Attrition in Blow-by-Blow Combat, Part II

This post continues my analysis of blow-versus-blow combat, begun in a previous post. The question at hand is how do damage bonuses and to-hit bonuses affect how characters fight many small monsters.

To be honest, the topic is pretty dry and the math gets more tedious, so I am going to try to wrap the topic up quickly.

This post: the art of fighting gnome hordes!

Hordes of Little Monsters

For smaller monsters kills per round (KPR) reflects combat effectiveness more accurately than damage per round. 

Kills per round is the product of kills per hit (KPH) and hits per round, which is just the probability of hitting (HITP) for characters with one attack.

Let's define smaller monsters as monsters who are with very high probability reduced to 0 HP by 1 or 2 hits. KPH depends on the probability that the first hit kills the monster outright (KILLP). 

Hits per kill (HPK) is

HPK = 1 * KILLP + 2 * (1 - KILLP) = 2 - KILLP

It ranges from 2 if KILLP=0 to 1 if KILLP=1. Inverting it gives KPH,

KPH = 1 / (2 - KILLP).

Putting these together,

KPR = KPH * HITP  = HITP / (2 - KILLP).

One can derive algebraic expressions for these in terms of more the fundamental values of attack bonus, AC, monsters' hit dice, damage dice, and damage modifier, but it is probably easier to get some insight into how an increase in attack bonus (AB) versus damage bonus (DB) affect KPH with a couple of examples.

First, unlike with big monsters where the effect of DB was linear -- each increase in DB increased damage per round by the same amount -- with small monster groups DB initially is somewhat important but after enough increases a further +1 DB does not affect KPR. For instance, consider a warrior with base d8 damage fighting orcs with d8 HP. Once her damage is d8+7, a further increase in DB does not help her KPR.

How important is DB initially? Pushing the example further, if orcs are AC 6 and the warrior has AB +3, then her HITP=9/20. Increasing AB by one brings her HITP to 10/20, or an 11% increase in HITP. Because HITP is one of the terms in KPR, this also increases KPR by 11%.

If the warrior has d8 damage which would increase by d8+1, KILLP goes from 36/64 to 43/64, so KPH goes from 64/92 to 64/85, or an 8% increase in KPH. This is also an 8% increase in KPR.

Going from d8 to d8+1 damage affects KILLP more than the move from d8+1 to d8+2. The figure below depicts this graphically. On the x-axis is every possible damage roll, and on the y-axis every possible roll for orc hp. There are 64 possible combinations, 36 of which (the green) defeat the orc if there is no damage bonus, and 43 of which (the green and the red) defeat the orc if there is a +1 damage bonus. Moving to d8+2 increases KILLP to 49/64, and so on.

Hence at least in this example, to-hit bonuses appear to be superior to damage bonuses. For most typical combinations of attack bonus and damage bonus, it is likely to be the same.

Danger, a Further Complication

A final consideration is that low AC monsters tend to be more powerful and are more likely to hit and do higher damage. With low monster AC, to-hit bonuses become relatively more important. Thus, maybe the conventional wisdom was correct all along that to-hit bonuses are superior to damage bonuses in most situations.

Summary: to-hit bonuses usually increase combat effectiveness more than damage bonuses. The one exception is large monsters with high AC (i.e., easy to hit monsters).