## Wednesday, March 2, 2011

### 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.

1. I dig this breakdown, Rob, but I think in your desire to keep the math manageable you left out area attacks. Controllers tend to skew the damage curve down as strikers bump it up, but I think that's only true when they're going after single targets. Using a power like Thunderwave, they can multiply their smaller damage numbers across a handful of monsters. I don't know if this exactly fills the damage gap you identified, but I think it goes a long way towards it.

2. @Zack Yeah, I filed that under the 30/60/90 bump but pretty much handwaved it because, man, it's hard to quantify. That said, when I sit down and do the class-based breakout, I'll probably adopt some sort of rubric like treating any area attack as attacking 2-3 targets and see what that does to the output numbers.

3. I don't know where I encountered the 5 rounds number, but it always felt about right to me.

In the Player's Strategy Guide, they suggest 7 rounds from their example on the effectiveness of focused fire. Of course, they use a party of 5 for their example. In it, they assume the average party deals about 75% of a standard monster's hit points each round. Underlying assumptions are that it takes 4 average attacks to kill a monster and that the average PC hits about 60% of the time. But I'm not sure how closely the example reflects reality.

4. It might be worth noting that the CharOp "baseline" DPR for strikers is 20/40/60 at levels 10/20/30. Assuming a linear scale (and it's not at all that smooth), that would mean that you're doing 10/20/30 DPR at your assumed levels. This might make your math work out better, at least considering strikers. My intuition says that non-strikers deal 50-75% of striker damage, so your numbers might be more on target.

You also might consider levels 6, 16, and 26, which let the 1/2level numbers round up.

I've heard the 5-round average (not necessarily by design) thrown around for years. I don't remember the source, but the idea seems ubiquitous enough that I would imagine play experience supports it, at least in Heroic tier, which is basically the same conclusion you came to.

5. Off-hand note: As of MM2 and DMG2, Elites and solos no longer get those built-in defense bumps - except for hp, you stat them the same as normal monsters.

Defense bumps made specials tougher, but only by dragging out the fight. These days, the focus seems to be on making specials more dangerous late in the fight (after bloodied, usually), and the only defensive tricks they get are things to help them deal with dazed, stun, etc.

6. "I'd be surprised if it was much more generous than my 30%/60%/90% progression."

Be surprised. :)

First off, let's look at a barbarian's numbers instead of the generic PC. A barbarian using a fullblade is doing 1d12 + 1d6 with his bread and butter at-will. Also, add in +4 damage from a magic item -- Iron Armbands of Power, say. This puts us up to 26 damage on a hit. We won't assume anything special for the magic weapon, but a fullblade is a high crit weapon, so on a crit his damage is 18 (power) + 6 (stat) + 4 (magic weapon) + 2 (feat) + 4 (misc. magic) + 4d6 + 1d12 = 54.5. So he's doing almost 100 points of damage over 5 rounds with at-wills, assuming no synergy. Back down to a 40% bump.

Mind you, the barbarian will have 4 encounter powers at this point, and there is a barbarian encounter power at every level capable of doing 3[W] damage. If he uses those throughout the encounter, with only one at-will, he's up to 133.5 damage. Still no synergy -- he's just using his most damaging powers and non-optimized magic items.

E.g., he could be using a jagged weapon to get crits on a 19-20, or a vicious weapon to get 4d12 bonus damage on a crit, and/or a war ring to get a bonus die of critical damage, and/or he could have Devastating Critical to get another 1d10 of damage on a crit.

And there's more barbarian damage which I didn't account for -- Rampage, potentially Swift Charge, etc.

So OK. Now figure in that some attacks come as immediate interrupts or reactions, so you get an extra wad of damage without chewing up a round doing it. You'll have an action point in half of the fights; there's another unit of damage. You have an action point feature which often increases damage as well.

Now, synergy: this is huge. I often play with a warlord who provides a +6 attack bonus on an action point attack. She also gives, um, I wanna say +6 damage bonuses for a whole turn if she heals you, and she gives a +2 or so to attacks on the first round of any combat.

My sorcerer (level 19 currently) always plays with my SO's bard. Between the two of us, we can potentially make at least one monster provide combat advantage for literally four rounds straight in every encounter. Sometimes it's more creatures, because some of the attacks that inflict that effect are bursts and blasts. This is not counting a daily or two of hers that also have the same effect.

I have a fighter who specializes in pushing people; he sometimes plays with a wizard with Stinking Cloud. I position myself a square or so from the edge of the cloud, wait for things to come out of it, and knock them back in for automatic damage.

I am probably belaboring the point, but it is really easy to just look at the raw numbers and assume that paragon and epic combat are too slow. I am hopefully not falling prey to argument from authority here -- it's just that after a year+ of paragon level LFR play, I have some experience under my belt that shows me how powerful synergy can be.

Now, this raises an interesting question about the role of system mastery. I have totally lost perspective on how easy it is for a newcomer to figure this stuff out. I suspect it's not easy, and I suspect this is a problem, but I don't know for sure.

7. If 5 rounds of combat is the magic number. It would be interesting to use this as a basis for deciding if the combat is a wash (the player are going to easily win) or if fighting should continue.

8. Why are the Solos and Elites getting an AC bump? That got eliminated in MM2.

9. This is the curse of using the core books as my data set. Change to Solos and elites is duly notes.

@Bryant The striker thing is going to be a big bump (almost as big as multi-target) but it'll be curious to amortize that across a group. That said, you're totally right that there's more to it (one of the reasons it's so hard to measure the value of controllers as anything but minion-killers) but I accepted the damage uber alles premise a the outset solely because it's really the only way to come to a standard.

That said, the 4 man band breakdown is definitely looking necessary.

10. Yeah, taking the striker as typical is also wrong. :) The question is how much the loss of damage from the leader and the defender matters. And there's such a huge spread there -- I know a couple of leaders who do 1d12 + modifiers on their at-wills, and I know a couple who do no damage at all.

I think you're right about the breakdown. I'd do it as a five man, though, since WotC's targeting five PC parties; the difference between 25% of the damage being striker damage and 40% of the damage being striker damage is important.

11. @Bryant well, if nothing else, the joy of busting out the spreadsheet is that I'll be able to do both.

12. Congratulations on reinventing the Monstermark system for D&D. <grin> [cf early White Dwarf]

One thing that extends the length of combat is that usually there is a round or two of manoeuvre before the actual combat is engaged in.

At higher levels you are going to get powers that affect manoeuvre and position which interfere with the ability to deal damage - The non-damaging side-effects of powers is going to have a greater effect, again lengthening combat slightly.