## The Mathematics of Fixed Damage

Ever since I conceived the idea for creating ** A Night in Seyvoth Manor**, I've been debating whether to use fixed damage values for the monsters and traps or not.

The general consensus is that using fixed damage values speeds up the encounter, primarily because mathematics becomes simpler. You don't have to read a handful of dice, add the modifier and figure out the total; it's one nice round number. No more fistfuls of dice!

The problem with using that premise in 4E is that the number is pretty much always the same. First off, when converting a monster's damage roll to a fixed number you kind of have to use the average of the damage; using anything other than the average wouldn't make sense (this is not taking "difficulty" in to consideration... we'll get to that later). But an inherent aspect of 4E is that, for any given level, all the monsters average the same amount of damage. That's *by design* in 4th Edition.

#### Damage, By the Numbers

As a point of reference, here are the average die values as presented in my 4E Dice Roller and in Sly Flourish's Die Roller:

**Minion:** 7 damage

**Low:** 10 damage

**Medium:** 14 damage

**High:** 17 damage

**Limited High:** 21 damage

**Hardcore:** 28 damage

Based on that, the first problem I have with using fixed damage would be that every monster would cause 14 damage, so for argument's sake let's make it a nice round number like 15.

*Everything* becomes 15 damage. Again, that's by design in 4th Edition; doing anything contrary to that risks unbalancing things. Sure there might be some exceptions to the rule - some things might do 10, some things might do 20 - but all in all it's the same average damage, over and over again. While creating my monsters for ** A Night in Seyvoth Manor**, I actually got sick of writing "15 damage" in almost every stat block.

### Critical Hits

One issue is how to handle critical hits. Gone are the days of the "ghetto crit" that could happen now; statistically speaking, a player is over three times more likely to roll maximum damage on a d6 (16.7%) than rolling a natural 20 (5%), so at low levels and players who may not have magical items a critical hit borders on being meaningless. So what do you do with critical hits and fixed damage? Double it? Add a die?

I thought about doubling the damage, but is that really "special"? With fixed damage, that's not much different than hitting twice. I tried to mitigate it by adding critical hit effects to monster powers, but sometimesÂ it feels like a lot of excess work.

### Characters, By the Numbers

To figure out the effectiveness of fixed damage I decided to compare the above values to two different characters that would be playing such an adventure. Considering that it's a 6th level adventure, I created the weakest character I could think of (level 6 human wizard w/ 12 Constitution = 42 HP) and the toughest I could muster (level 6 dwarf battlemind w/ 21 Constitution and Toughness feat = 71 HP).

Using 15 as the base damage, it would take three shots to knock the wizard unconscious and five hits to drop the dwarf. That's arguably acceptable, but the problem becomes apparent when you realize that the wizard doesn't need a 12 Constitution... He could have a ONE as his Constitution (31 HP) and it would have the very same effect: he'd be at 1 HP after the 2nd attack and drop on the 3rd.

This means that the Constitution value - unless it's your primary stat - isn't as important. In the above case, if I knew the end result would be the same I'd consider leaving my Constitution as an 8 and boosting all my other attributes. After all, what's the point? It's not like a "lucky" damage roll might hurt... Spending the points to up my CON by 4 could be considered a waste.

I also compared the damage to the players Fourthcore game I'm currently participating in, who happen to be 6th level: we all have 60, 48, 51, 50 and 63 HP (average 54.5 HP). Using the 15 damage base rule, four out of five of us would get dropped in 4 hits. Despite the disparity in HP (12 points between the highest and lowest of the four that would be dr0pped), it all boils down to AC and not HP.

### Time, By the Numbers

So, strictly from a mathematical perspective, I'm starting to not like fix damaged. But the question is: does it really improve play speed? With fixed damage do encounters blow by so quickly that it's worth the lack of randomness?

Many have pointed out that fixed damage improves combat speed considerably in epic tier, where you can expect pretty much any attack to use no less than 4 dice. And god help you if you crit, 'cause then you might be wheeling out five more dice. Oh and *Sneak Attack*, so there's four more. Or* Hunter's Quarry*. Or *Warlock's Curse*. Or assassin shrouds. Or this... or that... Bonuses galore. The days of throwing a bucketful of dice to determine damage are not forgotten. And it's not like past editions where the mage might have to roll 40d6s worth of damage every now and then... that's pretty much *on every attack*.

But we're not talking about epic tier here; we're talking about level 6. For the most part, all damage rolls end up bring at most two dice, maybe three. If everyone does what I do and rolls damage at the same time as the attack, does the math really take that long to do? I'm a mathematics and computer science major, so that math is pretty easy for me, but I can't speak for everyone else at any given gaming table. Help me out here... How long does it take you?

So I've been trying to think of how to quantify this. Let's try to figure this out; please let me know if my logic is horribly , horribly wrong:

- I assume the average combat takes four full rounds, and there are five players against three DM controlled monsters. That's, on average, eight attacks going on in any single round (I'm balancing lack of attacks - such as for Second Wind or other support duties - to compensate for opportunity attacks, granted attacks and minor action attacks from monsters). That's 32 possible attacks.

- Let's assume 70% of those attacks hit. Given the attack bonuses, that seems like a reasonable expectation. Rounding up, that gives us 23 attacks that require a damage roll.

So time to do a little testing.

For testing purposes, I'm assuming the attack and damage rolls are made separately. And here's what I did:

**Time Spent Finding Dice:** If your weapon always deals the same die's worth of damage, that die should not be far away. So I'm estimating the time to reach over and grab one or more of these dice, separate from getting the necessary d20, at about 5 seconds.

That, in my opinion, is high - I spend that amount of time looking for the dice inside of my bag, so I can only assume you don't have an idiot player that stows his dice after every roll - but we'll roll with it.

**Rolling Dice and Adding:** I grabbed my box of dice and picked some test subjects; I chose d4s because they are the hardest to read when thrown (they can't technically be read from directly above), and to compensate for the low number I got six of them. I also got one d20 that I will roll prior to starting the clock to determine the modifier I would be adding to the die roll; for example, if I roll an 11 on the d20 my damage roll will be 6d4+11.

After twenty rolls using this style, I averaged 8 seconds per roll. Because, let's be honest, math comes easy for me I'll double that number for the purposes of this test. So let's leave it at 16 seconds.

**Applying Damage:** The time spent to apply the damage does not change whether the damage itself is fixed or not, so it has no bearing on this test.

So let's round things off and say that for each damage roll the player spends 20 seconds. Let's add 50% and make it an even 30 seconds. Multiplied by the 23 attacks that hit you're looking at 460 seconds, or just under 8 minutes in every encounter.

Eight minutes an encounter, and that's a high estimate in my opinion... Is that really such a big deal, especially considering how long 4th Edition encounters take now?

Personally, I don't see that as enough time to justify it, but that's just me. I'd love to hear if anyone out there has had different experiences.

### Conclusion

In the end, I'm still not sure. I'll continue to use fixed damage simply because I started that way, but it's not all that hard to change at this point.

What do you think? How do you feel about it, good or bad? Do you use it? Is it simply a matter that combat just *feels* quicker when there are less dice, even though it might not actually be significantly faster?

Christopher MathieuOctober 21st, 2012 - 00:27

There’s something else you’re not accounting for in your estimates: the time it takes for players to decide on, and declare their actions, and the time the DM needs for bookkeeping.

It’s probably best to assume that each player needs about 30 seconds on their turn to figure out what they’re going to do and convey this to the DM. For inexperienced players, most of this time is spent looking at their powers and the battlemat; veterans can do these parts quicker, but tend to spend more time embellishing their descriptions, so it probably evens out.

Since this is roughly the same time you estimated for damage rolls, that brings your total for a single encounter to 920 seconds, or a little over 15 minutes.

This assumes that the DM is just as quick at math, or has a program or app for tracking effects and such — and, in the latter case, has used the software enough to be able to navigate it quickly. If the DM lacks any of this, he probably needs a little extra time to run the numbers and get everything ready for the next turn. As an optimistic estimate, I’d say that this would increase the per-encounter time to about 20 minutes.

That’s still pretty good compared to most 4E encounters. I’m not sure if I’d play this way, though, because it discourages some of the fun things that slow down the game — weird strategies (“Can I do this?”), side-jokes, gloating over a really good roll, that sort.

NighthawkOctober 21st, 2012 - 00:42

But way I see it is that the bookkeeping you describe is going to happen whether the damage rolls are made or not. Whether the power says “14 damage” or “2d6 +7 damage” is irrelevant; the player will spend the equal amount of time deciding what they’re going to do. The fact that the damage is fixed or not does not change the difficulty of the decision.

Furthermore, the bookkeeping doesn’t change much either; once the damage is determined, from then on the time spent bookkeeping is the same. It’s the time that it takes to reach that damage value that I’m analyzing… whether that damage value is a fixed 14 or any number from 9 to 19 (the aforementioned “2d6 +7”) doesn’t change how long it takes to subtract two numbers to determine how much HP the target has left.

Christopher MathieuOctober 21st, 2012 - 09:44

That’s understood. I just wanted to point out that your time estimate only told part of the story.

GrimnirOctober 21st, 2012 - 16:07

One of the reasons I read your posts is because it gets me thinking in an outside the box sort of manner. I’ve got a game to run on Monday, and I’ve been having trouble getting balance right without totally destroying the party or making the creatures too weak. I’m an old school 2e player, so this concept of fixed damage isn’t something we come across.

Right, so it made me think. Players really like their dice, I would never force fixed damage on them. But maybe for the sake of simplification, and to avoid excessive bookkeeping, I can come at this in reverse as far as enemy HP goes. I would still totally let the players have their big numbers and maybe even record them on the whiteboard because they do enjoy that sort of attachment to big numbers. I just don’t need to tell them I’m working the backend on a totally different system.

For example, take a relatively tough creature like an Ogre. I would say an Ogre should be able to take at least 4 hits from the hardest hitting player in the group. Twice that many from the average amount of damage done in the group, and twice that many again for weak hits. Instead of having actual HP for creatures and have to suffer through how much gobs of damage players are actually doing with I could then take how much they do and classify it as a light, medium, or strong hit.

(just realized, this is totally like the hearts in Zelda games, full-half-quarter)

Ex. an Ogre could take

O O O O – 4 Strong or

oo oo oo oo – 8 Medium or

iiii iiii iiii iiii – 16 light

This seems like it would give the ability to quickly consider what type of hit occurred and be easier to give an assessment of what creatures in combat look like. Taking a lot of light hits (small cuts all over the place) or a few strong hits (large open wounds with bone exposed) if you add flavor to your combat. It also removes the need for the DM to do a lot of small simple math (more a chore than a challenge) and allow for crits to have more significant effect by stepping them up a damage class or counting as two hits. The players on the other hand, at least all the ones I’ve run with, are obsessed with their numbers, and if you keep them in the dark how you run things, it can fill that need for them as well without going too far outside the realm of “are you actually keeping track of things?”.

Jacob ZimmermanOctober 21st, 2012 - 20:15

Healing surges add to the variety of the number of hits a character can take as well. Just calculating one surge value being added to the characters, and the character with 48 HP can’t even survive an extra hit. Plus there’s the variable amounts added with the surges that really make a difference. Not as flat as you were saying.

ChadNovember 28th, 2012 - 13:44

Are you worried about DM time, player time, or both? There are ways to speed up either one, but the DM time is probably the easiest to address, since you’re the DM. đź™‚ Even if dice addition feels really easy, it imposes non-zero cognitive load on the DM, who often has the most things to mentally juggle.

If you’re a math/CS guy, you probably won’t have any trouble rewriting the damage expressions to make the math simpler. Allow each monster one or two dice plus a static number. (Truly special named bad guys can break this rule, especially if it increases their table presence. Breaking out the sneak (quarry, curse, etc) dice for a monster gets their attention.) Figure out which monsters should be swingier, and give them the bigger variance. For simplicity, I almost always stick to one die-type, but simple multiplication is easy for me, so that doesn’t mean that the variance of the d4 is low – 1d4*3 +7, 2d10+3, and 1d12+8 are all good damage expressions for 6th level monsters.

If you’d like to speed up things past simplifying the damage expressions, stop rolling them yourself. This is especially helpful when you’re running many monsters. When you hit a player with a normal monster, tell them “The bloodied minotaur charges you for 2d8+8 damage.” and move on. This also makes it easy to give damage dice to minions, without increasing your overhead. You could instead offload this to another player, but be careful about creating subtle divisions between the players – you nearly always want to increase camaraderie, letting the story and the roleplaying create the divisions.

Give nearly everything crit damage. Minions get +50%, basic monsters get +1d6/half-tier, and things with big weapons get d10s instead of d6s. Crits are exciting no matter who rolls them, and while your players might feel like they’d rather their own character take a low-damage crit than a high one, they’ll usually be more excited if crits are important at the table no matter who rolls them. It’s surprisingly (if cruel) fun to make a player roll crit damage against their own character.

If you’re preping monsters ahead of time, pre-roll some damage using the complex expressions, and write them in or next to the stat block. Crits bypass these pre-rolls, of course. If you’re comfortable winging it, you can tweak these numbers slightly based on the circumstances (bump the near-crit up 1 or 2, drop the glancing blow behind cover by 1) to obscure the fact that there are only a few damage outcomes.

Finally, if you’d rather not give away all of the secrets (some tables really get into the optimizing part of the math when it’s visible, and some of *those* tables can really break the mood in a bad way), then start out with normal dice expressions, perhaps even rolled in secret, and migrate to another technique over time. When I’m running epic, I almost always switch to average damage after the first round, although I do try to keep an eye out for PCs near unconscious, so I can have a dramatic roll.

I hope that helps!