5 Rounds of Nerdy Math

Someone made an assertion online that a 4e fight is designed to last 5 rounds. That’s an interesting assertion, and I’ve had people express that it both sounds too long and too short. If it’s true, it’s a very interesting point that allows you to crunch the numbers a bit harder, but it’s unsourced, so it’s pretty suspect. So I’m going to crunch the numbers a bit here and see how that holds up with the reality. To do this, I’m going to focus on damage dealt, and I’m buying into two strongly held assumptions. First, that a +1 to hit is always better than a +1 to damage, and second that damage is what ends fights. Thus, for illustration, I’ll be focusing on damage output. These assumptions are not absolute certainties, but accepting them makes decision making much easier.

Ok, assume a level 5 D&D character with an 18 in whatever stat we happen to care about. Erring on the side of generosity we’ll assume a +2 weapon, so with a basic attack we’re looking at, what, +2 for level, +2 for magic, +4 for stat, +2 or 3 for weapon accuracy, plus some random +1 for a feat. Baseline weapon is going to be +2/1d10 (battleaxe) or +3/d8 (Longsword). For illustration, I’m going with longsword because, hey, accuracy.

Given all that, that means a Basic Attack with an attack bonus of +12 and 1d8+7 damage (+2 for magic, +4 for stat, +1 for misc), for an average of 11.5 HP damage per round. However, on a crit, that’s 15+2d6, which we’ll call 22. However, Crits only happen 1 time in 20, so that contribution depends a lot on the hit range.

Ok, so given that, let’s look at a level 5 monster. default ac is roughly Level + 14, so that’s a 19 AC, so our hypothetical basic attack will hit on a 7 or better. Pretty good odds, 70%. That means that the real damage output (assuming basic attacks) is ((13*11.5) + 22)/20 = 8.575, so call it average of 8.6 damage per round. This, over our hypothetical 5 rounds, that’s 43 points of damage (which we will generously assume to be perfectly distributed). How does that stack up? Baseline for monsters is (8 + Con) + Level * 8. Since we’re talking level 5 monsters, then that’s about 58 HP, which is to say we’re about 15 points short of our hypothetical 5 round fight.

Still, since we’ve just been using Basic attacks to reach this number, that 5 round guideline does not seem too far out of reach with additions bonus damage from strikers, encounter and daily powers, action points, multiple targets and other random factors. There’s a lot of extra math I could do here, but I’m comfortable with a gut read here – that it’s not too hard to get up to the ~11 DPR necessary for a 5 round fight without excessively depleting resources.

I’m a little concerned at how well that scales though. Let’s look at levels 15 and 25.

At 15, we’ve gotten our key stat up to 22 (so, +6), we’ve got a +4 weapon, a +7 level bonus, and the feat bonus is now +2 to hit and damage, +3 since we still have a longsword, so +22 to hit for an average damage of D8+12 (average 16.5, 20+4d6=34 on a crit). Monster AC at this point is 29, so we’ve kept pace – we still hit on a 7+ so once again we calculate damage average as ((13 * 16.5)+34)/20=12.425, call it 12.4.

I’m already kind of worried. That’s 62 damage over 5 rounds. In contrast, our average monster is looking at 138 hit points. Where the level 5 fight required only about a 30% bump to make 5 rounds, this is more than 100%. I accept that we’ll be looking at a bigger bump (since the encounter and daily powers are more potent, and we’re seeing more feat synergy) but even if that’s 60% (which would be ~20 DPR) that’s a 7 round fight.

By the same math at 25 it’s +31 to hit and 2d8+16(25) on average and 32+6d6(53) on a crit vs monsters with an AC of 39, so we’ve dropped a little, hitting on an 8+. Damage output is ((12*25) + 53)/20=17.65 (or 88 in 5 rounds). Critters are looking at 218 HP, so the gap is even greater. With a 90% bump (to about 33), that’s about 6 rounds.

Obviously, this is pretty approximate, and I’m pondering the takeaways. It would be possible to crunch this further – assume a canonical party of 4, add in the bonus for fighter accuracy and rogue sneak attack, assume all encounter powers are used and re-run the numbers to see if it changes the result, but I’d be surprised if it was much more generous than my 30%/60%/90% progression. I admit, I was surprised that the distribution is as tight as it is – if you’d asked me, I’d have expected that Epic tier fights might be at least 2 to 3 times longer than regular ones given the HP totals.

More than anything, I think this gets me thinking about the impact of minions and elites/Solos on fight duration (and it also makes me all the more leery of high level monsters designed to exceed the specs). Elites seem the nastiest, since they’ve effectively got double HP for double XP, but also have an AC bump that stretches things out. Solos are not so bad, with effectively 4x HP for 5x XP, but the further bump in AC offsets that. To crunch the numbers a little, let’s look at the level 5 Elite. It takes 10-11 “man rounds” to drop 2 level 5 critters (that is, to do ~120 points of damage at 11 points per round). For the elite. we’re looking at an extra man-round as that +2 bump to AC about a 10% drop in damage, so now we’re looking at 12-13 man-rounds. For a comparable solo, we’re looking to do 240 damage against an even better AC, something like 29-30 man-rounds (more than doubling, though increasing in line with XP increase).

Minions, on the other hand, speed things up more and more as you level up, despite the fact that the amount of “wasted” effort increases. Consider, with 4 minions making a normal critter, the comparison for effort is to doing 25% damage to a normal critter in one hit. Thus, for example, if a level 5 monster should have 58 HP, then each minion is ~15 HP. Since average damage output is 12.4, that’s a good deal. It’ smote pronounces at level 25, when the monster might have 218hp – each minion hit is roughly equivalent to a 55 point hit. That’s a VERY good deal. It also backs up the intuitive sense that the pricing of minions may be a little askew, but that’s another topic.

Anyway, this is all back of a napkin math without any MM3 changes, so I welcome corrections. I’m not yet sure what to think, but I feel like I’m a little bit better armed to go forward. It occurs to me that one advantage of taking the deeper math plunge is that it might provide greater insight into what the “right” damage expressions are for powers. If fights are open ended, there’s no good answer, but if there’s a target duration (and, by extension, a target damage output) the suddenly there’s a potential for a real yardstick.

I’ll dig into this more when I have a combination of a spreadsheet and copious spare time, but it’ll probably be a few days.