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On another thread, Random and I have been in a bit of a snit over the relative merits and mathematics of Whitebox hit point generation. In S&W:WB, upon advancing a level, hit points are rolled anew based on the hit dice of the new level. If the result is less than the prior level's total, the old total is kept. If higher, the new total is kept. This is different from much of the past practice of Gygax & Arneson's game, certainly from Supplement I onward. In this more accustomed method, all previous hit point totals are kept as one advances, and one merely adds a die roll (or set amount after name level) to the running hit point total.

What is the snit, you wonder? Well, the question is whether the two methods produce a significant difference in hit point distribution over time.

To save you all number crunching, here are the facts. With the more accustomed method, the bell curve remains in effect, with a balanced distribution at the extremes and in the middle, with a growing likelihood to fall in the middle and a decreasing likelihood to fall at the extremes as levels progress. In the WB method, over time (i.e. levels), and relatively quickly at that, the lowest scores for hit points (the bottom 1/5 of the range) become very highly unlikely (being effectively zero chance by 7th level if not before), and the lower range (under average) increasingly less likely, while still possible. The average range becomes increasingly probable over time, in fact the middle range becomes more likely than for the more accustomed method. At the highest range of hit points, the odds are identical in both methods, and those highest hit points become increasingly improbable (with the caveat that, in the WB reroll method, it is always at least possible for someone to end up with maximum hit points upon advancing a level).

The sticking point is the above average range. Over time, while the accustomed method keeps a balance such that below average and above average are equally likely, in the WB method, the above average method becomes generally more likely. The net result is, e.g. that the average hit points of a group of 8th level WB fighters will be 1-2 more than using the accustomed method. Another quirk is that characters with the same hit dice, but received at higher level (say, an 8 HD cleric at 10th or an 8 HD magic user at 16th), will be statistically more likely to have one or two more hit points than another character with the same hit dice received at a lower level.

Why? Well, since the WB method allows the "erasing" of bad rolls in the past (so that a 2 hit point 1st level fighting man can be 10 hit points at 2nd level), and each new roll from leveling up provides one more occasion to "repair" the old roll, the net result is a small — but here is the dispute — either significant or insignificant "gain" (at least considered in terms of probability and averages).

I think there are perfectly good reasons to be in favor of either approach. The accustomed method allows for consistency, such that a HP is a HP is a HP, and HD are HD are HD. So, there would be no difference statistically between rolling the hit points of an 8 HD monster or character all at once, advancing a fighting man to 8 HD at 8th level, or advancing a magic user to 8 HD at 16th level. This not only makes comparisons easier, it also means that there is at least no statistical oddity, viz. that an 8 HD archmage is likely slightly more hardy than his 8 HD superhero body guard!

The WB method allows for the difference of "history" if you will. It means that having worked a character up to 8 hit dice, e.g., has at least the slight odds of producing a minor advantage. It also means that a single bad roll for hit points in the past will not, if you survive to the next level, dog you forever. Indeed, it means that most mid- to high-level characters will have average or slightly above average hit points, and effectively none will fall in the low or lowest range. On the other hand, at most the difference for the average score will be one or two points, such that, for any given gaming group playing with regular frequency even over a number of years, the difference will be effectively invisible in actual play, and adventures or other products produced presuming the accustomed method will not be undone by those using the WB method as there is no powerfully disproportionate difference between the two methods.

As I said above, I like the latter, but am perfectly comfortable with the former.

Thoughts?

I am very pleased with your summary, Llenlleawg. Our main disagreement was whether or not the difference was significant, as I think our definitions of "significant" are slightly different.

For me it's usually 3 or 4 figures

Here's a quick chart I made up using Random's numbers for 5th level to help show the difference in the curves:

Nifty chart. I'm not so tech-savvy to do that sort of stuff myself; I just happen to know how to program. Thank you for the visual aid, Lord Kilgore.

Man, this post brings back memories. It's 1983. My friends and I are about 16 or so and play D&D. We're all smart, we all DM and we all like to argue.

I hope you don't mind, but I hot-linked your chart over on the OD&D board. Not much discussion yet, as I linked to these threads rather than restating all the data over there.

For what it's worth, if we look at the median rather than the mean (since "average" is not precise, and we have been using the mean in much of our discussion about averages), the set of a million fifth-level fighting men you generated yields the following — Random, median = 17 (500,092 @ 17 or below; 499,908 @ 18 or above); Reroll, median = 18 (542,140 @ 18 or below with 123,430 @ 18; 457,860 @ 19 or above with 115,505 @ 19). In another set of a million characters, the numbers above might have been flipped, with the median for random at 18. The reroll method looks rather solidly in the 18 category.

What about our (in)famous 10th level fighting man? For the random method, the results are not surprising. The median is 35 (536,196 @ 35 or below with 72,417 @ 35; 463,804 @ 36 or above with 71,355 @ 36). For reroll, the median is 37 (578,229 @ 37 or below with 93,506 @ 37; 421,771 @ 38 or above with 87,067 @ 38).

In short, the median and the mean in this case more or less coincide.

Llenlleawg wrote:For what it's worth, if we look at the median rather than the mean (since "average" is not precise, and we have been using the mean in much of our discussion about averages), the set of a million fifth-level fighting men you generated yields the following — Random, median = 17 (500,092 @ 17 or below; 499,908 @ 18 or above); Reroll, median = 18 (542,140 @ 18 or below with 123,430 @ 18; 457,860 @ 19 or above with 115,505 @ 19). In another set of a million characters, the numbers above might have been flipped, with the median for random at 18. The reroll method looks rather solidly in the 18 category.

What about our (in)famous 10th level fighting man? For the random method, the results are not surprising. The median is 35 (536,196 @ 35 or below with 72,417 @ 35; 463,804 @ 36 or above with 71,355 @ 36). For reroll, the median is 37 (578,229 @ 37 or below with 93,506 @ 37; 421,771 @ 38 or above with 87,067 @ 38).

In short, the median and the mean in this case more or less coincide.

When I say average, I am talking solely about the arithmetic mean of my data (or sample mean here, if you prefer). I don't give a crap about the median at all because it suppresses the effect of outliers, which I am actually trying to highlight. The outliers are important because they are valid rolls and not random flukes or measurement errors. The fact that they come up with such low probability is taken care of by the fact that not many of them are being considered because they already have appeared with such low probability.

Also, medians are not precise enough to show the subtle hit point bonus given by re-rolling each level, which likely factors into why you're really pushing for me to consider them, that it might somehow dull my argument. It does not.

Consider the set {0, 0, 0, 1, 5000, 5000, 5000} as an extreme example. The median is, of course, 1, but it isn't a very good figure for the "average" of this set. The arithmetic mean is actually 2143. In this case, you can't really call 0 and 5000 outliers as they compose the majority of the set! Perhaps 1 was a fluke, but perhaps not. What if this were a statistical sample, and by increasing the sample size to 7,000,000,000 you observed the same frequency of values? The median is simply inferior to the arithmetic mean for analyzing the primary effects of these dice rolling methods.

To explain why I am less than worried about a one point leaning by fifth and a two point leaning by tenth level, and likewise am less than concerned that a sixteenth level character might be slightly better off statistically than an eighth level character with the same hit dice, consider the following scenario. Suppose the referee has planned an expedition to the glacial rift of a frost giant jarl ( ) and he places 75 frost giants throughout their icy mountain fastness. Now, if the referee is a stickler about hit point distribution and worried about the effect of even one or two hit points askew, he will surely be as worried about the hit point distribution of the monsters he places as he is of the player characters. That is, he would surely not advantage the monsters while begrudging that a player might have a two-point advantage above their theoretical, statistical peers, especially since the frost giants will be dealing out 2d6 per blow.

Now, frost giants have 10+1 hit dice, so the distributions will be the same as a 10th level fighting man (so long as we simply add one). Using the numbers generated for our discussion, but bringing the giant population down to 75, and allowing that anything at 0.5000 or more yields one frost giant, this would mean that there would be no frost giants in the whole lair with greater than 47 hit points. [For the mathematically inclined, there were 0.9577% or 0.718275 frost giants at 46, plus one making 47 hit points; there are only 0.6312% or 0.4734, i.e., none at 47, plus one, or 48 hit points.] Likewise, there are no frost giants less than 25 hit points. Moreover, there are only four frost giants in the whole lair with 40 hit points, and a like number at 32 hit points.

Now, is any referee going to stock his glacial rift this way? Will there really be no frost giants with anything between 48 and 61 hit points? Maybe, but if published and otherwise circulated material is right, I suspect not. Higher-level adventures generally skew the opponents upward, with most around or just above average, some near the top, and almost none to speak of with below average, much less minimal, hit points. While having a number of kobolds or goblins at 1 or 2 hit points is as common as dirt, do we ever see fully-grown frost giants with 11, 12, 21, or 22 hit points? (Don't count the young, since they have lower hit dice; I am speaking of the fully-grown, 10+1 hit dice frost giants.) Moreover, using the older (Monsters & Treasure) rules, even a goblin lair of 40-400 goblins (1-1 HD) will have a goblin king and 5-30 bodyguards who count as hobgoblins (1+1 HD), skewing the numbers to the above average side.

Now, maybe some of you referees really let the dice fall as they may when generating monsters and NPCs. Or, maybe you never roll for the general mess but happily place significant "big bad evil" characters here and there with painfully assigned (non-randomly generated) scores (Lareth the Beautiful, I'm looking at you!). This is not a bad thing! It makes for interesting adventures and the stuff of challenge. However, to the extent that it is true of our own practices of dungeon-stocking and monster/NPC generating, might the slight skewing (to a point or two) of the PC hit point totals be reflective of the environments in which we as referees put them?

Just a thought and query, not a challenge or critique!