Tobold's Blog
Wednesday, August 06, 2014
Combat math

A surprisingly large variety of very different games of the general role-playing genre use the same basic approach to combat: The combatants exchange blows, each blow has a certain chance to hit, and if it hits it does a certain numerical value of "damage". The damage per se doesn't do anything, but if you accumulate more damage than you have "health", you fall unconscious or die. This description covers games from pen & paper Dungeons & Dragons, over single-player computer RPGs, to the latest MMORPGs. What varies is the probability to hit, how damage is determined, and how much health the combatants have.

Now imagine the most simple version of this system: Whether you hit is determined by flipping a coin, and the damage you deal is equal to the health of the enemy. It would work; with some minor modifications (each hit deals 1 hitpoint of damage, but combatants have more than 1 health) you could actually play an improvised tabletop RPG with your kids if you got stuck somewhere without dice or electronic entertainment. The reason you would have to modify it is that the extreme "1 hit you die" version isn't much fun. It will appear to the participants to be very random. And in a group vs. group combat the people eliminated in the first round will sit around bored while the others continue playing.

On the other extreme of the combat math are MMORPGs: In a MMORPG your chance to hit is very close to 100%, attacks rarely miss completely. The amount of damage from each attack is small compared to the health pools, and the random variation of damage is small. In games like World of Warcraft the damage doesn't even depend on the actions of the adversary, so you can fight against a training dummy and determine your "dps", your damage per second. At that point the math becomes rather predictable: If two combatants try to kill each other, you just have to divide their health by the dps of the opponent, and see who of the two will kill the other first. And then the game has to introduce other factors, like a "dance" of evasion moves around a boss mob, because if it was just that simple math, just tank & spank, the fight wouldn't be all that interesting either.

Tabletop role-playing games tend to work with hit chances around 50%. What differs a lot between different systems, and even between different editions of Dungeons & Dragons, is how many hits it typically takes to bring down an opponent. Some situations in 5th edition resemble a bit my coin-flip basic model: A duel between two level 1 wizards would be decided by the initiative roll, because a wizard at that level has between 6 and 8 hit points, and his magic missile does 3d4+3 damage without any attack roll or saving throw. So unless you roll only 1s, the first magic missile going off pretty much certainly kills the other wizard. 4th edition is pretty much the other extreme among the D&D systems: It it pretty much impossible for any character to kill a mirror image of himself in just 1 round.

High damage compared to health makes combat faster, because it doesn't take so many rounds before one side is dead. For people who don't want to spend too much time in combat, that is definitively an advantage. But in a high damage system questions like who gets to strike first suddenly become very important. An ambush, where one side strikes before being detected by the other and possibly with some advantage is a lot more lethal in a high damage system than in a system where damage is lower compared to health pools. Like in the coin-flipping combat example, high damage combat risks feeling somewhat random, because if combat only takes very few rounds, every low roll or high roll has a huge impact.

Low damage systems are inherently slower, which is not to everybody's liking. But fundamentally they are more predictable: Because there are more rounds of combat, there are more dice rolls for hits and damage, and then statistics kicks in and averages things out. For a dungeon master it is a lot easier to design interesting fights without constantly being close to an accidental total party kill for no good reason. But a tabletop system will never have as many blows per combat exchanged as a real-time MMORPG does, so you won't find your tabletop players calculating their dps anytime soon. There is still random variation, and you can still die from a series of bad rolls, but it doesn't come down a single bad roll any more. In a low damage system factors like who gets to strike first are less important, and people even sometimes voluntarily forego initiative because other factors like tactical positioning become more important. So low damage system work better for people who like tactical combat.

Because all of these systems use the rather unrealistic assumption that damage doesn't matter unless it kills you, the combat math also tells you what tactics to use. For example if a group of players fights against a group of monsters, it is always better to concentrate fire on a single enemy. Half your enemies dead is a lot better than all of your enemies half dead. The same of course is true in the other direction, which gives the DM a subtle tool to influence combat by letting his monsters concentrate fire or not. Monsters spreading their damage can appear a lot scarier than they actually are. And if you consider that creating dramatic combat situations is one of the goals of many pen & paper roleplaying systems, while constantly killing all players usually isn't, that can become quite a useful trick in the DM's dramatic arsenal.

In any case, the combat math has a profound effect on how combat feels to the players. Therefore it is worth knowing a bit about it, and being aware of the advantages and disadvantages of different parameters.

If 4e is a low damage system, and 5e is a high damage system, is it fair to say that Pathfinder is a "medium damage system"?
is it fair to say that Pathfinder is a "medium damage system"?

Let's find out. Three level 1 wizards, each with a constitution score of 14, walk into a bar and start firing magic missiles around.

5th edition: The wizard has 8 hit points, his magic missiles do 3d4+3 damage. One magic missile spell kills the wizard.

Pathfinder: The wizard also has 8 hit points, but his magic missile only does 1d4+1 damage, and thus needs on average just over 2 magic missile spells to kill.

4th edition: The wizard has 24 hit points, hit magic missile deals 2d4+3 (at 16 INT) damage. It takes 3 magic missiles on average to kill the wizard.

So yes, Pathfinder seems to be somewhere in the middle between 4E and 5E here.
Its interesting that 5e was an attempt to hybridize 3e and 4e and yet ended up with a damage system that is more lethal than either of them.

I think your analysis needs to also consider what happens when you get to 0 hits points in each system. I noticed in reviews of 5e that while its easier to get to 0 HP, it seems that is it harder to actually die in 5e than in either 4e or 3e. When damage reduces you to 0 hit points and there is damage remaining, you die if the remaining damage equals or exceeds your hit point maximum. That shouldn't happen very often. Once you are at 0, you are most likely "safe" for at least a round or two. As I understand it, once you hit zero HP you must spend your turns making death saving throws. This is not a Constitution check, but a flat d20 roll. If you roll 10 or higher, you succeed. Otherwise you fail.

Im not sure exactly how the math works out but it seems like you will be around for at least 3 rounds after you hit 0 and most likely a few more after that. Is that more forgiving than 4e?
The death systems between 4E and 5E are VERY similar. You die after 3 death saving throw failures. In 4E you can only self-stabilize on a 20. In 5E you can also self-stabilize by three successes, so that is slightly better. But then 5E gives you additional death save failures for taking damage or rolling a 1, so I wouldn't say 5E is more forgiving overall.
I really liked the system in Shadowrun 2nd edition. The target number system was a little wonky in that target numbers like six and seven were essentially the same. But I liked that not only hits were based on chance but so was defending those hits, damage could be resisted in whole or in parts. And you were forced to make tactical decisions regarding how much of your dice resources to put into attack and defense. And one of the best bits was that pretty much all characters and vehicles had 10 hit boxes, instead of hit points. The hit boxes were separated into ranges and when your damage crossed the thresholds between ranges your ability to act was hampered because all target numbers attempted by that entity were increased by some amount. We made it particularly fun/difficult by making those numbers cumulative. So when you had light damage you got a +1 to target numbers, moderate damage meant +3, and serious was +6.
I was thinking some more about hit points in 5E and was wondering how you would feel about all the classes starting with 50 hp, but then gaining 2 less possible hp per level. Right now a fighter starts with 10hp and gains a maximum of 10 per level for 19 levels, an overall max of 200. If the character started with 50 and added a cap of 8 per level, the maximum would be 202, which is almost the same. The math seems to work well for any class, if you just knock 2hp off their max gain per level.
My rationale is that most characters are not complete noobs when you role them. They have backgrounds as warriors or thieves or whatever and this would reflect that they bring some survivability to the table from the get go. It does not make sense that they would suddenly be twice as tough to kill after taking out 17-18 beetles (roughly the amount to advance to level 2). It seems like you could do more interesting adventures from the beginning and not have so many party members unconscious during early fights.
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