Power-creep, OSR-style

While preparing yesterday’s post I stumbled on a discussion of house rules and hit points at Grognardia, where James the Grognard describes his planned house rules for Swords and Wizardry. It’s old and I don’t know whether he ended up using them, but I noticed that his hit points rules induce a very strange, and in my opinion definitely not old-school, distribution of hit points. His proposed house rule is:

Hit Dice are re-rolled upon gaining a new level, but maximum hit points never decrease as a result of a re-roll, although they may not increase.

Example: Brother Candor of Tyche is a 3rd-level cleric; he has 15 hit points. Upon gaining 4th level, he rolls 4D6+4 for hit points. If the result is below 15 hit points, he gains no new hit points this level.

and one of the explanations given in comments is:

One of the goals of this campaign is to keep things as “middling” as possible. I think D&D works best when characters are fairly mediocre mechanically and the hit dice house rule is part of the plan to encourage that.

Now in fact, the hit dice house rule doesn’t do that at all – it does quite the opposite. It’s a fun[1] example of how a little bit of house-rule fiddling with dice can produce a result that is counter-intuitive and/or goes strongly against the original intention of the house-rule. In other parts of his post James makes it pretty clear that he is aware of the basic debate about the way hit points are defined (see e.g. comment 1) and shows an admirable commitment to the concept of abstracted rounds (see e.g. his rules for dual wielding). However, the hit dice design goes against the principles he espouses, and the way that it does this won’t be clear until his PCs reach higher levels. His hit dice rule essentially serves as a hit point boost for early edition characters, and is remarkably generous in the context of those rules. Here I will explain how. This isn’t a criticism of James’s rule,  or of the principles underlying his house rules, just an example of how fiddling with distributions isn’t always a good idea.

Basically, James’s hit dice rule is equivalent to granting all the players the chance to reroll all hit dice that fell below maximum every level. There is still, in theory, a chance of getting minimum hit points but this chance is so vanishingly small at higher levels that it is essentially zero. If you roll for HPs at level 6, you keep your previous roll unless the new roll is higher; but you did the same thing at level 5, and the same at level 4, and so on. This means, essentially, that at level k your hit points are the maximum of (k-1)dX and kdX. But at level 2 your HPs are max(dX,2dX). By induction it’s clear: your hit points at level k are:

  •  max(dX,2dX,…,kdX)

This does not have a central distribution: it reduces the probability of getting small numbers rapidly, and drives the weight of the probability distribution towards the maximum. By the time one reaches very high levels, the most likely roll will be kX+1dX, with a roughly uniform distribution within the maximum range.This is essentially equivalent to giving the players a chance to reroll their level 1 hit point roll k-1 times, their level 2 roll k-2 times, and so on. The chance of a 6 on a d6 is 1/6; if you give a 9th level cleric 8 rerolls of their first level hit dice, they are going to have a very very low chance of getting anything but a 6. This is going to push HP values to the right.

An example distribution is shown at the top of this page: the black line is the empirical distribution for 12d4 rolled classically, while the red line is for James’s version of 12d4. The minimum observed value in 100,000 simulations for 12d4 is 15; for James’s distribution, it’s 21. That’s the equivalent of nearly 3 hit dice for a wizard. The kind of effect this induces is visible even at low levels: Figure 1 shows how James’s system for 2d10 shifts all the probability weight for the fighter from the lower end of the distribution to the middle, with small increases in the probability of higher values. Note particularly that values of 2 or 3 are much less likely just at 2nd level.

Figure 2: Empirical Distribution for 2d10 using two systems (100,000 simulations, kernel density smoother with optimal bandwidth selection)

This power-creep grows with levels. Figure 3 shows how by 9d10 this weight is shifting to the right of the distribution, i.e. increasing the chance of very high values. Of course the effect stops at level 9, but by this time it’s powerful: the minimum value observed in 100,000 rolls is 15 under the classic distribution, and 31 under James’s distribution – the weight is seriously moving towards the right, to the tune of just over 3 hit dice for a fighter[2]. Probabilities of observing values over 60 are significantly higher in James’s distribution, and the most likely values are also shifted to the right compared to the classic distribution.

Figure 3: HP distributions for 9d10 hit dice

I think that the distributions shown here are not what James had in mind when he talked about “middling” values – the method he has proposed creates skewed distributions and shifts the entire distribution to the right, rather than narrowing the distribution and placing it in the middle. The best way to create middling values is to use large numbers of d4s: make Wizard hit dice 3d4/3, and fighters 3d4. Fiddling with maxima and exploding dice is not a good way to create a family-friendly distribution. I’m no OSR expert but giving players a chance to reroll all their hit points every level seems fundamentally at odds with the basic principles of old school play, and thus this house-rule is out of step with its intentions. I guess James was thinking that his method gives a high chance of HPs not increasing at any given level (due to the risk of rolling below your previous HPs) but this is only true if you’ve got average or above-average values to start with. Quite the opposite happens if you started with poor hit points. I think it’s one of those examples where the intuition about dice rolls and the effects are quite different.

On this note, I should point out that although the OSR likes this idea of pushing people into “average” values, the fundamental mechanic of AD&D – the d20 roll – is completely inconsistent with this. It forces high probability into the tails of the distribution, as do the uniform distributions of most damage rolls. It’s also inconsistent with the natural world – almost any experimental system you care to think of has normally distributed experimental error, not uniform distributed. If one is concerned with a “natural” approach to conflict resolution and encouraging middling results, one’s very first act should be to swap d20 for 2d10. It’s surprising that the d20 system has persisted through all the incarnations of D&D given its fundamentally unnatural and abhorrent distribution.

Finally, I’d be interested to find out if James is still using this hit dice rule, or whether he dropped it ages ago when he realized what it was doing.

Methods Note: the empirical distributions shown here were generated in R using 100,000 simulations. All charts are kernel density smooths, using R’s default kernel and optimal bandwidth calculation. Histograms are for losers.

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fn1: for statistician-based definitions of “fun”

fn2: I don’t know exactly what the hit dice rules are for S&W – it could be fighters are also d6, but because all dice have a uniform distribution the effect is consistent – it’s only the exact magnitude of the power creep that changes.