586884_84070111 A minor kafuffle went on in the RPG blogging community recently over the concept of the 7.5 hit point orcish standard. I don’t think that’s what anyone else called it. That just seems like a good name to me.  The original premise is this: If you’re an orc, or some other common low level fodder one hit die wonder, and you roll a one for hit points, you’re screwed. At some point you’re going to trip and hit your head on a rock, get into a brawl over whose mother did what with a goat or some such and you are going to die. (Think back if you’re old enough to the “first level wizard vs a housecat” arguments from pre-web days. This is sort of like that.) That being as it is, why do PCs ever meet one hit point monsters? Why haven’t their lives been cut short long before the PCs ever encounter them? The logical assumption is then that all orcs that PCs meet should have seven or eight hit points (and a bunch of math was presented to back this up). Taking this a step further, if only seven or eight hit point orcs survive consistently, then only seven and eight hit point orcs breed, and so after generations of this “survival of the fittest”, only strong orcs survive and only strong orcs are born. This is obviously an old system DnD concern, but the concept translates well into most games, and I find I have a lot to say about it.

First, you may need or want the required reading. It’s not necessary for understanding, enjoying, or employing the concept, but if the math is going to mean anything it may help to know where I got the assumptions and models I’m working with. The whole thing started out of The Tao of D&D here, here, here and finally here. I picked it up from a reply posted at Save Vs all Wands here, and here.  There is probably a lot more floating out there in the web, so if you know of more extra reading, feel free to call it out.

First, I love this idea and it’s supported by early game lore. Early games often included monster lair information with the numbers of additional non-combatants. It’s easy to imagine that the weaker monsters are the majority of the ones that stay home, mending torn armor, sweeping up, convalescing, etc… Now, this doesn’t mean that the PCs will ONLY fight strong monsters. Certainly there are monsters having a bad day and young bravos looking to prove themselves in battle, and those in support roles like bow fire, scouts, etc… may be less tough than the average toe-to-toe warrior, but overall, it makes sense that weaker monsters are weeded out or stay home. This may change the tone of your game by making low levels a little more gritty and dangerous, but that’s sort of the mainstay of old school play anyway, and it’s always possible to balance the increased challenge with fewer monsters in a given encounter or with more experience or treasure. By the same token, the same lair information often included a section on the stats of the “elite veterans” and how many could be expected to be protecting the chief. It make sense that while the common orc is tough, these guys are the cream of the crop. Generally they had extra hit dice, better equipment, the whole nine yards.

I do suggest letting your players know about this (or any other rule modification) so they can plan accordingly. Finding out about a change like this AFTER you charge a motley group of orcs wouldn’t be a pleasant surprise. (As an aside, I had a GM once who decided that all monsters should have 10 hit points per hit die, only we couldn’t figure out why we were getting our clock cleaned on a regular basis and why battles were these long painful slogs until about third level when someone finally stabbed an orc for exactly eight points and he didn’t go down. There may have been a little mutiny.)

But… There’s always a but isn’t there? In this case, there are several (none of which should in any way stop you from using the system if you like it): First, the part of the model that this modification was supposed to fix isn’t as clear cut as it appears to be. It’s actually part of a larger model, and as it’s an abstraction, it’s not really clear which parts of the abstraction apply to which parts of the “game reality”. Second, the math and scientific premise upon which the entire thing sits has some problems (as far as I can tell) and finally it has a few really interesting additional implications.

image   Let’s start with that first “but”. We’re discussing hit points and the number of them an orc has, and thus the number of spear pokes, and/or sword swings said orc can withstand before he drops. This is modeled by giving the orc a random number of hit points and the attack a random amount of damage it does, but the problem is that these random numbers are modeling a binary condition. Is the orc alive or not? The entire mechanic is supposedly invisible and changes this single binary state. In fact, changing a collection of these binary states is more or less the entirety of plenty of games.

But, because of that there’s no real reason that the number of hit points we give an orc has anything at all to do with the orc in question. You could get (roughly) the same result from saying that all orcs have one hit point and that spears do 1d6 – 1d8 damage per hit. But you could really go further than that because 1d6-1d8 has a roughly 31% chance to net a positive result, killing our orc. Further, we know that we have an approximately 35% chance to hit an orc. So you could get the same result from the entire system, soup to nuts with the following system: “To attack, roll a d10. On a result of 10, the orc dies.” Now that’s fine, and actually approximates the rules as written in probability (not perfectly, but we could get closer if we fiddled. This is mainly for an example, not a suggested system) but it isn’t easily able to be generalized, so we don’t use it but the point is that many different systems give similar results and the only reason we’re arguing about what hit points mean with regards to orcs is because Gygax decided to design his system the way he did, not because the number of hit points is actually integral to the system or actually tied to the orc itself any more than the sword swing that kills him. It just looks that way superficially.

To that end, the entire point of the 7.5 orcish standard is “I think this binary variable should get flipped slightly less quickly” and while the article addresses this via adjustment of hit points, it could have just as well addressed it by giving them some armor or a dex or con bonus, or whatever. Heck, you could just give the orcs better weapons so while they still die just as fast, they effectively dish out more punishment before they do, and really since it’s not the orcs’ binary variable that are important, but the players, that has the same effect too.

So, there really was no reason for the entire mess to devolve into a screaming match over hit points (a point the original blogger makes repeatedly) but of course it did. So that’s the first “but”. Just ignore that whole mess. It has very little to do with the intent of the system, and while you’re free to argue about the merits and meaning of hit points here, that’s not really the point.

image   I said the math and science had problems, so here they are: Tao did the math twice, Wands once, and they eventually came up with the same results but I wanted to do the math myself. My numbers more or less matched their final agreed upon numbers, but it’s not the fact that one hit point orcs die faster than 8 hit point orcs I disagree with. I’ll discuss the math problem I see in a bit. First, to determine how an orc fares in a battle, I used a Markov chain. This is a matrix that shows all possible states the orc could enter a round of battle on the left, and all possible states at the end of a round on top. Each cell then has the probability of starting on the state at the left and leaving on the state at the top. Here’s my orc combat matrix:

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Note that orcs can be in any state from 0 to 8 hit points (yeah, it’s weird to consider them starting the fight dead, but it’s necessary to make the matrix algebra work). So, for example, our 1 hit point orc has a 35% chance to be killed (the chance he’s hit) and a 65% chance to stay at 1 hit point. A 5 hit point orc, has a 12% chance to be killed, 6% chance to be reduced to 1, 2, 3, or 4 hit points, and a 65% chance to stay at 5. The big deal with 7 and 8 hit point orcs is that they have a 0% chance of being killed! That’s a pretty big deal. They do however, have a chance of being turned into an orc that can be killed. Mostly by liberal application of stabbing.

(Note that these probabilities are based on givens from the original article: 1d8 hps per orc, 1d6 hps of damage per hit, 35% chance of getting hit. Given that another article claimed those numbers should be 1d8, 1d8, 40% I made my open office sheet with a pair of cells “P(hit)” and “Dam die” that can be changed and will auto update all the calculations I have here. You can click that link to download it, so if you want to know what happens if we arm orcs with daggers, halberds, or hurled turnips you can just change those fields)

Here’s where using a Markov chain for this application shines: If I want to see what the probabilities look like after 2 rounds, or 3 rounds, etc… I just take the above matrix to that power and the results read exactly the same. On the left is the hit points the orc started with. On top is how many they have after this many rounds. Here are the matrices for round 2 and 3.

(the document goes up to round 5. If anyone knows how to make open office take a matrix to a power without repeated multiplication though, I’d be interested in knowing.)

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image This gives us the following probabilities of survival after a 3 round fight (a given in the original article).

Looking at these results, we see that it’s absolutely true that 8 hit point orcs survive much more often than 1 hit point orcs. These numbers are much kinder to the 1 hit point orcs than the numbers from the original article, and (mostly) match those finally agreed upon and while I’d like to definitively declare them correct, I have been wrong before. So if you’d like to debate them, by all means do so.

From here, we assume that not only have these poor beleaguered orcs been in a battle for 3 rounds, but that they go home, rest up, then do it twice more. In total, they’ve survived 9 rounds of combat, but they got to heal up twice in the middle. That’s reasonable. If you’re in the orc army, you have to fight sometimes. What’s more, the math is much simpler and doesn’t involve any matrix algebra. If a 1 hit point orc has a 27% chance to survive a 3 round combat, he has a .27 x .27 x .27 = .02 chance to survive 3 of them. Extending that to a table we have the following survival proportions:

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Sure enough, after 3 battles of 3 rounds each, 65% of our 8 hit point orcs are still alive, while only 2% of our 1 hit point orcs are. This is looking good. Even though these numbers don’t match the original article exactly, the trend is clear. 1 hit point orcs die a lot easier than 8 hit point orcs, and as our number of battles becomes large, the proportion of our survivors who are 8 hit point orcs will slowly approach 100%.

And here’s where it all falls down. Here’s an extension of the table above. It shows the distribution of the orcs at the start of each of the three fights and the distribution after.

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After those three battles, we’ve lost 74% of our starting orc population! If we had 100 to begin with, after these three fights, we have 26. Our survival rate is actually improving over time (because we have tougher and tougher orcs) but as the number of fights approaches infinity, our survival rate will only be 86%. We’ll still lose almost a sixth of our orcs every fight!

So why does this ruin the whole thing? Because our orc tribe is going to be wiped out in no time flat unless they get more orcs, and those orcs are coming in with the same uniform distribution of hit points that our original orcs came in with. True, we’ll see a slight increase in the number of tougher orcs (because there are always a few veterans around), but given the rate at which they’re dying compared to the rate at which new warriors must be coming of age just to keep their numbers stable, it’s clear which is the overwhelming force. Put simply, the more distinct difference this effect creates between 1 hit point and 8 hit point orcs, the fewer and fewer orcs it impacts. Either it has little to no impact and effects a lot of orcs, or it has a profound impact… on a small handful of orcs. Figuring out the optimal result achievable is doable with a lot more info and some matrix calculus, but I don’t feel like punishing myself that much today.

But what of the genetics argument? We have a tough pool of survivor orcs who are the parents of the next generation. Won’t this make the next generation of orcs bigger, tougher and so the new ones won’t really come in with a uniform distribution? Well, not really because that’s not how genetics work. We’ve all been taught genetics with very simple applications and punnett squares where there’s one blue eyes gene and one brown eyes gene, but continuous variables like height, intelligence, and toughness don’t work that way. Instead they have a large number of genes and environmental variables that “add up” to the “level” of your final trait. These follow what’s called a binomial distribution but with so many variables, they’re just expressed as a normal distribution and though a tall, smart or tough parent (or two) may give their child a slightly better chance to be above average, there’s a very strong pull towards “average” in these traits. So again, the tougher orcs reproducing does in fact create a slightly higher number of above average orcs, but not enough to change the distribution significantly. I have a biochemist verifying this information, because I’m not a scientist, so my understanding here is amateur at best, but it’s unlikely I hear back by my publication date.

Instead, the best way for orcs to get reliably bigger and tougher is probably the way that humans did it during the Victorian era (for one example): improve nutrition and environmental conditions. It’s amazing how much healthier as a whole your people becomes when they get 3 squares a day and aren’t living in a dark dank smoke and mold filled warren full of excrement and rats… which actually points towards “civilized” races like humans and demihumans being in much better average condition than orcs. This may be why in old DnD versions, elves got 1+1HD instead of 1 like everyone else.

Of note is the fact that if we had more information, like the rate at which warriors come of age, the specifics of orcish genetics, and the frequency and length of orcish battles, we could create a Markov chain like above and find it’s limiting distribution (the long term distribution of orc hit points after infinite generations of orcs that orc tribes will eventually get closer and closer to) this could take into effect those diminishing returns from “orc fight club” and small impacts from genetics but it seemed like a lot of assumptions to make and too much work for this project.

image image   Fun tidbit #1  With our Markov chain from above, we can find our fundamental matrix (a matrix showing the average number of rounds spent at each amount of hit points before an orc is killed) and from that we can calculate the mean survival rounds by starting HP. ie: How long do we expect a 1 hp or 5 hp orc to survive in one of these combats? We can see here, that we expect the average 1 hit point orc to hang around 2.86 rounds, while the average 5 hit point orc lasts 5.29 rounds, and the average 8 hit point orc lasts 7.93 rounds. This is interesting because it turns out that in terms of rounds per hit point, 8 hit point orcs have the worst stamina. This means that large groups of weak orcs are more dangerous than much smaller groups of veterans. Orc leaders should be well aware of this and probably throw waves of slaves and weaklings at threats, using better armed and armored veteran elites to fight particularly difficult enemies (like PCs that have managed to slaughter a pack of fodder).

image   Fun tidbit #2 I don’t have charts and math for this part, I consider it pretty self evident. If anyone desperately wants to see some numbers, say so in the comments and I’ll put them in a future article if I can boil them down to something manageable. It stands to reason, that the higher a monster’s hit dice the higher it’s minimum hit points, but also the higher it’s average hit points and the closer to average hit points we expect it to be (thanks to the law of large numbers). Also, higher level monsters have fewer equal or higher level challenges. Thus the effects that we’re assuming impact orcs and weed out weaklings apply less and less to monsters the more hit dice they have. Therefore, not only do monsters have closer to average hit points on average at higher hit dice, but they have less “evolutionary pressure”. If we assume rules as written, ignore monster resistances and special abilities so they’re just meat bags with hit dice, and pit them against a warrior wielding a 1d8 weapon with no bonuses and let them flail away at each other, the higher a monster’s hit dice, the closer it gets to average hit points per hit die and the closer the warrior gets to average damage per swing (again, law of large numbers) so as n (number of hit dice) goes to infinity, the number of swings necessary to defeat the monster goes to n. ie: asymptotically, a monster requires one sword swing per hit die to kill. This applies the same way across the averages of large numbers of fights with smaller monsters too. Over time, the average number of swings you take to kill a 4 hit die monster with a d8 weapon is 4. (as a side note, a 1HD monster that requires “on average” one hit with a sword to kill seems impossible. Remember that “less than one hit” could be a fractional hit with a sword, ie: this guy could be killed with a dagger.)

That is, until we apply those “evolutionary pressures” that weed out weak monsters. Since the weaker the average monster is, the harder these pressures impact them, the weaker the base monster is, the more they deviate from the average. This means that using these rules, hit die for hit die, orcs are the most badass creatures in the world.