - that's the 4.5 average of a 1d8 roll plus his bonus of 2 points.  Joe does only the sword's base 1d8 damage, averaging 4.5 points per successful attack.  Joe's average damage output per combat round overall, factoring in both hits and misses, is 35% times 4.5 points, or 1.575 points per round.  Brawny Bob, meanwhile, is dishing out 45% times 6.5 points, or 2.925 per round.  That's just a bit less than double what Joe can do.

Here's the spread for damage per round for each level of Strength, to complete the picture:
(Assume target is AC5, and attacker is wielding a standard 1d8-damage sword and using the 1st level line of the attack roll tables, with THAC0=19.  Remember also, penalties cannot adjust damage below 1 point; thus the wonky averages per hit for low Strength scores.  All results are rounded to the nearest thousandth.)

Str 3        -3 penalty   Hits 4 in 20 (20%)    Avg. damage per hit 2.25     Avg. per round 0.450
Str 4-5     -2 penalty   Hits 5 in 20 (25%)    Avg. damage per hit 2.875   Avg. per round 0.719
Str 6-8     -1 penalty   Hits 6 in 20 (30%)    Avg. damage per hit 3.625   Avg. per round 1.089
Str 9-12          0          Hits 7 in 20 (35%)    Avg. damage per hit 4.5       Avg. per round 1.575
Str 13-15  +1 bonus   Hits 8 in 20 (40%)    Avg. damage per hit 5.5       Avg. per round 2.200
Str 16-17  +2 bonus   Hits 9 in 20 (45%)    Avg. damage per hit 6.5       Avg. per round 2.925
Str 18       +3 bonus   Hits 10 in 20 (50%)  Avg. damage per hit 7.5       Avg. per round 3.750

As is clearly demonstrated here, an 18 Strength is not just a nice perk for a fighter; it's a monstrous advantage.  The guy with the 18 is dishing out, on average, 238% of what a person of ordinary might can do.  He's dealing out 170% of the punishment that a 15-Strength fighter - clearly no weakling himself - can manage.

Of course, there's sound reasoning behind both applications of the Strength adjustment; my question is whether the reason for applying them both at once is as sound.  The logic behind the damage adjustment is pretty obvious - the more muscle power you put behind your swing, the harder it hits, and the more it hurts.  The reasoning for the attack bonus isn't hard to grasp either - more force helps to penetrate armor.  The problem is that, applying the bonus both ways actually compounds it, and the result is essentially just a bigger bonus to damage per round.

Without a damage bonus, each bonus to the attack roll equals an additional 0.225 points of damage per round, on average - just under a quarter of a point.  This rate is constant within the limits of the d20 attack roll, assuming a minimum chance to hit of 1 in 20 and a maximum of 19 in 20; as long as a 1d8 damage weapon is used, each +1 to hit works out to 0.225 points of damage per round on average.  If only a natural 20 hits, a fighter with Str 10 and a sword averages 0.225 points of damage per round, and one who only misses on a natural 1 averages 4.275 points per round. 

Without a bonus to hit, each +1 bonus to the damage roll increases average damage per round by 0.35 points, or a little more than a third of a point (for a 1st level fighter attacking AC5.)  Moreover, the greater the chance of scoring a hit, whether due to a better base THAC0 of the attacker or a poorer AC of the defender, the more the average damage bonus per round per point of Strength bonus increases.  Whatever the attacker's chance to hit is, expressed as a decimal, that's the increase to its average damage output per round per point of Strength bonus.  A fighter who hits 50% of the time adds 0.5 points average per round with a +1 bonus, a full point per round for a +2 Str bonus, and 1.5 points per round for a +3 Str bonus.

That's a lot of math, and I wouldn't blame you if you skimmed or skipped it.  The upshot of all this is that, while there are some significant differences in how they play out mathematically, both attack roll bonuses and direct damage bonuses increase the characters potential for damage per round.  This means that the Strength bonus applied both ways is a double-dip advantage. 

Now, just for fun, let's imagine an ultimate championship fight between two 3rd level fighters: Mongo the Mauler, a muscular bruiser of 18 Strength, and Nimble Norman, fencing master with a Dexterity of 18.  Once again, we'll assume that both are clad in AC 5 mail.  Norman's adjusted AC is 2, his damage per hit is 4.5 points with a normal sword, and he needs a roll of 14 or better to hit Mongo.  On average, he dishes out 1.575 points of damage per round.  Mongo also needs a roll of 14 to hit Norman, because his +3 bonus from Strength completely cancels out Norman's -3 AC bonus from his amazing Dexterity.  However, his mighty blows deliver 7.5 points of damage per hit, or an average of 2.625 per round.   Mongo is clearly the favorite in this fight, dishing out the punishment at approximately 167% of the rate at which Norman can give it back to him.


Of course, combat is just about the swingiest (no pun intended, though in hindsight perhaps it should have been) part of D&D, and so those averages are averages of a very wide range of possible outcomes. Mongo's big theoretical advantage in average damage per round could very easily not pan out for him if only a few attack rolls don't go his way.

I'm curious, though, how removing the attack roll bonus from Strength, and retaining only the damage bonus, would affect things.  Revisiting Mongo and Norman's match-up, Norman's average damage per round doesn't change.  He still averages 1.575 points of damage per round.  However, without Mongo's Strength attack bonus canceling out Norman's enhanced skills of evasion, Mongo needs a 17 to hit, and his average damage per round drops to 1.5.  Advantage, Norman!  Not by much, mind you, but it's a significant turnaround from the large advantage Mongo enjoyed when he got to apply his Strength bonus twice. 

Despite the fairly even damage per round averages, there are still significant differences.  Norman will hit more often, so on any given round he's more likely to inflict some damage.  Mongo misses more, but when he does hit, he makes it count in a bigger way.  Since attack rolls are far more swingy than damage rolls, Mongo stands to gain or lose more from the luck of the dice.  He could put a quick end to the ruckus with a couple fortuitous attack rolls coupled with his heavily augmented damage dice, or he could have a frustrating time as Norman methodically nickel-and-dimes him to death in a drawn-out battle.  As it turns out, this is actually a pretty good representation of what I'd expect a fight between a masher and a speedster to look like.

Is it worth changing a long-standing rule of D&D for what might amount to a minor impact in the game?  Are these ruminations anything more than rank pedantry?  I really don't know.  It was just on my mind, so I decided to crunch some numbers.  Make of them whatever you will.