Better attacks, more damage, or multiple attacks?

 When it comes to doing better in combat, there are a few ways a character or monster's battle effectiveness can be scaled up. For purposes of this post, I'm talking about the character or creature's actual inherent combat effectiveness and improvements in combat ability from gaining levels of experience, not bonuses from magical weapons and the like.

Better attack rolls: The character "hits" more often. Each 1 point of attack bonus or decrease in THAC0 results in 5% greater average damage per round. The average damage per successful attack and the maximum possible damage in a round are unchanged. This is how it works for all characters in B/X D&D.

More damage: Damage per successful attack is augmented, either by adding a flat number or by increasing the size or number of dice rolled for each attack. The chance to succeed at an attack is not affected. This method is used in BECMI's Weapon Mastery rules and to a lesser degree in AD&D's weapon specialization.

More attack rolls: A character is allowed to make additional attack rolls. The chance of any single attack being successful are unchanged, as is the damage rolled per successful attack. AD&D uses multiple attacks for fighter types, as does BECMI at high levels (though the implementation is a bit wonky, such as having it activate only when an attack roll of 2 is enough to beat the target's AC,) and virtually every edition for various monsters.

Let's take a hypothetical B/X fighter, level 1, wielding a weapon that does base damage of 1d6. To hit a target of AC5 (supposing no Strength or magic bonuses) he normally needs to roll a 14; that's a 35% of success. On any successful attack, he rolls a d6 for damage. That makes his expected average damage per round 3.5 (the average result of a d6) times 35%, or 1.225 points per round. Of course, 65% of the time, he'll do no damage at all, and he'll never do more than 6 points of damage in any given round. 

 Now, let's suppose his attack rolls are improved by 2 points. This means his attacks will succeed 45% of the time. His average damage per round is now 1.575 points. He still does 1d6 damage per successful attack, and is capped at a maximum of 6 points, but he's rolling that d6 a little more often. The increased average per round comes from fewer zeros dragging it down.

Increasing the size of damage dice has an inverse effect that ultimately averages out the same. How about if instead of improving our fighter's odds of successfully attacking, we increase his damage potential with his weapon: instead of 1d6, our fighter's skills allow him to roll 1d8. With his original odds of success (35%) he deals damage at a clip of 4.5 (average roll of 1d8) times 35%, or 1.575. His average damage per round is exactly the same as if his attack rolls were improved instead. His attack succeeds less often, but his average damage per successful attack is slightly higher, and he's able to do up to 8 points of damage in a round if he rolls well. (Interestingly, within the usual bounds of D&D combat, a bonus to attack rolls and an increase in die size by the same number of pips will produce the same change in average damage per round. This only breaks down when the numbers push the attack roll past the point where it only fails on a natural 1.) 

If we increase either his attack rolls by +6 (equivalent to a 10th level fighter in B/X) or his damage die by six pips, to d12, his average damage per round is 2.275. The difference is that the first method hits more often for less damage per attack, while the second hits less often but with a chance to do up to twice as much damage in a single round. 

Multiple attacks are mathematically more complex. Let's revert our fighter to 1d6 damage and 35% chance of successfully attacking, but give him a second attack at those same odds. Below are the chances that:

Both attacks succeed: 12.25%

Both attacks fail: 42.25%

One attack succeeds, one fails: 45.5%

Overall, that equates to average damage per round of 2.45, with a maximum damage of 12 points. That's roughly equivalent to the hypothetical character above who uses a d12 for damage, except this fighter is going to do at least some damage a lot more often, with a 57.75% chance that at least one attack roll succeeds vs. the other's 35%. Compared to the fighter who gets +6 to attack, this fighter succeeds in his attacks less often (57.75% vs. 65%) but has a chance of doing twice as much damage. 

Above is a graph of these three methods showing the likelihood of damage results from 0 to 12 points. The most striking difference is in the 0 column, with a straight bonus to the attack roll resulting in the lowest number of outright failures to do damage. The model of d12 damage but no additional bonus to the attack roll fails a whopping 65% of the time, but allows up to twice the potential damage, so it's very "swingy;" much more of a "feast or famine" situation. Between those two extremes falls the two attack roll method, which provides a solid chance of doing at least some damage and a small chance of adding a second damage die. Statistically, it seems like a nice happy medium. It also averts the typical all-or-nothing combat results, where armor either blocks all damage from an attacker or none of it. Its primary drawback is that it's a huge jump in damage output (and B/X fighter level equivalent) from one attack roll to two. There's not much room for a steady graduation from raw 1st level fighter to mighty 10th level lord using that method alone. 

Of course, a combination of two or all three of these methods might prove the most satisfying. AD&D, as far as I know, has always used a combination of increasing attack rolls and multiple attacks for fighters. It definitely warrants some more mathematical analysis, not only of combination methods, but of the multiple attack roll method with respect to different ACs and possibly adding a third or fourth attack. In any case, make of it whatever you will.