Sunday, March 12, 2017

Some Math on Removed Iterative Attacks

Recently, a Pathfinder DM I play with instituted the Removing Iterative Attacks rule from Pathfinder Unchained. A player immediately objected on the basis of a perceived negative effect on critical hits. (I don't have a dog in this fight because I'm playing a sorcerer who won't get iteratives until level 12 and won't use them even then.) Let us now analyze its actual objective effect on critical hits (and fumbles), using the power of math!

(I'm not super-great at probability these days, as high school algebra was a long time ago, so feel free to correct my math.)

I don't know if we're using the "half minimum damage if you miss by 5 or less" rule, it didn't come up, and seems an unnecessary complication, but it shouldn't really affect critical hits or fumbles. We definitely don't seem to be using the part of the rules regarding natural attacks.

We're using a fumble rule where a natural 1 is a critical fumble threat, which you need to confirm like any critical threat: if the confirmation roll would hit, it's only a miss; if the confirmation roll would miss, it's a critical fumble.

In this setting, where critical hits and critical fumbles are precisely mirrors of one another, our first and most obvious conclusion will be that any effect the Removed Iterative Attacks rule has on critical hits, it will have pretty much the same effect on critical fumbles. (Human psychology is such that we will tend to want to avoid risk, so anything that reduces both critical hits and critical fumbles should ultimately be considered more desirable than something that increases both. But that's not math, that's psychology.)

Now, let us consider the variant's actual critical hit rule:
When you threaten a critical hit, roll to confirm at your full bonus and apply the effects of the critical hit to any one of your hits. If your original attack roll scored multiple hits and the critical confirmation roll also falls within your weapon’s critical threat range, you score two critical hits and can apply them to any two hits.
 Jeez, that actually makes the math way complicated. This will be harder than I thought.

Okay, so: under normal rules, you can potentially score up to n critical hits, where n is the number of attacks you make. Under the variant rule, you can potentially score up to 2 critical hits, if you are making at least two attacks. It's starting to not look good for the variant rule, at least in situations where you have more than 2 attacks.


Okay, to make it as simple as possible, let's imagine a situation where you're getting two iterative attacks (so you're a 6th-level fighter or the equivalent) and your opponent's AC stacks up against your total tohit such that you hit (or confirm) on an 11-20 on the die, and your weapon is a 20/x2 crit range. (Increased critical multiplier, such as  a scythe's 20/x4, won't have much effect on the numbers, though it makes critical hits more desirable; increased critical threat range, such as a rapier's 18-20/x2, may have significant effect on the numbers and will be scrutinized second.)

Under the regular rules, you have a 5% chance -- 1/20 -- per die roll to threaten a critical hit. On two dice, therefore, you have a 9.75% chance -- 39/400 -- to threaten at least one critical hit, and 1/400 -- 0.25% -- of threatening two. But since you only confirm 50% (10/20) of the threats, that's a 4.87% chance of confirming one critical hit and 0.12% chance of confirming two.

Under the removed iterative rules, you have a 5% chance -- 1/20 -- to threaten one critical hit. Of those 5% of rolls that will be critical threats, you will confirm 50% and confirm an additional critical hit on 5%. So that's a 2.5% chance of confirming one critical hit and a 0.25% chance of confirming two (the case where you roll a 20 and then roll a 20 to confirm -- 1/400).

So, in this situation, you're a bit better than half as likely to confirm one critical hit but twice as likely to confirm two.


Consider two attacks (as a 6th-level fighter or equivalent) with a rapier, with its 18-20/x2 crit range.

Under the regular rules, you now have an impressive 27.75% -- 111/400 -- chance of threatening at least one critical hit (getting at least an 18 on at least one die), and a 2.25% -- 9/400 -- chance of threatening two (getting at least an 18 on two dice). Again, halved for the 50% chance of confirming the critical hit, that's 13.87% chance of one confirmed critical hit and 1.12% chance of two.

Under the removed iterative rules, you have a 15% chance -- 3/20 -- of threatening one critical hit. This has a 50% chance of confirming (7.5% chance of one confirmed critical hit), and a 15% chance of confirming a second critical hit (2.25% chance of two confirmed critical hits).

Again, you've got a bit better than half the chance of one confirmed critical, and twice the chance of two confirmed criticals.


Now a harder one: a 16th-level fighter (or equivalent) getting 4 iteratives with a 20/x2 weapon.

I don't know that I understand probability enough to do this, but... there are 130321/160000 ways to not get any 20s on a die roll of 4d20. That's an 81.45% chance of no critical threats, meaning an 18.55% chance of at least one. If one of your dice is a 20, there's 6859/8000 -- 85.73% -- ways for the other three to turn up no 20s, so of the 18.55% of the time you get one critical threat, 14.26% of the time you'll get a second -- so 2.64% of total rolls, you'll get at least 2 critical threats. Of those times, there are 361/400 -- 90.25% -- ways to not have any 20s, so 9.75% of the times you get two 20s, you'll get a third -- 0.26% of the time you'll get 3 20s. And there are of course 19/20 -- 95% -- ways for the remaining die to not be 20, 5% chance of 20, for a total of 0.0129% chance for 4 20s. And then halve all the numbers for the 50% chance of not confirming.

The math is the same as the first example for the variant rule, because you can only get at most two critical hits. 2.5% chance of confirming one critical hit, 0.25% chance of confirming two, 0% chance of more than two.

  • Chance of one confirmed critical hit: 9.27% vs 2.5%
  • Chance of two confirmed critical hits: 1.32% vs 0.25%
  • Chance of three confirmed critical hits: 0.13% vs 0%
  • Chance of four confirmed critical hits: negligible vs 0%
Now the player who objected is right, it's looking much more in favor of the old way.


However, as mentioned above, everything that applies to critical hits also applies to critical fumbles.


An additional concern: consider how you do damage for a critical hit. Some DMs want you to roll once and multiply, other DMs want you to roll multiple times. Rolling once sucks because you could get a 1 (woo, my critical hit did 2 damage!) or you could get max (woo, pretty much instant kill!) -- it's way too swingy. Rolling multiple dice gives you a nice bell curve, and bell curves are always more pleasant than straight lines.

The same applies here: if you roll a 20 under the removed iteratives rule, you've hit four times; if you roll a 1, you've missed four times. If you're using the base rules, you're much more likely to hit some of the time and miss some of the time, which is much better.


So for that last reason, not the critical hit/fumble reason, I ultimately side with using the base rule instead of the variant rule.