Gnome in Chief keeps his own independent blog over at www.martinralya.com and of course it’s on our slack feed. He often writes about all things OSRÂ and recently wrote an article about the various ways one could roll for hps in 0e. To sum up, there were four methods discussed:
- Official method 1: roll a d6 and add it to your hps each level
- Official method 2: re-roll all your hit dice when you leveled up and kept the new roll (if it was lower than your old total, sucked to be you)
- Method from Empire of the Petal Throne: re-roll all your hit dice every level, keep the better of your old and new totals
- Hedgehobbit’s method from the comments: start with hp equal to your constitution, then use EPT rules going forward
Being the stats geek I am, this was irresistible to me. Did these methods make a difference? So I spent a chunk of my weekend figuring it out for myself (not that this hasn’t been done prior mind you, I just wanted to check the numbers for myself).
First, because we’re discussing this in terms of 0e, I limited myself to die 6s (which all characters used) and stopped at 10 of them (since that was the maximum anyone got). If you’re looking at a similar problem with different numbers or types of dice, your results will vary, but the general principles below should hold.
The difference between the two official methods is simple: the expected value of a single die is (max + 1)/2 (in the case of a d6, 3.5). With no prior knowledge, the expected value of X dice is X times the value of an individual die, so the expected value of, for example, 5d6 is 17.5. Since all dice rolls are independent it doesn’t matter if you roll them one per level or all at once. This means that if you faithfully use one or the other official method, there is no real difference in the expected value. BUT, in the official rules you’re free to mix and match the two methods at will, so any given level you get to decide if you want to roll one more hit die or re-roll all of them and keep the new total. Fortunately this is an easy choice. Your expected value of rolling a single new die is current hp + 3.5, while your expected value for re-rolling all of them is new level X 3.5, which means that if your current hps are > current level X 3.5, you want to roll a single die, if they’re less than current level X 3.5, you want to re-roll them all, and if they’re equal to current level X 3.5, it doesn’t really matter which you do.
But how does the EPT method compare to the two official methods? The quickest, non-cheating way to compare the two is with a series of markov chains. It’s not difficult, but it is tedious. Fortunately for us, there is a much quicker and easier way to cheat: www.anydice.com. The standard method is easy, the re-roll keep highest method requires nested [highest of …] statements. All of the graphs below are linked to anydice pages so if you’d like to check how I got them, click through.
As you can see from the below charts, the EPT method does have an advantage over the official method, but it’s a very small one. Over the course of a character’s career from 1d6 hp to 10d6 hps, using the EPT method will net you an average of 1.88 hps. Since, using both methods, the tail values become very rare very quickly, I have also listed the range into which ~95% of characters will fall (for the traditional method which is symmetrical this is done as you would expect. For the EPT method this is done by removing the least likely possibilities until ~95% probability remains). This shows that while the average difference is fairly low, characters using the EPT method are less likely to have low HP values (by 1-5 hps depending on level)
However, the entire reason the EPT method is championed is the assumption that it provides increased survivability at low levels with minimal impact on higher ones. While the minimal impact at high levels appears to bear out, the increased survivability at lower ones does not, so the method is of questionable usefulness.
Hedgehobbit’s method of starting with hps equal to your con and then using the EPT method after actually seems to meet those criteria better. It starts characters with a mean of 10.5 hps (common range of 5-16) so it should definitely provide the increased survivability at low levels, but how does it stack up to the other methods at high levels? I haven’t provided charts, but they’re not really necessary. We know that the highest likely starting hps are 16, and checking down the EPT charts we see that at level 7 the chance that the highest roll of 1d6, 2d6, … 7d6 is 16 or less is less than a percent, so while this method provides a HP bump in lower levels, it disappears into the normal small bump provided by the EPT method somewhere between levels 1 and 7 (essentially you’d have to skunk your level 4, 5, and 6 rolls for it to even persist that long)
Final conclusion: using only the official methods it’s possible to game the system for a few extra hp, but it’s neither sustainable nor anything to get excited about. Using the EPT method provides nominal benefit to higher level hps but again nothing to get excited over. If you really want to increase survivability at low levels with minimal high level impact, a variant of Hedgehobbit’s method is the one to use.
| 1/lvl or reroll keep new | reroll keep highest | |||
| 1d6 |  |
Mean: 3.5
CI: 1-6 (100%) |
NA | NA |
| 2d6 |  |
Mean: 7
CI: 3-11 (94%) |
 |
Mean: 7.16
CI: 4-11 |
| 3d6 |  |
Mean: 10.5
CI: 5-16 (96%) |
 |
Mean: 10.87
CI: 7-16 |
| 4d6 |  |
Mean: 14
CI: 8-20 (95%) |
 |
Mean: 14.59
CI: 10-20 |
| 5d6 |  |
Mean: 17.5
CI: 11-24 (94%) |
 |
Mean: 18.31
CI: 13-24 |
| 6d6 |  |
Mean: 21
CI: 13-29 (96%) |
 |
Mean: 22.03
CI: 16-28 |
| 7d6 |  |
Mean: 24.5
CI: 16-33 (96%) |
 |
Mean: 25.75
CI: 19-32 |
| 8d6 |  |
Mean: 28
CI: 19-37 (95%) |
 |
Mean: 29.46
CI: 23-37 |
| 9d6 |  |
Mean: 31.5
CI: 22-41 (95%) |
 |
Mean: 33.17
CI: 26-41 |
| 10d6 |  |
Mean: 35
CI: 25-45 (95%) |
 |
Mean: 36.88
CI: 30-45 |
Thanks for crunching the numbers, Matt! I wouldn’t have known where to start with this, and I got to learn about Markov chains.
It’s interesting to me that the basic trend for the EPT method is slightly higher HP overall. Is there a way to model the likelihood of average HP vs. extremely low/high HP across both basic models (cumulative roll and keep vs. reroll every level)?
Across the two basic models there’s no real difference in distribution. They both use the left set of tables.
Neat! Thanks again.
Interesting discussion as always, Matt.
I would like to know when the standard grammatical form for introducing matter below the discussion became “the below [whatever]” rather than “the [whatever] below”.
My Indian colleagues use the first form, but I’ve always assumed that’s because their native Hindi grammar does it that way. Obviously the norm changed while I wasn’t paying attention.
Also: Having plumped for a method, if the GM then says “Good. I wasn’t going to let you use [scheme x] anyway”, aren’t you better off changing to one of the other remaining allowable systems?
8oD
To your first point, I have no idea. Below is an adjective, so it can technically go either place. Which sounds right to each of us is probably a dialect issue.
To your second, this seems like something you should hammer out pre-game and then inform your players of.
I think you may be missing the point. If I’m a 3rd level character with 5 or fewer hit points I’m probably going to die since I can’t take one good hit. Characters with more hit points will at least have a turn to react. By your numbers the probability of getting a 5 or less is:
Standard = 0.46+1.39+2.78 = 4.63
EPT method = 0.02+0.19+0.87 = 1.08
So the standard method is over 4 times more likely to produce these unplayable characters. You point out that it changes little else but I take this to mean it fixes a problem without seriously altering the rest of the game.
Well, I’ve talked about this here on the stew before. By comparing those in a ratio you’re artificially inflating the significance of the occurrence of that event.
In the EPT method, a little over 1 in 100 characters will have 5 or less hit points by level 3.
In the standard method, a little less than 1 in 20 will have 5 or less hit points by level 3.
That is a statistically significant difference, which you have pointed out, but it is not really of any practical significance. Both chances are low enough to discount. It is unlikely that over the course of several games you will see a character with this problem using either system.