Since of course there’s nothing more important to write about, such as states full of people who’d refuse to nominate a black HARVARD LAW GRADUATE for President of the United States just because he’s black (and therefore inferior to all the white people who’ve GRADUATED FROM HARVARD LAW SCHOOL magna cum laude–oh, well there ya go, a white person would’a graduated suma…)
Anyway, since there’s nothing like that to blog about this week, I’ll move on to how to weigh critical hit properties of weapons in d20. Is a broader threat range better or a higher multiplier? I’ll denote critical hit properties in d20 like so:
{w|×n}
where w is the width of the threat range, such as 18-20, 19-20, or just 20, and n is the threat multiplier. The threat chance is the width of the threat range divide by 20, or the chance to hit–whichever is smaller. Once you roll a critical threat, you roll d20 again to “confirm” the critical, with the goal to simply roll a hit this time. If you fail this roll, you hit the target, but don’t get the critical multiplier to your damage. Therefore, the expected damage on an attack looks like:
<D> = ∑ P(D) × D = d×P(regular hit) + nd×P(crit) = d[P(hit) - P(threat) + P(threat)P(miss) + nP(threat)P(hit)]
= d×P(hit)[ 1 + (n - 1) P(threat) ]
where d is the expected value of the damage roll, d = (min damage + max damage)/2, and P(threat) = w/20 (or hit probability if it’s lower) is the probability of rolling a critical threat. Note that for typical threat ranges (19-20 for sword, 20 for axe) and multipliers (2 for sword, 3 for axe), you get about 10% of your damage from critical hits; while with some feats, keen weapons, weapon master perks, it might get up to 20% or more (unless you fight a lot of undead, etc. with critical hit immunity).
So, we have
(1/d×P(hit))<D+> = w(n-1)/20
(1/d×P(hit))Δ<D> = [w/20] × Δn
(1/d×P(hit))Δ<D> = [(n-1)/20] × Δw
with <D+> the extra damage that comes from critical hits, which depends on 1 less than n because you were going to do some damage anyway. When weighing critical hit properties, you are mostly concerned about the product (width of threat range)×(crit multiplier – 1). A sword with {19-20|×2} is worth 2×1=2, an axe with {20|3} is worth 1×2=2, and a falchion with {18-20|×2} is the same as a scythe with {20|×4}, both equal 1.5. Keen weapon or improved critical both double the threat range, which doubles the value of critical hits. Adding one the the critical damage multiplier doubles the value of a sword, but only increases the value of a scythe by 1/3.
The exception is if you expect to be fighting enemies with ACs so high you’ll only hit them on rolls of 20, or if you are acquiring feats or magic weapons that give extra damage on critical hits. Extra critical threat range is just wasted in that situation. Also, since you’ll rarely hit such a beast, you’ll want to get the most out of every hit. Therefore, the scythe, if available, is the way to go…except that really, you’ll only be hitting one in every 20 swings, and critical hitting one in every 400 swings, so maybe you should just run in that situation. Except that, of course, advancing the plot probably depends on winning the fight, and the designers have probably locked you in a room with him. /sigh
Anyway, there are some annoying discrepancies in my notation, which I’ll straighten out in a future edit.
