Hit Dice Debate

[Random]

www.swordsandwizardry.com/forum/viewtopic.php?p=18973#p18973 (and continuing as a thread derailment on the next couple of pages)

I would love to hear everyone's opinion on this. It seems that no one is able to get that re-rolling hit points at each level does not keep hit dice equivalent to just plain rolling a higher level character unless you allow for lower totals when gaining a level (which sucks, but hey!).

It's not that re-rolling is silly; it's that believing the characters with each method gain the same hit points on average that is silly. That just can't happen when you take the higher of the old and new rolls. I specify this because this exact notion has been tossed around these boards as a cool idea (which is most likely why it's in S&W:WB).

And I have no idea why I seem so cranky in that thread. Maybe I'm just in the mood to flex my math muscles. (Although I haven't proven anything, but rather stated my seemingly obvious conjecture.)

[Random]

www.swordsandwizardry.com/forum/viewtopic.php?p=19047#p19047

I've written a computer program to illustrate my point, in case anyone is curious.

[verhaden]

If there's a problem, let me be the first to blame Philotomy.

[Random]

I don't know if I'd call it a problem, but I've discovered that fighting-men generated with the official S&W:WB method (re-rolling) have nearly two extra hit points by 10th level. The edge starts small but grows over time.

Here are the average hit points at each level:
www.swordsandwizardry.com/forum/viewtopic.php?p=19049#p19049

My complaint isn't the power creep but that the method is not strictly equivalent to just throwing the appropriate dice and writing down the result, which is exactly what you'll be doing for NPCs and monsters. Call me particular, but I lose sleep over this kind of stuff.

I have exchanged a couple of PMs with the guy I'm arguing with there, but I have yet to hear back to see what he thinks of my averages.

[coffee]

I can relate.

I have my own quirks, things that bug me but that normal people would find absolutely insignificant.

It's okay. Sometimes you just have to accept that other people are bound and determined to be wrong, and you can't help them.

You know you're right. So do I. That should be enough.

[Random]

Thanks for the vote of confidence, coffee.

A new observation, 8 HD produces a different average number of hit points depending on which class you are. An 8 HD magic-user has about 1 more hit point on average than an 8 HD fighting-man. (Remember, we're talking about S&W:WB here for the exact progressions, but something similar will happen for OD&D if you re-reroll HP at every level.)

Anyways, if anyone is interested enough to see some serious data, I could reset my program for OD&D progressions and crunch the averages for each class for maybe 100,000,000 characters of each. I'm convinced though, and doing such would be a waste of time if no one cared to know.

(By the way, I love computers. I can write a program to "prove" a point a lot faster than I can pen the mathematics.)

[geoffrey]

Your computer program with 1,000,000 men reminds me of these words from Gary's introduction to Warriors of Mars:

"As with any other part of these rules, however, the participants can do their own research and add whatever they wish as desired. If an array of a million synthetic men seems useful, for example, we wish you the best!"

;D

[Random]

Feb 13, 2010 14:44:39 GMT -6 geoffrey said:

Your computer program with 1,000,000 men reminds me of these words from Gary's introduction to Warriors of Mars:

"As with any other part of these rules, however, the participants can do their own research and add whatever they wish as desired. If an array of a million synthetic men seems useful, for example, we wish you the best!"

;D

Funny you should mention it; I've read the 2nd and 3rd book of the Barsoom series in the past week.

[waysoftheearth]

Random,

I'm a sucker for number crunching too, so I find your argument interesting.

It's clear that there is a subtle advantage to a PC's average hit-points when using the "re-roll each level but ignore if lower" (RRELBIIL) method, as you point out.

However, you don't describe the complete distribution of results in your argument, which skews the case somewhat.

Without doing any figures, I strongly suspect that the RRELBIIL method produces a increasing tenancy toward average results as PC level increases. This is simply because the more dice you roll, the more likely you are to approach an average result. The RRELBIIL method requires you to roll more dice.

So, even though the average may be a couple of hit-points higher by 10th level, this is a simplification of the overall distribution of outcomes. Using the RRELBIIL method, PCs with below average hit points will be much rarer, but also there will be fewer PCs with above average hit points.

One could argue that, depending upon the survivability of the particular game, the RRELBIIL method will generate fewer PCs that survive to level 10, because it produces fewer PCs with above average hit-points -- even though that average is slightly higher.

From a purely analytical perspective, it is worth mentioning also that computers cannot generate truly random numbers, so any computer generated sampling is unlikely to capture the whole picture. E.g., in reality there is no such thing as an exactly fair die. Thus it is very questionable whether it is valid to expect that hit-point rolls should come from a perfectly random source. They won't, and any bias in individual die could very well be at least as significant as the increased average described.

All that aside, I agree with your fundamental assertion; that given perfect randomness, the two methods produce different distributions of results.

I am not convinced, however, that the 2 in 37 increase in average hit-points at 10th level described is as advantageous in-game as it might seem on the face of it.

Just my take

[Random]

A few things, waysoftheearth:

1. You must not have followed the links and read everything closely (there is also another thread there opened about it). We are well aware that hit points are skewed heavily towards the average range. It's not just your suspicion; it's an absolute fact! I did not describe the distribution in detail since avoiding the mathematics was THE reason I coded it, and the difference in curve was visibly apparent in my data. For your pleasure, someone in the other thread posted a nice graph though for 5th Level Fighting-men.

2. You are 100% incorrect in stating that there are less characters with "above average" hit points (where average here is the average of simply rolling the hit dice at once). There are actually more! I recommend you count them. This is caused by the fact that you are ignoring rolls when they are lower than the previous total. That's the same as simply adding a bunch of hit points to many of the rolls. The Re-roll method invariably produces more above average characters at every single level past first.

3. My computer generates random numbers (likely a lot) better than your game dice, so pfft! You cannot discredit my findings on that basis, although you could discredit the exact numbers (they are only accurate to a few decimal places with this sample size). Were I to decrease the sample size, I'm positive the program would have more error due to randomness, and as my sample size increases, the averages tend to their theoretical values. In short, computer-generated numbers are sufficiently random for the purposes of this experiment.

4. On dice being biased enough to be as significant as the bonus attained by re-rolling hit points every level, this is not my problem but rather that of the owner of the crappy dice. Personally, I have some nice casino dice that I use. Regardless, assuming you're using the same dice to roll with each method, the re-roll method will still generate a bonus above and beyond that already given by a dice bias (or hey, cancel it out if your dice are biased in reverse), as that bias will be present during all the tons of re-rolls at each level.

5. As far as advantageous in-game, even a single hit point is a huge advantage in OD&D! If you don't agree, I'll be sure to dock your character a hit point when you play with me, just to see if you still feel that way. Consider also that the advantage is had not only at the highest levels where it is most pronounced, but at every level above first. If two million gaming groups played OD&D tonight and every Sunday for a year (what a world!), half rolling like me and half re-rolling upon their level ups, the million re-rolling will have a higher survival rate; I guarantee it!

6. Can you sleep at night knowing your 8 HD magic-users have more hit points on average than your 8 HD fighting-men?

7. You guys are all taking my word for all of these numbers, but you haven't seen my code. I wasn't going to share it unless someone doubted the integrity of it, as my extreme programming for personal use might insult someone's sense of style and good practice.

[Random]

I'll re-post Lord Kilgore's chart.

Consider the number of fighting-men of 5th level with more than 20 hit points (clearly above average). The re-roll method, as you can see, generates more such characters.
The only thing the standard method has more of is low HP characters.

By the way, the simple fix would be to allow the hit dice roll at level up to decrease hit points. Then the curves would be strictly identical.

[geoffrey]

I have no problem with re-rolled hit points resulting in a lower hit point score. Jeff Rients suggested that each PC re-roll hit points not merely with each level gained, but at the start of each game session! That often-varying number of hp would represent the fact that we all have "down days" and also days "in the zone".

I pushed that as far as it could go in the dice conventions included in Supplement V: CARCOSA, in which hp are re-rolled at the beginning of each combat!

[Random]

I too, am a fan of re-rolling hit points like that, geoffrey. It's one less number on the character sheet!
But, this debate was strictly about whether or not re-rolling at level up (ignoring lower rolls) had a significant impact on the survivability of D&D characters as compared to the "perfect" method*.

*That is, when going from (X HD) + 1 to (X + 1) HD, just subtract one hit point and add a die roll, or when going from (X HD) + 4 to (X + 1) HD + 1, just subtract three hit points and add a die roll (which could of course lower the amount). This results in identical hit point distribution no matter if you roll hit points at once or accumulate during level up.

[badger2305]

Random - dude, I really appreciate your effort here, and having taught research methods and stats for social sciences, I know what you're talking about.

But from a qualitative perspective, how much immediate salience does this have? If anything, just like with auto accidents, I doubt that people really notice, even if the overall result is statistically significant.

My recommendation: summarize your argument, make a House Rule, and then let people decide. Some people will agree; others won't.

Wish I had more to say about it, but think about what I'm pointing out about auto accidents.

[Random]

I am well aware that no one will ever notice the difference (and admitted this on the S&W boards).

I'm just saying that there is a difference, and given enough D&D games, it's going to make the difference of life or death in at least a few of them even if the participants don't notice.

Think about how many times your character has ever had just one measly hit point!

[delta]

Feb 12, 2010 7:44:54 GMT -6 Random said:

I would love to hear everyone's opinion on this. It seems that no one is able to get that re-rolling hit points at each level does not keep hit dice equivalent to just plain rolling a higher level character...

You are, of course, correct. To my mind it's pretty patently obvious. However (without even reading the linked thread), my experience is that people determined to disagree with you on an issue like this will liberally ignore any amount of statistics. I also find that a lot of people abuse the technical language and get a bit cranky when called out on it.

The roll-hit-points-each-level camp seems really crazy to me. It sounds like it's mostly radical response to wondering how the OD&D class tables can progress from 1+1 to 2HD. (Which I saw Gygax respond with basically "keep the +1 even after the 2HD are rolled", which itself would also skew new-vs-old-characters.)

M.A. Mathematics & Statistics, U. Maine 1995

[Random]

Keeping each +1 would be a different interpretation of the charts. That would be accepting that the +1's on the chart are accumulative, which I did not assume*, though it would make enough sense to do it that way. I have never actually seen that interpretation, and if it is indeed Gygax's original intention, he did a completely piss-poor job of indicating it (or rather actually no job at all of indicating it).

*The hit dice are not accumulative, so I have no reason to expect that the bonus hit points would be.

[waysoftheearth]

Random, as I said I wrote "without doing the figures".

LordKilgore's distribution chart is very nicely done, and quite convincing enough for me. I must admit, I expected to see a difference at the top end which wasn't shown in the distribution.

Feb 14, 2010 10:51:07 GMT -6 geoffrey said:

I have no problem with re-rolled hit points resulting in a lower hit point score. Jeff Rients suggested that each PC re-roll hit points not merely with each level gained, but at the start of each game session!

This is how I played my game all of last year too. As a ref, I really liked it for the variety it introduced. The fighting men had to decide who was on front line or missile support each session based on their hit-point rolls, and players often came up with entertaining in-game justifications for their high or low hit-point rolls.

[Random]

The top end of the distribution is totally unaffected because those rolls are impossible to have attained before reaching the level in question, so the most recent roll is always taken when those occur. There is no chance to keep the previous total in those cases.

[Elphilm]

I have to say that this thread makes great points in favor of the "roll HP at each level and keep highest result" method. Favoring PCs over NPCs and longstanding characters over newly rolled ones are both desireable results for me.

Regardless, I really like Random's number crunching, and Lord Kilgore's chart is especially neat!

[Finarvyn]

For me, I've played one way for decades and have only recently been "won over" to the other method.

For the vast majority of my gaming career, we've played that if a person went from 3 HD to 4 HD you simply roll one new HD and add it to the old total. I think this system is fine (since we played it for 30+ years that way) but have always been bothered by advancing from 3+1 HD to 4 HD, becasue you're kind of adding d6-1 at that next level. However, it never gave us problems and we liked it well enough.

It is only recently that the alternate interpretation was brought to my attention. I'm not sure who first mentioned it, and if it was here or DF or K&K, but the idea of re-rolling all HD on advancing to a new level really seemed to make sense and my players have embraced it enthusiastically. I only allow for HP to stay even or go up with advancement, never down, and in play we've had pleanty of cases where the player didn't gain any HP with level advancement. If a player's dice are hot early on, that character won't gain any HP for a while. On the other hand, if a player rolls badly early on, that character can get back on track faster. What it tends to do in my experience, is give characters the chance to overcome a bad die roll or two without introducing a huge imbalance into the game.

Of course, I tend to play low-to-middle level campaigns and it's possible that at high levels the system will break down somewhat as Random's data suggests. So far, it hasn't really had a big impact in my games.

[Finarvyn]

Feb 12, 2010 7:44:54 GMT -6 Random said:

It seems that no one is able to get that re-rolling hit points at each level does not keep hit dice equivalent to just plain rolling a higher level character unless you allow for lower totals when gaining a level (which sucks, but hey!).

It's not that re-rolling is silly; it's that believing the characters with each method gain the same hit points on average that is silly. That just can't happen when you take the higher of the old and new rolls. I specify this because this exact notion has been tossed around these boards as a cool idea (which is most likely why it's in S&W:WB).

To respond to the original post, Random, I have no idea why people can't (won't) understand this, either. Seems obvious to me that the averages differ each time you change any dice rolling system. And any system which keeps high rolls and tosses out low rolls has to put a mathematical advantage in over any system which keeps all rolls. Strange that people argue this point at all, except that most folks don't really seem to have a good feel for mathematics.

I thought it was a cool idea and my players like it.

[Finarvyn]

Feb 12, 2010 7:44:54 GMT -6 Random said:

I would love to hear everyone's opinion on this. It seems that no one is able to get that re-rolling hit points at each level does not keep hit dice equivalent to just plain rolling a higher level character...

I should also point out that I use this same system when I have players create a higher level character from scratch. I like to start characters off at 3rd level so that they have decent hit points and spells to work with, and my players are used to me saying "okay mister magic-user -- roll d6 for hit points and tell me the number ... now roll 1d6+1, is it higher? ... now roll 2d6, is it higher? ... okay ... let's play!"

No way that this could "be the same" as starting out with a single 2d6 roll.